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Foundations Of Modern Physics Steven Weinberg

Foundations Of Modern Physics Steven Weinberg Foundations Of Modern Physics Steven Weinberg Foundations Of Modern Physics Steven Weinberg

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Foundations of Modern Physics In addition to his ground-breaking research, Nobel Laureate Steven Weinberg is known for a series of highly praised texts on various aspects of physics, com- bining exceptional physical insight with a gift for clear exposition. Describing the foundations of modern physics in their historical context and with some new derivations, Weinberg introduces topics ranging from early applications of atomic theory through thermodynamics, statistical mechanics, transport theory, special relativity, quantum mechanics, nuclear physics, and quantum field theory. This volume provides the basis for advanced undergraduate and graduate physics courses as well as being a handy introduction to aspects of modern physics for working scientists. steven weinberg is a member of the Physics and Astronomy Departments at the University of Texas at Austin. He has been honored with numerous awards, including the Nobel Prize in Physics, the National Medal of Science, the Heinemann Prize in Mathematical Physics, and most recently a Special Breakthrough Prize in Fundamental Physics. He is a member of the US National Academy of Sciences, the UKā€™s Royal Society, and other academies in the US and internationally. The American Philosophical Society awarded him the Benjamin Franklin medal, with a citation that said he is ā€œconsidered by many to be the preeminent theoretical physicist alive in the world today.ā€ He has written several highly regarded books, including Gravitation and Cosmology, the three-volume work The Quantum Theory of Fields, Cosmology, Lectures on Quantum Mechanics, and Lectures on Astrophysics.
Foundations of Modern Physics Steven Weinberg University of Texas, Austin
University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314ā€“321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi ā€“ 110025, India 79 Anson Road, #06ā€“04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the Universityā€™s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108841764 DOI: 10.1017/9781108894845 Ā© Steven Weinberg 2021 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2021 Printed in the United Kingdom by TJ Books Limited, Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Weinberg, Steven, 1933ā€“ author. Title: Foundations of modern physics / Steven Weinberg, The University of Texas at Austin. Description: New York : Cambridge University Press, 2021. | Includes bibliographical references and indexes. Identifiers: LCCN 2020055431 (print) | LCCN 2020055432 (ebook) | ISBN 9781108841764 (hardback) | ISBN 9781108894845 (epub) Subjects: LCSH: Physics. Classification: LCC QC21.3 .W345 2021 (print) | LCC QC21.3 (ebook) | DDC 530ā€“dc23 LC record available at https://lccn.loc.gov/2020055431 LC ebook record available at https://lccn.loc.gov/2020055432 ISBN 978-1-108-84176-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
For Louise, Elizabeth, and Gabrielle
Contents PREFACE page xiii 1 EARLY ATOMIC THEORY 1 1.1 Gas Properties 2 Air pressure Boyleā€™s law Temperature Scales Charlesā€™ law Explanation of gas laws Ideal gas law 1.2 Chemistry 6 Elements Law of combining weights Daltonā€™s atomic weights Law of combining volumes Avogadroā€™s principle The gas constant Avogadroā€™s number 1.3 Electrolysis 10 Early electricity Early magnetism Electromagnetism Discovery of electrolysis Faradayā€™s theory The faraday 1.4 The Electron 14 Cathode rays Thomsonā€™s experiments Electrons as atomic constituents 2 THERMODYNAMICS AND KINETIC THEORY 16 2.1 Heat and Energy 16 Caloric Heat as energy Kinetic energy Specific heat Energy density and pressure Adiabatic changes vii
viii Contents 2.2 Absolute Temperature 21 Carnot cycles Theorems on efficiency Absolute temperature defined Relation to gas thermometers 2.3 Entropy 27 Definition of entropy Independence of path Increase of entropy Thermodynamic relations Entropy of ideal gases Neutral matter Radiation energy Laws of thermodynamics 2.4 Kinetic Theory and Statistical Mechanics 33 Maxwellā€“Boltzmann distribution General H-theorem Time reversal Canonical and grand-canonical distributions Connection with thermodynamics Compound systems Probability distribution in gases Equipartition of energy Entropy as disorder 2.5 Transport Phenomena 42 Conservation laws Galilean relativity Navierā€“Stokes equation Viscosity Mean free path Diffusion 2.6 The Atomic Scale 53 Nineteenth century estimates Electronic charge Brownian motion Consistency of constants Appendix: Einsteinā€™s diffusion constant rederived 3 EARLY QUANTUM THEORY 61 3.1 Black Body Radiation 61 Absorption and energy density Degrees of freedom of electromagnetic fields Rayleighā€“Jeans distribution Planck distribution Measurement of Boltzmann constant Radiation energy constant 3.2 Photons 67 Quantization of radiation energy Derivation of Planck distribution Photoelectric effect Particles of light 3.3 The Nuclear Atom 71 Radioactivity Alpha and beta rays Discovery of the nucleus Nuclear mass Nuclear size Scattering pattern Nuclear charge 3.4 Atomic Energy Levels 77 Spectral lines Electron orbits Combination principle Bohrā€™s quantization condition Correspondence principle Comparison with observed one-electron atomic spectra Reduced mass Atomic number Outstanding questions
Contents ix 3.5 Emission and Absorption of Radiation 84 Einstein A and B coefficients Equilibrium with black body radiation Relations among coefficients Lasers Suppressed absorption 4 RELATIVITY 88 4.1 Early Relativity 88 Motion of the Earth Relativity of motion Speed of light Michelsonā€“Morley experiment Lorentzā€“Fitzgerald contraction 4.2 Einsteinian Relativity 94 Postulate of invariance of electrodynamics Lorentz transformations Space inversion, time reversal The Galilean limit Maximum speed Boosts in general directions Special and general relativity 4.3 Clocks, Rulers, Light Waves 103 Clocks and time dilation Rulers and length contraction Transformation of frequency and wave number 4.4 Mass, Energy, Momentum, Force 106 Einsteinā€™s thought experiment Formulas for energy and momentum E = mc2 Force in relativistic dynamics 4.5 Photons as Particles 111 Photon momentum Compton scattering Other massless particles 4.6 Maxwellā€™s Equations 114 The inhomogeneous and homogeneous equations Density and current of electric charge Relativistic formulation of inhomogeneous Maxwell equations Indices upstairs and downstairs Relativistic formulation of homogeneous Maxwell equations Electric and magnetic forces 4.7 Causality 121 Causes precede effects Invariance of temporal order Maximum signal speed Light cone 5 QUANTUM MECHANICS 124 5.1 De Broglie Waves 125 Free-particle wave functions Group velocity Application to hydrogen Davissonā€“Germer experiment Electron microscopes Appendix: Derivation of the Bragg formula
x Contents 5.2 The Schrƶdinger Equation 129 Wave equation for particle in potential Boundary conditions Spherical symmetry Radial and angular wave functions Angular multiplicity Spherical harmonics Hydrogenic energy levels Degeneracy 5.3 General Principles of Quantum Mechanics 138 States and wave functions Observables and operators Hamiltonian Adjoints Expectation values Probabilities Continuum limit Momentum space Commutation relations Uncertainty principle Time dependent wave functions Conservation laws Heisenberg and Schrƶdinger pictures 5.4 Spin and Orbital Angular Momentum 151 Doubling of sodium D-line The idea of spin General action of rotations on wave functions Total angular momentum operator Commutation relations Spin and orbital angular momentum Multiplets Adding angular momenta Atomic fine structure and space inversion Hyperfine structure Appendix: Clebschā€“Gordan Coefficients 5.5 Bosons and Fermions 165 Identical particles Symmetric and antisymmetric wave functions Bosons and fermions in statistical mechanics Hartree approximation Slater determinant Pauli exclusion principle Periodic table of elements Diatomic molecules: para and ortho Astrophysical cooling 5.6 Scattering 175 Scattering wave function Representations of the delta function Calculation of the Greenā€™s function Scattering amplitude Probabilistic interpretation Cross section Born approximation Scattering by shielded Coulomb potential Appendix: General transition rates 5.7 Canonical Formalism 190 Hamiltonian formalism Canonical commutation relations Lagrangian formalism Action principle Connection of formalisms Noetherā€™s theorem: symmetries and conservation laws Space translation and momentum 5.8 Charged Particles in Electromagnetic Fields 195 Vector and scalar potential Charged particle Hamiltonian Equations of motion Gauge transformations Magnetic interactions Spin coupling 5.9 Perturbation Theory 199 Perturbative expansion First-order perturbation theory Dealing with degeneracy The Zeeman effect Second-order perturbation theory
Contents xi 5.10 Beyond Wave Mechanics 206 State vectors Linear operators First postulate: values of observables Second postulate: expectation values Probabilities Continuum limit Wave functions as vector components 6 NUCLEAR PHYSICS 210 6.1 Protons and Neutrons 210 Discovery of the proton Integer atomic weights Nuclei as protons and electrons? Trouble with diatomic nitrogen Discovery of the neutron Nuclear radius and binding energy Liquid drop model Stable valley and decay modes 6.2 Isotopic Spin Symmetry 216 Neutronā€“proton and protonā€“proton forces Isotopic spin rotations Isotopic spin multiplets Quark model Pions Appendix: The threeā€“three resonance 6.3 Shell Structure 224 Harmonic oscillator approximation Raising and lowering operators Degenerate multiplets Magic numbers Spinā€“orbit coupling 6.4 Alpha Decay 229 Coulomb barrier Barrier suppression factors Semi-classical estimate of alpha decay rate Level splitting Geigerā€“Nuttall law Radium alpha decay Appendix: Quantum theory of barrier penetration rates 6.5 Beta Decay 243 Electron energy distribution Neutrinos proposed Fermi theory Gamowā€“Teller modification Selection rules Strength of weak interactions Neutrinos discovered Violation of leftā€“right and matterā€“antimatter symmetries Neutrino helicities Varieties of neutrino 7 QUANTUM FIELD THEORY 251 7.1 Canonical Formalism for Fields 252 Action, Lagrangian, Lagrangian density Functional derivatives Eulerā€“Lagrange field equations Commutation relations Energy and momentum of fields 7.2 Free Real Scalar Field 255 Lagrangian density Field equation Creation and annihilation operators Energy and momentum Vacuum state Multiparticle states
xii Contents 7.3 Interactions 261 Time-ordered perturbation theory Requirements for Lorentz invariance Example: Scattering of neutral spinless particles Feynman diagram Calculation of the propagator Yukawa potential 7.4 Antiparticles, Spin, Statistics 270 Antiparticles needed Complex scalar field General fields Lorentz transformation Spinā€“statistics connection Appendix: Dirac fields 7.5 Quantum Theory of Electromagnetism 280 Lagrangian density for electrodynamics Four-vector potential Gauge transformations Coulomb gauge Commutation relations Free fields Photon momentum and helicity Radiative decay rates Selection rules Gauge invariance and charge conservation Local phase invariance Standard model ASSORTED PROBLEMS 296 BIBLIOGRAPHY 301 AUTHOR INDEX 303 SUBJECT INDEX 307
Preface This book grew out of the notes for a course I gave for undergraduate physics students at the University of Texas. In this book I think I go farther forward than is usual in undergraduate courses, giving readers a taste of nuclear physics and quantum field theory. I also go farther back than is usual, starting with the struggle in the nineteenth century to establish the existence and properties of atoms, including the development of thermodynamics that both aided in this struggle and offered an alternative program. I fear that some readers may want to skim through this early part and hurry on to what they regard as the good stuff, quantum mechanics and relativity. That would be a pity. In my experience physics students who aim at a career in atomic or nuclear or elementary particle physics often manage to get through their formal education without ever becoming familiar with entropy, or equipartition, or viscosity, or diffusion. That was true in my own case. This book, or a course based on it, may provide some students with their last chance to learn about these and other matters needed to understand the macroscopic world. Readers may find this book unusual also in its strong emphasis on history. I make a point of saying a little about the welter of theoretical guesswork and ill- understood experiments out of which modern physics emerged in the twentieth century. This, it seems to me, is a help in understanding what otherwise may seem an arbitrary set of postulates for relativity and quantum mechanics. It is also a matter of personal taste. Research in physics seems to me to lose some of its excitement if we do not see it as part of a great historical progression. Some valuable historical works are listed in a bibliography, along with collections of original articles that I have found most helpful. But this is not a work of history. Historians aim at uncovering how the scien- tists of the past thought about their own problems ā€“ for instance, how Einstein in 1905 thought about the measurement of space and time separations in de- veloping the special theory of relativity. For this aim of historical writing it is necessary to go deeply into personal accounts, institutional development, and xiii
xiv Preface false starts, and to put aside our knowledge of subsequent progress. I try to be accurate in describing the state of physics in past times, but the aim of this book in discussing the problems of the past is different: it is to make clear how physicists think about these things today. This book is intended chiefly for physics students who are well into their time as undergraduates, and for working scientists who want a brief introduction to some area of modern physics. I have therefore not hesitated to use calculus and matrix algebra, though not in advanced versions. As required by the subject matter, the mathematical level here slopes upwards through the book. Where possible I have chosen concrete rather than abstract formulations of physical theories. For instance, in Chapter 5, on quantum mechanics, I mostly represent physical states as wave functions, only coming at the end of the chapter to their representation as vectors in Hilbert space. In some sections detailed material that can be skipped without losing the thread of the theory is put into appendices. Two of these appendices present what in my unbiased opinion are improved derivations of important results: the appendix to Section 2.6 gives a revised version of Einsteinā€™s derivation of his formula for the diffusion constant in Brownian motion, and the appendix to Section 6.4 presents a revision of Fermiā€™s calculation of the rate of alpha decay. In my experience, with some judicious pruning, the material of the book up to about the middle of Chapter 5 can be covered in a one-term undergraduate course. But I think that to go over the whole book would take a full two-term academic year. This book treats such a broad range of topics that it is impossible to go very far into any of them. Certainly its treatment of quantum mechanics, statistical mechanics, transport theory, nuclear physics, and quantum field theory is no substitute for graduate-level courses on these topics, any one of which would occupy at least a whole year. This book presents what I think, in an ideal world, the ambitious physics student would already know when he or she enters graduate school. At least, it is what I wish that I had known when I entered graduate school. In any case, I hope that the student or reader may be sufficiently interested in what I do discuss that they will want to go into these topics in greater detail in more specialized books or courses, and that they will find in this book a good preparation for such further studies. I am grateful to many students and colleagues for pointing out errors in the lecture notes on which this book is based and for the expert and friendly assistance I have received from Simon Capelin and Vince Higgs, the editors at Cambridge University Press who guided the publication of this book. STEVEN WEINBERG
1 Early Atomic Theory It is an old idea that matter consists of atoms, tiny indivisible particles moving in empty space. This theory can be traced to Democritus, working in the Greek city of Abdera, on the north shore of the Aegean sea. In the late 400s BC Democritus proclaimed that ā€œatoms and void alone exist in reality.ā€ He offered neither evidence for this hypothesis nor calculations on which to base predic- tions that could confirm it. Nevertheless, this idea was tremendously influential, if only as an example of how it might be possible to account for natural phe- nomena without invoking the gods. Atoms were brought into the materialistic philosophy of Epicurus of Samos, who a little after 300 BC founded one of the four great schools of Athens, the Garden. In turn, the idea of atoms and the philosophy of Epicurus were invoked in the poem On the Nature of Things by the Roman Lucretius. After this poem was rediscovered in 1417 it influ- enced Machiavelli, More, Shakespeare, Montaigne, and Newton, among others. Newton in his Opticks speculated that the properties of matter arise from the clustering of atoms into larger particles, which themselves cluster into larger particles, and so on. As we will see, Newton made a stab at an atomic theory of air pressure, but without significant success. The serious scientific application of the atomic theory began in the eighteenth century, with calculations of the properties of gases, which had been studied experimentally since the century before. This is the topic with which we begin this chapter. Applications to chemistry and electrolysis followed in the nine- teenth century and will be considered in subsequent sections. The final section of this chapter describes how the nature of atoms began to be clarified with the discovery of the electron. In the following chapter we will see how it became possible to estimate the atomsā€™ masses and sizes.1 1 Further historical details about some of these matters can be found in Weinberg, The Discovery of Subatomic Particles, listed in the bibliography. 1
2 1 Early Atomic Theory 1.1 Gas Properties Experimental Relations The upsurge of enthusiasm for experiment in the seventeenth century was largely concentrated on the properties of air. The execution and reports of these experiments did not depend on hypotheses regarding atoms, but we need to recall them here because their results provided the background for later theories of gas properties that did rely on assumptions about atoms. It had been thought by Aristotle and his followers that the suction observed in pumps and bellows arises from natureā€™s abhorrence of a vacuum. This notion was challenged in the 1640s by the invention of the barometer by the Florentine polymath Evangelista Torricelli (1608ā€“1647). If nature abhors a vacuum, then when a long glass tube with one end closed is filled with mercury and set upright with the closed end on top, why does the mercury flow out of the bottom until the column is only 760 mm high, with empty space appearing above the mercury? Is there a limit to how much nature abhors a vacuum? Torricelli argued that the mercury is held up instead by the pressure of the air acting on the open end of the glass tube (or on the surface of a bath of mercury in which the open end of the tube is immersed), which is just sufficient to support a column of mercury 760 mm high. If so, then it should be possible to measure variations in air pressure using a column of mercury in a vertical glass tube, a device that we know as a barometer. Such measurements were made from 1648 to 1651 by Blaise Pascal (1623ā€“1662), who found that the height of mercury in a barometer is decreased by moving to the top of a mountain, where less air extends above the barometer. The quantitative properties of air pressure soon began to be studied experimentally, before there was any correct theoretical understanding of gas properties. In 1662, in the second edition of his book New Experiments Physico- Mechanical Concerning the Spring of the Air and its Effects, the Anglo-Irish aristocrat Robert Boyle (1627ā€“1691) described experiments relating the pres- sure (the ā€œspring of the airā€) and volume of a fixed mass of air. He studied a sample of air enclosed at the end of a glass tube by a column of mercury in the tube. The air was compressed at constant temperature by pushing on the mercuryā€™s surface, revealing what came to be known as Boyleā€™s law, that for constant temperature the volume of a gas of fixed mass and composition is inversely proportional to the pressure, now defined by Boyle as the force per area exerted on the gas. Temperature Scales A word must be said about the phrase ā€œat constant temperature.ā€ Boyle lived before the establishment of our modern Fahrenheit and Celsius scales, whose
1.1 Gas Properties 3 forerunners go back respectively to 1724 and 1742. But, although in Boyleā€™s time no meaningful numerical value could be given to the temperature of any given body, it was nevertheless possible to speak with precision of two bodies being at the same temperature: they are at the same temperature if when put in contact neither body is felt to grow appreciably hotter or colder. Boyleā€™s glass tube could be kept at constant temperature by immersing it in a large bath, say of water from melting ice. Later the Fahrenheit temperature scale was established by defining the temperature of melting ice as 32 ā—¦F and the temperature of boiling water at mean atmospheric pressure as 212 ā—¦F, and defining a 1 ā—¦F increase of temperature by etching 212 āˆ’ 32 equal divisions between 32 and 212 on the glass tube of a mercury thermometer. Likewise, in the Celsius scale, the temperatures of melting ice and boiling water are 0 ā—¦C and 100 ā—¦C, and 1 ā—¦C is the temperature difference required to increase the volume of mercury in a thermometer by 1% of the volume change in heating from melting ice to boiling water. As we will see in the next chapter, there is a more sophis- ticated universal definition of temperature, to which scales based on mercury thermometers provide only a good approximation. After the temperature scale was established it became possible to carry out a quantitative study of the relation between volume and temperature, with pres- sure and mass kept fixed by enclosing the air in a vessel with flexible walls, which expand or contract to keep the pressure inside equal to the air pressure outside. This relation was announced in an 1802 lecture by Joseph Louis Gay- Lussac (1775ā€“1850), who attributed it to unpublished work in the 1780s by Jacques Charles (1746ā€“1823). The relation, subsequently known as Charlesā€™ Law, is that at constant pressure and mass the volume of gas is proportional to T āˆ’ T0, where T is the temperature measured for instance with a mercury thermometer and T0 is a constant whose numerical value naturally depends on the units used for temperature: T0 = āˆ’459.67 ā—¦F = āˆ’273.15 ā—¦C. Thus T0 is absolute zero, the minimum possible temperature, at which the gas vol- ume vanishes. Using Celsius units for temperature differences, the absolute temperature T ā‰” T āˆ’ T0 is known today as the temperature in degrees Kelvin, denoted K. Theoretical Explanations In Proposition 23 of his great book, the Principia, Isaac Newton (1643ā€“1727) made an attempt to account for Boyleā€™s law by considering air to consist of particles repelling each other at a distance. Using little more than dimensional analysis, he showed that the pressure p of a fixed mass of air is inversely proportional to the volume V if the repulsive force between particles separated by a distance r falls off as 1/r. But as he pointed out, if the repulsive force goes as 1/r2, then p āˆ V āˆ’4/3. He did not claim to offer any reason why the repulsive force should go as 1/r and, as we shall see, it is not forces that go as 1/r but
4 1 Early Atomic Theory rather forces of very short range that act only in collisions that mostly account for the properties of gases. It was the Swiss mathematical physicist Daniel Bernoulli (1700ā€“1782) who made the first attempt to understand the properties of gases theoretically, on the assumption that a gas consists of many tiny particles moving freely except in very brief collisions. In 1738, in the chapter, ā€œOn the Properties and Motions of Elastic Fluids, Especially Airā€ of his book Hydrodynamics, he argued that in a gas (then called an ā€œelastic fluidā€) with n particles per unit volume moving with a velocity v that is the same (because of collisions) in all directions, the pressure is proportional to n and to v2, because the number of particles that hit any given area of the wall in a given time is proportional to the number in any given volume, to the rate at which they hit the wall, which is proportional to v, and to the force that each particle exerts on the wall, which is also proportional to v. For a fixed mass of gas n is inversely proportional to the volume V , so pV is proportional to v2. If (as Bernoulli thought) v2 depends only on the temperature, this explains Boyleā€™s law. If v2 is proportional to the absolute temperature, it also gives Charlesā€™ law. Bernoulli did not give much in the way of mathematical details, and did not try to say to what else the pressure might be proportional besides nv2, a matter crucial for the history of chemistry. These details were provided by Rudolf Clausius (1822ā€“1888) in 1857, in an article entitled ā€œThe Nature of the Motion which We Call Heat.ā€ Below is a more-or-less faithful description of Clausiusā€™ derivation, in a somewhat different notation. Suppose a particle hits the wall of a vessel and remains in contact with it for a small time t, during which it exerts a force with component F along the inward normal to the wall. Its momentum in the direction of the inward normal to the wall will decrease by an amount Ft, so if the component of the velocity of the particle before it strikes the wall is vāŠ„ 0, and it bounces back elastically with normal velocity component āˆ’vāŠ„, the change in the inward normal component of momentum is āˆ’2mvāŠ„, where m is the particle mass, so F = 2mvāŠ„/t . Now, suppose that this goes on with many particles hitting the wall over a time interval T t, all particles with the same velocity vector v. The number N of particles that will hit an area A of the wall in this time is the number of particles in a cylinder with base A and height vāŠ„T , or N = nAvāŠ„T , where n is the number density, the number of particles per volume. Each of these particles is in contact with the wall for a fraction t/T of the time T , so the total force exerted on the wall is FN(t/T ) = 2mvāŠ„/t Ɨ nAvāŠ„T Ɨ (t/T ) = 2nmv2 āŠ„A .
1.1 Gas Properties 5 We see that all dependence on the times t and T cancels. The pressure p is defined as the force per area, so this gives the relation p = 2nmv2 āŠ„ . (1.1.1) This is for the unphysical case in which every particle has the same value of vāŠ„, positive in the sense that the particles are assumed to be going toward the wall. In the real world, different particles will be moving with different speeds in different directions, and Eq. (1.1.1) should be replaced with p = 2nm Ɨ 1 2 v2 āŠ„ = nmv2 āŠ„ , (1.1.2) the brackets indicating an average over all gas particles, with the factor 1/2 inserted in the first expression because only 50% of these particles will be going toward any given wall area. To express v2 āŠ„ in terms of the root mean square velocity, Clausius assumed without proof that ā€œon the average each direction [of the particle velocities] is equally represented.ā€ In this case, the average square of each component of velocity equals v2 āŠ„, and the average of the squared velocity vector is then v2 = v2 1 + v2 2 + v2 3 = 3v2 āŠ„ and therefore Eq. (1.1.2) reads p = nmv2 /3 . (1.1.3) This is essentially the result p āˆ nv2 of Bernoulli, except that, with the factor m/3, Eq. (1.1.3) is now an equality, not just a statement of proportion- ality. For a fixed mass M of gas occupying a volume V , the number density is n = M/mV , so Clausius could use Boyleā€™s law (which he called Mariotteā€™s law), which states that pV is constant for fixed temperature, to conclude that for a given gas v2 depends only on the temperature. Further, as Clausius remarked, Eq. (1.1.3) together with Charlesā€™ law (which Clausius called the law of Gay-Lussac) indicates that v2 is proportional to the absolute temper- ature T . If we like, we can adopt a modern notation and write the constant of proportionality as 3k/m, so that mv2 /3 = kT , (1.1.4) and therefore Eq. (1.1.3) reads p = nkT , (1.1.5) where k is a constant, in the sense of being independent of p, n, and T . But the choice of notation does not tell us whether k varies from one type of gas to another or whether it depends on the molecular mass m. Clausius could not answer this question, and did not offer any theoretical justification for Boyleā€™s
6 1 Early Atomic Theory law or Charlesā€™ law. Clausius deserves to be called the founder of thermo- dynamics, discussed in Sections 2.2 and 2.3, but these are not questions that can be answered by thermodynamics alone. As we will see in the following section, experiments in the chemistry of gases indicated that k is the same for all gases, a universal constant now known as Boltzmannā€™s constant, but the theoretical explanation for this and for Boyleā€™s law and Charlesā€™ law had to wait for the development of kinetic theory and statistical mechanics, the subject of Section 2.4. As indicated by the title of his article, ā€œThe Nature of the Motion which We Call Heat,ā€ Clausius was concerned to show that, at least in gases, the phe- nomenon of heat is explained by the motion of the particles of which gases are composed. He defended this view by using his theory to calculate the specific heat of gases, a topic to be considered in the next chapter. 1.2 Chemistry Elements The idea that all matter is composed of a limited number of elements goes back to the earliest speculations about the nature of matter. At first, in the century before Socrates, it was supposed that there is just one element: water (Thales) or air (Anaximenes) or fire (Heraclitus) or earth (perhaps Xenophanes). The idea of four elements was proposed around 450 BC by Empedocles of Acragas (modern Agrigento). In On Nature he identified the elements as ā€œfire and water and earth and the endless height of air.ā€ Classical Chinese sources list five elements: water, fire, earth, wood, and metal. Like the theory of atoms, these early proposals of elements did not come accompanied with any evidence that these really are elements, or any suggestion how such evidence might be gained. Plato in Timaeus even doubled down and stated that the difference between one element and another arises from the shapes of the atoms of which the elements are composed: earth atoms are tiny cubes, while the atoms of fire, air, and water are other regular polyhedra ā€“ solids bounded respectively by 4, 8, or 20 identical regular polygons, with every edge and every vertex of each solid the same as every other edge or vertex of that solid. By the end of the middle ages this list of elements had come to seem implau- sible. It is difficult to identify any particular sample of dirt as the element earth, and fire seems more like a process than a substance. Alchemists narrowed the list of elements to just three: mercury, sulfur, and salt. Modern chemistry began around the end of the eighteenth century, with careful experiments by Joseph Priestley (1733ā€“1804), Henry Cavendish (1743ā€“ 1810), Antoine Lavoisier (1743ā€“1794), and others. By 1787 Lavoisier had
1.2 Chemistry 7 worked out a list of 55 elements. In place of air there were several gases: hydrogen, oxygen, and nitrogen; air was identified as a mixture of nitrogen and oxygen. There were other non-metals on the list of elements: sulfur, carbon, and phosphorus, and a number of common metals: iron, copper, tin, lead, silver, gold, mercury. Lavoisier also listed as elements some chemicals that we now know are tightly bound compounds: lime, soda, and potash. And the list also included heat and light, which of course are not substances at all. Law of Combining Weights Chemistry was first used to provide quantitative information about atoms by John Dalton (1766ā€“1844), the son of a poor weaver. His laboratory notebooks from 1802 to 1804 describe careful measurements of the weights of elements combining in compounds. He discovered that these weights are always in fixed ratios. For instance, he found that when hydrogen burns in oxygen, 1 gram of hydrogen combines with 5.5 grams of oxygen, giving 6.5 grams of water, with nothing left over. Under the assumption that one particle of water consists of one atom of hydrogen and one atom of oxygen, one oxygen atom must weigh 5.5 times as much as one hydrogen atom. As we will see, water was soon discovered to be H2O: two atoms of hydrogen to each atom of oxygen. If Dalton had known this, he would have concluded that an oxygen atom weighs 5.5 times as much as two hydrogen atoms, i.e., 11 times the weight of one hydrogen atom. Of course, more accurate measurements later revealed that 1 gram of hydrogen combines with about 8 grams of oxygen, so one oxygen atom weighs eight times the weight of two hydrogen atoms, or 16 times as much as one hydrogen atom. Atomic weights soon became defined as the weights of atoms relative to the weight of one hydrogen atom, so the atomic weight of oxygen is 16. (This is only approximate. Today the atomic weight of the atoms of the most common isotope of carbon is defined to be precisely 12; with this definition, the atomic weights of the most common isotopes of hydrogen and oxygen are measured to be 1.007825 and 15.99491.) The following table compares Daltonā€™s assumed formulas for a few common compounds with the correct formulas: Compound Dalton formula True formula Water HO H2O Carbon dioxide CO2 CO2 Ammonia NH NH3 Sulfuric acid SO2 H2SO4
8 1 Early Atomic Theory Here is a list of the approximate true atomic weights for a few elements, the weights deduced by Dalton, and (in the column marked with an asterisk) the weights Dalton would have calculated if he had known the true chemical formulas. Element True Dalton Dalton* H 1 1 1 C 12 4.3 8.6 N 14 4.2 12.6 O 16 5.5 11 S 32 14.4 57.6 To make progress in measuring atomic weights, it was evidently necessary to find some way of working out the correct formulas for various chemical compounds. This was provided by the study of chemical reactions in gases. Law of Combining Volumes On December 31, 1808, Gay-Lussac read a paper to the Societe Philomathique in Paris, in which he announced his observation that gases at the same tem- perature and pressure always combine in definite proportions of volumes. For instance, two liters of hydrogen combine with one liter of oxygen to give water vapor, with no hydrogen or oxygen left over. Likewise, one liter of nitrogen combines with three liters of hydrogen to give ammonia gas, with nothing left over. And so on. The correct interpretation of this experimental result was given in 1811 by Count Amadeo Avogadro (1776ā€“1856) in Turin. Avogadroā€™s principle states that equal volumes of gases at the same temperature and pressure always con- tain equal numbers of the gas particles, which Avogadro called ā€œmolecules,ā€ particles that may consist of single atoms or of several atoms of the same or different elements joined together. The observation that water vapor is formed from a volume of oxygen combined with a volume of hydrogen twice as large shows, according to Avogadroā€™s principle, that molecules of water are formed from twice as many molecules of hydrogen as molecules of oxygen, which is not what Dalton had assumed. There was a further surprise in the data. Two liters of hydrogen combined with one liter of oxygen give not one but two liters of water vapor. This is not what one would expect if oxygen and hydrogen molecules consist of single atoms and water molecules consist of two atoms of hydrogen and one atom of oxygen. In that case two liters of hydrogen plus one liter of oxygen would produce one liter of water vapor. Avogadro could conclude that if, as seemed
1.2 Chemistry 9 plausible, molecules of water contain two atoms of hydrogen and one atom of oxygen, the molecules of oxygen and hydrogen must each contain two atoms. That is, taking water molecules as H2O, the reaction for producing molecules of water is 2H2 + O2 ā†’ 2H2O . The use of Avogadroā€™s principle rapidly provided the correct formulas for gases such as CO2, NH3, NO, and so on. Knowing these formulas and measuring the weights of gases participating in various reactions, it was possible to cor- rect Daltonā€™s atomic weights and calculate more reliable values for the atomic weights of the atoms in gas molecules, relative to any one of them. Taking the atomic weight of hydrogen as unity, this gave atomic weights close to 12 for carbon, 14 for nitrogen, 16 for oxygen, 32 for sulfur, and so on. Then, knowing these atomic weights, it became possible to find atomic weights for many other elements, not just those commonly found in gases, by measuring the weights of elements combining in various chemical reactions. The Gas Constant As we saw in the previous section, in 1857 Clausius had shown that in a gas consisting of n particles of mass m per volume with mean square velocity v2, the pressure is p = nmv2/3. Using Charlesā€™ law, he concluded that v2 is proportional to absolute temperature. Writing this relation as mv2/3 = kT with k some constant gives Eq. (1.1.5), p = nkT . But this in itself does not tell us how k varies from one gas to another. This is answered by Avogadroā€™s principle. With N particles in a volume V , the number density is n = N/V , so Eq. (1.1.5) can be written pV = NkT . (1.2.1) If as stated by Avogadro the number of molecules in a gas with a given pressure, volume, and temperature is the same for any gas, then k = pV/NT must be the same for any gas. Clausius did not draw this conclusion, perhaps because there was then no known theoretical basis for Avogadroā€™s principle. The universality of the constant k, and hence Avogadroā€™s principle, were explained later by kinetic theory, to be covered in the next chapter. The constant k came to be called Boltzmannā€™s constant, after Ludwig Boltzmann, who as we shall see was one of the chief founders of kinetic theory. The molecular weight Ī¼ of any compound is defined as the sum of the atomic weights of the atoms in a single molecule. The actual mass m of a molecule is its molecular weight times the mass m1 of a hypothetical atom with atomic weight unity: m = Ī¼m1 . (1.2.2)
10 1 Early Atomic Theory In the modern system of atomic weights, with the atomic weight of the most common isotope of carbon defined as precisely 12, m1 = 1.660539 Ɨ 10āˆ’24 g, which of course was not known in Avogadroā€™s time. A mass M contains N = M/m = M/m1Ī¼ molecules, so the ideal gas law (1.2.1) can be written pV = MkT/m1Ī¼ = (M/Ī¼)RT (1.2.3) where R is the gas constant R = k/m1 . (1.2.4) Physicists in the early nineteenth century could use Eq. (1.2.3) to measure R, and they found a value close to the modern value R = 8.314 J/K. This would have allowed a determination of m1 and hence of the masses of all atoms of known atomic weight if k were known, but k did not become known until the developments described in Section 2.6. Avogadroā€™s Number Incidentally, a mole of any element or compound of molecular weight Ī¼ is defined as Ī¼ grams, so in Eq. (1.2.3) the ratio M/Ī¼ expressed in grams equals the number of moles of gas. Since N = M/m1Ī¼, one mole contains a number of molecules equal to 1/m1 with m1 given in grams. This is known as Avogadroā€™s number. But of course Avogadro did not know Avogadroā€™s number. It is now known to be 6.02214 Ɨ 1023 molecules per mole, corresponding to unit molec- ular weight m1 = 1.66054 Ɨ 10āˆ’24 grams. The measurement of Avogadroā€™s number was widely recognized in the late nineteenth century as one of the great challenges facing physics. 1.3 Electrolysis Early Electricity Electricity was known in the ancient world, as what we now call static electricity. Amber rubbed with fur was seen to attract or repel small bits of light material. Plato in Timaeus mentions ā€œmarvels concerning the attraction of amber.ā€ (This is where the word electricity comes from; the Greek word for amber is ā€œelektron.ā€) Electricity began to be studied scientifically in the eighteenth century. Two kinds of electricity were distinguished: resinous electricity is left on an amber rod when rubbed with fur, while vitreous electricity is left on a glass rod when rubbed with silk. Unlike charges were found to attract each other, while like charges repel each other. Benjamin Franklin (1706ā€“1790) gave our modern terms positive and negative to vitreous and resinous electricity, respectively.
1.3 Electrolysis 11 In 1785 Charles-Augustin de Coulomb (1736ā€“1806) reported that the force F between two bodies carrying charges q1 and q2 separated by a distance r is F = keq1q2 r2 (1.3.1) where ke is a universal constant. For like and unlike charges the product q1q2 is positive or negative, respectively, indicating a repulsive or attractive force. Coulomb had no way of actually measuring these charges, but he could reduce the charge on a body by a factor 2 by touching it to an uncharged body of the same material and size, and observe that this reduces the force between it and any other charged body by the same factor 2. The introduction of our modern units of electric charge had to wait until the quantitative study of magnetism. Early Magnetism Magnetism too was known in the ancient world, as what we now call per- manent magnetism. The Greeks knew of naturally occurring lodestones that could attract or repel small bits of iron. Platoā€™s Timaeus refers to lodestones as ā€œHeraclean stones.ā€ (Our word magnet comes from the city Magnesia in Asia Minor, near where lodestones were commonly found.) Very early the Chinese also discovered the lodestone and used it as a magnetic compass (a ā€œsouth-seeking stoneā€) for purposes of geomancy and navigation. Each lodestone has a south-seeking pole at one end, attracted to a point near the South Pole of the Earth, and a north-seeking pole at the other end, attracted to a point near the Earthā€™s North Pole. Magnetism was first studied scientifically by William Gilbert (1544ā€“1603), court physician to Elizabeth I. It was observed that the south-seeking poles of different lodestones repel each other, and like- wise for the north-seeking poles, while the south-seeking pole of one lodestone attracts the north-seeking pole of another lodestone. Gilbert concluded that one pole of a lodestone is pulled toward the north and the other toward the south because the Earth itself is a magnet, with what in a lodestone would be its south-seeking and north-seeking poles respectively near the Earthā€™s North Pole and South Pole. Electromagnetism It began to be possible to explore the relations between electricity and mag- netism quantitatively with the invention in 1809 of electric batteries by Count Alessandro Volta (1745ā€“1827). These were stacks of disks of two different metals separated by cardboard disks soaked in salt water. Such batteries drive steady currents of electricity through wires attached to the ends of the stacks, with positive and negative terminals identified respectively as the ends of the stacks from which and towards which electric current flows.
12 1 Early Atomic Theory In July 1820 Hans Christian Oersted (1777ā€“1851) in Copenhagen noticed that turning on an electric current deflected a nearby compass needle, and concluded that electric currents exert force on magnets. Conversely, he found also that magnets exert force on wires carrying electric currents. These discoveries were carried further in Paris a few months later by AndrĆØ- Marie AmpĆØre (1775ā€“1836), who found that wires carrying electric current exert force on each other. For two parallel wires of length L carrying electric currents (charge per second) I1 and I2, and separated by a distance r L, the force is F = kmI1I2L r , (1.3.2) where km is another universal constant. The force is repulsive if the currents are in the same direction; attractive if in opposite directions. One ampere is defined so that F = 10āˆ’7 Ɨ L/r newtons if I1 = I2 = 1 ampere. (That is, km ā‰” 10āˆ’7 N/ampere2.) The electromagnetic unit of electric charge, the coulomb, is defined as the electric charge carried in one second by a current of one ampere. A modern ammeter measures electric currents by observing the magnetic force produced by current flowing through a wire loop. The connection between electricity and magnetism was strengthened in 1831 by Michael Faraday (1791ā€“1867), at the Royal Institution in London. He dis- covered that changing magnetic fields generate electric forces that can drive currents in conducting wires. This is the principle underlying the generation of electric currents today. Electricity began soon after to have important practical applications, with the invention in 1831 of the electric telegraph by the Ameri- can painter Samuel F. B. Morse (1791ā€“1872). Finally, in the 1870s, the great Scottish physicist James Clerk Maxwell (1831ā€“1879) showed that the consistency of the equation for the generation of magnetic fields by electric currents required that magnetic fields are also generated by changing electric fields. In particular, while oscillating magnetic fields produce oscillating electric fields, also oscillating electric fields produce oscillating magnetic fields, so a self-sustaining oscillation in both electric and magnetic fields can propagate in apparently empty space. Maxwell calculated the speed of its propagation and found it to equal āˆš 2ke/km,2 numerically about equal to the measured speed of light, suggesting strongly that light is such a self-sustaining oscillation in electric and magnetic fields. We will see more of Maxwellā€™s equations in subsequent chapters, especially in Chapters 4 and 5. 2 This quantity is independent of the units used for electric charge as long as the currents appearing in Eq. (1.3.2) are defined as the rates of flow of charge in the same units as used in Eq. (1.3.1). It is obviously also independent of the units used for force, as long as the same force units are used in Eqs. (1.3.1) and (1.3.2).
1.3 Electrolysis 13 Discovery of Electrolysis Electrolysis was discovered in 1800 by the chemist William Nicholson (1753ā€“ 1815) and the surgeon Anthony Carlisle (1768ā€“1840). They found that bubbles of hydrogen and oxygen would be produced where wires attached respectively to the negative and positive terminals of a Volta-style battery were inserted in water. Sir Humphrey Davy (1778ā€“1829), Faradayā€™s boss at the Royal Institution, carried out extensive experiments on the electrolysis of molten salts, finding for instance that, in the electrolysis of molten table salt, sodium, a previously unknown metal, was produced at the wire attached to the negative terminal of the battery and a greenish gas, chlorine, was produced at the wire attached to the other, positive, terminal. Davyā€™s electrolysis experiments added several metals aside from sodium to Lavoisierā€™s list of elements, including aluminum, potassium, calcium, and magnesium. A theory of electrolysis was worked out by Faraday. In modern terms, a small fraction (1.8 Ɨ 10āˆ’9 at room temperature) of water molecules are nor- mally dissociated into positive hydrogen ions (H+), which are attracted to the wire attached to the negative terminal of a battery, and negative hydroxyl ions (OHāˆ’), which are attracted to the wire attached to the positive terminal. At the wire attached to the negative terminal, two H+ ions combine with two units of negative charge from the battery to form a neutral H2 molecule. At the wire attached to the positive terminal, four OHāˆ’ ions give one O2 molecule plus two H2O molecules plus four units of negative charge, which flow through the battery to the negative terminal.3 Likewise, a small fraction of molten table salt (NaCl) molecules are normally dissociated into Na+ ions and Clāˆ’ ions. At the wire attached to the negative terminal of a battery, one Na+ ion plus one unit of negative charge gives one atom of metallic sodium (Na); at the wire attached to the positive terminal, two Clāˆ’ ions give one chlorine (Cl2) molecule and two units of negative charge, which flow through the battery to the negative terminal. In Faradayā€™s theory, it takes one unit of electric charge to convert a singly charged ion such as H+ or Clāˆ’ to a neutral atom or molecule, so since molecules of molecular weight Ī¼ have mass Ī¼m1, it takes M/m1Ī¼ units of electric charge to convert a mass M of singly charged ions to a mass M of neutral atoms or molecules of molecular weight Ī¼. Experiment showed that it takes about 96 500 coulombs (e.g., one ampere for about 96 500 seconds) to convert Ī¼ grams (that is, one mole) of singly charged ions to neutral atoms or molecules. (This is called a faraday; the modern value is 96 486.3 coulombs/mole.) Hence 3 We now know that it is negative charge, i.e., electrons, that flows through a battery. As far as Faraday knew, it was equally possible that positive charges flow through a battery, in which case at the wire attached to the negative terminal two H+ ions would give an H2 molecule plus two units of positive charge, which would flow though the battery to the wire attached to the positive terminal, where four OHāˆ’ ions plus four units of positive charge would give an O2 molecule and two H2O molecules.
14 1 Early Atomic Theory Faraday knew that e/m1 96 500 coulombs/gram, where e is the unit of electric charge, which was called an ā€œelectrineā€ in 1874 by the Irish physi- cist George Johnstone Stoney (1826ā€“1911). Having measured the faraday, if physicists knew the value of e then they would know m1, but they didnā€™t have this information until later. Also, no one then knew that e is the charge of an actual particle. 1.4 The Electron As sometimes happens, in 1858 a new path in fundamental physics was opened with the invention of a practical device, in this case an improved air pump. In his pump the Bonn craftsman Heinrich Johann Geissler (1814ā€“1879) used a column of mercury as a piston, in this way greatly reducing the leakage of air through the piston that had troubled all previous air pumps. With his pump Geissler was able to reduce the pressure in a closed glass tube to about a ten-thousandth of the typical air pressure on the Earthā€™s surface. With such a near vacuum in a glass tube, electric currents could travel without wires through the tube. It was discovered that an electric current would flow from a cathode, a metal plate attached to the negative terminal of a powerful electric battery, fly through a hole in an anode, another metal plate attached to the positive pole of the battery, and light up a spot on the far wall of the tube. Adding small amounts of various gases to the interior of the tube caused these cathode rays to light up, with orange or pink or blue-green light emitted along the path of the ray, when neon, helium, or mercury vapor was added. Using Geisslerā€™s pumps, Julius PlĆ¼cker (1801ā€“1868) in 1858ā€“1859 found that cathode rays could be deflected by magnetic fields, thus moving the spot of light where the ray hits the glass at the tube end. In 1897 Joseph John Thomson (1856ā€“1940), the successor to Maxwell as Cavendish Professor at Cambridge, began a series of measurements of the deflection of cathode rays. In his experiments, after the ray particles pass through the anode they feel an electric or magnetic force F exerted at a right angle to their direction of motion for a distance d along the ray. They then drift in a force-free region for a distance D d until they hit the end of the tube. If a ray particle has velocity v along the direction of the ray, it feels the electric or magnetic force for a time d/v and then drifts for a longer time D/v. A force F normal to the ray gives ray particles of mass m a component of velocity perpendicular to the ray that is equal to the acceleration F/m times the time d/v, so by the time they hit the end of the tube they have been displaced by an amount displacement = (F/m) Ɨ (d/v) Ɨ (D/v) = FdD mv2 .
1.4 The Electron 15 The forces exerted on a charge e by an electric field E or a magnetic field B at right angles to the ray are Felec = eE , Fmag = evB so electric displacement = eEdD mv2 , magnetic displacement = eBdD mv . Thomson wanted to measure e/m. He knew D, d, E, and B, but not v. He could eliminate v from these equations if he could measure both the electric and magnetic displacements, but the electric displacement was difficult to measure. A strong electric field tends to ionize any residual air in the tube, with positive and negative ions pulled to the negatively and positively charged plates that produce the electric fields, neutralizing their charges. Finally Thomson suc- ceeded in measuring the electric as well as the magnetic deflection by using a cathode ray tube with very low air pressure. (Both the electric and magnetic displacements were only a few inches.) This gave results for the ratio of charge to mass ranging from 6 Ɨ 107 to 108 coulombs per gram. Thomson compared this with the result that Faraday had found in measure- ments of electrolysis, that e/m1 ā‰ˆ 105 coulombs per gram, where e is the electric charge of a singly ionized atom or molecule (such as a sodium ion in the electrolysis of NaCl) and m1 is the mass of a hypothetical atom of atomic weight unity, close to the mass of the hydrogen atom. He reasoned that if the particles in his cathode rays are the same as those transferred in electrolysis, then their charge must be the same as e, so their mass must be about 10āˆ’3m1. Thomson concluded that since the cathode ray particles are so much lighter than ions or atoms, they must be the basic constituents of ions and atoms. Thomson had still not measured e or m. He had not even shown that cathode rays are streams of particles; they might be streams of electrically charged fluid, with any volume of fluid having a ratio of charge to mass equal to his measured e/m. Nevertheless, in the following decade it became widely accepted that Thomson had indeed discovered a particle present in atoms, and the particle came to be called the electron.
2 Thermodynamics and Kinetic Theory The successful uses of atomic theory described in the previous chapter did not settle the existence of atoms in all scientistsā€™ minds. This was in part because of the appearance in the first half of the nineteenth century of an attractive competitor, the physical theory of thermodynamics. As we shall see in the first three sections of this chapter, with thermodynamics one may derive powerful results of great generality without ever committing oneself to the existence of atoms or molecules. But thermodynamics could not do everything. Section 2.4 will describe the advent of kinetic theory, which is based on the assumption that matter consists of very large numbers of particles, and its generalization to sta- tistical mechanics. From these thermodynamics could be derived, and together with the atomic hypothesis it yielded results far more powerful than could be obtained from thermodynamics alone. Even so, it was not until the appearance of direct evidence for the graininess of matter, described in Section 2.5, that the existence of atoms became almost universally accepted. 2.1 Heat and Energy The first step in the development of thermodynamics was the recognition that heat is a form of energy. Though so familiar to us today, this was far from obvious to the physicists and chemists of the early nineteenth century. Until the 1840s heat was widely regarded as a fluid, named caloric by Lavoisier. Caloric theory was used to calculate the speed of sound by Pierre-Simon Laplace (1749ā€“1827) in 1816, the conduction of heat by Joseph Fourier (1768ā€“1830) in 1807 and 1822, and the efficiency of steam engines by Sadi Carnot (1796ā€“1832) in 1824, whose work as we will see in the next section became a foundation of thermodynamics. Adding to the confusion, other scientists considered heat as some sort of wave. This reflected uncertainty regarding the nature of what is now called infrared radiation, discovered by William Herschel (1738ā€“1822) in 1800. 16
2.1 Heat and Energy 17 Heat as Energy In 1798 Benjamin Thompson (1753ā€“1814), an American expatriate in Eng- land, offered evidence against the idea that heat is a fluid. (Thompson is also known as Count Rumford, a title he was given when he later served as military adviser in Austria.) It was well known that boring a cannon produces heat, which might be supposed to be due to the liberation of caloric from the iron, but Rumford observed that if the heat is carried away by immersing the cannon in running water while it is being bored there is no limit to the heat that can be produced. The first measurement of the energy in heat was provided in the mid-1840s by James Prescott Joule (1818ā€“1889). In his apparatus a falling weight turned paddles in a tank of water, heating the water. The gravitational force on a mass m kilograms is m times the acceleration of gravity, 9.8 meters/sec2 or 9.8 newtons per kilogram. Work is force times distance, so dropping one kilogram a distance of one meter gave it an energy equal to 9.8 newton meters, now also known as 9.8 joules. Joule found that the paddles driven by this dropping weight would raise the temperature of 100 grams of water by 0.023 ā—¦C, so the paddles pro- duced heat equal to 0.023 Ɨ 100 calories, the calorie being defined as the heat required to raise the temperature of one gram of water by one degree Celsius. Hence Joule could conclude that 9.8 joules is equivalent to 2.3 calories, so one calorie is equivalent to 9.8/2.3 = 4.3 joules. The modern value is 4.184 joules. In 1847 the Prussian physician and physicist Hermann von Helmholtz (1821ā€“1894) put forward the idea of the universal conservation of energy, whether in the form of kinetic or potential or chemical energy or heat. But what sort of energy is heat? For some nineteenth century physicists the question was irrelevant. They developed the science of heat known as thermodynamics, which did not depend on any detailed model of heat energy. But there was one context in which the nature of heat energy seemed evident. In his great 1857 paper, The Nature of the Motion which We Call Heat, Clausius found that at least part of the heat energy of gases is the kinetic energy of their molecules. Kinetic Energy The concept of kinetic energy was long familiar. If a steady force F is exerted on a particle of mass m, it produces an acceleration F/m, so after a time t the velocity of the body is v = Ft/m. The distance traveled in this time is t times the average velocity v/2, and the work done on the particle is the force times this distance: F Ɨ t Ɨ Ft/2m = F2 t2 /2m = mv2 /2 . Instead of this work going into heating a tub of water, as in the experiment of Joule, it goes into giving the particle an energy mv2/2.
18 2 Thermodynamics and Kinetic Theory This energy has the special property of being conserved when bodies come into contact in collisions. Consider a collision between two rigid balls A and B with initial vector velocities vA and vB. For the moment suppose that the time interval t over which this force acts is sufficiently brief that the forces acting on the balls do not change appreciably during this time. The force that A exerts on B is equal and opposite to the force F that B exerts on A, so Newtonā€™s second law tells us that the final velocities of A and B are v A = vA + Ft/mA and v B = vB āˆ’ Ft/mB. Hence, as Newton showed, momentum is conserved: mAv A + mBv B = mAvA + mBvB . (2.1.1) Neglecting changes in acceleration during the brief time t, the vector displace- ments traveled by A and B equal t times the average velocities, [vA + v A]/2 and [vB + v B]/2, respectively. If the balls remain in contact during this time interval, then these displacements must be the same, so vA + v A = vB + v B . (2.1.2) To derive a second conservation law, rewrite Eq. (2.1.2) as v B āˆ’ v A = vA āˆ’ vB and square this, giving v2 B āˆ’ 2v B Ā· v A + v2 A = v2 B āˆ’ 2vB Ā· vA + v2 A . Multiply this with mAmB and add the square of Eq. (2.1.1), so that the scalar products cancel. Dividing by 2(mA+mB), the result is another conservation law, mA 2 v2 A + mB 2 v2 B = mA 2 v2 A + mB 2 v2 B . (2.1.3) Equations (2.1.1) and (2.1.3) have been derived here only for the case in which the particles are in contact only for a brief time interval during which the force acting between the bodies is constant, but this is not an essential requirement for we can break up any time interval into a large number of brief intervals in each of which the change in the force is negligible, Then, since mAvA + mBvB and mAv2 A/2+mBv2 B/2 do not change in each interval, they do not change at all, as long as the bodies exert forces on each other only when they are in contact. In 1669 Christiaan Huygens (1629ā€“1695) reported in Journal des SƧavans that he had confirmed the conservation of the total of mv2/2, probably by observing collisions of pendulum bobs, for which initial and final velocities could be precisely determined. Newton in the Principia called the conserved quantity mv the quantity of motion, while Huygens gave the name vis viva (ā€œliving forceā€) to the conserved quantity mv2/2. These two quantities have since become known as momentum and kinetic energy. On the other hand, it was essential in deriving the conservation of kinetic energy that we assumed that particles interact only when in contact. This is generally a good approximation in gases, but it is not valid in the presence of
2.1 Heat and Energy 19 long range forces, such as electromagnetic or gravitational forces. In such cases kinetic energy is not conserved ā€“ it is only the sum of kinetic energy plus some sort of potential energy that does not change. Specific Heat The total kinetic energy of N molecules of gas of mass m and mean square velocity v2 is Nmv2/2. Clausius had found the relation (1.1.4) between mean square velocity and absolute temperature, according to which mv2/2 = 3kT/2, where k is some constant (later identified as a universal constant of nature), so the total kinetic energy is 3NkT/2. A mass M of gas of molecu- lar weight Ī¼ contains N = M/Ī¼m1 molecules, so the total kinetic energy is 3MRT/2Ī¼, where R = k/m1 is the gas constant (1.2.4). Clausius concluded that to raise the temperature of a mass M of gas of molecular weight Ī¼ by an amount dT at constant volume, so that the gas does no work on its con- tainer, requires an energy dE = 3MRdT/2Ī¼. The ratio dE/MdT is known as the specific heat, so Clausius found that the specific heat of a gas at constant volume is Cv = 3R/2Ī¼ . (2.1.4) This result must be distinguished from the value for a different sort of specific heat, measured at constant pressure, such as when the gas is in a container with an expandable wall, for which the volume V can change to keep the pressure p equal to the pressure of the surrounding air or other medium. When pressure pushes a surface of area A a small distance dL, the work done is the force pA times dL, which equals pdV where dV = AdL is the change in volume. According to the ideal gas law (1.2.3), pV = RT M/Ī¼, so if the temperature is increased by an amount dT , then at constant pressure the gas does work pdV = RMdT/Ī¼, and this temperature increase therefore requires an energy 3MRdT/2Ī¼ + MRdT/Ī¼ = 5MRdT/3. In other words, the specific heat at constant pressure is Cp = 5R/2Ī¼ . (2.1.5) This result is often expressed in terms of the ratio of specific heats, Ī³ ā‰” Cp/Cv . (2.1.6) So Clausius found that if all the heat of a gas is contained in the kinetic energy of its molecules then Ī³ = 5/3. This did not agree with measurements of the specific heats of common diatomic gases, such as oxygen or hydrogen, which Clausius cited as giving Ī³ = 1.421. Later, it was found that Ī³ does indeed equal 5/3 for a monatomic gas like mercury vapor, but this left the question, in what form is the energy in ordinary gases that are not monatomic?
20 2 Thermodynamics and Kinetic Theory To deal with this issue, Clausius suggested that the internal energy of a gas is larger than the kinetic energy of the molecules, say by a factor 1 + f , with f some positive number. Then instead of Eq. (2.1.4) we have Cv = (1 + f ) Ɨ 3R/2Ī¼ , (2.1.7) and in place of Eq. (2.1.5), Cp = (1 + f )3R/2Ī¼ + R/Ī¼ . (2.1.8) The specific heat ratio is then Ī³ = 1 + 2 3(1 + f ) . (2.1.9) This is often expressed (especially in astrophysics) as a formula for the internal energy density E in terms of the pressure and Ī³ : E = 3RT M(1 + f )/2Ī¼V = 3(1 + f )p/2 = p/(Ī³ āˆ’ 1) . (2.1.10) The observation that Ī³ 1.4 for diatomic gases like O2 and H2 indicated that the internal energy of these gases is larger than the kinetic energy of its molecules by a factor 1 + f 5/3. Measurements gave values of Ī³ for more complicated molecules like H2O or CO2 even closer to unity, indicating that f is even larger for these molecules. The reason for these values for f and Ī³ did not become clear until the formulation of the equipartition of energy, to be discussed in Section 2.4. Adiabatic Changes It often happens that work is done adiabatically, that is, without the transfer of heat. In this case the conservation of energy tells us that the work done by an expanding fluid must be balanced by a decrease in its internal energy EV : 0 = p dV + d(EV ) = (p + E)dV + V dE . (2.1.11) For an ideal gas, the internal energy per unit volume E is given by Eq. (2.1.10), so this tells us that 0 = Ī³p dV + V dp and so, in an adiabatic process, p āˆ V āˆ’Ī³ āˆ ĻĪ³ , (2.1.12) or, since for a fixed mass p āˆ T/V , T āˆ V 1āˆ’Ī³ āˆ ĻĪ³ āˆ’1 . (2.1.13) This is in contrast with an isothermal process, for which T is constant and p āˆ V āˆ’1.
2.2 Absolute Temperature 21 Equation (2.1.12) has an immediate consequence for the speed of sound. At audible frequencies the conduction of heat is typically too slow to be effective, so the expansion and compression of a fluid carrying a sound wave is adiabatic. It is a standard result of hydrodynamics, proved by Newton, that the speed of sound is cs = āˆ‚p āˆ‚Ļ . Newton thought that p would be proportional to Ļ in a sound wave, which would give cs = āˆš p/Ļ, but in fact at audible frequencies the pressure is given by the adiabatic relation (2.1.12), and cs is larger than Newtonā€™s value by a factor āˆš Ī³ . 2.2 Absolute Temperature We have been casually discussing temperature, but what precisely do we mean by this? It is not hard to give a precise meaning to a statement that one body has a higher temperature than another, by a generalization of common experience that is sometimes known as the second law of thermodynamics (the first law being the conservation of energy). Observation of heat flow shows that if heat can flow spontaneously from a body A to a body B, then it cannot flow spontaneously from B to A. We can say then that the temperature tA of A is higher than the temperature tB of B. Likewise, we say that two bodies are at the same temperature if heat cannot flow from either one to the other spontaneously, without work being done on these bodies. Temperature defined in this way is observed to be transitive: If heat can flow spontaneously from a body A to a body B, and from B to a body C, then it can flow spontaneously from A to C. This is a property shared with real numbers ā€“ if a number a is larger than number b, and if b is larger than c, then a is larger than c ā€“ and is a necessary condition for temperatures to be represented by real numbers. But this does not give a precise meaning to any particular numerical value of temperature, or even to numerical ratios of temperatures. If, for some definition of temperature t, a comparison of values of t tells us the direction of heat flow then the same would be true of any monotonic function T (t). Conventionally temperatures are defined by thermometers. With a column of some liquid such as mercury or alcohol in a glass tube, we mark off the heights of the column when the tube is placed in freezing or boiling water, and for a Celsius tem- perature scale etch on the tube marks that divide the distance from freezing to boiling to a hundred equal parts. The trouble is that different liquids expand differently with increasing temperature, and the temperatures measured in this way with a mercury or alcohol thermometer will not be precisely equal. We can try instead to give significance to numerical values of temperature by using a
22 2 Thermodynamics and Kinetic Theory gas thermometer, relying on the ideal gas law pV = MRT/Ī¼, but this law is approximate, holding precisely only for molecules of negligible size that interact only in contact in collisions. How can we give precise meaning to numerical values of temperature without relying on approximate relations? Surprisingly, as shown by Rudolf Clausius in his 1850 paper1 ā€œOn the Mov- ing Force of Heat,ā€ it is possible by find a definition of temperature T with absolute significance by the study of thermodynamic engines known as Carnot cycles. Sadi Carnot (1796ā€“1832) was a French military engineer, the son of Lazare Carnot, organizer of military victory in the French Revolution, and uncle of a later president of the Third Republic. In 1824 Carnot in Reflections on the Motive Power of Fire set out to study the efficiency of steam engines, explaining that ā€œAlready the steam engine works our mines, impels our ships, excavates our ports and our rivers, forges iron, fashions wood, grinds grains, spins and weaves our clothes, transports our heaviest burdens, etc.ā€ (A few years later he might also have mentioned the beginning of steam-propelled locomotives, with the opening of the Liverpoolā€“Manchester railroad in 1830.) Carnot invented an idealized engine, known as a Carnot cycle, which as we shall see is maximally efficient and provides a natural definition of absolute temperature. In the Carnot cycle, a working fluid (such as steam in a cylinder fitted with a piston) goes through four frictionless steps: 1. Isothermal: The working fluid does work on its environment, for instance by pushing a piston against external pressure, but keeping a constant tem- perature by absorbing heat Q2 from a hot reservoir at temperature t2. (We will continue to use lower case t to indicate temperature defined in any way that indicates the direction of heat flow, without specifying any physical significance to its particular numerical values.) 2. Adiabatic: The working fluid, perfectly insulated from its environment and with no internal friction, does more work, with its temperature dropping to the temperature t1 of a cold reservoir but with no heat flowing in or out. 3. Isothermal: Work is done on the fluid, for instance by pushing in the piston, with its temperature kept constant by its giving up heat Q1 to the cold reservoir. 4. Adiabatic: With the working fluid again completely insulated from its envi- ronment, work is done on it, bringing its volume back down to its original value and its temperature back up to the temperature t2 of the hot reservoir. 1 This paper is reprinted in Brush, The Kinetic Theory of Gases ā€“ An Anthology of Classic Papers with Historical Commentary, listed in the bibliography.
2.2 Absolute Temperature 23 C B 1 2 3 4 p D A V Figure 2.1 A Carnot cycle (not drawn to scale). A graph of the pressure versus the volume of the working fluid in this cycle is a closed curve, with the net work W done on the environment equal to pdV ā€“ that is, to the area enclosed by the curve. (See Figure 2.1.) As long as steps 2 and 4 are truly adiabatic, the conservation of energy tells us that this work is W = Q2 āˆ’ Q1 (2.2.1) and the efficiency of this cycle is W/Q2 = Q2 āˆ’ Q1 Q2 1 . (2.2.2) (We call this the efficiency, having in mind that, as for a steam engine, we have to pay for the heat Q2 taken up at the higher temperature t2, while the heat Q1 given up at the lower temperature t1 is wasted.) Any Carnot cycle is reversible, because any frictionless adiabatic or isother- mal process follows the same track, depending only on its endpoints, whichever direction the process takes. But not all thermodynamic cycles, which take a working fluid through a series of steps back to the original temperature and volume, are reversible even though of course they all conserve energy. For reversibility it is not enough that all steps be either isothermal or adiabatic ā€“ there also should be no friction, which if present would provide an internal source of heat that is not available to do work.
24 2 Thermodynamics and Kinetic Theory The importance of Carnot cycles in thermodynamics rests on the following theorem: I. The efficiency of the Carnot cycle C described above is at least as great as that for any general thermodynamical cycle C, not necessarily reversible, which begins with the working fluid absorbing a heat Q 2 from a reservoir at the same high temperature t2, then emitting heat at the same lower temperature t1, and then returning to its original temperature and volume, in the process doing net work W. That is, W/Q2 ā‰„ W /Q 2 . (2.2.3) II. All Carnot cycles that take heat from a reservoir at the same temperature t2, using it to do work, and giving up waste heat to a reservoir at the same lower temperature t1, have the same efficiency, which depends only on t2 and t1. Proof:2 Like any positive real number, the ratio of the work done in the Carnot cycle C and in a general cycle C can be approximated to an arbitrary accuracy by a ratio of positive real integers N and N: W/W = N /N . (2.2.4) Since any Carnot cycle by definition is reversible, the cycle C has an inverse Cāˆ’1. This is a refrigeration cycle, following the same steps as for C but in the opposite order, so that by doing work W on the fluid an amount of heat Q1 is taken from the reservoir at temperature t1 and heat Q2 Q1 is delivered to the reservoir at temperature t2 t1. Suppose we perform a compound cycle Cāˆ—, consisting of N repetitions of Cāˆ’1 and N repetitions of C. According to Eq. (2.2.4), the net work done by the working fluid is Wāˆ— = N W āˆ’ NW = 0 . Also, the net heat taken from the hot reservoir at temperature t2 is Qāˆ— 2 = N Q 2 āˆ’ NQ2 . Now, since no work is done in the compound cycle, according to the fundamen- tal property of temperature t, it is not possible for positive-definite net heat to be transferred to a reservoir at temperature t2 from a lower temperature t1, so the net heat Qāˆ— 2 taken from the hot reservoir in the cycle Cāˆ— must be positive-definite or zero. Hence, using Eq. (2.2.4), 0 ā‰¤ Qāˆ— 2 NW = NQ 2 āˆ’ NQ2 NW = Q 2 W āˆ’ Q2 W , 2 This treatment and that of the following section is based on that given by Fermi, Thermodynamics, listed in the bibliography.
2.2 Absolute Temperature 25 and therefore W Q2 ā‰„ W Q 2 (2.2.5) as was to be proved in the first part of the theorem. As to the second part of the theorem, note that if C is also a Carnot cycle then, by the same reasoning, W Q 2 ā‰„ W Q2 , so the efficiencies are equal: W Q2 = W Q 2 . (2.2.6) This has now been proved for any pair of Carnot cycles, operating between the same temperatures t2 and t1, whatever the values of the heat taken from the reservoir at temperature t2 and given up to the reservoir at temperature t1, so the common efficiency can only depend on t2 and t1, as was to be proved. We shall write this relation in terms of the inefficiency: 1 āˆ’ W Q2 = Q1 Q2 ā‰” F(t1, t2) (2.2.7) with F the same function for all Carnot cycles. We next prove that the function F(t1, t2) takes the form F(t1, t2) = T (t1)/T (t2) (2.2.8) for some function T (t). For this purpose we consider a compound cycle consist- ing of a Carnot cycle operating between the temperatures t2 and t0 ā‰¤ t2 followed by a Carnot cycle operating between the temperatures t0 and t1 ā‰¤ t0, with all the waste heat that is given to the reservoir at temperature t0 in the first cycle taken up from this reservoir in the second cycle. Since (Q0/Q2)(Q1/Q0) = Q1/Q2, the inefficiency (2.2.7) of the compound cycle is the product of the inefficiencies of the individual cycles, so F(t1, t2) = F(t1, t0)F(t0, t2) . (2.2.9) From Eq. (2.2.7) it is evident that F(t2, t0)F(t0, t2) = 1, so Eq. (2.2.9) may be written F(t1, t2) = F(t1, t0) F(t2, t0) . (2.2.10) This holds for any t0 with t2 ā‰„ t0 ā‰„ t1, so we can define T (t) ā‰” F(t, t0) with an arbitrary choice of t0 in this range, and then Eq. (2.2.10) is the desired result (2.2.8).
26 2 Thermodynamics and Kinetic Theory Now, efficiencies are never greater than 100%, so the ratio F(t1, t2) = T (t1)/T (t2) in Eq. (2.2.7) must be positive, and so T (t) has the same sign for all temperatures. Since only the ratios of the T s appear in the efficiency, we are free to choose this sign to be positive, so that T (t) ā‰„ 0 for all t. Also, inefficiencies are never greater than 100%, so Eq. (2.2.8) shows that T (t1) ā‰¤ T (t2) for any t1 and t2 with t1 ā‰¤ t2. That is, T (t) is a monotonically increasing function of t and can therefore be used to judge the direction of spontaneous heat flow as well as t itself. We can therefore define the absolute temperature T by just using T (t) as the temperature in place of t. That is, using Eqs. (2.2.7) and (2.2.8), we define absolute temperature T by the statement that a Carnot cycle running between any two temperatures T2 and T1 has Q1 Q2 = T1 T2 . (2.2.11) A Carnot cycle running between an upper temperature T2 and a lower tempera- ture T1 has an efficiency W Q2 = Q2 āˆ’ Q1 Q2 = T2 āˆ’ T1 T2 . (2.2.12) Of course, this only defines T up to a constant factor, leaving us free to use what units we like for temperature. But we are not free to shift T (t) by adding a constant term. Indeed, since in this Carnot cycle heat flows from a reservoir at temperature T2 to one at temperature T1, we must have T2 T1, and therefore in order for the efficiency (2.2.11) to be a positive quantity, the lower temperature must have T1 0. Because any heat reservoir must have T positive-definite, we see that T is the absolute temperature, in the same sense as was found for gases by Charles. The temperature defined by Carnot cycles is identical (up to a choice of units) to the temperature given by a gas thermometer, which for the moment we will call T g, in the approximation that the gas is ideal. To see this, let us label the states of the gas as A at the start of the isothermal expansion 1 (and at the end of the adiabatic compression 4); as B at the start of the adiabatic expansion 2 (and the end of the isothermal expansion 1); as C at the start of the isothermal compression 3 (and the end of the adiabatic expansion 2); and as D at the start of the adiabatic compression 4 (and the end of the isothermal compression 3). Since the expansion from A to B is isothermal, during this phase the internal energy of the gas, which is given by Eqs. (1.2.3) and (2.1.10) as EV = RT M/(Ī³ āˆ’ 1)Ī¼, does not change, and so the heat drawn from the hot reservoir is the work done: Q2 = B A pdV = MRT g 2 Ī¼ B A dV V = MRT g 2 Ī¼ ln VB VA .
2.3 Entropy 27 Likewise, the heat given up to the cold reservoir in the isothermal compression from C to D is Q1 = MRT g 1 Ī¼ ln VC VD . Further, since the expansion from B to C and the contraction from D to A are adiabatic, Eq. (2.1.13) gives V āˆ (T g)āˆ’1/(Ī³ āˆ’1), and so during these parts of the cycle VC/VB = VD/VA = T g 2 T g 1 1/(Ī³ āˆ’1) , and therefore VB/VA = VC/VD, and the logarithmic factors in Q2 and Q1 are equal. The efficiency is then Q2 āˆ’ Q1 Q2 = T g 2 āˆ’ T g 1 T g 2 in agreement with Eq. (2.2.12) if T = T g, up to a possible constant factor. 2.3 Entropy In macroscopic classical thermodynamics we characterize the state of a system by a set of variables that can be specified independently. For instance, for a fluid of fixed mass and chemical composition, in a vessel with adjustable volume (say, with a movable piston) the state is specified by giving the values of any two of the thermodynamic variables ā€“ pressure, volume, temperature, energy, etc. ā€“ the remaining variables being determined in equilibrium or in adiabatic variations by these two values and some equation of state, such as the ideal gas law (1.2.3). Many of the consequences of macroscopic classical thermody- namics can be deduced from the existence of another thermodynamic variable, known as the entropy, introduced in 1854 by Rudolf Clausius, that like other thermodynamic variables depends only on the state of the system, although its definition seems to indicate that it also depends on the way that the system is prepared. Suppose a system is prepared in a given state 1 by starting with it in a standard state (labeled 0 below) and then taken to 1 on a path P through the space of independent variables used to define thermodynamic states, in which by a series of small reversible changes at varying absolute temperatures T it picks up small net amounts dQ of heat energy from the environment. (The heat energy increment dQ is taken as positive if the system takes heat from the environment and negative if it gives up heat.) Then the entropy of this state is defined by
28 2 Thermodynamics and Kinetic Theory S1 = S0 + P dQ T , (2.3.1) where the integral is taken over any reversible path from state 0 to state 1, and S0 is whatever entropy we choose to ascribe to the standard state 0. The remarkable thing is that the integral here is independent of the particular reversible path chosen, so that this really defines the entropy S1 up to a common constant term S0 as a function of the state of the system, not of how it is prepared, provided that T is the absolute temperature defined, as in the previous section, by the efficiency of Carnot cycles. Furthermore, with the entropy defined in this way, for any path P from state 0 to state 1 that may or may not be reversible, we have P dQ T ā‰¤ S1 āˆ’ S0 . (2.3.2) Proof: The first step in proving these results is to prove the following lemma: for an arbitrary cycle, reversible or irreversible, that takes a system from any state back to the same state, taking in and giving up heat at various temperatures, we have dQ T ā‰¤ 0 . (2.3.3) After establishing this lemma, the rest of the proof will be straightforward. To prove this lemma we can approximate the cycle by a sequence of brief isothermal steps, in each of which the system takes in heat (if dQ is positive) or gives heat up (if dQ is negative) at a momentary temperature T . We can imagine that, at each step, the heat taken in or given up is given up or taken in by another system, which undergoes a Carnot cycle between the momentary temperature T and a fixed temperature T0. In this Carnot cycle, the ratio of the heat dQ given up by the Carnot cycle to and the heat dQ0 taken by the Carnot cycle from the reservoir at temperature T0 is given by Eq. (2.2.11): dQ dQ0 = T T0 , or in other words dQ0 = T0 dQ T . Hence in the complete cycle the Carnot cycles take in a total net heat T0 dQ/T from the reservoir at temperature T0. Since the system and each of the Carnot cycles return to their original states, if this heat taken in at temperature T0 were positive-definite then it would have to go into work, which is impossible since work cannot be done by taking heat from a reservoir at a fixed temperature with no changes elsewhere. (If it could, then this work by producing friction could
2.3 Entropy 29 be used to transfer some heat to any body, even one at a temperature higher than T0.) So we conclude that the integral dQ/T is at most zero, as was to be shown. The rest is easy. Note that if two paths P and P are both reversible paths that go from state 0 to state 1, then PP āˆ’1 is a closed cycle, where P āˆ’1 is path P taken in reverse, from state 1 to state 0. It follows then from the inequality (2.3.3) that 0 ā‰„ PPāˆ’1 dQ T = P dQ T āˆ’ P dQ T . But P Pāˆ’1 is also a closed cycle, so 0 ā‰„ PPāˆ’1 dQ T = P dQ T āˆ’ P dQ T . These two results are consistent only if for reversible paths both cyclic integrals vanish, in which case P dQ T = P dQ T . (2.3.4) We can therefore define the entropy up to an additive constant as in Eq. (2.3.1), where P is any reversible path. Finally, if P is a general path from state 0 to state 1, reversible or irreversible, while P is a reversible path from state 0 to state 1, then P Pāˆ’1 (but not neces- sarily PPāˆ’1 !) is a closed cycle, so the inequality (2.3.3) gives 0 ā‰„ PPāˆ’1 dQ T = P dQ T āˆ’ P dQ T , and therefore, using Eq. (2.3.1), P dQ T ā‰¤ S1 āˆ’ S0 as was to be shown. In the special case of a completely isolated system , no heat can be taken into or given up by , so the integrand in the integral on the left-hand side of Eq. (2.3.2) must vanish and therefore S1 ā‰„ S0. In isolated systems the entropy can only increase. On the other hand, if an isolated system is undergoing only reversible changes, then according to Eq. (2.3.1) the entropy is constant. There is another definition of entropy, used in information theory as well as in physics. If a system can be in any one of a number of states characterized by a continuous (generally multidimensional) parameter Ī±, with a probability P(Ī±) dĪ± of being in states with this parameter in a narrow range dĪ± around Ī±, then the entropy is
30 2 Thermodynamics and Kinetic Theory S = āˆ’k P(Ī±) ln P(Ī±) dĪ± , (2.3.5) where k is the universal constant, known as Boltzmannā€™s constant, appearing in Eq. (1.2.1). As we shall see in the next section, according to kinetic the- ory, with a suitable choice of S0 the thermodynamic entropy (2.3.1) equals the information-theoretic entropy (2.3.5). The mere fact that the entropy S defined by (2.3.1) depends only on the ther- modynamic state has far-reaching consequences. Consider a fixed mass of fluid in a vessel with variable volume. The independent thermodynamic variables here can be taken as the volume V and the temperature T , with pressure p, internal energy E, and entropy S all functions of V and T . The work done by the fluid pressure p in increasing the fluid volume by a small amount dV is pdV , so the heat required to change the temperature by an infinitesimal amount dT and the volume by an infinitesimal amount dV is dQ = dE + pdV , so according to Eq. (2.3.1), the change in the entropy is given by T dS = dE + pdV . (2.3.6) In other words āˆ‚S(V , T ) āˆ‚T = 1 T āˆ‚E(V , T ) āˆ‚T (2.3.7) āˆ‚S(V , T ) āˆ‚V = 1 T āˆ‚E(V , T ) āˆ‚V + p(V , T ) T . (2.3.8) To squeeze information about pressure and internal energy from these formulas, we use the fact that partial derivatives commute. From Eq. (2.3.7) we have āˆ‚ āˆ‚V āˆ‚S(V , T ) āˆ‚T = 1 T āˆ‚2E(V , T ) āˆ‚T āˆ‚V while, from Eq. (2.3.8), āˆ‚ āˆ‚T āˆ‚S(V , T ) āˆ‚V = 1 T āˆ‚2E(V , T ) āˆ‚T āˆ‚V āˆ’ 1 T 2 āˆ‚E(V , T ) āˆ‚V + āˆ‚(p(V , T )/T ) āˆ‚T . Setting these equal gives a relation between the derivatives of E and p: 0 = āˆ’ 1 T 2 āˆ‚E(V , T ) āˆ‚V + āˆ‚ p(V , T )/T āˆ‚T . (2.3.9) This is for a fixed mass. Since E(V , T ) is an extensive variable, it must be proportional to this mass but does not otherwise have to depend on volume. In fact, it is frequently a good approximation to suppose that, apart from its proportionality to mass, E(V , T ) is independent of volume. This is the case if
2.3 Entropy 31 the fluid consists of infinitesimal particles that interact only in contact in colli- sions; since there is nothing with the dimensions of length that can enter in the calculation of the energy, E(V , T ) cannot here depend on volume. In this case Eq. (2.3.9) yields Charlesā€™ law, that for fixed volume V the pressure p(V , T ) is proportional to T . This shows again that the absolute temperature T in the ideal gas law (1.2.3) is the same up to a constant factor as the temperature T defined by the efficiency (2.2.11) of Carnot cycles. Although this result was obtained without having a formula for the entropy, for some purposes it is useful actually to know what the entropy is. In a homo- geneous medium, the entropy S of any mass M of matter may conveniently be written as S = Ms, where s is the entropy per unit mass, a function of temper- ature and various densities known as the specific entropy. Dividing Eq. (2.3.6) by M, we have then T ds = d(E/Ļ) + pd(1/Ļ) , (2.3.10) where as before E ā‰” E/V is the internal energy density and Ļ ā‰” M/V is the mass density. We consider an ideal gas, for which T = pĪ¼/RĻ while E and p are related by Eq. (2.1.10): E = p/(Ī³ āˆ’ 1). Then Eq. (2.3.10) gives pĪ¼ RĻ ds = 1 Ī³ āˆ’ 1 dp Ļ + Ī³pd 1 Ļ = ĻĪ³ āˆ’1 Ī³ āˆ’ 1 d p ĻĪ³ , so ds = R/Ī¼ Ī³ āˆ’ 1 ĻĪ³ p d p ĻĪ³ . The solution is s = R/Ī¼ Ī³ āˆ’ 1 ln p ĻĪ³ + constant . (2.3.11) We see that the result of Section 2.2 that p āˆ ĻĪ³ for adiabatic processes is just the statement that s is constant in these processes, which of course it must be since in an adiabatic process the heat input dQ vanishes. In many stars there are regions in which convection effectively mixes matter from various depths. Since heat conduction is usually ineffective in stars, little heat flows into or out of a bit of matter as it rises or falls, and so it keeps the same specific entropy. These regions therefore have a uniform specific entropy, and therefore a uniform value for the ratio p/ĻĪ³ . For instance, this is the case in the Sun for distances from the center greater than about 65% of the Sunā€™s radius out to a thin surface layer. Neutral Matter We have been mostly concerned with matter in which in each mass there is a non-vanishing conserved quantity, the number of particles. There is a different
32 2 Thermodynamics and Kinetic Theory context, with no similar conserved numbers, in which thermodynamics yields more detailed information about pressure and energy. In the early universe, at temperatures above about 1010 K, there is so much energy in radiation and electronā€“antielectron pairs that the contribution to the energy of the excess of matter over antimatter may be neglected. Here there is no number density on which the pressure and energy density E ā‰” E/V can significantly depend, so here E(V , T ) = V E(T ) and p(V , T ) = p(T ); thus here Eq. (2.3.9) is an ordinary differential equation for p(T ): 0 = āˆ’ E(T ) T 2 + d dT p(T ) T or, in other words, p (T ) = E(T ) + p(T ) T . (2.3.12) Thermodynamics alone does not fix any relation between E(T ) and p(T ), but given such a relation this result gives both as functions of temperature. For instance, as an example of the power of thermodynamics, it was known in the nineteenth century as a consequence of Maxwellā€™s theory of electromagnetism that the pressure of electromagnetic radiation is one-third of its energy density. Setting p = E/3 in Eq. (2.3.12) gives E(T ) = 4E(T ), so E(T ) = 3p(T ) = aT 4 , (2.3.13) where a is a constant, known as the radiation energy constant. But, as we shall see in Section 3.1, it was not possible to understand the value of a until the advent of quantum mechanics in the early twentieth century. The Laws of Thermodynamics It is common to summarize the content of classical thermodynamics in three laws. As already mentioned, the first law is just the conservation of energy, discussed in the context of heat energy in Section 2.1, and the second, usually attributed to Clausius, on which the discussion of thermodynamic efficiency in Section 2.2 is based, can be stated as the principle that without doing work it is not possible to transfer heat from a cold reservoir to one at higher temperature. We have seen that this leads to the existence of a quantity, the entropy, which depends only on the thermodynamic state and satisfies Eq. (2.3.1) when reversible changes are made in this state. This can instead be taken as the second law of thermodynamics. There are several formulations of the third law, some given by Walther Nernst (1864ā€“1941) in 1906ā€“1912. The most fruitful, it seems, is that it is possible to assign a common value to the entropy (conventionally taken as zero) for all systems at absolute zero temperature, so that at absolute zero the integral
2.4 Kinetic Theory and Statistical Mechanics 33 in Eq. (2.3.1) must converge. This has the consequence, in particular, that the specific heat dQ/dT must vanish for T ā†’ 0. This seems to contradict the results of Section 2.1 for ideal gases, which give a temperature-independent specific heat whether for fixed volume or fixed pressure. The contradiction is avoided in practice because no substance remains close to an ideal gas as the temperature approaches absolute zero. We will see when we come to quantum mechanics that if an otherwise free particle is confined in any fixed volume, then it cannot have precisely zero momentum, as required for a classical ideal gas at absolute zero temperature. On the other hand, solids can exist at absolute zero temperature, and in that limit their specific heats do approach zero. 2.4 Kinetic Theory and Statistical Mechanics We saw in the previous chapter how by the mid nineteenth century the ideal gas law had been established through the work especially of Bernoulli and Clausius. But, though derived by considering the motions of individual gas molecules, in its conclusions it dealt only with bulk gas properties such as pressure, tem- perature, mass density, and energy density. For many purposes, including the calculation of chemical or transport processes, it was necessary to go further and work out the detailed probability distribution of the motion of individual gas particles. This was done in the kinetic theory of James Clerk Maxwell and Ludwig Boltzmann (1844ā€“1906). Kinetic theory was later generalized to the formalism known as statistical mechanics, especially by the American theorist Josiah Willard Gibbs (1839ā€“1906). As it turned out, these methods went a long way toward not only establishing a correspondence with thermodynamics but also explaining the principles of thermodynamics on the assumption that macroscopic matter is composed of very many particles, and thereby helping to establish the reality of atoms. The Maxwellā€“Boltzmann Distribution Maxwell in 1860 considered the form of the probability distribution function P(vx, vy, vz) for the x, y, and z components of the velocity of any molecule in a gas in equilibrium.3 The probability distribution function is defined so that the probability that these components are respectively between vx and vx + dvx, between vy and vy + dvy, and between vz and vz + dvz, is of the form P(vx, vy, vz)dvxdvydvz . 3 J. C. Maxwell, Phil. Mag. 19, 19; 20, 21 (1860). This article is included in Brush, The Kinetic Theory of Gases ā€“ An Anthology of Classic Papers with Historical Commentary, listed in the bibliography.
34 2 Thermodynamics and Kinetic Theory He assumed (without offering a real justification) that the probability that any component of velocity of a particle is in a particular range is not correlated with the other components of the velocity. Then P(vx, vy, vz) must be proportional to a function of vx alone, with a coefficient that depends only on vy and vz, and likewise for vy and vz, so P(vx, vy, vz) must take the form of a product: P(vx, vy, vz) = f (vx)g(vy)h(vz) . Rotational symmetry requires further that P can depend only on the magnitude of the velocity, not on its direction, and hence only on v2 x + v2 y + v2 z . The only function of v2 x + v2 y + v2 z that takes the form f (vx)g(vy)h(vz) is proportional to an exponential: P(vx, vy, vz) āˆ exp āˆ’ C(v2 x + v2 y + v2 z ) . The constant C must be positive in order that P should not blow up for large velocity, which would make it impossible to set the total probability equal to unity, as it must be. Taking C to be positive, and setting the total probability (the integral of P over all velocities) for each particle equal to one, gives the factor of proportionality: P(vx, vy, vz) = C Ļ€ 3/2 exp āˆ’ C(v2 x + v2 y + v2 z ) . We can use this to calculate the mean square velocity components: v2 x = v2 y = v2 z = 1 2C . Clausius had introduced an absolute temperature T by setting mv2 āŠ„ = kT , where k is a constant to be determined experimentally and vāŠ„ is the component of the velocity in a direction normal to the container wall, which for an isotropic velocity distribution can be taken as any direction, so the constant C must be given by C = m/2kT and the Maxwell distribution takes the form P(vx, vy, vz) = m 2Ļ€kT 3/2 exp āˆ’ m(v2 x + v2 y + v2 z )/2kT . (2.4.1) As we saw at the end of Section 2.2, the requirement that in an ideal gas mvāŠ„ 2 = kT , which led here to C = m/2kT , also ensures that, up to an arbitrary constant factor, T is the absolute temperature defined by the efficiency of Carnot cycles. The formula for the probability distribution P was derived in 1868 in a more convincing way by Boltzmann.4 He defined a quantity 4 L. Boltzmann, Sitz. Ber. Akad. Wiss. (Vienna), part II, 66, 875 (1872). A translation into English of this article is included in Brush, The Kinetic Theory of Gases ā€“ An Anthology of Classic Papers with Historical Commentary, listed in the bibliography.
2.4 Kinetic Theory and Statistical Mechanics 35 H ā‰” ln P = +āˆž āˆ’āˆž dvx +āˆž āˆ’āˆž dvy +āˆž āˆ’āˆž dvz P(v) ln P(v) , and showed that collisions of gas particles always lead to a decrease in H until a minimum is reached, at which P(v) is the Maxwellā€“Boltzmann distribution function. A generalization of this H-theorem was given in 1901 by Gibbs.5 The generalization and proof are given below, along with the application to gases. The General H-Theorem Consider a large system with many degrees of freedom, such as a gas with many molecules (but not necessarily a gas). The states of the system are parameterized by many variables, which we summarize with a symbol Ī±. (For instance, for a monatomic gas Ī± stands for the set of positions x1, x2, etc. and momenta p1, p2, etc. of atoms 1, 2, . . . For a gas of multi-atom molecules, Ī± would also include the orientations and their rates of change for each molecule.) We denote an infinitesimal range of these parameters by dĪ±. (For instance, for a monatomic gas dĪ± stands for the product d3x1d3p1d3x2d3p2 . . ., known as the phase space volume.) We define P(Ī±) so that the probability that the parameters of the system are in an infinitesimal range dĪ± around Ī± is P(Ī±)dĪ±, with P normalized so that dĪ±P(Ī±) = 1. Define H ā‰” ln P = P(Ī±) dĪ± ln P(Ī±) . (2.4.2) Gibbs showed that H always decreases until it reaches a minimum value, at which P(Ī±) is proportional to the exponential of a linear combination of con- served quantities, such as the total energy. Proof: Define a differential rate (Ī± ā†’ Ī²) such that the rate at which a system in state Ī± makes a transition to a state within a range dĪ² around state Ī² is (Ī± ā†’ Ī²) dĪ². The probability P(Ī±)dĪ± can either increase because the system in a range dĪ² of states around Ī² makes a transition to the range dĪ± of states around Ī±, or decrease because the system in the range of states dĪ± around Ī± makes a transition to some other state in a range dĪ² around Ī², so dP(Ī±)dĪ± dt = dĪ² [P(Ī²) (Ī² ā†’ Ī±)dĪ± āˆ’ P(Ī±) dĪ±(Ī± ā†’ Ī²)] , or, cancelling the differentials dĪ±, dP(Ī±) dt = dĪ² [P(Ī²) (Ī² ā†’ Ī±) āˆ’ P(Ī±)(Ī± ā†’ Ī²)] . (2.4.3) 5 J. W. Gibbs, Elementary Principles of Statistical Mechanics, Developed with Especial Reference to The Rational Foundation of Thermodynamics (Scribner, New York, 1902).
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CHAPTER XI GALILEE Neither old ā€œsacredā€ [123] himself, nor any of his helpers, knew the road which I meant to take from Nazareth to the Sea of Galilee and from thence to Jerusalem, so I was forced to add another to my party by hiring a guide. The associations of Nazareth, as well as my kind feeling towards the hospitable monks, whose guest I had been, inclined me to set at naught the advice which I had received against employing Christians. I accordingly engaged a lithe, active young Nazarene, who was recommended to me by the monks, and who affected to be familiar with the line of country through which I intended to pass. My disregard of the popular prejudices against Christians was not justified in this particular instance by the result of my choice. This you will see by and by. I passed by Cana and the house in which the water had been turned into wine; I came to the field in which our Saviour had rebuked the Scotch Sabbath-keepers of that period, by suffering His disciples to pluck corn on the Lordā€™s Day; I rode over the ground on which the fainting multitude had been fed, and they showed me some massive fragmentsā€”the relics, they said, of that wondrous banquet, now turned into stone. The petrifaction was most complete. I ascended the height on which our Lord was standing when He wrought the miracle. The hill was lofty enough to show me the fairness of the land on all sides, but I have an ancient love for the mere features of a lake, and so forgetting all else when I reached the summit, I looked away eagerly to the eastward. There she lay,
the Sea of Galilee. Less stern than Wast Water, less fair than gentle Windermere, she had still the winning ways of an English lake; she caught from the smiling heavens unceasing light and changeful phases of beauty, and with all this brightness on her face, she yet clung so fondly to the dull he-looking mountain at her side, as though she would ā€œSoothe him with her finer fancies, Touch him with her lighter thought.ā€ [124] If one might judge of menā€™s real thoughts by their writings, it would seem that there are people who can visit an interesting locality and follow up continuously the exact train of thought that ought to be suggested by the historical associations of the place. A person of this sort can go to Athens and think of nothing later than the age of Pericles; can live with the Scipios as long as he stays in Rome; can go up in a balloon, and think how resplendently in former times the now vacant and desolate air was peopled with angels, how prettily it was crossed at intervals by the rounds of Jacobā€™s ladder! I donā€™t possess this power at all; it is only by snatches, and for few moments together, that I can really associate a place with its proper history. ā€œThere at Tiberias, and along this western shore towards the north, and upon the bosom too of the lake, our Saviour and His disciplesā€ā€” away flew those recollections, and my mind strained eastward, because that that farthest shore was the end of the world that belongs to man the dweller, the beginning of the other and veiled world that is held by the strange race, whose life (like the pastime of Satan) is a ā€œgoing to and fro upon the face of the earth.ā€ From those grey hills right away to the gates of Bagdad stretched forth the mysterious ā€œdesertā€ā€”not a pale, void, sandy tract, but a land abounding in rich pastures, a land without cities or towns, without any ā€œrespectableā€ people or any ā€œrespectableā€ things, yet yielding its eighty thousand cavalry to the beck of a few old men. But once moreā€”ā€œTiberiasā€”the plain of Gennesarethā€”the very earth on which
I stoodā€”that the deep low tones of the Saviourā€™s voice should have gone forth into eternity from out of the midst of these hills and these valleys!ā€ā€”Ay, ay, but yet again the calm face of the lake was uplifted, and smiled upon my eyes with such familiar gaze, that the ā€œdeep low tonesā€ were hushed, the listening multitudes all passed away, and instead there came to me a dear old memory from over the seas in England, a memory sweeter than Gospel to that poor wilful mortal, me. I went to Tiberias, and soon got afloat upon the water. In the evening I took up my quarters in the Catholic church, and the building being large enough, the whole of my party were admitted to the benefit of the same shelter. With portmanteaus and carpet bags, and books and maps, and fragrant tea, Mysseri soon made me a home on the southern side of the church. One of old Shereefā€™s helpers was an enthusiastic Catholic, and was greatly delighted at having so sacred a lodging. He lit up the altar with a number of tapers, and when his preparations were complete, he began to perform his orisons in the strangest manner imaginable. His lips muttered the prayers of the Latin Church, but he bowed himself down and laid his forehead to the stones beneath him after the manner of a Mussulman. The universal aptness of a religious system for all stages of civilisation, and for all sorts and conditions of men, well befits its claim of divine origin. She is of all nations, and of all times, that wonderful Church of Rome! Tiberias is one of the four holy cities, [126] according to the Talmud, and it is from this place, or the immediate neighbourhood of it, that the Messiah is to arise. Except at Jerusalem, never think of attempting to sleep in a ā€œholy city.ā€ Old Jews from all parts of the world go to lay their bones upon the sacred soil, and as these people never return to their homes, it follows that any domestic vermin which they may bring with them are likely to become permanently resident, so that the population is continually increasing. No recent census had been taken when I
was at Tiberias, but I know that the congregation of fleas which attended at my church alone must have been something enormous. It was a carnal, self-seeking congregation, wholly inattentive to the service which was going on, and devoted to the one object of having my blood. The fleas of all nations were there. The smug, steady, importunate flea from Holywell Street; the pert, jumping puce from hungry France, the wary, watchful pulce with his poisoned stiletto; the vengeful pulga of Castile with his ugly knife; the German floh with his knife and fork, insatiate, not rising from table; whole swarms from all the Russias, and Asiatic hordes unnumberedā€”all these were there, and all rejoiced in one great international feast. I could no more defend myself against my enemies than if I had been pain Ć  discretion in the hands of a French patriot, or English gold in the claws of a Pennsylvanian Quaker. After passing a night like this you are glad to pick up the wretched remains of your body long, long before morning dawns. Your skin is scorched, your temples throb, your lips feel withered and dried, your burning eyeballs are screwed inwards against the brain. You have no hope but only in the saddle and the freshness of the morning air.
CHAPTER XII MY FIRST BIVOUAC The course of the Jordan is from the north to the south, and in that direction, with very little of devious winding, it carries the shining waters of Galilee straight down into the solitudes of the Dead Sea. Speaking roughly, the river in that meridian is a boundary between the people living under roofs and the tented tribes that wander on the farther side. And so, as I went down in my way from Tiberias towards Jerusalem, along the western bank of the stream, my thinking all propended to the ancient world of herdsmen and warriors that lay so close over my bridle arm. If a man, and an Englishman, be not born of his mother with a natural Chiffney-bit in his mouth, there comes to him a time for loathing the wearisome ways of society; a time for not liking tamed people; a time for not dancing quadrilles, not sitting in pews; a time for pretending that Milton and Shelley, and all sorts of mere dead people, were greater in death than the first living Lord of the Treasury; a time, in short, for scoffing and railing, for speaking lightly of the very opera, and all our most cherished institutions. It is from nineteen to two or three and twenty perhaps that this war of the man against men is like to be waged most sullenly. You are yet in this smiling England, but you find yourself wending away to the dark sides of her mountains, climbing the dizzy crags, exulting in the fellowship of mists and clouds, and watching the storms how they gather, or proving the mettle of your mare upon the broad and dreary downs, because that you feel congenially with the yet unparcelled earth. A little while you are free and unlabelled, like the
ground that you compass; but civilisation is coming and coming; you and your much-loved waste lands will be surely enclosed, and sooner or later brought down to a state of mere usefulness; the ground will be curiously sliced into acres and roods and perches, and you, for all you sit so smartly in your saddle, you will be caught, you will be taken up from travel as a colt from grass, to be trained and tried, and matched and run. All this in time, but first come Continental tours and the moody longing for Eastern travel. The downs and the moors of England can hold you no longer; with large strides you burst away from these slips and patches of free land; you thread your path through the crowds of Europe, and at last, on the banks of Jordan, you joyfully know that you are upon the very frontier of all accustomed respectabilities. There, on the other side of the river (you can swim it with one arm), there reigns the people that will be like to put you to death for not being a vagrant, for not being a robber, for not being armed and houseless. There is comfort in thatā€”health, comfort, and strength to one who is dying from very weariness of that poor, dear, middle-aged, deserving, accomplished, pedantic, and painstaking governess, Europe. I had ridden for some hours along the right bank of Jordan when I came to the Djesr el MedjamĆ© (an old Roman bridge, I believe), which crossed the river. My Nazarene guide was riding ahead of the party, and now, to my surprise and delight, he turned leftwards, and led on over the bridge. I knew that the true road to Jerusalem must be mainly by the right bank of Jordan, but I supposed that my guide was crossing the bridge at this spot in order to avoid some bend in the river, and that he knew of a ford lower down by which we should regain the western bank. I made no question about the road, for I was but too glad to set my horseā€™s hoofs upon the land of the wandering tribes. None of my party except the Nazarene knew the country. On we went through rich pastures upon the eastern side of the water. I looked for the expected bend of the river, but far as I could see it kept a straight southerly course; I still left my guide unquestioned.
The Jordan is not a perfectly accurate boundary betwixt roofs and tents, for soon after passing the bridge I came upon a cluster of huts. Some time afterwards the guide, upon being closely questioned by my servants, confessed that the village which we had left behind was the last that we should see, but he declared that he knew a spot at which we should find an encampment of friendly Bedouins, who would receive me with all hospitality. I had long determined not to leave the East without seeing something of the wandering tribes, but I had looked forward to this as a pleasure to be found in the desert between El Arish and Egypt; I had no idea that the Bedouins on the east of Jordan were accessible. My delight was so great at the near prospect of bread and salt in the tent of an Arab warrior, that I wilfully allowed my guide to go on and mislead me. I saw that he was taking me out of the straight route towards Jerusalem, and was drawing me into the midst of the Bedouins; but the idea of his betraying me seemed (I know not why) so utterly absurd, that I could not entertain it for a moment. I fancied it possible that the fellow had taken me out of my route in order to attempt some little mercantile enterprise with the tribe for which he was seeking, and I was glad of the opportunity which I might thus gain of coming in contact with the wanderers. Not long after passing the village a horseman met us. It appeared that some of the cavalry of Ibrahim Pasha had crossed the river for the sake of the rich pastures on the eastern bank, and that this man was one of the troopers. He stopped and saluted; he was obviously surprised at meeting an unarmed, or half-armed, cavalcade, and at last fairly told us that we were on the wrong side of the river, and that if we proceeded we must lay our account with falling amongst robbers. All this while, and throughout the day, my Nazarene kept well ahead of the party, and was constantly up in his stirrups, straining forward and searching the distance for some objects which still remained unseen. For the rest of the day we saw no human being; we pushed on eagerly in the hope of coming up with the Bedouins before nightfall.
Night came, and we still went on in our way till about ten oā€™clock. Then the thorough darkness of the night, and the weariness of our beasts (which had already done two good daysā€™ journey in one), forced us to determine upon coming to a standstill. Upon the heights to the eastward we saw lights; these shone from caves on the mountain-side, inhabited, as the Nazarene told us, by rascals of a low sortā€”not real Bedouins, men whom we might frighten into harmlessness, but from whom there was no willing hospitality to be expected. We heard at a little distance the brawling of a rivulet, and on the banks of this it was determined to establish our bivouac. We soon found the stream, and following its course for a few yards, came to a spot which was thought to be fit for our purpose. It was a sharply cold night in February, and when I dismounted I found myself standing upon some wet rank herbage that promised ill for the comfort of our resting-place. I had bad hopes of a fire, for the pitchy darkness of the night was a great obstacle to any successful search for fuel, and, besides, the boughs of trees or bushes would be so full of sap in this early spring, that they would not be easily persuaded to burn. However, we were not likely to submit to a dark and cold bivouac without an effort, and my fellows groped forward through the darkness, till after advancing a few paces they were happily stopped by a complete barrier of dead prickly bushes. Before our swords could be drawn to reap this welcome harvest it was found to our surprise that the fuel was already hewn and strewed along the ground in a thick mass. A spot for the fire was found with some difficulty, for the earth was moist and the grass high and rank. At last there was a clicking of flint and steel, and presently there stood out from darkness one of the tawny faces of my muleteers, bent down to near the ground, and suddenly lit up by the glowing of the spark which he courted with careful breath. Before long there was a particle of dry fibre or leaf that kindled to a tiny flame; then another was lit from that, and then another. Then small crisp twigs, little bigger than bodkins, were laid athwart the glowing fire. The swelling cheeks of the muleteer, laid level with the
earth, blew tenderly at first and then more boldly upon the young flame, which was daintily nursed and fed, and fed more plentifully when it gained good strength. At last a whole armful of dry bushes was piled up over the fire, and presently, with a loud cheery crackling and crackling, a royal tall blaze shot up from the earth and showed me once more the shapes and faces of my men, and the dim outlines of the horses and mules that stood grazing hard by. My servants busied themselves in unpacking the baggage as though we had arrived at an hotelā€”Shereef and his helpers unsaddled their cattle. We had left Tiberias without the slightest idea that we were to make our way to Jerusalem along the desolate side of the Jordan, and my servants (generally provident in those matters) had brought with them only, I think, some unleavened bread and a rocky fragment of goatā€™s-milk cheese. These treasures were produced. Tea and the contrivances for making it were always a standing part of my baggage. My men gathered in circle round the fire. The Nazarene was in a false position from having misled us so strangely, and he would have shrunk back, poor devil, into the cold and outer darkness, but I made him draw near and share the luxuries of the night. My quilt and my pelisse were spread, and the rest of my party had all their capotes or pelisses, or robes of some sort, which furnished their couches. The men gathered in circle, some kneeling, some sitting, some lying reclined around our common hearth. Sometimes on one, sometimes on another, the flickering light would glare more fiercely. Sometimes it was the good Shereef that seemed the foremost, as he sat with venerable beard the image of manly pietyā€”unknowing of all geography, unknowing where he was or whither he might go, but trusting in the goodness of God and the clinching power of fate and the good star of the Englishman. Sometimes, like marble, the classic face of the Greek Mysseri would catch the sudden light, and then again by turns the ever-perturbed Dthemetri, with his old Chinamanā€™s eye and bristling, terrier-like moustache, shone forth illustrious.
I always liked the men who attended me on these Eastern travels, for they were all of them brave, cheery-hearted fellows; and although their following my career brought upon them a pretty large share of those toils and hardships which are so much more amusing to gentlemen than to servants, yet not one of them ever uttered or hinted a syllable of complaint, or even affected to put on an air of resignation. I always liked them, but never perhaps so much as when they were thus grouped together under the light of the bivouac fire. I felt towards them as my comrades rather than as my servants, and took delight in breaking bread with them, and merrily passing the cup. The love of tea is a glad source of fellow-feeling between the Englishman and the Asiatic. In Persia it is drunk by all, and although it is a luxury that is rarely within the reach of the Osmanlees, there are few of them who do not know and love the blessed tchƤi. Our camp-kettle, filled from the brook, hummed doubtfully for a while, then busily bubbled under the sidelong glare of the flames; cups clinked and rattled; the fragrant steam ascended, and soon this little circlet in the wilderness grew warm and genial as my ladyā€™s drawing- room. And after this there came the tchibouqueā€”great comforter of those that are hungry and wayworn. And it has this virtueā€”it helps to destroy the gĆŖne and awkwardness which one sometimes feels at being in company with oneā€™s dependants; for whilst the amber is at your lips, there is nothing ungracious in your remaining silent, or speaking pithily in short inter-whiff sentences. And for us that night there was pleasant and plentiful matter of talk; for the where we should be on the morrow, and the wherewithal we should be fed, whether by some ford we should regain the western bank of Jordan, or find bread and salt under the tents of a wandering tribe, or whether we should fall into the hands of the Philistines, and so come to see deathā€”the last and greatest of all ā€œthe fine sightsā€ that there beā€”these were questionings not dull nor wearisome to us, for we were all concerned in the answers. And it was not an all-imagined
morrow that we probed with our sharp guesses, for the lights of those low Philistines, the men of the caves, still hung over our heads, and we knew by their yells that the fire of our bivouac had shown us. At length we thought it well to seek for sleep. Our plans were laid for keeping up a good watch through the night. My quilt and my pelisse and my cloak were spread out so that I might lie spokewise, with my feet towards the central fire. I wrapped my limbs daintily round, and gave myself positive orders to sleep like a veteran soldier. But I found that my attempt to sleep upon the earth that God gave me was more new and strange than I had fancied it. I had grown used to the scene which was before me whilst I was sitting or reclining by the side of the fire, but now that I laid myself down at length it was the deep black mystery of the heavens that hung over my eyesā€”not an earthly thing in the way from my own very forehead right up to the end of all space. I grew proud of my boundless bedchamber. I might have ā€œfound sermonsā€ in all this greatness (if I had I should surely have slept), but such was not then my way. If this cherished self of mine had built the universe, I should have dwelt with delight on ā€œthe wonders of creation.ā€ As it was, I felt rather the vain-glory of my promotion from out of mere rooms and houses into the midst of that grand, dark, infinite palace. And then, too, my head, far from the fire, was in cold latitudes, and it seemed to me strange that I should be lying so still and passive, whilst the sharp night breeze walked free over my cheek, and the cold damp clung to my hair, as though my face grew in the earth and must bear with the footsteps of the wind and the falling of the dew as meekly as the grass of the field. Besides, I got puzzled and distracted by having to endure heat and cold at the same time, for I was always considering whether my feet were not over-devilled and whether my face was not too well iced. And so when from time to time the watch quietly and gently kept up the languishing fire, he seldom, I think, was unseen to my restless eyes. Yet, at last, when they called me and said that the morn would soon be dawning, I
rose from a state of half-oblivion not much unlike to sleep, though sharply qualified by a sort of vegetableā€™s consciousness of having been growing still colder and colder for many and many an hour.
CHAPTER XIII THE DEAD SEA The grey light of the morning showed us for the first time the ground which we had chosen for our resting-place. We found that we had bivouacked upon a little patch of barley plainly belonging to the men of the caves. The dead bushes which we found so happily placed in readiness for our fire had been strewn as a fence for the protection of the little crop. This was the only cultivated spot of ground which we had seen for many a league, and I was rather sorry to find that our night fire and our cattle had spread so much ruin upon this poor solitary slip of corn-land. The saddling and loading of our beasts was a work which generally took nearly an hour, and before this was half over daylight came. We could now see the men of the caves. They collected in a body, amounting, I should think, to nearly fifty, and rushed down towards our quarters with fierce shouts and yells. But the nearer they got the slower they went; their shouts grew less resolute in tone, and soon ceased altogether. The fellows, however, advanced to a thicket within thirty yards of us, and behind this ā€œtook up their position.ā€ My men without premeditation did exactly that which was best; they kept steadily to their work of loading the beasts without fuss or hurry; and whether it was that they instinctively felt the wisdom of keeping quiet, or that they merely obeyed the natural inclination to silence which one feels in the early morning, I cannot tell, but I know that, except when they exchanged a syllable or two relative to the work they were about, not a word was said. I now believe that this quietness of our party created an undefined terror in the minds
of the cave-holders and scared them from coming on; it gave them a notion that we were relying on some resources which they knew not of. Several times the fellows tried to lash themselves into a state of excitement which might do instead of pluck. They would raise a great shout and sway forward in a dense body from behind the thicket; but when they saw that their bravery thus gathered to a head did not even suspend the strapping of a portmanteau or the tying of a hatbox, their shout lost its spirit, and the whole mass was irresistibly drawn back like a wave receding from the shore. These attempts at an onset were repeated several times, but always with the same result. I remained under the apprehension of an attack for more than half an hour, and it seemed to me that the work of packing and loading had never been done so slowly. I felt inclined to tell my fellows to make their best speed, but just as I was going to speak I observed that every one was doing his duty already; I therefore held my peace and said not a word, till at last Mysseri led up my horse and asked me if I were ready to mount. We all marched off without hindrance. After some time we came across a party of Ibrahimā€™s cavalry, which had bivouacked at no great distance from us. The knowledge that such a force was in the neighbourhood may have conduced to the forbearance of the cave-holders. We saw a scraggy-looking fellow nearly black, and wearing nothing but a cloth round the loins; he was tending flocks. Afterwards I came up with another of these goatherds, whose helpmate was with him. They gave us some goatā€™s milk, a welcome present. I pitied the poor devil of a goat-herd for having such a very plain wife. I spend an enormous quantity of pity upon that particular form of human misery. About midday I began to examine my map and to question my guide, who at last fell on his knees and confessed that he knew nothing of the country in which we were. I was thus thrown upon
my own resources, and calculating that on the preceding day we had nearly performed a two daysā€™ journey, I concluded that the Dead Sea must be near. In this I was right, for at about three or four oā€™clock in the afternoon I caught a first sight of its dismal face. I went on and came near to those waters of death. They stretched deeply into the southern desert, and before me, and all around, as far away as the eye could follow, blank hills piled high over hills, pale, yellow, and naked, walled up in her tomb for ever the dead and damned Gomorrah. There was no fly that hummed in the forbidden air, but instead a deep stillness; no grass grew from the earth, no weed peered through the void sand; but in mockery of all life there were trees borne down by Jordan in some ancient flood, and these, grotesquely planted upon the forlorn shore, spread out their grim skeleton arms, all scorched and charred to blackness by the heats of the long silent years. I now struck off towards the dĆ©bouchure of the river; but I found that the country, though seemingly quite flat, was intersected by deep ravines, which did not show themselves until nearly approached. For some time my progress was much obstructed; but at last I came across a track which led towards the river, and which might, as I hoped, bring me to a ford. I found, in fact, when I came to the riverā€™s side that the track reappeared upon the opposite bank, plainly showing that the stream had been fordable at this place. Now, however, in consequence of the late rains the river was quite impracticable for baggage-horses. A body of waters about equal to the Thames at Eton, but confined to a narrower channel, poured down in a current so swift and heavy, that the idea of passing with laden baggage-horses was utterly forbidden. I could have swum across myself, and I might, perhaps, have succeeded in swimming a horse over; but this would have been useless, because in such case I must have abandoned not only my baggage, but all my attendants, for none of them were able to swim, and without that resource it would have been madness for them to rely upon the swimming of their beasts across such a powerful stream. I still hoped, however,
that there might be a chance of passing the river at the point of its actual junction with the Dead Sea, and I therefore went on in that direction. Night came upon us whilst labouring across gullies and sandy mounds, and we were obliged to come to a standstill quite suddenly upon the very edge of a precipitous descent. Every step towards the Dead Sea had brought us into a country more and more dreary; and this sandhill, which we were forced to choose for our resting-place, was dismal enough. A few slender blades of grass, which here and there singly pierced the sand, mocked bitterly the hunger of our jaded beasts, and with our small remaining fragment of goatā€™s-milk rock by way of supper, we were not much better off than our horses. We wanted, too, the great requisite of a cheery bivouac- fire. Moreover, the spot on which we had been so suddenly brought to a standstill was relatively high and unsheltered, and the night wind blew swiftly and cold. The next morning I reached the dĆ©bouchure of the Jordan, where I had hoped to find a bar of sand that might render its passage possible. The river, however, rolled its eddying waters fast down to the ā€œseaā€ in a strong, deep stream that shut out all hope of crossing. It now seemed necessary either to construct a raft of some kind, or else to retrace my steps and remount the banks of the Jordan. I had once happened to give some attention to the subject of military bridgesā€”a branch of military science which includes the construction of rafts and contrivances of the like sortā€”and I should have been very proud indeed if I could have carried my party and my baggage across by dint of any idea gathered from Sir Howard Douglas or Robinson Crusoe. But we were all faint and languid from want of food, and besides there were no materials. Higher up the river there were bushes and river plants, but nothing like timber; and the cord with which my baggage was tied to the pack-saddles amounted altogether to a very small quantity, not nearly enough to haul any sort of craft across the stream.
And now it was, if I remember rightly, that Dthemetri submitted to me a plan for putting to death the Nazarene, whose misguidance had been the cause of our difficulties. There was something fascinating in this suggestion, for the slaying of the guide was of course easy enough, and would look like an act of what politicians call ā€œvigour.ā€ If it were only to become known to my friends in England that I had calmly killed a fellow-creature for taking me out of my way, I might remain perfectly quiet and tranquil for all the rest of my days, quite free from the danger of being considered ā€œslowā€; I might ever after live on upon my reputation, like ā€œsingle-speech Hamiltonā€ in the last century, or ā€œsingle sinā€”ā€ in this, without being obliged to take the trouble of doing any more harm in the world. This was a great temptation to an indolent person, but the motive was not strengthened by any sincere feeling of anger with the Nazarene. Whilst the question of his life and death was debated he was riding in front of our party, and there was something in the anxious writhing of his supple limbs that seemed to express a sense of his false position, and struck me as highly comic. I had no crotchet at that time against the punishment of death, but I was unused to blood, and the proposed victim looked so thoroughly capable of enjoying life (if he could only get to the other side of the river), that I thought it would be hard for him to die merely in order to give me a character for energy. Acting on the result of these considerations, and reserving to myself a free and unfettered discretion to have the poor villain shot at any future moment, I magnanimously decided that for the present he should live, and not die. I bathed in the Dead Sea. The ground covered by the water sloped so gradually, that I was not only forced to ā€œsneak in,ā€ but to walk through the water nearly a quarter of a mile before I could get out of my depth. When at last I was able to attempt to dive, the salts held in solution made my eyes smart so sharply, that the pain which I thus suffered, together with the weakness occasioned by want of food, made me giddy and faint for some moments, but I soon grew better. I knew beforehand the impossibility of sinking in this buoyant
water, but I was surprised to find that I could not swim at my accustomed pace; my legs and feet were lifted so high and dry out of the lake, that my stroke was baffled, and I found myself kicking against the thin air instead of the dense fluid upon which I was swimming. The water is perfectly bright and clear; its taste detestable. After finishing my attempts at swimming and diving, I took some time in regaining the shore, and before I began to dress I found that the sun had already evaporated the water which clung to me, and that my skin was thickly encrusted with salts.
CHAPTER XIV THE BLACK TENTS My steps were reluctantly turned towards the north. I had ridden some way, and still it seemed that all life was fenced and barred out from the desolate ground over which I was journeying. On the west there flowed the impassable Jordan, on the east stood an endless range of barren mountains, and on the south lay that desert sea that knew not the plashing of an oar; greatly therefore was I surprised when suddenly there broke upon my ear the long, ludicrous, persevering bray of a donkey. I was riding at this time some few hundred yards ahead of all my party except the Nazarene (who by a wise instinct kept closer to me than to Dthemetri), and I instantly went forward in the direction of the sound, for I fancied that where there were donkeys, there too most surely would be men. The ground on all sides of me seemed thoroughly void and lifeless, but at last I got down into a hollow, and presently a sudden turn brought me within thirty yards of an Arab encampment. The low black tents which I had so long lusted to see were right before me, and they were all teeming with live Arabsā€”men, women, and children. I wished to have let my party behind know where I was, but I recollected that they would be able to trace me by the prints of my horseā€™s hoofs in the sand; and having to do with Asiatics, I felt the danger of the slightest movement which might be looked upon as a sign of irresolution. Therefore, without looking behind me, without looking to the right or to the left, I rode straight up towards the foremost tent. Before this was strewed a semi-circular fence of dead boughs, through which there was an opening opposite to the front
of the tent. As I advanced, some twenty or thirty of the most uncouth-looking fellows imaginable came forward to meet me. In their appearance they showed nothing of the Bedouin blood; they were of many colours, from dingy brown to jet black, and some of these last had much of the negro look about them. They were tall, powerful fellows, but awfully ugly. They wore nothing but the Arab shirts, confined at the waist by leathern belts. I advanced to the gap left in the fence, and at once alighted from my horse. The chief greeted me after his fashion by alternately touching first my hand and then his own forehead, as if he were conveying the virtue of the touch like a spark of electricity. Presently I found myself seated upon a sheepskin, which was spread for me under the sacred shade of Arabian canvas. The tent was of a long, narrow, oblong form, and contained a quantity of men, women, and children so closely huddled together, that there was scarcely one of them who was not in actual contact with his neighbour. The moment I had taken my seat the chief repeated his salutations in the most enthusiastic manner, and then the people having gathered densely about me, got hold of my unresisting hand and passed it round like a claret jug for the benefit of everybody. The women soon brought me a wooden bowl full of buttermilk, and welcome indeed came the gift to my hungry and thirsty soul. After some time my party, as I had expected, came up, and when poor Dthemetri saw me on my sheepskin, ā€œthe life and soulā€ of this ragamuffin party, he was so astounded, that he even failed to check his cry of horror; he plainly thought that now, at last, the Lord had delivered me (interpreter and all) into the hands of the lowest Philistines. Mysseri carried a tobacco-pouch slung at his belt, and as soon as its contents were known the whole population of the tent began begging like spaniels for bits of the beloved weed. I concluded from the abject manner of these people that they could not possibly be thoroughbred Bedouins, and I saw, too, that they must be in the very last stage of misery, for poor indeed is the man in these climes
who cannot command a pipeful of tobacco. I began to think that I had fallen amongst thorough savages, and it seemed likely enough that they would gain their very first knowledge of civilisation by ravishing and studying the contents of my dearest portmanteaus, but still my impression was that they would hardly venture upon such an attempt. I observed, indeed, that they did not offer me the bread and salt which I had understood to be the pledges of peace amongst wandering tribes, but I fancied that they refrained from this act of hospitality, not in consequence of any hostile determination, but in order that the notion of robbing me might remain for the present an ā€œopen question.ā€ I afterwards found that the poor fellows had no bread to offer. They were literally ā€œout at grass.ā€ It is true that they had a scanty supply of milk from goats, but they were living almost entirely upon certain grass stems, which were just in season at that time of the year. These, if not highly nourishing, are pleasant enough to the taste, and their acid juices come gratefully to thirsty lips.
CHAPTER XV PASSAGE OF THE JORDAN And now Dthemetri began to enter into a negotiation with my hosts for a passage over the river. I never interfered with my worthy dragoman upon these occasions, because from my entire ignorance of the Arabic I should have been quite unable to exercise any real control over his words, and it would have been silly to break the stream of his eloquence to no purpose. I have reason to fear, however, that he lied transcendently, and especially in representing me as the bosom friend of Ibrahim Pasha. The mention of that name produced immense agitation and excitement, and the Sheik explained to Dthemetri the grounds of the infinite respect which he and his tribe entertained for the Pasha. A few weeks before Ibrahim had craftily sent a body of troops across the Jordan. The force went warily round to the foot of the mountains on the east, so as to cut off the retreat of this tribe, and then surrounded them as they lay encamped in the vale; their camels, and indeed all their possessions worth taking, were carried off by the soldiery, and moreover the then Sheik, together with every tenth man of the tribe, was brought out and shot. You would think that this conduct on the part of the Pasha might not procure for his ā€œfriendā€ a very gracious reception amongst the people whom he had thus despoiled and decimated; but the Asiatic seems to be animated with a feeling of profound respect, almost bordering upon affection, for all who have done him any bold and violent wrong; and there is always, too, so much of vague and undefined apprehension mixed up with his really well-
founded alarms, that I can see no limit to the yielding and bending of his mind when it is wrought upon by the idea of power. After some discussion the Arabs agreed, as I thought, to conduct me to a ford, and we moved on towards the river, followed by seventeen of the most able-bodied of the tribe, under the guidance of several grey-bearded elders, and Sheik Ali Djoubran at the head of the whole detachment. Upon leaving the encampment a sort of ceremony was performed, for the purpose, it seemed, of ensuring, if possible, a happy result for the undertaking. There was an uplifting of arms, and a repeating of words that sounded like formulƦ, but there were no prostrations, and I did not understand that the ceremony was of a religious character. The tented Arabs are looked upon as very bad Mahometans. [149] We arrived upon the banks of the riverā€”not at a ford, but at a deep and rapid part of the stream, and I now understood that it was the plan of these men, if they helped me at all, to transport me across the river by some species of raft. But a reaction had taken place in the opinions of many, and a violent dispute arose upon a motion which seemed to have been made by some honourable member with a view to robbery. The fellows all gathered together in circle, at a little distance from my party, and there disputed with great vehemence and fury for nearly two hours. I canā€™t give a correct report of the debate, for it was held in a barbarous dialect of the Arabic unknown to my dragoman. I recollect I sincerely felt at the time that the arguments in favour of robbing me must have been almost unanswerable, and I gave great credit to the speakers on my side for the ingenuity and sophistry which they must have shown in maintaining the fight so well. During the discussion I remained lying in front of my baggage, which had all been taken from the pack-saddles and placed upon the ground. I was so languid from want of food, that I had scarcely animation enough to feel as deeply interested as you would suppose in the result of the discussion. I thought, however, that the
pleasantest toys to play with during this interval were my pistols, and now and then, when I listlessly visited my loaded barrels with the swivel ramrods, or drew a sweet, musical click from my English firelocks, it seemed to me that I exercised a slight and gentle influence on the debate. Thanks to Ibrahim Pashaā€™s terrible visitation the men of the tribe were wholly unarmed, and my advantage in this respect might have counterbalanced in some measure the superiority of numbers. Mysseri (not interpreting in Arabic) had no duty to perform, and he seemed to be faint and listless as myself. Shereef looked perfectly resigned to any fate. But Dthemetri (faithful terrier!) was bristling with zeal and watchfulness. He could not understand the debate, which indeed was carried on at a distance too great to be easily heard, even if the language had been familiar; but he was always on the alert, and now and then conferring with men who had straggled out of the assembly. At last he found an opportunity of making a proposal, which at once produced immense sensation; he offered, on my behalf, that if the tribe should bear themselves loyally towards me, and take my party and my baggage in safety to the other bank of the river, I should give them a teskeri, or written certificate of their good conduct, which might avail them hereafter in the hour of their direst need. This proposal was received and instantly accepted by all the men of the tribe there present with the utmost enthusiasm. I was to give the men, too, a baksheish, that is, a present of money, which is usually made upon the conclusion of any sort of treaty; but although the people of the tribe were so miserably poor, they seemed to look upon the pecuniary part of the arrangement as a matter quite trivial in comparison with the teskeri. Indeed the sum which Dthemetri promised them was extremely small, and not the slightest attempt was made to extort any further reward. The council now broke up, and most of the men rushed madly towards me, and overwhelmed me with vehement gratulations; they
caressed my boots with much affection, and my hands were severely kissed. The Arabs now went to work in right earnest to effect the passage of the river. They had brought with them a great number of the skins which they use for carrying water in the desert; these they filled with air, and fastened several of them to small boughs which they cut from the banks of the river. In this way they constructed a raft not more than about four or five feet square, but rendered buoyant by the inflated skins which supported it. On this a portion of my baggage was placed, and was firmly tied to it by the cords used on my pack-saddles. The little raft with its weighty cargo was then gently lifted into the water, and I had the satisfaction to see that it floated well. Twelve of the Arabs now stripped, and tied inflated skins to their loins; six of the men went down into the river, got in front of the little raft, and pulled it off a few feet from the bank. The other six then dashed into the stream with loud shouts, and swam along after the raft, pushing it from behind. Off went the craft in capital style at first, for the stream was easy on the eastern side; but I saw that the tug was to come, for the main torrent swept round in a bend near the western bank of the river. The old men, with their long grey grisly beards, stood shouting and cheering, praying and commanding. At length the raft entered upon the difficult part of its course; the whirling stream seized and twisted it about, and then bore it rapidly downwards; the swimmers flagged, and seemed to be beaten in the struggle. But now the old men on the bank, with their rigid arms uplifted straight, sent forth a cry and a shout that tore the wide air into tatters, and then to make their urging yet more strong they shrieked out the dreadful syllables, ā€œā€™Brahim Pasha!ā€ The swimmers, one moment before so blown and so weary, found lungs to answer the cry, and shouting back the name of their great destroyer, they dashed on through the torrent, and bore the raft in safety to the western bank.
Afterwards the swimmers returned with the raft, and attached to it the rest of my baggage. I took my seat upon the top of the cargo, and the raft thus laden passed the river in the same way, and with the same struggle as before. The skins, however, not being perfectly air-tight, had lost a great part of their buoyancy, so that I, as well as the luggage that passed on this last voyage, got wet in the waters of Jordan. The raft could not be trusted for another trip, and the rest of my party passed the river in a different and (for them) much safer way. Inflated skins were fastened to their loins, and thus supported, they were tugged across by Arabs swimming on either side of them. The horses and mules were thrown into the water and forced to swim over. The poor beasts had a hard struggle for their lives in that swift stream; and I thought that one of the horses would have been drowned, for he was too weak to gain a footing on the western bank, and the stream bore him down. At last, however, he swam back to the side from which he had come. Before dark all had passed the river except this one horse and old Shereef. He, poor fellow, was shivering on the eastern bank, for his dread of the passage was so great, that he delayed it as long as he could, and at last it became so dark that he was obliged to wait till the morning. I lay that night on the banks of the river, and at a little distance from me the Arabs kindled a fire, round which they sat in a circle. They were made most savagely happy by the tobacco with which I supplied them, and they soon determined that the whole night should be one smoking festival. The poor fellows had only a cracked bowl, without any tube at all, but this morsel of a pipe they handed round from one to the other, allowing to each a fixed number of whiffs. In that way they passed the whole night. The next morning old Shereef was brought across. It was a strange sight to see this solemn old Mussulman, with his shaven head and his sacred beard, sprawling and puffing upon the surface of the water. When at last he reached the bank the people told him that by his baptism in Jordan he had surely become a mere Christian. Poor
Shereef!ā€”the holy man! the descendant of the Prophet!ā€”he was sadly hurt by the taunt, and the more so as he seemed to feel that there was some foundation for it, and that he really might have absorbed some Christian errors. When all was ready for departure I wrote the teskeri in French and delivered it to Sheik Ali Djoubran, together with the promised baksheish; he was exceedingly grateful, and I parted in a very friendly way from this ragged tribe. In two or three hours I gained Rihah, a village said to occupy the site of ancient Jericho. There was one building there which I observed with some emotion, for although it may not have been actually standing in the days of Jericho, it contained at this day a most interesting collection ofā€”modern loaves. Some hours after sunset I reached the convent of Santo Saba, and there remained for the night.
CHAPTER XVI TERRA SANTA The enthusiasm that had glowed, or seemed to glow, within me for one blessed moment when I knelt by the shrine of the Virgin at Nazareth, was not rekindled at Jerusalem. In the stead of the solemn gloom and the deep stillness that of right belonged to the Holy City, there was the hum and the bustle of active life. It was the ā€œheight of the season.ā€ The Easter ceremonies drew near. The pilgrims were flocking in from all quarters; and although their objects were partly at least of a religious character, yet their ā€œarrivalsā€ brought as much stir and liveliness to the city as if they had come up to marry their daughters. The votaries who every year crowd to the Holy Sepulchre are chiefly of the Greek and Armenian Churches. They are not drawn into Palestine by a mere sentimental longing to stand upon the ground trodden by our Saviour, but rather they perform the pilgrimage as a plain duty strongly inculcated by their religion. A very great proportion of those who belong to the Greek Church contrive at some time or other in the course of their lives to achieve the enterprise. Many in their infancy and childhood are brought to the holy sites by their parents, but those who have not had this advantage will often make it the main object of their lives to save money enough for this holy undertaking. The pilgrims begin to arrive in Palestine some weeks before the Easter festival of the Greek Church. They come from Egypt, from all parts of Syria, from Armenia and Asia Minor, from Stamboul, from
Roumelia, from the provinces of the Danube, and from all the Russias. Most of these people bring with them some articles of merchandise, but I myself believe (notwithstanding the common taunt against pilgrims) that they do this rather as a mode of paying the expenses of their journey, than from a spirit of mercenary speculation. They generally travel in families, for the women are of course more ardent than their husbands in undertaking these pious enterprises, and they take care to bring with them all their children, however young; for the efficacy of the rites does not depend upon the age of the votary, so that people whose careful mothers have obtained for them the benefit of the pilgrimage in early life, are saved from the expense and trouble of undertaking the journey at a later age. The superior veneration so often excited by objects that are distant and unknown shows not perhaps the wrongheadedness of a man, but rather the transcendent power of his imagination. However this may be, and whether it is by mere obstinacy that they poke their way through intervening distance, or whether they come by the winged strength of fancy, quite certainly the pilgrims who flock to Palestine from the most remote homes are the people most eager in the enterprise, and in number too they bear a very high proportion to the whole mass. The great bulk of the pilgrims make their way by sea to the port of Jaffa. A number of families charter a vessel amongst them, all bringing their own provisions, which are of the simplest and cheapest kind. On board every vessel thus freighted there is, I believe, a priest, who helps the people in their religious exercises, and tries (and fails) to maintain something like order and harmony. The vessels employed in this service are usually Greek brigs or brigantines and schooners, and the number of passengers stowed in them is almost always horribly excessive. The voyages are sadly protracted, not only by the land-seeking, storm-flying habits of the Greek seamen, but also by their endless schemes and speculations, which are for ever tempting them to touch at the nearest port. The voyage, too, must be made in winter, in order that Jerusalem may be reached some weeks before the Greek Easter, and thus by the time
they attain to the holy shrines the pilgrims have really and truly undergone a very respectable quantity of suffering. I once saw one of these pious cargoes put ashore on the coast of Cyprus, where they had touched for the purpose of visiting (not Paphos, but) some Christian sanctuary; I never saw (no, never even in the most horridly stuffy ballroom) such a discomfortable collection of human beings. Long huddled together in a pitching and rolling prison, fed on beans, exposed to some real danger and to terrors without end, they had been tumbled about for many wintry weeks in the chopping seas of the Mediterranean. As soon as they landed they stood upon the beach and chanted a hymn of thanks; the chant was morne and doleful, but really the poor people were looking so miserable that one could not fairly expect from them any lively outpouring of gratitude. When the pilgrims have landed at Jaffa they hire camels, horses, mules, or donkeys, and make their way as well as they can to the Holy City. The space fronting the Church of the Holy Sepulchre soon becomes a kind of bazaar, or rather, perhaps, reminds you of an English fair. On this spot the pilgrims display their merchandise, and there too the trading residents of the place offer their goods for sale. I have never, I think, seen elsewhere in Asia so much commercial animation as upon this square of ground by the church door; the ā€œmoney-changersā€ seemed to be almost as brisk and lively as if they had been within the temple. When I entered the church I found a babel of worshippers. Greek, Roman, and Armenian priests were performing their different rites in various nooks and corners, and crowds of disciples were rushing about in all directions, some laughing and talking, some begging, but most of them going round in a regular and methodical way to kiss the sanctified spots, and speak the appointed syllables, and lay down the accustomed coin. If this kissing of the shrines had seemed as though it were done at the bidding of enthusiasm, or of any poor sentiment even feebly approaching to it, the sight would have been less odd to English eyes; but as it was, I stared to see
grown men thus steadily and carefully embracing the sticks and the stones, not from love or from zeal (else God forbid that I should have stared!), but from a calm sense of duty; they seemed to be not ā€œworking out,ā€ but transacting the great business of salvation. Dthemetri, however, who generally came with me when I went out, in order to do duty as interpreter, really had in him some enthusiasm. He was a zealous and almost fanatical member of the Greek Church, and had long since performed the pilgrimage, so now great indeed was the pride and delight with which he guided me from one holy spot to another. Every now and then, when he came to an unoccupied shrine, he fell down on his knees and performed devotion; he was almost distracted by the temptations that surrounded him; there were so many stones absolutely requiring to be kissed, that he rushed about happily puzzled and sweetly teased, like ā€œJack among the maidens.ā€ A Protestant, familiar with the Holy Scriptures, but ignorant of tradition and the geography of modern Jerusalem, finds himself a good deal ā€œmazedā€ when he first looks for the sacred sites. The Holy Sepulchre is not in a field without the walls, but in the midst, and in the best part of the town, under the roof of the great church which I have been talking about. It is a handsome tomb of oblong form, partly subterranean and partly above ground, and closed in on all sides except the one by which it is entered. You descend into the interior by a few steps, and there find an altar with burning tapers. This is the spot which is held in greater sanctity than any other at Jerusalem. When you have seen enough of it you feel perhaps weary of the busy crowd, and inclined for a gallop; you ask your dragoman whether there will be time before sunset to procure horses and take a ride to Mount Calvary. Mount Calvary, signor?ā€” eccolo! it is upstairsā€”on the first floor. In effect you ascend, if I remember rightly, just thirteen steps, and then you are shown the now golden sockets in which the crosses of our Lord and the two thieves were fixed. All this is startling, but the truth is, that the city having gathered round the Sepulchre, which is the main point of
interest, has crept northward, and thus in great measure are occasioned the many geographical surprises that puzzle the ā€œBible Christian.ā€ The Church of the Holy Sepulchre comprises very compendiously almost all the spots associated with the closing career of our Lord. Just there, on your right, He stood and wept; by the pillar, on your left, He was scourged; on the spot, just before you, He was crowned with the crown of thorns; up there He was crucified, and down here He was buried. A locality is assigned to every, the minutest, event connected with the recorded history of our Saviour; even the spot where the cock crew when Peter denied his Master is ascertained, and surrounded by the walls of an Armenian convent. Many Protestants are wont to treat these traditions contemptuously, and those who distinguish themselves from their brethren by the appellation of ā€œBible Christiansā€ are almost fierce in their denunciation of these supposed errors. It is admitted, I believe, by everybody that the formal sanctification of these spots was the act of the Empress Helena, the mother of Constantine, but I think it is fair to suppose that she was guided by a careful regard to the then prevailing traditions. Now the nature of the ground upon which Jerusalem stands is such, that the localities belonging to the events there enacted might have been more easily, and permanently, ascertained by tradition than those of any city that I know of. Jerusalem, whether ancient or modern, was built upon and surrounded by sharp, salient rocks intersected by deep ravines. Up to the time of the siege Mount Calvary of course must have been well enough known to the people of Jerusalem; the destruction of the mere buildings could not have obliterated from any manā€™s memory the names of those steep rocks and narrow ravines in the midst of which the city had stood. It seems to me, therefore, highly probable that in fixing the site of Calvary the Empress was rightly guided. Recollect, too, that the voice of tradition at Jerusalem is quite unanimous, and that Romans, Greeks, Armenians, and Jews, all hating each other sincerely, concur in assigning the same
localities to the events told in the Gospel. I concede, however, that the attempt of the Empress to ascertain the sites of the minor events cannot be safely relied upon. With respect, for instance, to the certainty of the spot where the cock crew, I am far from being convinced. Supposing that the Empress acted arbitrarily in fixing the holy sites, it would seem that she followed the Gospel of St. John, and that the geography sanctioned by her can be more easily reconciled with that history than with the accounts of the other Evangelists. The authority exercised by the Mussulman Government in relation to the holy sites is in one view somewhat humbling to the Christians, for it is almost as an arbitrator between the contending sects (this always, of course, for the sake of pecuniary advantage) that the Mussulman lends his contemptuous aid; he not only grants, but enforces toleration. All persons, of whatever religion, are allowed to go as they will into every part of the Church of the Holy Sepulchre, but in order to prevent indecent contests, and also from motives arising out of money payments, the Turkish Government assigns the peculiar care of each sacred spot to one of the ecclesiastic bodies. Since this guardianship carries with it the receipt of the coins which the pilgrims leave upon the shrines, it is strenuously fought for by all the rival Churches, and the artifices of intrigue are busily exerted at Stamboul in order to procure the issue or revocation of the firmans by which the coveted privilege is granted. In this strife the Greek Church has of late years signally triumphed, and the most famous of the shrines are committed to the care of their priesthood. They possess the golden socket in which stood the cross of our Lord, whilst the Latins are obliged to content themselves with the apertures in which were inserted the crosses of the two thieves. They are naturally discontented with that poor privilege, and sorrowfully look back to the days of their former gloryā€”the days when Napoleon was Emperor, and Sebastiani ambassador at the Porte. It seems that the ā€œcitizenā€ sultan, old Louis Philippe, has done very little indeed for Holy Church in Palestine.
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Foundations Of Modern Physics Steven Weinberg

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    Foundations of ModernPhysics In addition to his ground-breaking research, Nobel Laureate Steven Weinberg is known for a series of highly praised texts on various aspects of physics, com- bining exceptional physical insight with a gift for clear exposition. Describing the foundations of modern physics in their historical context and with some new derivations, Weinberg introduces topics ranging from early applications of atomic theory through thermodynamics, statistical mechanics, transport theory, special relativity, quantum mechanics, nuclear physics, and quantum field theory. This volume provides the basis for advanced undergraduate and graduate physics courses as well as being a handy introduction to aspects of modern physics for working scientists. steven weinberg is a member of the Physics and Astronomy Departments at the University of Texas at Austin. He has been honored with numerous awards, including the Nobel Prize in Physics, the National Medal of Science, the Heinemann Prize in Mathematical Physics, and most recently a Special Breakthrough Prize in Fundamental Physics. He is a member of the US National Academy of Sciences, the UKā€™s Royal Society, and other academies in the US and internationally. The American Philosophical Society awarded him the Benjamin Franklin medal, with a citation that said he is ā€œconsidered by many to be the preeminent theoretical physicist alive in the world today.ā€ He has written several highly regarded books, including Gravitation and Cosmology, the three-volume work The Quantum Theory of Fields, Cosmology, Lectures on Quantum Mechanics, and Lectures on Astrophysics.
  • 8.
    Foundations of ModernPhysics Steven Weinberg University of Texas, Austin
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    University Printing House,Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314ā€“321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi ā€“ 110025, India 79 Anson Road, #06ā€“04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the Universityā€™s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108841764 DOI: 10.1017/9781108894845 Ā© Steven Weinberg 2021 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2021 Printed in the United Kingdom by TJ Books Limited, Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Weinberg, Steven, 1933ā€“ author. Title: Foundations of modern physics / Steven Weinberg, The University of Texas at Austin. Description: New York : Cambridge University Press, 2021. | Includes bibliographical references and indexes. Identifiers: LCCN 2020055431 (print) | LCCN 2020055432 (ebook) | ISBN 9781108841764 (hardback) | ISBN 9781108894845 (epub) Subjects: LCSH: Physics. Classification: LCC QC21.3 .W345 2021 (print) | LCC QC21.3 (ebook) | DDC 530ā€“dc23 LC record available at https://lccn.loc.gov/2020055431 LC ebook record available at https://lccn.loc.gov/2020055432 ISBN 978-1-108-84176-4 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
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    Contents PREFACE page xiii 1EARLY ATOMIC THEORY 1 1.1 Gas Properties 2 Air pressure Boyleā€™s law Temperature Scales Charlesā€™ law Explanation of gas laws Ideal gas law 1.2 Chemistry 6 Elements Law of combining weights Daltonā€™s atomic weights Law of combining volumes Avogadroā€™s principle The gas constant Avogadroā€™s number 1.3 Electrolysis 10 Early electricity Early magnetism Electromagnetism Discovery of electrolysis Faradayā€™s theory The faraday 1.4 The Electron 14 Cathode rays Thomsonā€™s experiments Electrons as atomic constituents 2 THERMODYNAMICS AND KINETIC THEORY 16 2.1 Heat and Energy 16 Caloric Heat as energy Kinetic energy Specific heat Energy density and pressure Adiabatic changes vii
  • 13.
    viii Contents 2.2 AbsoluteTemperature 21 Carnot cycles Theorems on efficiency Absolute temperature defined Relation to gas thermometers 2.3 Entropy 27 Definition of entropy Independence of path Increase of entropy Thermodynamic relations Entropy of ideal gases Neutral matter Radiation energy Laws of thermodynamics 2.4 Kinetic Theory and Statistical Mechanics 33 Maxwellā€“Boltzmann distribution General H-theorem Time reversal Canonical and grand-canonical distributions Connection with thermodynamics Compound systems Probability distribution in gases Equipartition of energy Entropy as disorder 2.5 Transport Phenomena 42 Conservation laws Galilean relativity Navierā€“Stokes equation Viscosity Mean free path Diffusion 2.6 The Atomic Scale 53 Nineteenth century estimates Electronic charge Brownian motion Consistency of constants Appendix: Einsteinā€™s diffusion constant rederived 3 EARLY QUANTUM THEORY 61 3.1 Black Body Radiation 61 Absorption and energy density Degrees of freedom of electromagnetic fields Rayleighā€“Jeans distribution Planck distribution Measurement of Boltzmann constant Radiation energy constant 3.2 Photons 67 Quantization of radiation energy Derivation of Planck distribution Photoelectric effect Particles of light 3.3 The Nuclear Atom 71 Radioactivity Alpha and beta rays Discovery of the nucleus Nuclear mass Nuclear size Scattering pattern Nuclear charge 3.4 Atomic Energy Levels 77 Spectral lines Electron orbits Combination principle Bohrā€™s quantization condition Correspondence principle Comparison with observed one-electron atomic spectra Reduced mass Atomic number Outstanding questions
  • 14.
    Contents ix 3.5 Emissionand Absorption of Radiation 84 Einstein A and B coefficients Equilibrium with black body radiation Relations among coefficients Lasers Suppressed absorption 4 RELATIVITY 88 4.1 Early Relativity 88 Motion of the Earth Relativity of motion Speed of light Michelsonā€“Morley experiment Lorentzā€“Fitzgerald contraction 4.2 Einsteinian Relativity 94 Postulate of invariance of electrodynamics Lorentz transformations Space inversion, time reversal The Galilean limit Maximum speed Boosts in general directions Special and general relativity 4.3 Clocks, Rulers, Light Waves 103 Clocks and time dilation Rulers and length contraction Transformation of frequency and wave number 4.4 Mass, Energy, Momentum, Force 106 Einsteinā€™s thought experiment Formulas for energy and momentum E = mc2 Force in relativistic dynamics 4.5 Photons as Particles 111 Photon momentum Compton scattering Other massless particles 4.6 Maxwellā€™s Equations 114 The inhomogeneous and homogeneous equations Density and current of electric charge Relativistic formulation of inhomogeneous Maxwell equations Indices upstairs and downstairs Relativistic formulation of homogeneous Maxwell equations Electric and magnetic forces 4.7 Causality 121 Causes precede effects Invariance of temporal order Maximum signal speed Light cone 5 QUANTUM MECHANICS 124 5.1 De Broglie Waves 125 Free-particle wave functions Group velocity Application to hydrogen Davissonā€“Germer experiment Electron microscopes Appendix: Derivation of the Bragg formula
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    x Contents 5.2 TheSchrƶdinger Equation 129 Wave equation for particle in potential Boundary conditions Spherical symmetry Radial and angular wave functions Angular multiplicity Spherical harmonics Hydrogenic energy levels Degeneracy 5.3 General Principles of Quantum Mechanics 138 States and wave functions Observables and operators Hamiltonian Adjoints Expectation values Probabilities Continuum limit Momentum space Commutation relations Uncertainty principle Time dependent wave functions Conservation laws Heisenberg and Schrƶdinger pictures 5.4 Spin and Orbital Angular Momentum 151 Doubling of sodium D-line The idea of spin General action of rotations on wave functions Total angular momentum operator Commutation relations Spin and orbital angular momentum Multiplets Adding angular momenta Atomic fine structure and space inversion Hyperfine structure Appendix: Clebschā€“Gordan Coefficients 5.5 Bosons and Fermions 165 Identical particles Symmetric and antisymmetric wave functions Bosons and fermions in statistical mechanics Hartree approximation Slater determinant Pauli exclusion principle Periodic table of elements Diatomic molecules: para and ortho Astrophysical cooling 5.6 Scattering 175 Scattering wave function Representations of the delta function Calculation of the Greenā€™s function Scattering amplitude Probabilistic interpretation Cross section Born approximation Scattering by shielded Coulomb potential Appendix: General transition rates 5.7 Canonical Formalism 190 Hamiltonian formalism Canonical commutation relations Lagrangian formalism Action principle Connection of formalisms Noetherā€™s theorem: symmetries and conservation laws Space translation and momentum 5.8 Charged Particles in Electromagnetic Fields 195 Vector and scalar potential Charged particle Hamiltonian Equations of motion Gauge transformations Magnetic interactions Spin coupling 5.9 Perturbation Theory 199 Perturbative expansion First-order perturbation theory Dealing with degeneracy The Zeeman effect Second-order perturbation theory
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    Contents xi 5.10 BeyondWave Mechanics 206 State vectors Linear operators First postulate: values of observables Second postulate: expectation values Probabilities Continuum limit Wave functions as vector components 6 NUCLEAR PHYSICS 210 6.1 Protons and Neutrons 210 Discovery of the proton Integer atomic weights Nuclei as protons and electrons? Trouble with diatomic nitrogen Discovery of the neutron Nuclear radius and binding energy Liquid drop model Stable valley and decay modes 6.2 Isotopic Spin Symmetry 216 Neutronā€“proton and protonā€“proton forces Isotopic spin rotations Isotopic spin multiplets Quark model Pions Appendix: The threeā€“three resonance 6.3 Shell Structure 224 Harmonic oscillator approximation Raising and lowering operators Degenerate multiplets Magic numbers Spinā€“orbit coupling 6.4 Alpha Decay 229 Coulomb barrier Barrier suppression factors Semi-classical estimate of alpha decay rate Level splitting Geigerā€“Nuttall law Radium alpha decay Appendix: Quantum theory of barrier penetration rates 6.5 Beta Decay 243 Electron energy distribution Neutrinos proposed Fermi theory Gamowā€“Teller modification Selection rules Strength of weak interactions Neutrinos discovered Violation of leftā€“right and matterā€“antimatter symmetries Neutrino helicities Varieties of neutrino 7 QUANTUM FIELD THEORY 251 7.1 Canonical Formalism for Fields 252 Action, Lagrangian, Lagrangian density Functional derivatives Eulerā€“Lagrange field equations Commutation relations Energy and momentum of fields 7.2 Free Real Scalar Field 255 Lagrangian density Field equation Creation and annihilation operators Energy and momentum Vacuum state Multiparticle states
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    xii Contents 7.3 Interactions261 Time-ordered perturbation theory Requirements for Lorentz invariance Example: Scattering of neutral spinless particles Feynman diagram Calculation of the propagator Yukawa potential 7.4 Antiparticles, Spin, Statistics 270 Antiparticles needed Complex scalar field General fields Lorentz transformation Spinā€“statistics connection Appendix: Dirac fields 7.5 Quantum Theory of Electromagnetism 280 Lagrangian density for electrodynamics Four-vector potential Gauge transformations Coulomb gauge Commutation relations Free fields Photon momentum and helicity Radiative decay rates Selection rules Gauge invariance and charge conservation Local phase invariance Standard model ASSORTED PROBLEMS 296 BIBLIOGRAPHY 301 AUTHOR INDEX 303 SUBJECT INDEX 307
  • 18.
    Preface This book grewout of the notes for a course I gave for undergraduate physics students at the University of Texas. In this book I think I go farther forward than is usual in undergraduate courses, giving readers a taste of nuclear physics and quantum field theory. I also go farther back than is usual, starting with the struggle in the nineteenth century to establish the existence and properties of atoms, including the development of thermodynamics that both aided in this struggle and offered an alternative program. I fear that some readers may want to skim through this early part and hurry on to what they regard as the good stuff, quantum mechanics and relativity. That would be a pity. In my experience physics students who aim at a career in atomic or nuclear or elementary particle physics often manage to get through their formal education without ever becoming familiar with entropy, or equipartition, or viscosity, or diffusion. That was true in my own case. This book, or a course based on it, may provide some students with their last chance to learn about these and other matters needed to understand the macroscopic world. Readers may find this book unusual also in its strong emphasis on history. I make a point of saying a little about the welter of theoretical guesswork and ill- understood experiments out of which modern physics emerged in the twentieth century. This, it seems to me, is a help in understanding what otherwise may seem an arbitrary set of postulates for relativity and quantum mechanics. It is also a matter of personal taste. Research in physics seems to me to lose some of its excitement if we do not see it as part of a great historical progression. Some valuable historical works are listed in a bibliography, along with collections of original articles that I have found most helpful. But this is not a work of history. Historians aim at uncovering how the scien- tists of the past thought about their own problems ā€“ for instance, how Einstein in 1905 thought about the measurement of space and time separations in de- veloping the special theory of relativity. For this aim of historical writing it is necessary to go deeply into personal accounts, institutional development, and xiii
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    xiv Preface false starts,and to put aside our knowledge of subsequent progress. I try to be accurate in describing the state of physics in past times, but the aim of this book in discussing the problems of the past is different: it is to make clear how physicists think about these things today. This book is intended chiefly for physics students who are well into their time as undergraduates, and for working scientists who want a brief introduction to some area of modern physics. I have therefore not hesitated to use calculus and matrix algebra, though not in advanced versions. As required by the subject matter, the mathematical level here slopes upwards through the book. Where possible I have chosen concrete rather than abstract formulations of physical theories. For instance, in Chapter 5, on quantum mechanics, I mostly represent physical states as wave functions, only coming at the end of the chapter to their representation as vectors in Hilbert space. In some sections detailed material that can be skipped without losing the thread of the theory is put into appendices. Two of these appendices present what in my unbiased opinion are improved derivations of important results: the appendix to Section 2.6 gives a revised version of Einsteinā€™s derivation of his formula for the diffusion constant in Brownian motion, and the appendix to Section 6.4 presents a revision of Fermiā€™s calculation of the rate of alpha decay. In my experience, with some judicious pruning, the material of the book up to about the middle of Chapter 5 can be covered in a one-term undergraduate course. But I think that to go over the whole book would take a full two-term academic year. This book treats such a broad range of topics that it is impossible to go very far into any of them. Certainly its treatment of quantum mechanics, statistical mechanics, transport theory, nuclear physics, and quantum field theory is no substitute for graduate-level courses on these topics, any one of which would occupy at least a whole year. This book presents what I think, in an ideal world, the ambitious physics student would already know when he or she enters graduate school. At least, it is what I wish that I had known when I entered graduate school. In any case, I hope that the student or reader may be sufficiently interested in what I do discuss that they will want to go into these topics in greater detail in more specialized books or courses, and that they will find in this book a good preparation for such further studies. I am grateful to many students and colleagues for pointing out errors in the lecture notes on which this book is based and for the expert and friendly assistance I have received from Simon Capelin and Vince Higgs, the editors at Cambridge University Press who guided the publication of this book. STEVEN WEINBERG
  • 20.
    1 Early Atomic Theory Itis an old idea that matter consists of atoms, tiny indivisible particles moving in empty space. This theory can be traced to Democritus, working in the Greek city of Abdera, on the north shore of the Aegean sea. In the late 400s BC Democritus proclaimed that ā€œatoms and void alone exist in reality.ā€ He offered neither evidence for this hypothesis nor calculations on which to base predic- tions that could confirm it. Nevertheless, this idea was tremendously influential, if only as an example of how it might be possible to account for natural phe- nomena without invoking the gods. Atoms were brought into the materialistic philosophy of Epicurus of Samos, who a little after 300 BC founded one of the four great schools of Athens, the Garden. In turn, the idea of atoms and the philosophy of Epicurus were invoked in the poem On the Nature of Things by the Roman Lucretius. After this poem was rediscovered in 1417 it influ- enced Machiavelli, More, Shakespeare, Montaigne, and Newton, among others. Newton in his Opticks speculated that the properties of matter arise from the clustering of atoms into larger particles, which themselves cluster into larger particles, and so on. As we will see, Newton made a stab at an atomic theory of air pressure, but without significant success. The serious scientific application of the atomic theory began in the eighteenth century, with calculations of the properties of gases, which had been studied experimentally since the century before. This is the topic with which we begin this chapter. Applications to chemistry and electrolysis followed in the nine- teenth century and will be considered in subsequent sections. The final section of this chapter describes how the nature of atoms began to be clarified with the discovery of the electron. In the following chapter we will see how it became possible to estimate the atomsā€™ masses and sizes.1 1 Further historical details about some of these matters can be found in Weinberg, The Discovery of Subatomic Particles, listed in the bibliography. 1
  • 21.
    2 1 EarlyAtomic Theory 1.1 Gas Properties Experimental Relations The upsurge of enthusiasm for experiment in the seventeenth century was largely concentrated on the properties of air. The execution and reports of these experiments did not depend on hypotheses regarding atoms, but we need to recall them here because their results provided the background for later theories of gas properties that did rely on assumptions about atoms. It had been thought by Aristotle and his followers that the suction observed in pumps and bellows arises from natureā€™s abhorrence of a vacuum. This notion was challenged in the 1640s by the invention of the barometer by the Florentine polymath Evangelista Torricelli (1608ā€“1647). If nature abhors a vacuum, then when a long glass tube with one end closed is filled with mercury and set upright with the closed end on top, why does the mercury flow out of the bottom until the column is only 760 mm high, with empty space appearing above the mercury? Is there a limit to how much nature abhors a vacuum? Torricelli argued that the mercury is held up instead by the pressure of the air acting on the open end of the glass tube (or on the surface of a bath of mercury in which the open end of the tube is immersed), which is just sufficient to support a column of mercury 760 mm high. If so, then it should be possible to measure variations in air pressure using a column of mercury in a vertical glass tube, a device that we know as a barometer. Such measurements were made from 1648 to 1651 by Blaise Pascal (1623ā€“1662), who found that the height of mercury in a barometer is decreased by moving to the top of a mountain, where less air extends above the barometer. The quantitative properties of air pressure soon began to be studied experimentally, before there was any correct theoretical understanding of gas properties. In 1662, in the second edition of his book New Experiments Physico- Mechanical Concerning the Spring of the Air and its Effects, the Anglo-Irish aristocrat Robert Boyle (1627ā€“1691) described experiments relating the pres- sure (the ā€œspring of the airā€) and volume of a fixed mass of air. He studied a sample of air enclosed at the end of a glass tube by a column of mercury in the tube. The air was compressed at constant temperature by pushing on the mercuryā€™s surface, revealing what came to be known as Boyleā€™s law, that for constant temperature the volume of a gas of fixed mass and composition is inversely proportional to the pressure, now defined by Boyle as the force per area exerted on the gas. Temperature Scales A word must be said about the phrase ā€œat constant temperature.ā€ Boyle lived before the establishment of our modern Fahrenheit and Celsius scales, whose
  • 22.
    1.1 Gas Properties3 forerunners go back respectively to 1724 and 1742. But, although in Boyleā€™s time no meaningful numerical value could be given to the temperature of any given body, it was nevertheless possible to speak with precision of two bodies being at the same temperature: they are at the same temperature if when put in contact neither body is felt to grow appreciably hotter or colder. Boyleā€™s glass tube could be kept at constant temperature by immersing it in a large bath, say of water from melting ice. Later the Fahrenheit temperature scale was established by defining the temperature of melting ice as 32 ā—¦F and the temperature of boiling water at mean atmospheric pressure as 212 ā—¦F, and defining a 1 ā—¦F increase of temperature by etching 212 āˆ’ 32 equal divisions between 32 and 212 on the glass tube of a mercury thermometer. Likewise, in the Celsius scale, the temperatures of melting ice and boiling water are 0 ā—¦C and 100 ā—¦C, and 1 ā—¦C is the temperature difference required to increase the volume of mercury in a thermometer by 1% of the volume change in heating from melting ice to boiling water. As we will see in the next chapter, there is a more sophis- ticated universal definition of temperature, to which scales based on mercury thermometers provide only a good approximation. After the temperature scale was established it became possible to carry out a quantitative study of the relation between volume and temperature, with pres- sure and mass kept fixed by enclosing the air in a vessel with flexible walls, which expand or contract to keep the pressure inside equal to the air pressure outside. This relation was announced in an 1802 lecture by Joseph Louis Gay- Lussac (1775ā€“1850), who attributed it to unpublished work in the 1780s by Jacques Charles (1746ā€“1823). The relation, subsequently known as Charlesā€™ Law, is that at constant pressure and mass the volume of gas is proportional to T āˆ’ T0, where T is the temperature measured for instance with a mercury thermometer and T0 is a constant whose numerical value naturally depends on the units used for temperature: T0 = āˆ’459.67 ā—¦F = āˆ’273.15 ā—¦C. Thus T0 is absolute zero, the minimum possible temperature, at which the gas vol- ume vanishes. Using Celsius units for temperature differences, the absolute temperature T ā‰” T āˆ’ T0 is known today as the temperature in degrees Kelvin, denoted K. Theoretical Explanations In Proposition 23 of his great book, the Principia, Isaac Newton (1643ā€“1727) made an attempt to account for Boyleā€™s law by considering air to consist of particles repelling each other at a distance. Using little more than dimensional analysis, he showed that the pressure p of a fixed mass of air is inversely proportional to the volume V if the repulsive force between particles separated by a distance r falls off as 1/r. But as he pointed out, if the repulsive force goes as 1/r2, then p āˆ V āˆ’4/3. He did not claim to offer any reason why the repulsive force should go as 1/r and, as we shall see, it is not forces that go as 1/r but
  • 23.
    4 1 EarlyAtomic Theory rather forces of very short range that act only in collisions that mostly account for the properties of gases. It was the Swiss mathematical physicist Daniel Bernoulli (1700ā€“1782) who made the first attempt to understand the properties of gases theoretically, on the assumption that a gas consists of many tiny particles moving freely except in very brief collisions. In 1738, in the chapter, ā€œOn the Properties and Motions of Elastic Fluids, Especially Airā€ of his book Hydrodynamics, he argued that in a gas (then called an ā€œelastic fluidā€) with n particles per unit volume moving with a velocity v that is the same (because of collisions) in all directions, the pressure is proportional to n and to v2, because the number of particles that hit any given area of the wall in a given time is proportional to the number in any given volume, to the rate at which they hit the wall, which is proportional to v, and to the force that each particle exerts on the wall, which is also proportional to v. For a fixed mass of gas n is inversely proportional to the volume V , so pV is proportional to v2. If (as Bernoulli thought) v2 depends only on the temperature, this explains Boyleā€™s law. If v2 is proportional to the absolute temperature, it also gives Charlesā€™ law. Bernoulli did not give much in the way of mathematical details, and did not try to say to what else the pressure might be proportional besides nv2, a matter crucial for the history of chemistry. These details were provided by Rudolf Clausius (1822ā€“1888) in 1857, in an article entitled ā€œThe Nature of the Motion which We Call Heat.ā€ Below is a more-or-less faithful description of Clausiusā€™ derivation, in a somewhat different notation. Suppose a particle hits the wall of a vessel and remains in contact with it for a small time t, during which it exerts a force with component F along the inward normal to the wall. Its momentum in the direction of the inward normal to the wall will decrease by an amount Ft, so if the component of the velocity of the particle before it strikes the wall is vāŠ„ 0, and it bounces back elastically with normal velocity component āˆ’vāŠ„, the change in the inward normal component of momentum is āˆ’2mvāŠ„, where m is the particle mass, so F = 2mvāŠ„/t . Now, suppose that this goes on with many particles hitting the wall over a time interval T t, all particles with the same velocity vector v. The number N of particles that will hit an area A of the wall in this time is the number of particles in a cylinder with base A and height vāŠ„T , or N = nAvāŠ„T , where n is the number density, the number of particles per volume. Each of these particles is in contact with the wall for a fraction t/T of the time T , so the total force exerted on the wall is FN(t/T ) = 2mvāŠ„/t Ɨ nAvāŠ„T Ɨ (t/T ) = 2nmv2 āŠ„A .
  • 24.
    1.1 Gas Properties5 We see that all dependence on the times t and T cancels. The pressure p is defined as the force per area, so this gives the relation p = 2nmv2 āŠ„ . (1.1.1) This is for the unphysical case in which every particle has the same value of vāŠ„, positive in the sense that the particles are assumed to be going toward the wall. In the real world, different particles will be moving with different speeds in different directions, and Eq. (1.1.1) should be replaced with p = 2nm Ɨ 1 2 v2 āŠ„ = nmv2 āŠ„ , (1.1.2) the brackets indicating an average over all gas particles, with the factor 1/2 inserted in the first expression because only 50% of these particles will be going toward any given wall area. To express v2 āŠ„ in terms of the root mean square velocity, Clausius assumed without proof that ā€œon the average each direction [of the particle velocities] is equally represented.ā€ In this case, the average square of each component of velocity equals v2 āŠ„, and the average of the squared velocity vector is then v2 = v2 1 + v2 2 + v2 3 = 3v2 āŠ„ and therefore Eq. (1.1.2) reads p = nmv2 /3 . (1.1.3) This is essentially the result p āˆ nv2 of Bernoulli, except that, with the factor m/3, Eq. (1.1.3) is now an equality, not just a statement of proportion- ality. For a fixed mass M of gas occupying a volume V , the number density is n = M/mV , so Clausius could use Boyleā€™s law (which he called Mariotteā€™s law), which states that pV is constant for fixed temperature, to conclude that for a given gas v2 depends only on the temperature. Further, as Clausius remarked, Eq. (1.1.3) together with Charlesā€™ law (which Clausius called the law of Gay-Lussac) indicates that v2 is proportional to the absolute temper- ature T . If we like, we can adopt a modern notation and write the constant of proportionality as 3k/m, so that mv2 /3 = kT , (1.1.4) and therefore Eq. (1.1.3) reads p = nkT , (1.1.5) where k is a constant, in the sense of being independent of p, n, and T . But the choice of notation does not tell us whether k varies from one type of gas to another or whether it depends on the molecular mass m. Clausius could not answer this question, and did not offer any theoretical justification for Boyleā€™s
  • 25.
    6 1 EarlyAtomic Theory law or Charlesā€™ law. Clausius deserves to be called the founder of thermo- dynamics, discussed in Sections 2.2 and 2.3, but these are not questions that can be answered by thermodynamics alone. As we will see in the following section, experiments in the chemistry of gases indicated that k is the same for all gases, a universal constant now known as Boltzmannā€™s constant, but the theoretical explanation for this and for Boyleā€™s law and Charlesā€™ law had to wait for the development of kinetic theory and statistical mechanics, the subject of Section 2.4. As indicated by the title of his article, ā€œThe Nature of the Motion which We Call Heat,ā€ Clausius was concerned to show that, at least in gases, the phe- nomenon of heat is explained by the motion of the particles of which gases are composed. He defended this view by using his theory to calculate the specific heat of gases, a topic to be considered in the next chapter. 1.2 Chemistry Elements The idea that all matter is composed of a limited number of elements goes back to the earliest speculations about the nature of matter. At first, in the century before Socrates, it was supposed that there is just one element: water (Thales) or air (Anaximenes) or fire (Heraclitus) or earth (perhaps Xenophanes). The idea of four elements was proposed around 450 BC by Empedocles of Acragas (modern Agrigento). In On Nature he identified the elements as ā€œfire and water and earth and the endless height of air.ā€ Classical Chinese sources list five elements: water, fire, earth, wood, and metal. Like the theory of atoms, these early proposals of elements did not come accompanied with any evidence that these really are elements, or any suggestion how such evidence might be gained. Plato in Timaeus even doubled down and stated that the difference between one element and another arises from the shapes of the atoms of which the elements are composed: earth atoms are tiny cubes, while the atoms of fire, air, and water are other regular polyhedra ā€“ solids bounded respectively by 4, 8, or 20 identical regular polygons, with every edge and every vertex of each solid the same as every other edge or vertex of that solid. By the end of the middle ages this list of elements had come to seem implau- sible. It is difficult to identify any particular sample of dirt as the element earth, and fire seems more like a process than a substance. Alchemists narrowed the list of elements to just three: mercury, sulfur, and salt. Modern chemistry began around the end of the eighteenth century, with careful experiments by Joseph Priestley (1733ā€“1804), Henry Cavendish (1743ā€“ 1810), Antoine Lavoisier (1743ā€“1794), and others. By 1787 Lavoisier had
  • 26.
    1.2 Chemistry 7 workedout a list of 55 elements. In place of air there were several gases: hydrogen, oxygen, and nitrogen; air was identified as a mixture of nitrogen and oxygen. There were other non-metals on the list of elements: sulfur, carbon, and phosphorus, and a number of common metals: iron, copper, tin, lead, silver, gold, mercury. Lavoisier also listed as elements some chemicals that we now know are tightly bound compounds: lime, soda, and potash. And the list also included heat and light, which of course are not substances at all. Law of Combining Weights Chemistry was first used to provide quantitative information about atoms by John Dalton (1766ā€“1844), the son of a poor weaver. His laboratory notebooks from 1802 to 1804 describe careful measurements of the weights of elements combining in compounds. He discovered that these weights are always in fixed ratios. For instance, he found that when hydrogen burns in oxygen, 1 gram of hydrogen combines with 5.5 grams of oxygen, giving 6.5 grams of water, with nothing left over. Under the assumption that one particle of water consists of one atom of hydrogen and one atom of oxygen, one oxygen atom must weigh 5.5 times as much as one hydrogen atom. As we will see, water was soon discovered to be H2O: two atoms of hydrogen to each atom of oxygen. If Dalton had known this, he would have concluded that an oxygen atom weighs 5.5 times as much as two hydrogen atoms, i.e., 11 times the weight of one hydrogen atom. Of course, more accurate measurements later revealed that 1 gram of hydrogen combines with about 8 grams of oxygen, so one oxygen atom weighs eight times the weight of two hydrogen atoms, or 16 times as much as one hydrogen atom. Atomic weights soon became defined as the weights of atoms relative to the weight of one hydrogen atom, so the atomic weight of oxygen is 16. (This is only approximate. Today the atomic weight of the atoms of the most common isotope of carbon is defined to be precisely 12; with this definition, the atomic weights of the most common isotopes of hydrogen and oxygen are measured to be 1.007825 and 15.99491.) The following table compares Daltonā€™s assumed formulas for a few common compounds with the correct formulas: Compound Dalton formula True formula Water HO H2O Carbon dioxide CO2 CO2 Ammonia NH NH3 Sulfuric acid SO2 H2SO4
  • 27.
    8 1 EarlyAtomic Theory Here is a list of the approximate true atomic weights for a few elements, the weights deduced by Dalton, and (in the column marked with an asterisk) the weights Dalton would have calculated if he had known the true chemical formulas. Element True Dalton Dalton* H 1 1 1 C 12 4.3 8.6 N 14 4.2 12.6 O 16 5.5 11 S 32 14.4 57.6 To make progress in measuring atomic weights, it was evidently necessary to find some way of working out the correct formulas for various chemical compounds. This was provided by the study of chemical reactions in gases. Law of Combining Volumes On December 31, 1808, Gay-Lussac read a paper to the Societe Philomathique in Paris, in which he announced his observation that gases at the same tem- perature and pressure always combine in definite proportions of volumes. For instance, two liters of hydrogen combine with one liter of oxygen to give water vapor, with no hydrogen or oxygen left over. Likewise, one liter of nitrogen combines with three liters of hydrogen to give ammonia gas, with nothing left over. And so on. The correct interpretation of this experimental result was given in 1811 by Count Amadeo Avogadro (1776ā€“1856) in Turin. Avogadroā€™s principle states that equal volumes of gases at the same temperature and pressure always con- tain equal numbers of the gas particles, which Avogadro called ā€œmolecules,ā€ particles that may consist of single atoms or of several atoms of the same or different elements joined together. The observation that water vapor is formed from a volume of oxygen combined with a volume of hydrogen twice as large shows, according to Avogadroā€™s principle, that molecules of water are formed from twice as many molecules of hydrogen as molecules of oxygen, which is not what Dalton had assumed. There was a further surprise in the data. Two liters of hydrogen combined with one liter of oxygen give not one but two liters of water vapor. This is not what one would expect if oxygen and hydrogen molecules consist of single atoms and water molecules consist of two atoms of hydrogen and one atom of oxygen. In that case two liters of hydrogen plus one liter of oxygen would produce one liter of water vapor. Avogadro could conclude that if, as seemed
  • 28.
    1.2 Chemistry 9 plausible,molecules of water contain two atoms of hydrogen and one atom of oxygen, the molecules of oxygen and hydrogen must each contain two atoms. That is, taking water molecules as H2O, the reaction for producing molecules of water is 2H2 + O2 ā†’ 2H2O . The use of Avogadroā€™s principle rapidly provided the correct formulas for gases such as CO2, NH3, NO, and so on. Knowing these formulas and measuring the weights of gases participating in various reactions, it was possible to cor- rect Daltonā€™s atomic weights and calculate more reliable values for the atomic weights of the atoms in gas molecules, relative to any one of them. Taking the atomic weight of hydrogen as unity, this gave atomic weights close to 12 for carbon, 14 for nitrogen, 16 for oxygen, 32 for sulfur, and so on. Then, knowing these atomic weights, it became possible to find atomic weights for many other elements, not just those commonly found in gases, by measuring the weights of elements combining in various chemical reactions. The Gas Constant As we saw in the previous section, in 1857 Clausius had shown that in a gas consisting of n particles of mass m per volume with mean square velocity v2, the pressure is p = nmv2/3. Using Charlesā€™ law, he concluded that v2 is proportional to absolute temperature. Writing this relation as mv2/3 = kT with k some constant gives Eq. (1.1.5), p = nkT . But this in itself does not tell us how k varies from one gas to another. This is answered by Avogadroā€™s principle. With N particles in a volume V , the number density is n = N/V , so Eq. (1.1.5) can be written pV = NkT . (1.2.1) If as stated by Avogadro the number of molecules in a gas with a given pressure, volume, and temperature is the same for any gas, then k = pV/NT must be the same for any gas. Clausius did not draw this conclusion, perhaps because there was then no known theoretical basis for Avogadroā€™s principle. The universality of the constant k, and hence Avogadroā€™s principle, were explained later by kinetic theory, to be covered in the next chapter. The constant k came to be called Boltzmannā€™s constant, after Ludwig Boltzmann, who as we shall see was one of the chief founders of kinetic theory. The molecular weight Ī¼ of any compound is defined as the sum of the atomic weights of the atoms in a single molecule. The actual mass m of a molecule is its molecular weight times the mass m1 of a hypothetical atom with atomic weight unity: m = Ī¼m1 . (1.2.2)
  • 29.
    10 1 EarlyAtomic Theory In the modern system of atomic weights, with the atomic weight of the most common isotope of carbon defined as precisely 12, m1 = 1.660539 Ɨ 10āˆ’24 g, which of course was not known in Avogadroā€™s time. A mass M contains N = M/m = M/m1Ī¼ molecules, so the ideal gas law (1.2.1) can be written pV = MkT/m1Ī¼ = (M/Ī¼)RT (1.2.3) where R is the gas constant R = k/m1 . (1.2.4) Physicists in the early nineteenth century could use Eq. (1.2.3) to measure R, and they found a value close to the modern value R = 8.314 J/K. This would have allowed a determination of m1 and hence of the masses of all atoms of known atomic weight if k were known, but k did not become known until the developments described in Section 2.6. Avogadroā€™s Number Incidentally, a mole of any element or compound of molecular weight Ī¼ is defined as Ī¼ grams, so in Eq. (1.2.3) the ratio M/Ī¼ expressed in grams equals the number of moles of gas. Since N = M/m1Ī¼, one mole contains a number of molecules equal to 1/m1 with m1 given in grams. This is known as Avogadroā€™s number. But of course Avogadro did not know Avogadroā€™s number. It is now known to be 6.02214 Ɨ 1023 molecules per mole, corresponding to unit molec- ular weight m1 = 1.66054 Ɨ 10āˆ’24 grams. The measurement of Avogadroā€™s number was widely recognized in the late nineteenth century as one of the great challenges facing physics. 1.3 Electrolysis Early Electricity Electricity was known in the ancient world, as what we now call static electricity. Amber rubbed with fur was seen to attract or repel small bits of light material. Plato in Timaeus mentions ā€œmarvels concerning the attraction of amber.ā€ (This is where the word electricity comes from; the Greek word for amber is ā€œelektron.ā€) Electricity began to be studied scientifically in the eighteenth century. Two kinds of electricity were distinguished: resinous electricity is left on an amber rod when rubbed with fur, while vitreous electricity is left on a glass rod when rubbed with silk. Unlike charges were found to attract each other, while like charges repel each other. Benjamin Franklin (1706ā€“1790) gave our modern terms positive and negative to vitreous and resinous electricity, respectively.
  • 30.
    1.3 Electrolysis 11 In1785 Charles-Augustin de Coulomb (1736ā€“1806) reported that the force F between two bodies carrying charges q1 and q2 separated by a distance r is F = keq1q2 r2 (1.3.1) where ke is a universal constant. For like and unlike charges the product q1q2 is positive or negative, respectively, indicating a repulsive or attractive force. Coulomb had no way of actually measuring these charges, but he could reduce the charge on a body by a factor 2 by touching it to an uncharged body of the same material and size, and observe that this reduces the force between it and any other charged body by the same factor 2. The introduction of our modern units of electric charge had to wait until the quantitative study of magnetism. Early Magnetism Magnetism too was known in the ancient world, as what we now call per- manent magnetism. The Greeks knew of naturally occurring lodestones that could attract or repel small bits of iron. Platoā€™s Timaeus refers to lodestones as ā€œHeraclean stones.ā€ (Our word magnet comes from the city Magnesia in Asia Minor, near where lodestones were commonly found.) Very early the Chinese also discovered the lodestone and used it as a magnetic compass (a ā€œsouth-seeking stoneā€) for purposes of geomancy and navigation. Each lodestone has a south-seeking pole at one end, attracted to a point near the South Pole of the Earth, and a north-seeking pole at the other end, attracted to a point near the Earthā€™s North Pole. Magnetism was first studied scientifically by William Gilbert (1544ā€“1603), court physician to Elizabeth I. It was observed that the south-seeking poles of different lodestones repel each other, and like- wise for the north-seeking poles, while the south-seeking pole of one lodestone attracts the north-seeking pole of another lodestone. Gilbert concluded that one pole of a lodestone is pulled toward the north and the other toward the south because the Earth itself is a magnet, with what in a lodestone would be its south-seeking and north-seeking poles respectively near the Earthā€™s North Pole and South Pole. Electromagnetism It began to be possible to explore the relations between electricity and mag- netism quantitatively with the invention in 1809 of electric batteries by Count Alessandro Volta (1745ā€“1827). These were stacks of disks of two different metals separated by cardboard disks soaked in salt water. Such batteries drive steady currents of electricity through wires attached to the ends of the stacks, with positive and negative terminals identified respectively as the ends of the stacks from which and towards which electric current flows.
  • 31.
    12 1 EarlyAtomic Theory In July 1820 Hans Christian Oersted (1777ā€“1851) in Copenhagen noticed that turning on an electric current deflected a nearby compass needle, and concluded that electric currents exert force on magnets. Conversely, he found also that magnets exert force on wires carrying electric currents. These discoveries were carried further in Paris a few months later by AndrĆØ- Marie AmpĆØre (1775ā€“1836), who found that wires carrying electric current exert force on each other. For two parallel wires of length L carrying electric currents (charge per second) I1 and I2, and separated by a distance r L, the force is F = kmI1I2L r , (1.3.2) where km is another universal constant. The force is repulsive if the currents are in the same direction; attractive if in opposite directions. One ampere is defined so that F = 10āˆ’7 Ɨ L/r newtons if I1 = I2 = 1 ampere. (That is, km ā‰” 10āˆ’7 N/ampere2.) The electromagnetic unit of electric charge, the coulomb, is defined as the electric charge carried in one second by a current of one ampere. A modern ammeter measures electric currents by observing the magnetic force produced by current flowing through a wire loop. The connection between electricity and magnetism was strengthened in 1831 by Michael Faraday (1791ā€“1867), at the Royal Institution in London. He dis- covered that changing magnetic fields generate electric forces that can drive currents in conducting wires. This is the principle underlying the generation of electric currents today. Electricity began soon after to have important practical applications, with the invention in 1831 of the electric telegraph by the Ameri- can painter Samuel F. B. Morse (1791ā€“1872). Finally, in the 1870s, the great Scottish physicist James Clerk Maxwell (1831ā€“1879) showed that the consistency of the equation for the generation of magnetic fields by electric currents required that magnetic fields are also generated by changing electric fields. In particular, while oscillating magnetic fields produce oscillating electric fields, also oscillating electric fields produce oscillating magnetic fields, so a self-sustaining oscillation in both electric and magnetic fields can propagate in apparently empty space. Maxwell calculated the speed of its propagation and found it to equal āˆš 2ke/km,2 numerically about equal to the measured speed of light, suggesting strongly that light is such a self-sustaining oscillation in electric and magnetic fields. We will see more of Maxwellā€™s equations in subsequent chapters, especially in Chapters 4 and 5. 2 This quantity is independent of the units used for electric charge as long as the currents appearing in Eq. (1.3.2) are defined as the rates of flow of charge in the same units as used in Eq. (1.3.1). It is obviously also independent of the units used for force, as long as the same force units are used in Eqs. (1.3.1) and (1.3.2).
  • 32.
    1.3 Electrolysis 13 Discoveryof Electrolysis Electrolysis was discovered in 1800 by the chemist William Nicholson (1753ā€“ 1815) and the surgeon Anthony Carlisle (1768ā€“1840). They found that bubbles of hydrogen and oxygen would be produced where wires attached respectively to the negative and positive terminals of a Volta-style battery were inserted in water. Sir Humphrey Davy (1778ā€“1829), Faradayā€™s boss at the Royal Institution, carried out extensive experiments on the electrolysis of molten salts, finding for instance that, in the electrolysis of molten table salt, sodium, a previously unknown metal, was produced at the wire attached to the negative terminal of the battery and a greenish gas, chlorine, was produced at the wire attached to the other, positive, terminal. Davyā€™s electrolysis experiments added several metals aside from sodium to Lavoisierā€™s list of elements, including aluminum, potassium, calcium, and magnesium. A theory of electrolysis was worked out by Faraday. In modern terms, a small fraction (1.8 Ɨ 10āˆ’9 at room temperature) of water molecules are nor- mally dissociated into positive hydrogen ions (H+), which are attracted to the wire attached to the negative terminal of a battery, and negative hydroxyl ions (OHāˆ’), which are attracted to the wire attached to the positive terminal. At the wire attached to the negative terminal, two H+ ions combine with two units of negative charge from the battery to form a neutral H2 molecule. At the wire attached to the positive terminal, four OHāˆ’ ions give one O2 molecule plus two H2O molecules plus four units of negative charge, which flow through the battery to the negative terminal.3 Likewise, a small fraction of molten table salt (NaCl) molecules are normally dissociated into Na+ ions and Clāˆ’ ions. At the wire attached to the negative terminal of a battery, one Na+ ion plus one unit of negative charge gives one atom of metallic sodium (Na); at the wire attached to the positive terminal, two Clāˆ’ ions give one chlorine (Cl2) molecule and two units of negative charge, which flow through the battery to the negative terminal. In Faradayā€™s theory, it takes one unit of electric charge to convert a singly charged ion such as H+ or Clāˆ’ to a neutral atom or molecule, so since molecules of molecular weight Ī¼ have mass Ī¼m1, it takes M/m1Ī¼ units of electric charge to convert a mass M of singly charged ions to a mass M of neutral atoms or molecules of molecular weight Ī¼. Experiment showed that it takes about 96 500 coulombs (e.g., one ampere for about 96 500 seconds) to convert Ī¼ grams (that is, one mole) of singly charged ions to neutral atoms or molecules. (This is called a faraday; the modern value is 96 486.3 coulombs/mole.) Hence 3 We now know that it is negative charge, i.e., electrons, that flows through a battery. As far as Faraday knew, it was equally possible that positive charges flow through a battery, in which case at the wire attached to the negative terminal two H+ ions would give an H2 molecule plus two units of positive charge, which would flow though the battery to the wire attached to the positive terminal, where four OHāˆ’ ions plus four units of positive charge would give an O2 molecule and two H2O molecules.
  • 33.
    14 1 EarlyAtomic Theory Faraday knew that e/m1 96 500 coulombs/gram, where e is the unit of electric charge, which was called an ā€œelectrineā€ in 1874 by the Irish physi- cist George Johnstone Stoney (1826ā€“1911). Having measured the faraday, if physicists knew the value of e then they would know m1, but they didnā€™t have this information until later. Also, no one then knew that e is the charge of an actual particle. 1.4 The Electron As sometimes happens, in 1858 a new path in fundamental physics was opened with the invention of a practical device, in this case an improved air pump. In his pump the Bonn craftsman Heinrich Johann Geissler (1814ā€“1879) used a column of mercury as a piston, in this way greatly reducing the leakage of air through the piston that had troubled all previous air pumps. With his pump Geissler was able to reduce the pressure in a closed glass tube to about a ten-thousandth of the typical air pressure on the Earthā€™s surface. With such a near vacuum in a glass tube, electric currents could travel without wires through the tube. It was discovered that an electric current would flow from a cathode, a metal plate attached to the negative terminal of a powerful electric battery, fly through a hole in an anode, another metal plate attached to the positive pole of the battery, and light up a spot on the far wall of the tube. Adding small amounts of various gases to the interior of the tube caused these cathode rays to light up, with orange or pink or blue-green light emitted along the path of the ray, when neon, helium, or mercury vapor was added. Using Geisslerā€™s pumps, Julius PlĆ¼cker (1801ā€“1868) in 1858ā€“1859 found that cathode rays could be deflected by magnetic fields, thus moving the spot of light where the ray hits the glass at the tube end. In 1897 Joseph John Thomson (1856ā€“1940), the successor to Maxwell as Cavendish Professor at Cambridge, began a series of measurements of the deflection of cathode rays. In his experiments, after the ray particles pass through the anode they feel an electric or magnetic force F exerted at a right angle to their direction of motion for a distance d along the ray. They then drift in a force-free region for a distance D d until they hit the end of the tube. If a ray particle has velocity v along the direction of the ray, it feels the electric or magnetic force for a time d/v and then drifts for a longer time D/v. A force F normal to the ray gives ray particles of mass m a component of velocity perpendicular to the ray that is equal to the acceleration F/m times the time d/v, so by the time they hit the end of the tube they have been displaced by an amount displacement = (F/m) Ɨ (d/v) Ɨ (D/v) = FdD mv2 .
  • 34.
    1.4 The Electron15 The forces exerted on a charge e by an electric field E or a magnetic field B at right angles to the ray are Felec = eE , Fmag = evB so electric displacement = eEdD mv2 , magnetic displacement = eBdD mv . Thomson wanted to measure e/m. He knew D, d, E, and B, but not v. He could eliminate v from these equations if he could measure both the electric and magnetic displacements, but the electric displacement was difficult to measure. A strong electric field tends to ionize any residual air in the tube, with positive and negative ions pulled to the negatively and positively charged plates that produce the electric fields, neutralizing their charges. Finally Thomson suc- ceeded in measuring the electric as well as the magnetic deflection by using a cathode ray tube with very low air pressure. (Both the electric and magnetic displacements were only a few inches.) This gave results for the ratio of charge to mass ranging from 6 Ɨ 107 to 108 coulombs per gram. Thomson compared this with the result that Faraday had found in measure- ments of electrolysis, that e/m1 ā‰ˆ 105 coulombs per gram, where e is the electric charge of a singly ionized atom or molecule (such as a sodium ion in the electrolysis of NaCl) and m1 is the mass of a hypothetical atom of atomic weight unity, close to the mass of the hydrogen atom. He reasoned that if the particles in his cathode rays are the same as those transferred in electrolysis, then their charge must be the same as e, so their mass must be about 10āˆ’3m1. Thomson concluded that since the cathode ray particles are so much lighter than ions or atoms, they must be the basic constituents of ions and atoms. Thomson had still not measured e or m. He had not even shown that cathode rays are streams of particles; they might be streams of electrically charged fluid, with any volume of fluid having a ratio of charge to mass equal to his measured e/m. Nevertheless, in the following decade it became widely accepted that Thomson had indeed discovered a particle present in atoms, and the particle came to be called the electron.
  • 35.
    2 Thermodynamics and KineticTheory The successful uses of atomic theory described in the previous chapter did not settle the existence of atoms in all scientistsā€™ minds. This was in part because of the appearance in the first half of the nineteenth century of an attractive competitor, the physical theory of thermodynamics. As we shall see in the first three sections of this chapter, with thermodynamics one may derive powerful results of great generality without ever committing oneself to the existence of atoms or molecules. But thermodynamics could not do everything. Section 2.4 will describe the advent of kinetic theory, which is based on the assumption that matter consists of very large numbers of particles, and its generalization to sta- tistical mechanics. From these thermodynamics could be derived, and together with the atomic hypothesis it yielded results far more powerful than could be obtained from thermodynamics alone. Even so, it was not until the appearance of direct evidence for the graininess of matter, described in Section 2.5, that the existence of atoms became almost universally accepted. 2.1 Heat and Energy The first step in the development of thermodynamics was the recognition that heat is a form of energy. Though so familiar to us today, this was far from obvious to the physicists and chemists of the early nineteenth century. Until the 1840s heat was widely regarded as a fluid, named caloric by Lavoisier. Caloric theory was used to calculate the speed of sound by Pierre-Simon Laplace (1749ā€“1827) in 1816, the conduction of heat by Joseph Fourier (1768ā€“1830) in 1807 and 1822, and the efficiency of steam engines by Sadi Carnot (1796ā€“1832) in 1824, whose work as we will see in the next section became a foundation of thermodynamics. Adding to the confusion, other scientists considered heat as some sort of wave. This reflected uncertainty regarding the nature of what is now called infrared radiation, discovered by William Herschel (1738ā€“1822) in 1800. 16
  • 36.
    2.1 Heat andEnergy 17 Heat as Energy In 1798 Benjamin Thompson (1753ā€“1814), an American expatriate in Eng- land, offered evidence against the idea that heat is a fluid. (Thompson is also known as Count Rumford, a title he was given when he later served as military adviser in Austria.) It was well known that boring a cannon produces heat, which might be supposed to be due to the liberation of caloric from the iron, but Rumford observed that if the heat is carried away by immersing the cannon in running water while it is being bored there is no limit to the heat that can be produced. The first measurement of the energy in heat was provided in the mid-1840s by James Prescott Joule (1818ā€“1889). In his apparatus a falling weight turned paddles in a tank of water, heating the water. The gravitational force on a mass m kilograms is m times the acceleration of gravity, 9.8 meters/sec2 or 9.8 newtons per kilogram. Work is force times distance, so dropping one kilogram a distance of one meter gave it an energy equal to 9.8 newton meters, now also known as 9.8 joules. Joule found that the paddles driven by this dropping weight would raise the temperature of 100 grams of water by 0.023 ā—¦C, so the paddles pro- duced heat equal to 0.023 Ɨ 100 calories, the calorie being defined as the heat required to raise the temperature of one gram of water by one degree Celsius. Hence Joule could conclude that 9.8 joules is equivalent to 2.3 calories, so one calorie is equivalent to 9.8/2.3 = 4.3 joules. The modern value is 4.184 joules. In 1847 the Prussian physician and physicist Hermann von Helmholtz (1821ā€“1894) put forward the idea of the universal conservation of energy, whether in the form of kinetic or potential or chemical energy or heat. But what sort of energy is heat? For some nineteenth century physicists the question was irrelevant. They developed the science of heat known as thermodynamics, which did not depend on any detailed model of heat energy. But there was one context in which the nature of heat energy seemed evident. In his great 1857 paper, The Nature of the Motion which We Call Heat, Clausius found that at least part of the heat energy of gases is the kinetic energy of their molecules. Kinetic Energy The concept of kinetic energy was long familiar. If a steady force F is exerted on a particle of mass m, it produces an acceleration F/m, so after a time t the velocity of the body is v = Ft/m. The distance traveled in this time is t times the average velocity v/2, and the work done on the particle is the force times this distance: F Ɨ t Ɨ Ft/2m = F2 t2 /2m = mv2 /2 . Instead of this work going into heating a tub of water, as in the experiment of Joule, it goes into giving the particle an energy mv2/2.
  • 37.
    18 2 Thermodynamicsand Kinetic Theory This energy has the special property of being conserved when bodies come into contact in collisions. Consider a collision between two rigid balls A and B with initial vector velocities vA and vB. For the moment suppose that the time interval t over which this force acts is sufficiently brief that the forces acting on the balls do not change appreciably during this time. The force that A exerts on B is equal and opposite to the force F that B exerts on A, so Newtonā€™s second law tells us that the final velocities of A and B are v A = vA + Ft/mA and v B = vB āˆ’ Ft/mB. Hence, as Newton showed, momentum is conserved: mAv A + mBv B = mAvA + mBvB . (2.1.1) Neglecting changes in acceleration during the brief time t, the vector displace- ments traveled by A and B equal t times the average velocities, [vA + v A]/2 and [vB + v B]/2, respectively. If the balls remain in contact during this time interval, then these displacements must be the same, so vA + v A = vB + v B . (2.1.2) To derive a second conservation law, rewrite Eq. (2.1.2) as v B āˆ’ v A = vA āˆ’ vB and square this, giving v2 B āˆ’ 2v B Ā· v A + v2 A = v2 B āˆ’ 2vB Ā· vA + v2 A . Multiply this with mAmB and add the square of Eq. (2.1.1), so that the scalar products cancel. Dividing by 2(mA+mB), the result is another conservation law, mA 2 v2 A + mB 2 v2 B = mA 2 v2 A + mB 2 v2 B . (2.1.3) Equations (2.1.1) and (2.1.3) have been derived here only for the case in which the particles are in contact only for a brief time interval during which the force acting between the bodies is constant, but this is not an essential requirement for we can break up any time interval into a large number of brief intervals in each of which the change in the force is negligible, Then, since mAvA + mBvB and mAv2 A/2+mBv2 B/2 do not change in each interval, they do not change at all, as long as the bodies exert forces on each other only when they are in contact. In 1669 Christiaan Huygens (1629ā€“1695) reported in Journal des SƧavans that he had confirmed the conservation of the total of mv2/2, probably by observing collisions of pendulum bobs, for which initial and final velocities could be precisely determined. Newton in the Principia called the conserved quantity mv the quantity of motion, while Huygens gave the name vis viva (ā€œliving forceā€) to the conserved quantity mv2/2. These two quantities have since become known as momentum and kinetic energy. On the other hand, it was essential in deriving the conservation of kinetic energy that we assumed that particles interact only when in contact. This is generally a good approximation in gases, but it is not valid in the presence of
  • 38.
    2.1 Heat andEnergy 19 long range forces, such as electromagnetic or gravitational forces. In such cases kinetic energy is not conserved ā€“ it is only the sum of kinetic energy plus some sort of potential energy that does not change. Specific Heat The total kinetic energy of N molecules of gas of mass m and mean square velocity v2 is Nmv2/2. Clausius had found the relation (1.1.4) between mean square velocity and absolute temperature, according to which mv2/2 = 3kT/2, where k is some constant (later identified as a universal constant of nature), so the total kinetic energy is 3NkT/2. A mass M of gas of molecu- lar weight Ī¼ contains N = M/Ī¼m1 molecules, so the total kinetic energy is 3MRT/2Ī¼, where R = k/m1 is the gas constant (1.2.4). Clausius concluded that to raise the temperature of a mass M of gas of molecular weight Ī¼ by an amount dT at constant volume, so that the gas does no work on its con- tainer, requires an energy dE = 3MRdT/2Ī¼. The ratio dE/MdT is known as the specific heat, so Clausius found that the specific heat of a gas at constant volume is Cv = 3R/2Ī¼ . (2.1.4) This result must be distinguished from the value for a different sort of specific heat, measured at constant pressure, such as when the gas is in a container with an expandable wall, for which the volume V can change to keep the pressure p equal to the pressure of the surrounding air or other medium. When pressure pushes a surface of area A a small distance dL, the work done is the force pA times dL, which equals pdV where dV = AdL is the change in volume. According to the ideal gas law (1.2.3), pV = RT M/Ī¼, so if the temperature is increased by an amount dT , then at constant pressure the gas does work pdV = RMdT/Ī¼, and this temperature increase therefore requires an energy 3MRdT/2Ī¼ + MRdT/Ī¼ = 5MRdT/3. In other words, the specific heat at constant pressure is Cp = 5R/2Ī¼ . (2.1.5) This result is often expressed in terms of the ratio of specific heats, Ī³ ā‰” Cp/Cv . (2.1.6) So Clausius found that if all the heat of a gas is contained in the kinetic energy of its molecules then Ī³ = 5/3. This did not agree with measurements of the specific heats of common diatomic gases, such as oxygen or hydrogen, which Clausius cited as giving Ī³ = 1.421. Later, it was found that Ī³ does indeed equal 5/3 for a monatomic gas like mercury vapor, but this left the question, in what form is the energy in ordinary gases that are not monatomic?
  • 39.
    20 2 Thermodynamicsand Kinetic Theory To deal with this issue, Clausius suggested that the internal energy of a gas is larger than the kinetic energy of the molecules, say by a factor 1 + f , with f some positive number. Then instead of Eq. (2.1.4) we have Cv = (1 + f ) Ɨ 3R/2Ī¼ , (2.1.7) and in place of Eq. (2.1.5), Cp = (1 + f )3R/2Ī¼ + R/Ī¼ . (2.1.8) The specific heat ratio is then Ī³ = 1 + 2 3(1 + f ) . (2.1.9) This is often expressed (especially in astrophysics) as a formula for the internal energy density E in terms of the pressure and Ī³ : E = 3RT M(1 + f )/2Ī¼V = 3(1 + f )p/2 = p/(Ī³ āˆ’ 1) . (2.1.10) The observation that Ī³ 1.4 for diatomic gases like O2 and H2 indicated that the internal energy of these gases is larger than the kinetic energy of its molecules by a factor 1 + f 5/3. Measurements gave values of Ī³ for more complicated molecules like H2O or CO2 even closer to unity, indicating that f is even larger for these molecules. The reason for these values for f and Ī³ did not become clear until the formulation of the equipartition of energy, to be discussed in Section 2.4. Adiabatic Changes It often happens that work is done adiabatically, that is, without the transfer of heat. In this case the conservation of energy tells us that the work done by an expanding fluid must be balanced by a decrease in its internal energy EV : 0 = p dV + d(EV ) = (p + E)dV + V dE . (2.1.11) For an ideal gas, the internal energy per unit volume E is given by Eq. (2.1.10), so this tells us that 0 = Ī³p dV + V dp and so, in an adiabatic process, p āˆ V āˆ’Ī³ āˆ ĻĪ³ , (2.1.12) or, since for a fixed mass p āˆ T/V , T āˆ V 1āˆ’Ī³ āˆ ĻĪ³ āˆ’1 . (2.1.13) This is in contrast with an isothermal process, for which T is constant and p āˆ V āˆ’1.
  • 40.
    2.2 Absolute Temperature21 Equation (2.1.12) has an immediate consequence for the speed of sound. At audible frequencies the conduction of heat is typically too slow to be effective, so the expansion and compression of a fluid carrying a sound wave is adiabatic. It is a standard result of hydrodynamics, proved by Newton, that the speed of sound is cs = āˆ‚p āˆ‚Ļ . Newton thought that p would be proportional to Ļ in a sound wave, which would give cs = āˆš p/Ļ, but in fact at audible frequencies the pressure is given by the adiabatic relation (2.1.12), and cs is larger than Newtonā€™s value by a factor āˆš Ī³ . 2.2 Absolute Temperature We have been casually discussing temperature, but what precisely do we mean by this? It is not hard to give a precise meaning to a statement that one body has a higher temperature than another, by a generalization of common experience that is sometimes known as the second law of thermodynamics (the first law being the conservation of energy). Observation of heat flow shows that if heat can flow spontaneously from a body A to a body B, then it cannot flow spontaneously from B to A. We can say then that the temperature tA of A is higher than the temperature tB of B. Likewise, we say that two bodies are at the same temperature if heat cannot flow from either one to the other spontaneously, without work being done on these bodies. Temperature defined in this way is observed to be transitive: If heat can flow spontaneously from a body A to a body B, and from B to a body C, then it can flow spontaneously from A to C. This is a property shared with real numbers ā€“ if a number a is larger than number b, and if b is larger than c, then a is larger than c ā€“ and is a necessary condition for temperatures to be represented by real numbers. But this does not give a precise meaning to any particular numerical value of temperature, or even to numerical ratios of temperatures. If, for some definition of temperature t, a comparison of values of t tells us the direction of heat flow then the same would be true of any monotonic function T (t). Conventionally temperatures are defined by thermometers. With a column of some liquid such as mercury or alcohol in a glass tube, we mark off the heights of the column when the tube is placed in freezing or boiling water, and for a Celsius tem- perature scale etch on the tube marks that divide the distance from freezing to boiling to a hundred equal parts. The trouble is that different liquids expand differently with increasing temperature, and the temperatures measured in this way with a mercury or alcohol thermometer will not be precisely equal. We can try instead to give significance to numerical values of temperature by using a
  • 41.
    22 2 Thermodynamicsand Kinetic Theory gas thermometer, relying on the ideal gas law pV = MRT/Ī¼, but this law is approximate, holding precisely only for molecules of negligible size that interact only in contact in collisions. How can we give precise meaning to numerical values of temperature without relying on approximate relations? Surprisingly, as shown by Rudolf Clausius in his 1850 paper1 ā€œOn the Mov- ing Force of Heat,ā€ it is possible by find a definition of temperature T with absolute significance by the study of thermodynamic engines known as Carnot cycles. Sadi Carnot (1796ā€“1832) was a French military engineer, the son of Lazare Carnot, organizer of military victory in the French Revolution, and uncle of a later president of the Third Republic. In 1824 Carnot in Reflections on the Motive Power of Fire set out to study the efficiency of steam engines, explaining that ā€œAlready the steam engine works our mines, impels our ships, excavates our ports and our rivers, forges iron, fashions wood, grinds grains, spins and weaves our clothes, transports our heaviest burdens, etc.ā€ (A few years later he might also have mentioned the beginning of steam-propelled locomotives, with the opening of the Liverpoolā€“Manchester railroad in 1830.) Carnot invented an idealized engine, known as a Carnot cycle, which as we shall see is maximally efficient and provides a natural definition of absolute temperature. In the Carnot cycle, a working fluid (such as steam in a cylinder fitted with a piston) goes through four frictionless steps: 1. Isothermal: The working fluid does work on its environment, for instance by pushing a piston against external pressure, but keeping a constant tem- perature by absorbing heat Q2 from a hot reservoir at temperature t2. (We will continue to use lower case t to indicate temperature defined in any way that indicates the direction of heat flow, without specifying any physical significance to its particular numerical values.) 2. Adiabatic: The working fluid, perfectly insulated from its environment and with no internal friction, does more work, with its temperature dropping to the temperature t1 of a cold reservoir but with no heat flowing in or out. 3. Isothermal: Work is done on the fluid, for instance by pushing in the piston, with its temperature kept constant by its giving up heat Q1 to the cold reservoir. 4. Adiabatic: With the working fluid again completely insulated from its envi- ronment, work is done on it, bringing its volume back down to its original value and its temperature back up to the temperature t2 of the hot reservoir. 1 This paper is reprinted in Brush, The Kinetic Theory of Gases ā€“ An Anthology of Classic Papers with Historical Commentary, listed in the bibliography.
  • 42.
    2.2 Absolute Temperature23 C B 1 2 3 4 p D A V Figure 2.1 A Carnot cycle (not drawn to scale). A graph of the pressure versus the volume of the working fluid in this cycle is a closed curve, with the net work W done on the environment equal to pdV ā€“ that is, to the area enclosed by the curve. (See Figure 2.1.) As long as steps 2 and 4 are truly adiabatic, the conservation of energy tells us that this work is W = Q2 āˆ’ Q1 (2.2.1) and the efficiency of this cycle is W/Q2 = Q2 āˆ’ Q1 Q2 1 . (2.2.2) (We call this the efficiency, having in mind that, as for a steam engine, we have to pay for the heat Q2 taken up at the higher temperature t2, while the heat Q1 given up at the lower temperature t1 is wasted.) Any Carnot cycle is reversible, because any frictionless adiabatic or isother- mal process follows the same track, depending only on its endpoints, whichever direction the process takes. But not all thermodynamic cycles, which take a working fluid through a series of steps back to the original temperature and volume, are reversible even though of course they all conserve energy. For reversibility it is not enough that all steps be either isothermal or adiabatic ā€“ there also should be no friction, which if present would provide an internal source of heat that is not available to do work.
  • 43.
    24 2 Thermodynamicsand Kinetic Theory The importance of Carnot cycles in thermodynamics rests on the following theorem: I. The efficiency of the Carnot cycle C described above is at least as great as that for any general thermodynamical cycle C, not necessarily reversible, which begins with the working fluid absorbing a heat Q 2 from a reservoir at the same high temperature t2, then emitting heat at the same lower temperature t1, and then returning to its original temperature and volume, in the process doing net work W. That is, W/Q2 ā‰„ W /Q 2 . (2.2.3) II. All Carnot cycles that take heat from a reservoir at the same temperature t2, using it to do work, and giving up waste heat to a reservoir at the same lower temperature t1, have the same efficiency, which depends only on t2 and t1. Proof:2 Like any positive real number, the ratio of the work done in the Carnot cycle C and in a general cycle C can be approximated to an arbitrary accuracy by a ratio of positive real integers N and N: W/W = N /N . (2.2.4) Since any Carnot cycle by definition is reversible, the cycle C has an inverse Cāˆ’1. This is a refrigeration cycle, following the same steps as for C but in the opposite order, so that by doing work W on the fluid an amount of heat Q1 is taken from the reservoir at temperature t1 and heat Q2 Q1 is delivered to the reservoir at temperature t2 t1. Suppose we perform a compound cycle Cāˆ—, consisting of N repetitions of Cāˆ’1 and N repetitions of C. According to Eq. (2.2.4), the net work done by the working fluid is Wāˆ— = N W āˆ’ NW = 0 . Also, the net heat taken from the hot reservoir at temperature t2 is Qāˆ— 2 = N Q 2 āˆ’ NQ2 . Now, since no work is done in the compound cycle, according to the fundamen- tal property of temperature t, it is not possible for positive-definite net heat to be transferred to a reservoir at temperature t2 from a lower temperature t1, so the net heat Qāˆ— 2 taken from the hot reservoir in the cycle Cāˆ— must be positive-definite or zero. Hence, using Eq. (2.2.4), 0 ā‰¤ Qāˆ— 2 NW = NQ 2 āˆ’ NQ2 NW = Q 2 W āˆ’ Q2 W , 2 This treatment and that of the following section is based on that given by Fermi, Thermodynamics, listed in the bibliography.
  • 44.
    2.2 Absolute Temperature25 and therefore W Q2 ā‰„ W Q 2 (2.2.5) as was to be proved in the first part of the theorem. As to the second part of the theorem, note that if C is also a Carnot cycle then, by the same reasoning, W Q 2 ā‰„ W Q2 , so the efficiencies are equal: W Q2 = W Q 2 . (2.2.6) This has now been proved for any pair of Carnot cycles, operating between the same temperatures t2 and t1, whatever the values of the heat taken from the reservoir at temperature t2 and given up to the reservoir at temperature t1, so the common efficiency can only depend on t2 and t1, as was to be proved. We shall write this relation in terms of the inefficiency: 1 āˆ’ W Q2 = Q1 Q2 ā‰” F(t1, t2) (2.2.7) with F the same function for all Carnot cycles. We next prove that the function F(t1, t2) takes the form F(t1, t2) = T (t1)/T (t2) (2.2.8) for some function T (t). For this purpose we consider a compound cycle consist- ing of a Carnot cycle operating between the temperatures t2 and t0 ā‰¤ t2 followed by a Carnot cycle operating between the temperatures t0 and t1 ā‰¤ t0, with all the waste heat that is given to the reservoir at temperature t0 in the first cycle taken up from this reservoir in the second cycle. Since (Q0/Q2)(Q1/Q0) = Q1/Q2, the inefficiency (2.2.7) of the compound cycle is the product of the inefficiencies of the individual cycles, so F(t1, t2) = F(t1, t0)F(t0, t2) . (2.2.9) From Eq. (2.2.7) it is evident that F(t2, t0)F(t0, t2) = 1, so Eq. (2.2.9) may be written F(t1, t2) = F(t1, t0) F(t2, t0) . (2.2.10) This holds for any t0 with t2 ā‰„ t0 ā‰„ t1, so we can define T (t) ā‰” F(t, t0) with an arbitrary choice of t0 in this range, and then Eq. (2.2.10) is the desired result (2.2.8).
  • 45.
    26 2 Thermodynamicsand Kinetic Theory Now, efficiencies are never greater than 100%, so the ratio F(t1, t2) = T (t1)/T (t2) in Eq. (2.2.7) must be positive, and so T (t) has the same sign for all temperatures. Since only the ratios of the T s appear in the efficiency, we are free to choose this sign to be positive, so that T (t) ā‰„ 0 for all t. Also, inefficiencies are never greater than 100%, so Eq. (2.2.8) shows that T (t1) ā‰¤ T (t2) for any t1 and t2 with t1 ā‰¤ t2. That is, T (t) is a monotonically increasing function of t and can therefore be used to judge the direction of spontaneous heat flow as well as t itself. We can therefore define the absolute temperature T by just using T (t) as the temperature in place of t. That is, using Eqs. (2.2.7) and (2.2.8), we define absolute temperature T by the statement that a Carnot cycle running between any two temperatures T2 and T1 has Q1 Q2 = T1 T2 . (2.2.11) A Carnot cycle running between an upper temperature T2 and a lower tempera- ture T1 has an efficiency W Q2 = Q2 āˆ’ Q1 Q2 = T2 āˆ’ T1 T2 . (2.2.12) Of course, this only defines T up to a constant factor, leaving us free to use what units we like for temperature. But we are not free to shift T (t) by adding a constant term. Indeed, since in this Carnot cycle heat flows from a reservoir at temperature T2 to one at temperature T1, we must have T2 T1, and therefore in order for the efficiency (2.2.11) to be a positive quantity, the lower temperature must have T1 0. Because any heat reservoir must have T positive-definite, we see that T is the absolute temperature, in the same sense as was found for gases by Charles. The temperature defined by Carnot cycles is identical (up to a choice of units) to the temperature given by a gas thermometer, which for the moment we will call T g, in the approximation that the gas is ideal. To see this, let us label the states of the gas as A at the start of the isothermal expansion 1 (and at the end of the adiabatic compression 4); as B at the start of the adiabatic expansion 2 (and the end of the isothermal expansion 1); as C at the start of the isothermal compression 3 (and the end of the adiabatic expansion 2); and as D at the start of the adiabatic compression 4 (and the end of the isothermal compression 3). Since the expansion from A to B is isothermal, during this phase the internal energy of the gas, which is given by Eqs. (1.2.3) and (2.1.10) as EV = RT M/(Ī³ āˆ’ 1)Ī¼, does not change, and so the heat drawn from the hot reservoir is the work done: Q2 = B A pdV = MRT g 2 Ī¼ B A dV V = MRT g 2 Ī¼ ln VB VA .
  • 46.
    2.3 Entropy 27 Likewise,the heat given up to the cold reservoir in the isothermal compression from C to D is Q1 = MRT g 1 Ī¼ ln VC VD . Further, since the expansion from B to C and the contraction from D to A are adiabatic, Eq. (2.1.13) gives V āˆ (T g)āˆ’1/(Ī³ āˆ’1), and so during these parts of the cycle VC/VB = VD/VA = T g 2 T g 1 1/(Ī³ āˆ’1) , and therefore VB/VA = VC/VD, and the logarithmic factors in Q2 and Q1 are equal. The efficiency is then Q2 āˆ’ Q1 Q2 = T g 2 āˆ’ T g 1 T g 2 in agreement with Eq. (2.2.12) if T = T g, up to a possible constant factor. 2.3 Entropy In macroscopic classical thermodynamics we characterize the state of a system by a set of variables that can be specified independently. For instance, for a fluid of fixed mass and chemical composition, in a vessel with adjustable volume (say, with a movable piston) the state is specified by giving the values of any two of the thermodynamic variables ā€“ pressure, volume, temperature, energy, etc. ā€“ the remaining variables being determined in equilibrium or in adiabatic variations by these two values and some equation of state, such as the ideal gas law (1.2.3). Many of the consequences of macroscopic classical thermody- namics can be deduced from the existence of another thermodynamic variable, known as the entropy, introduced in 1854 by Rudolf Clausius, that like other thermodynamic variables depends only on the state of the system, although its definition seems to indicate that it also depends on the way that the system is prepared. Suppose a system is prepared in a given state 1 by starting with it in a standard state (labeled 0 below) and then taken to 1 on a path P through the space of independent variables used to define thermodynamic states, in which by a series of small reversible changes at varying absolute temperatures T it picks up small net amounts dQ of heat energy from the environment. (The heat energy increment dQ is taken as positive if the system takes heat from the environment and negative if it gives up heat.) Then the entropy of this state is defined by
  • 47.
    28 2 Thermodynamicsand Kinetic Theory S1 = S0 + P dQ T , (2.3.1) where the integral is taken over any reversible path from state 0 to state 1, and S0 is whatever entropy we choose to ascribe to the standard state 0. The remarkable thing is that the integral here is independent of the particular reversible path chosen, so that this really defines the entropy S1 up to a common constant term S0 as a function of the state of the system, not of how it is prepared, provided that T is the absolute temperature defined, as in the previous section, by the efficiency of Carnot cycles. Furthermore, with the entropy defined in this way, for any path P from state 0 to state 1 that may or may not be reversible, we have P dQ T ā‰¤ S1 āˆ’ S0 . (2.3.2) Proof: The first step in proving these results is to prove the following lemma: for an arbitrary cycle, reversible or irreversible, that takes a system from any state back to the same state, taking in and giving up heat at various temperatures, we have dQ T ā‰¤ 0 . (2.3.3) After establishing this lemma, the rest of the proof will be straightforward. To prove this lemma we can approximate the cycle by a sequence of brief isothermal steps, in each of which the system takes in heat (if dQ is positive) or gives heat up (if dQ is negative) at a momentary temperature T . We can imagine that, at each step, the heat taken in or given up is given up or taken in by another system, which undergoes a Carnot cycle between the momentary temperature T and a fixed temperature T0. In this Carnot cycle, the ratio of the heat dQ given up by the Carnot cycle to and the heat dQ0 taken by the Carnot cycle from the reservoir at temperature T0 is given by Eq. (2.2.11): dQ dQ0 = T T0 , or in other words dQ0 = T0 dQ T . Hence in the complete cycle the Carnot cycles take in a total net heat T0 dQ/T from the reservoir at temperature T0. Since the system and each of the Carnot cycles return to their original states, if this heat taken in at temperature T0 were positive-definite then it would have to go into work, which is impossible since work cannot be done by taking heat from a reservoir at a fixed temperature with no changes elsewhere. (If it could, then this work by producing friction could
  • 48.
    2.3 Entropy 29 beused to transfer some heat to any body, even one at a temperature higher than T0.) So we conclude that the integral dQ/T is at most zero, as was to be shown. The rest is easy. Note that if two paths P and P are both reversible paths that go from state 0 to state 1, then PP āˆ’1 is a closed cycle, where P āˆ’1 is path P taken in reverse, from state 1 to state 0. It follows then from the inequality (2.3.3) that 0 ā‰„ PPāˆ’1 dQ T = P dQ T āˆ’ P dQ T . But P Pāˆ’1 is also a closed cycle, so 0 ā‰„ PPāˆ’1 dQ T = P dQ T āˆ’ P dQ T . These two results are consistent only if for reversible paths both cyclic integrals vanish, in which case P dQ T = P dQ T . (2.3.4) We can therefore define the entropy up to an additive constant as in Eq. (2.3.1), where P is any reversible path. Finally, if P is a general path from state 0 to state 1, reversible or irreversible, while P is a reversible path from state 0 to state 1, then P Pāˆ’1 (but not neces- sarily PPāˆ’1 !) is a closed cycle, so the inequality (2.3.3) gives 0 ā‰„ PPāˆ’1 dQ T = P dQ T āˆ’ P dQ T , and therefore, using Eq. (2.3.1), P dQ T ā‰¤ S1 āˆ’ S0 as was to be shown. In the special case of a completely isolated system , no heat can be taken into or given up by , so the integrand in the integral on the left-hand side of Eq. (2.3.2) must vanish and therefore S1 ā‰„ S0. In isolated systems the entropy can only increase. On the other hand, if an isolated system is undergoing only reversible changes, then according to Eq. (2.3.1) the entropy is constant. There is another definition of entropy, used in information theory as well as in physics. If a system can be in any one of a number of states characterized by a continuous (generally multidimensional) parameter Ī±, with a probability P(Ī±) dĪ± of being in states with this parameter in a narrow range dĪ± around Ī±, then the entropy is
  • 49.
    30 2 Thermodynamicsand Kinetic Theory S = āˆ’k P(Ī±) ln P(Ī±) dĪ± , (2.3.5) where k is the universal constant, known as Boltzmannā€™s constant, appearing in Eq. (1.2.1). As we shall see in the next section, according to kinetic the- ory, with a suitable choice of S0 the thermodynamic entropy (2.3.1) equals the information-theoretic entropy (2.3.5). The mere fact that the entropy S defined by (2.3.1) depends only on the ther- modynamic state has far-reaching consequences. Consider a fixed mass of fluid in a vessel with variable volume. The independent thermodynamic variables here can be taken as the volume V and the temperature T , with pressure p, internal energy E, and entropy S all functions of V and T . The work done by the fluid pressure p in increasing the fluid volume by a small amount dV is pdV , so the heat required to change the temperature by an infinitesimal amount dT and the volume by an infinitesimal amount dV is dQ = dE + pdV , so according to Eq. (2.3.1), the change in the entropy is given by T dS = dE + pdV . (2.3.6) In other words āˆ‚S(V , T ) āˆ‚T = 1 T āˆ‚E(V , T ) āˆ‚T (2.3.7) āˆ‚S(V , T ) āˆ‚V = 1 T āˆ‚E(V , T ) āˆ‚V + p(V , T ) T . (2.3.8) To squeeze information about pressure and internal energy from these formulas, we use the fact that partial derivatives commute. From Eq. (2.3.7) we have āˆ‚ āˆ‚V āˆ‚S(V , T ) āˆ‚T = 1 T āˆ‚2E(V , T ) āˆ‚T āˆ‚V while, from Eq. (2.3.8), āˆ‚ āˆ‚T āˆ‚S(V , T ) āˆ‚V = 1 T āˆ‚2E(V , T ) āˆ‚T āˆ‚V āˆ’ 1 T 2 āˆ‚E(V , T ) āˆ‚V + āˆ‚(p(V , T )/T ) āˆ‚T . Setting these equal gives a relation between the derivatives of E and p: 0 = āˆ’ 1 T 2 āˆ‚E(V , T ) āˆ‚V + āˆ‚ p(V , T )/T āˆ‚T . (2.3.9) This is for a fixed mass. Since E(V , T ) is an extensive variable, it must be proportional to this mass but does not otherwise have to depend on volume. In fact, it is frequently a good approximation to suppose that, apart from its proportionality to mass, E(V , T ) is independent of volume. This is the case if
  • 50.
    2.3 Entropy 31 thefluid consists of infinitesimal particles that interact only in contact in colli- sions; since there is nothing with the dimensions of length that can enter in the calculation of the energy, E(V , T ) cannot here depend on volume. In this case Eq. (2.3.9) yields Charlesā€™ law, that for fixed volume V the pressure p(V , T ) is proportional to T . This shows again that the absolute temperature T in the ideal gas law (1.2.3) is the same up to a constant factor as the temperature T defined by the efficiency (2.2.11) of Carnot cycles. Although this result was obtained without having a formula for the entropy, for some purposes it is useful actually to know what the entropy is. In a homo- geneous medium, the entropy S of any mass M of matter may conveniently be written as S = Ms, where s is the entropy per unit mass, a function of temper- ature and various densities known as the specific entropy. Dividing Eq. (2.3.6) by M, we have then T ds = d(E/Ļ) + pd(1/Ļ) , (2.3.10) where as before E ā‰” E/V is the internal energy density and Ļ ā‰” M/V is the mass density. We consider an ideal gas, for which T = pĪ¼/RĻ while E and p are related by Eq. (2.1.10): E = p/(Ī³ āˆ’ 1). Then Eq. (2.3.10) gives pĪ¼ RĻ ds = 1 Ī³ āˆ’ 1 dp Ļ + Ī³pd 1 Ļ = ĻĪ³ āˆ’1 Ī³ āˆ’ 1 d p ĻĪ³ , so ds = R/Ī¼ Ī³ āˆ’ 1 ĻĪ³ p d p ĻĪ³ . The solution is s = R/Ī¼ Ī³ āˆ’ 1 ln p ĻĪ³ + constant . (2.3.11) We see that the result of Section 2.2 that p āˆ ĻĪ³ for adiabatic processes is just the statement that s is constant in these processes, which of course it must be since in an adiabatic process the heat input dQ vanishes. In many stars there are regions in which convection effectively mixes matter from various depths. Since heat conduction is usually ineffective in stars, little heat flows into or out of a bit of matter as it rises or falls, and so it keeps the same specific entropy. These regions therefore have a uniform specific entropy, and therefore a uniform value for the ratio p/ĻĪ³ . For instance, this is the case in the Sun for distances from the center greater than about 65% of the Sunā€™s radius out to a thin surface layer. Neutral Matter We have been mostly concerned with matter in which in each mass there is a non-vanishing conserved quantity, the number of particles. There is a different
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    32 2 Thermodynamicsand Kinetic Theory context, with no similar conserved numbers, in which thermodynamics yields more detailed information about pressure and energy. In the early universe, at temperatures above about 1010 K, there is so much energy in radiation and electronā€“antielectron pairs that the contribution to the energy of the excess of matter over antimatter may be neglected. Here there is no number density on which the pressure and energy density E ā‰” E/V can significantly depend, so here E(V , T ) = V E(T ) and p(V , T ) = p(T ); thus here Eq. (2.3.9) is an ordinary differential equation for p(T ): 0 = āˆ’ E(T ) T 2 + d dT p(T ) T or, in other words, p (T ) = E(T ) + p(T ) T . (2.3.12) Thermodynamics alone does not fix any relation between E(T ) and p(T ), but given such a relation this result gives both as functions of temperature. For instance, as an example of the power of thermodynamics, it was known in the nineteenth century as a consequence of Maxwellā€™s theory of electromagnetism that the pressure of electromagnetic radiation is one-third of its energy density. Setting p = E/3 in Eq. (2.3.12) gives E(T ) = 4E(T ), so E(T ) = 3p(T ) = aT 4 , (2.3.13) where a is a constant, known as the radiation energy constant. But, as we shall see in Section 3.1, it was not possible to understand the value of a until the advent of quantum mechanics in the early twentieth century. The Laws of Thermodynamics It is common to summarize the content of classical thermodynamics in three laws. As already mentioned, the first law is just the conservation of energy, discussed in the context of heat energy in Section 2.1, and the second, usually attributed to Clausius, on which the discussion of thermodynamic efficiency in Section 2.2 is based, can be stated as the principle that without doing work it is not possible to transfer heat from a cold reservoir to one at higher temperature. We have seen that this leads to the existence of a quantity, the entropy, which depends only on the thermodynamic state and satisfies Eq. (2.3.1) when reversible changes are made in this state. This can instead be taken as the second law of thermodynamics. There are several formulations of the third law, some given by Walther Nernst (1864ā€“1941) in 1906ā€“1912. The most fruitful, it seems, is that it is possible to assign a common value to the entropy (conventionally taken as zero) for all systems at absolute zero temperature, so that at absolute zero the integral
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    2.4 Kinetic Theoryand Statistical Mechanics 33 in Eq. (2.3.1) must converge. This has the consequence, in particular, that the specific heat dQ/dT must vanish for T ā†’ 0. This seems to contradict the results of Section 2.1 for ideal gases, which give a temperature-independent specific heat whether for fixed volume or fixed pressure. The contradiction is avoided in practice because no substance remains close to an ideal gas as the temperature approaches absolute zero. We will see when we come to quantum mechanics that if an otherwise free particle is confined in any fixed volume, then it cannot have precisely zero momentum, as required for a classical ideal gas at absolute zero temperature. On the other hand, solids can exist at absolute zero temperature, and in that limit their specific heats do approach zero. 2.4 Kinetic Theory and Statistical Mechanics We saw in the previous chapter how by the mid nineteenth century the ideal gas law had been established through the work especially of Bernoulli and Clausius. But, though derived by considering the motions of individual gas molecules, in its conclusions it dealt only with bulk gas properties such as pressure, tem- perature, mass density, and energy density. For many purposes, including the calculation of chemical or transport processes, it was necessary to go further and work out the detailed probability distribution of the motion of individual gas particles. This was done in the kinetic theory of James Clerk Maxwell and Ludwig Boltzmann (1844ā€“1906). Kinetic theory was later generalized to the formalism known as statistical mechanics, especially by the American theorist Josiah Willard Gibbs (1839ā€“1906). As it turned out, these methods went a long way toward not only establishing a correspondence with thermodynamics but also explaining the principles of thermodynamics on the assumption that macroscopic matter is composed of very many particles, and thereby helping to establish the reality of atoms. The Maxwellā€“Boltzmann Distribution Maxwell in 1860 considered the form of the probability distribution function P(vx, vy, vz) for the x, y, and z components of the velocity of any molecule in a gas in equilibrium.3 The probability distribution function is defined so that the probability that these components are respectively between vx and vx + dvx, between vy and vy + dvy, and between vz and vz + dvz, is of the form P(vx, vy, vz)dvxdvydvz . 3 J. C. Maxwell, Phil. Mag. 19, 19; 20, 21 (1860). This article is included in Brush, The Kinetic Theory of Gases ā€“ An Anthology of Classic Papers with Historical Commentary, listed in the bibliography.
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    34 2 Thermodynamicsand Kinetic Theory He assumed (without offering a real justification) that the probability that any component of velocity of a particle is in a particular range is not correlated with the other components of the velocity. Then P(vx, vy, vz) must be proportional to a function of vx alone, with a coefficient that depends only on vy and vz, and likewise for vy and vz, so P(vx, vy, vz) must take the form of a product: P(vx, vy, vz) = f (vx)g(vy)h(vz) . Rotational symmetry requires further that P can depend only on the magnitude of the velocity, not on its direction, and hence only on v2 x + v2 y + v2 z . The only function of v2 x + v2 y + v2 z that takes the form f (vx)g(vy)h(vz) is proportional to an exponential: P(vx, vy, vz) āˆ exp āˆ’ C(v2 x + v2 y + v2 z ) . The constant C must be positive in order that P should not blow up for large velocity, which would make it impossible to set the total probability equal to unity, as it must be. Taking C to be positive, and setting the total probability (the integral of P over all velocities) for each particle equal to one, gives the factor of proportionality: P(vx, vy, vz) = C Ļ€ 3/2 exp āˆ’ C(v2 x + v2 y + v2 z ) . We can use this to calculate the mean square velocity components: v2 x = v2 y = v2 z = 1 2C . Clausius had introduced an absolute temperature T by setting mv2 āŠ„ = kT , where k is a constant to be determined experimentally and vāŠ„ is the component of the velocity in a direction normal to the container wall, which for an isotropic velocity distribution can be taken as any direction, so the constant C must be given by C = m/2kT and the Maxwell distribution takes the form P(vx, vy, vz) = m 2Ļ€kT 3/2 exp āˆ’ m(v2 x + v2 y + v2 z )/2kT . (2.4.1) As we saw at the end of Section 2.2, the requirement that in an ideal gas mvāŠ„ 2 = kT , which led here to C = m/2kT , also ensures that, up to an arbitrary constant factor, T is the absolute temperature defined by the efficiency of Carnot cycles. The formula for the probability distribution P was derived in 1868 in a more convincing way by Boltzmann.4 He defined a quantity 4 L. Boltzmann, Sitz. Ber. Akad. Wiss. (Vienna), part II, 66, 875 (1872). A translation into English of this article is included in Brush, The Kinetic Theory of Gases ā€“ An Anthology of Classic Papers with Historical Commentary, listed in the bibliography.
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    2.4 Kinetic Theoryand Statistical Mechanics 35 H ā‰” ln P = +āˆž āˆ’āˆž dvx +āˆž āˆ’āˆž dvy +āˆž āˆ’āˆž dvz P(v) ln P(v) , and showed that collisions of gas particles always lead to a decrease in H until a minimum is reached, at which P(v) is the Maxwellā€“Boltzmann distribution function. A generalization of this H-theorem was given in 1901 by Gibbs.5 The generalization and proof are given below, along with the application to gases. The General H-Theorem Consider a large system with many degrees of freedom, such as a gas with many molecules (but not necessarily a gas). The states of the system are parameterized by many variables, which we summarize with a symbol Ī±. (For instance, for a monatomic gas Ī± stands for the set of positions x1, x2, etc. and momenta p1, p2, etc. of atoms 1, 2, . . . For a gas of multi-atom molecules, Ī± would also include the orientations and their rates of change for each molecule.) We denote an infinitesimal range of these parameters by dĪ±. (For instance, for a monatomic gas dĪ± stands for the product d3x1d3p1d3x2d3p2 . . ., known as the phase space volume.) We define P(Ī±) so that the probability that the parameters of the system are in an infinitesimal range dĪ± around Ī± is P(Ī±)dĪ±, with P normalized so that dĪ±P(Ī±) = 1. Define H ā‰” ln P = P(Ī±) dĪ± ln P(Ī±) . (2.4.2) Gibbs showed that H always decreases until it reaches a minimum value, at which P(Ī±) is proportional to the exponential of a linear combination of con- served quantities, such as the total energy. Proof: Define a differential rate (Ī± ā†’ Ī²) such that the rate at which a system in state Ī± makes a transition to a state within a range dĪ² around state Ī² is (Ī± ā†’ Ī²) dĪ². The probability P(Ī±)dĪ± can either increase because the system in a range dĪ² of states around Ī² makes a transition to the range dĪ± of states around Ī±, or decrease because the system in the range of states dĪ± around Ī± makes a transition to some other state in a range dĪ² around Ī², so dP(Ī±)dĪ± dt = dĪ² [P(Ī²) (Ī² ā†’ Ī±)dĪ± āˆ’ P(Ī±) dĪ±(Ī± ā†’ Ī²)] , or, cancelling the differentials dĪ±, dP(Ī±) dt = dĪ² [P(Ī²) (Ī² ā†’ Ī±) āˆ’ P(Ī±)(Ī± ā†’ Ī²)] . (2.4.3) 5 J. W. Gibbs, Elementary Principles of Statistical Mechanics, Developed with Especial Reference to The Rational Foundation of Thermodynamics (Scribner, New York, 1902).
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    Other documents randomlyhave different content
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    CHAPTER XI GALILEE Neither oldā€œsacredā€ [123] himself, nor any of his helpers, knew the road which I meant to take from Nazareth to the Sea of Galilee and from thence to Jerusalem, so I was forced to add another to my party by hiring a guide. The associations of Nazareth, as well as my kind feeling towards the hospitable monks, whose guest I had been, inclined me to set at naught the advice which I had received against employing Christians. I accordingly engaged a lithe, active young Nazarene, who was recommended to me by the monks, and who affected to be familiar with the line of country through which I intended to pass. My disregard of the popular prejudices against Christians was not justified in this particular instance by the result of my choice. This you will see by and by. I passed by Cana and the house in which the water had been turned into wine; I came to the field in which our Saviour had rebuked the Scotch Sabbath-keepers of that period, by suffering His disciples to pluck corn on the Lordā€™s Day; I rode over the ground on which the fainting multitude had been fed, and they showed me some massive fragmentsā€”the relics, they said, of that wondrous banquet, now turned into stone. The petrifaction was most complete. I ascended the height on which our Lord was standing when He wrought the miracle. The hill was lofty enough to show me the fairness of the land on all sides, but I have an ancient love for the mere features of a lake, and so forgetting all else when I reached the summit, I looked away eagerly to the eastward. There she lay,
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    the Sea ofGalilee. Less stern than Wast Water, less fair than gentle Windermere, she had still the winning ways of an English lake; she caught from the smiling heavens unceasing light and changeful phases of beauty, and with all this brightness on her face, she yet clung so fondly to the dull he-looking mountain at her side, as though she would ā€œSoothe him with her finer fancies, Touch him with her lighter thought.ā€ [124] If one might judge of menā€™s real thoughts by their writings, it would seem that there are people who can visit an interesting locality and follow up continuously the exact train of thought that ought to be suggested by the historical associations of the place. A person of this sort can go to Athens and think of nothing later than the age of Pericles; can live with the Scipios as long as he stays in Rome; can go up in a balloon, and think how resplendently in former times the now vacant and desolate air was peopled with angels, how prettily it was crossed at intervals by the rounds of Jacobā€™s ladder! I donā€™t possess this power at all; it is only by snatches, and for few moments together, that I can really associate a place with its proper history. ā€œThere at Tiberias, and along this western shore towards the north, and upon the bosom too of the lake, our Saviour and His disciplesā€ā€” away flew those recollections, and my mind strained eastward, because that that farthest shore was the end of the world that belongs to man the dweller, the beginning of the other and veiled world that is held by the strange race, whose life (like the pastime of Satan) is a ā€œgoing to and fro upon the face of the earth.ā€ From those grey hills right away to the gates of Bagdad stretched forth the mysterious ā€œdesertā€ā€”not a pale, void, sandy tract, but a land abounding in rich pastures, a land without cities or towns, without any ā€œrespectableā€ people or any ā€œrespectableā€ things, yet yielding its eighty thousand cavalry to the beck of a few old men. But once moreā€”ā€œTiberiasā€”the plain of Gennesarethā€”the very earth on which
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    I stoodā€”that thedeep low tones of the Saviourā€™s voice should have gone forth into eternity from out of the midst of these hills and these valleys!ā€ā€”Ay, ay, but yet again the calm face of the lake was uplifted, and smiled upon my eyes with such familiar gaze, that the ā€œdeep low tonesā€ were hushed, the listening multitudes all passed away, and instead there came to me a dear old memory from over the seas in England, a memory sweeter than Gospel to that poor wilful mortal, me. I went to Tiberias, and soon got afloat upon the water. In the evening I took up my quarters in the Catholic church, and the building being large enough, the whole of my party were admitted to the benefit of the same shelter. With portmanteaus and carpet bags, and books and maps, and fragrant tea, Mysseri soon made me a home on the southern side of the church. One of old Shereefā€™s helpers was an enthusiastic Catholic, and was greatly delighted at having so sacred a lodging. He lit up the altar with a number of tapers, and when his preparations were complete, he began to perform his orisons in the strangest manner imaginable. His lips muttered the prayers of the Latin Church, but he bowed himself down and laid his forehead to the stones beneath him after the manner of a Mussulman. The universal aptness of a religious system for all stages of civilisation, and for all sorts and conditions of men, well befits its claim of divine origin. She is of all nations, and of all times, that wonderful Church of Rome! Tiberias is one of the four holy cities, [126] according to the Talmud, and it is from this place, or the immediate neighbourhood of it, that the Messiah is to arise. Except at Jerusalem, never think of attempting to sleep in a ā€œholy city.ā€ Old Jews from all parts of the world go to lay their bones upon the sacred soil, and as these people never return to their homes, it follows that any domestic vermin which they may bring with them are likely to become permanently resident, so that the population is continually increasing. No recent census had been taken when I
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    was at Tiberias,but I know that the congregation of fleas which attended at my church alone must have been something enormous. It was a carnal, self-seeking congregation, wholly inattentive to the service which was going on, and devoted to the one object of having my blood. The fleas of all nations were there. The smug, steady, importunate flea from Holywell Street; the pert, jumping puce from hungry France, the wary, watchful pulce with his poisoned stiletto; the vengeful pulga of Castile with his ugly knife; the German floh with his knife and fork, insatiate, not rising from table; whole swarms from all the Russias, and Asiatic hordes unnumberedā€”all these were there, and all rejoiced in one great international feast. I could no more defend myself against my enemies than if I had been pain Ć  discretion in the hands of a French patriot, or English gold in the claws of a Pennsylvanian Quaker. After passing a night like this you are glad to pick up the wretched remains of your body long, long before morning dawns. Your skin is scorched, your temples throb, your lips feel withered and dried, your burning eyeballs are screwed inwards against the brain. You have no hope but only in the saddle and the freshness of the morning air.
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    CHAPTER XII MY FIRSTBIVOUAC The course of the Jordan is from the north to the south, and in that direction, with very little of devious winding, it carries the shining waters of Galilee straight down into the solitudes of the Dead Sea. Speaking roughly, the river in that meridian is a boundary between the people living under roofs and the tented tribes that wander on the farther side. And so, as I went down in my way from Tiberias towards Jerusalem, along the western bank of the stream, my thinking all propended to the ancient world of herdsmen and warriors that lay so close over my bridle arm. If a man, and an Englishman, be not born of his mother with a natural Chiffney-bit in his mouth, there comes to him a time for loathing the wearisome ways of society; a time for not liking tamed people; a time for not dancing quadrilles, not sitting in pews; a time for pretending that Milton and Shelley, and all sorts of mere dead people, were greater in death than the first living Lord of the Treasury; a time, in short, for scoffing and railing, for speaking lightly of the very opera, and all our most cherished institutions. It is from nineteen to two or three and twenty perhaps that this war of the man against men is like to be waged most sullenly. You are yet in this smiling England, but you find yourself wending away to the dark sides of her mountains, climbing the dizzy crags, exulting in the fellowship of mists and clouds, and watching the storms how they gather, or proving the mettle of your mare upon the broad and dreary downs, because that you feel congenially with the yet unparcelled earth. A little while you are free and unlabelled, like the
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    ground that youcompass; but civilisation is coming and coming; you and your much-loved waste lands will be surely enclosed, and sooner or later brought down to a state of mere usefulness; the ground will be curiously sliced into acres and roods and perches, and you, for all you sit so smartly in your saddle, you will be caught, you will be taken up from travel as a colt from grass, to be trained and tried, and matched and run. All this in time, but first come Continental tours and the moody longing for Eastern travel. The downs and the moors of England can hold you no longer; with large strides you burst away from these slips and patches of free land; you thread your path through the crowds of Europe, and at last, on the banks of Jordan, you joyfully know that you are upon the very frontier of all accustomed respectabilities. There, on the other side of the river (you can swim it with one arm), there reigns the people that will be like to put you to death for not being a vagrant, for not being a robber, for not being armed and houseless. There is comfort in thatā€”health, comfort, and strength to one who is dying from very weariness of that poor, dear, middle-aged, deserving, accomplished, pedantic, and painstaking governess, Europe. I had ridden for some hours along the right bank of Jordan when I came to the Djesr el MedjamĆ© (an old Roman bridge, I believe), which crossed the river. My Nazarene guide was riding ahead of the party, and now, to my surprise and delight, he turned leftwards, and led on over the bridge. I knew that the true road to Jerusalem must be mainly by the right bank of Jordan, but I supposed that my guide was crossing the bridge at this spot in order to avoid some bend in the river, and that he knew of a ford lower down by which we should regain the western bank. I made no question about the road, for I was but too glad to set my horseā€™s hoofs upon the land of the wandering tribes. None of my party except the Nazarene knew the country. On we went through rich pastures upon the eastern side of the water. I looked for the expected bend of the river, but far as I could see it kept a straight southerly course; I still left my guide unquestioned.
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    The Jordan isnot a perfectly accurate boundary betwixt roofs and tents, for soon after passing the bridge I came upon a cluster of huts. Some time afterwards the guide, upon being closely questioned by my servants, confessed that the village which we had left behind was the last that we should see, but he declared that he knew a spot at which we should find an encampment of friendly Bedouins, who would receive me with all hospitality. I had long determined not to leave the East without seeing something of the wandering tribes, but I had looked forward to this as a pleasure to be found in the desert between El Arish and Egypt; I had no idea that the Bedouins on the east of Jordan were accessible. My delight was so great at the near prospect of bread and salt in the tent of an Arab warrior, that I wilfully allowed my guide to go on and mislead me. I saw that he was taking me out of the straight route towards Jerusalem, and was drawing me into the midst of the Bedouins; but the idea of his betraying me seemed (I know not why) so utterly absurd, that I could not entertain it for a moment. I fancied it possible that the fellow had taken me out of my route in order to attempt some little mercantile enterprise with the tribe for which he was seeking, and I was glad of the opportunity which I might thus gain of coming in contact with the wanderers. Not long after passing the village a horseman met us. It appeared that some of the cavalry of Ibrahim Pasha had crossed the river for the sake of the rich pastures on the eastern bank, and that this man was one of the troopers. He stopped and saluted; he was obviously surprised at meeting an unarmed, or half-armed, cavalcade, and at last fairly told us that we were on the wrong side of the river, and that if we proceeded we must lay our account with falling amongst robbers. All this while, and throughout the day, my Nazarene kept well ahead of the party, and was constantly up in his stirrups, straining forward and searching the distance for some objects which still remained unseen. For the rest of the day we saw no human being; we pushed on eagerly in the hope of coming up with the Bedouins before nightfall.
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    Night came, andwe still went on in our way till about ten oā€™clock. Then the thorough darkness of the night, and the weariness of our beasts (which had already done two good daysā€™ journey in one), forced us to determine upon coming to a standstill. Upon the heights to the eastward we saw lights; these shone from caves on the mountain-side, inhabited, as the Nazarene told us, by rascals of a low sortā€”not real Bedouins, men whom we might frighten into harmlessness, but from whom there was no willing hospitality to be expected. We heard at a little distance the brawling of a rivulet, and on the banks of this it was determined to establish our bivouac. We soon found the stream, and following its course for a few yards, came to a spot which was thought to be fit for our purpose. It was a sharply cold night in February, and when I dismounted I found myself standing upon some wet rank herbage that promised ill for the comfort of our resting-place. I had bad hopes of a fire, for the pitchy darkness of the night was a great obstacle to any successful search for fuel, and, besides, the boughs of trees or bushes would be so full of sap in this early spring, that they would not be easily persuaded to burn. However, we were not likely to submit to a dark and cold bivouac without an effort, and my fellows groped forward through the darkness, till after advancing a few paces they were happily stopped by a complete barrier of dead prickly bushes. Before our swords could be drawn to reap this welcome harvest it was found to our surprise that the fuel was already hewn and strewed along the ground in a thick mass. A spot for the fire was found with some difficulty, for the earth was moist and the grass high and rank. At last there was a clicking of flint and steel, and presently there stood out from darkness one of the tawny faces of my muleteers, bent down to near the ground, and suddenly lit up by the glowing of the spark which he courted with careful breath. Before long there was a particle of dry fibre or leaf that kindled to a tiny flame; then another was lit from that, and then another. Then small crisp twigs, little bigger than bodkins, were laid athwart the glowing fire. The swelling cheeks of the muleteer, laid level with the
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    earth, blew tenderlyat first and then more boldly upon the young flame, which was daintily nursed and fed, and fed more plentifully when it gained good strength. At last a whole armful of dry bushes was piled up over the fire, and presently, with a loud cheery crackling and crackling, a royal tall blaze shot up from the earth and showed me once more the shapes and faces of my men, and the dim outlines of the horses and mules that stood grazing hard by. My servants busied themselves in unpacking the baggage as though we had arrived at an hotelā€”Shereef and his helpers unsaddled their cattle. We had left Tiberias without the slightest idea that we were to make our way to Jerusalem along the desolate side of the Jordan, and my servants (generally provident in those matters) had brought with them only, I think, some unleavened bread and a rocky fragment of goatā€™s-milk cheese. These treasures were produced. Tea and the contrivances for making it were always a standing part of my baggage. My men gathered in circle round the fire. The Nazarene was in a false position from having misled us so strangely, and he would have shrunk back, poor devil, into the cold and outer darkness, but I made him draw near and share the luxuries of the night. My quilt and my pelisse were spread, and the rest of my party had all their capotes or pelisses, or robes of some sort, which furnished their couches. The men gathered in circle, some kneeling, some sitting, some lying reclined around our common hearth. Sometimes on one, sometimes on another, the flickering light would glare more fiercely. Sometimes it was the good Shereef that seemed the foremost, as he sat with venerable beard the image of manly pietyā€”unknowing of all geography, unknowing where he was or whither he might go, but trusting in the goodness of God and the clinching power of fate and the good star of the Englishman. Sometimes, like marble, the classic face of the Greek Mysseri would catch the sudden light, and then again by turns the ever-perturbed Dthemetri, with his old Chinamanā€™s eye and bristling, terrier-like moustache, shone forth illustrious.
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    I always likedthe men who attended me on these Eastern travels, for they were all of them brave, cheery-hearted fellows; and although their following my career brought upon them a pretty large share of those toils and hardships which are so much more amusing to gentlemen than to servants, yet not one of them ever uttered or hinted a syllable of complaint, or even affected to put on an air of resignation. I always liked them, but never perhaps so much as when they were thus grouped together under the light of the bivouac fire. I felt towards them as my comrades rather than as my servants, and took delight in breaking bread with them, and merrily passing the cup. The love of tea is a glad source of fellow-feeling between the Englishman and the Asiatic. In Persia it is drunk by all, and although it is a luxury that is rarely within the reach of the Osmanlees, there are few of them who do not know and love the blessed tchƤi. Our camp-kettle, filled from the brook, hummed doubtfully for a while, then busily bubbled under the sidelong glare of the flames; cups clinked and rattled; the fragrant steam ascended, and soon this little circlet in the wilderness grew warm and genial as my ladyā€™s drawing- room. And after this there came the tchibouqueā€”great comforter of those that are hungry and wayworn. And it has this virtueā€”it helps to destroy the gĆŖne and awkwardness which one sometimes feels at being in company with oneā€™s dependants; for whilst the amber is at your lips, there is nothing ungracious in your remaining silent, or speaking pithily in short inter-whiff sentences. And for us that night there was pleasant and plentiful matter of talk; for the where we should be on the morrow, and the wherewithal we should be fed, whether by some ford we should regain the western bank of Jordan, or find bread and salt under the tents of a wandering tribe, or whether we should fall into the hands of the Philistines, and so come to see deathā€”the last and greatest of all ā€œthe fine sightsā€ that there beā€”these were questionings not dull nor wearisome to us, for we were all concerned in the answers. And it was not an all-imagined
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    morrow that weprobed with our sharp guesses, for the lights of those low Philistines, the men of the caves, still hung over our heads, and we knew by their yells that the fire of our bivouac had shown us. At length we thought it well to seek for sleep. Our plans were laid for keeping up a good watch through the night. My quilt and my pelisse and my cloak were spread out so that I might lie spokewise, with my feet towards the central fire. I wrapped my limbs daintily round, and gave myself positive orders to sleep like a veteran soldier. But I found that my attempt to sleep upon the earth that God gave me was more new and strange than I had fancied it. I had grown used to the scene which was before me whilst I was sitting or reclining by the side of the fire, but now that I laid myself down at length it was the deep black mystery of the heavens that hung over my eyesā€”not an earthly thing in the way from my own very forehead right up to the end of all space. I grew proud of my boundless bedchamber. I might have ā€œfound sermonsā€ in all this greatness (if I had I should surely have slept), but such was not then my way. If this cherished self of mine had built the universe, I should have dwelt with delight on ā€œthe wonders of creation.ā€ As it was, I felt rather the vain-glory of my promotion from out of mere rooms and houses into the midst of that grand, dark, infinite palace. And then, too, my head, far from the fire, was in cold latitudes, and it seemed to me strange that I should be lying so still and passive, whilst the sharp night breeze walked free over my cheek, and the cold damp clung to my hair, as though my face grew in the earth and must bear with the footsteps of the wind and the falling of the dew as meekly as the grass of the field. Besides, I got puzzled and distracted by having to endure heat and cold at the same time, for I was always considering whether my feet were not over-devilled and whether my face was not too well iced. And so when from time to time the watch quietly and gently kept up the languishing fire, he seldom, I think, was unseen to my restless eyes. Yet, at last, when they called me and said that the morn would soon be dawning, I
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    rose from astate of half-oblivion not much unlike to sleep, though sharply qualified by a sort of vegetableā€™s consciousness of having been growing still colder and colder for many and many an hour.
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    CHAPTER XIII THE DEADSEA The grey light of the morning showed us for the first time the ground which we had chosen for our resting-place. We found that we had bivouacked upon a little patch of barley plainly belonging to the men of the caves. The dead bushes which we found so happily placed in readiness for our fire had been strewn as a fence for the protection of the little crop. This was the only cultivated spot of ground which we had seen for many a league, and I was rather sorry to find that our night fire and our cattle had spread so much ruin upon this poor solitary slip of corn-land. The saddling and loading of our beasts was a work which generally took nearly an hour, and before this was half over daylight came. We could now see the men of the caves. They collected in a body, amounting, I should think, to nearly fifty, and rushed down towards our quarters with fierce shouts and yells. But the nearer they got the slower they went; their shouts grew less resolute in tone, and soon ceased altogether. The fellows, however, advanced to a thicket within thirty yards of us, and behind this ā€œtook up their position.ā€ My men without premeditation did exactly that which was best; they kept steadily to their work of loading the beasts without fuss or hurry; and whether it was that they instinctively felt the wisdom of keeping quiet, or that they merely obeyed the natural inclination to silence which one feels in the early morning, I cannot tell, but I know that, except when they exchanged a syllable or two relative to the work they were about, not a word was said. I now believe that this quietness of our party created an undefined terror in the minds
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    of the cave-holdersand scared them from coming on; it gave them a notion that we were relying on some resources which they knew not of. Several times the fellows tried to lash themselves into a state of excitement which might do instead of pluck. They would raise a great shout and sway forward in a dense body from behind the thicket; but when they saw that their bravery thus gathered to a head did not even suspend the strapping of a portmanteau or the tying of a hatbox, their shout lost its spirit, and the whole mass was irresistibly drawn back like a wave receding from the shore. These attempts at an onset were repeated several times, but always with the same result. I remained under the apprehension of an attack for more than half an hour, and it seemed to me that the work of packing and loading had never been done so slowly. I felt inclined to tell my fellows to make their best speed, but just as I was going to speak I observed that every one was doing his duty already; I therefore held my peace and said not a word, till at last Mysseri led up my horse and asked me if I were ready to mount. We all marched off without hindrance. After some time we came across a party of Ibrahimā€™s cavalry, which had bivouacked at no great distance from us. The knowledge that such a force was in the neighbourhood may have conduced to the forbearance of the cave-holders. We saw a scraggy-looking fellow nearly black, and wearing nothing but a cloth round the loins; he was tending flocks. Afterwards I came up with another of these goatherds, whose helpmate was with him. They gave us some goatā€™s milk, a welcome present. I pitied the poor devil of a goat-herd for having such a very plain wife. I spend an enormous quantity of pity upon that particular form of human misery. About midday I began to examine my map and to question my guide, who at last fell on his knees and confessed that he knew nothing of the country in which we were. I was thus thrown upon
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    my own resources,and calculating that on the preceding day we had nearly performed a two daysā€™ journey, I concluded that the Dead Sea must be near. In this I was right, for at about three or four oā€™clock in the afternoon I caught a first sight of its dismal face. I went on and came near to those waters of death. They stretched deeply into the southern desert, and before me, and all around, as far away as the eye could follow, blank hills piled high over hills, pale, yellow, and naked, walled up in her tomb for ever the dead and damned Gomorrah. There was no fly that hummed in the forbidden air, but instead a deep stillness; no grass grew from the earth, no weed peered through the void sand; but in mockery of all life there were trees borne down by Jordan in some ancient flood, and these, grotesquely planted upon the forlorn shore, spread out their grim skeleton arms, all scorched and charred to blackness by the heats of the long silent years. I now struck off towards the dĆ©bouchure of the river; but I found that the country, though seemingly quite flat, was intersected by deep ravines, which did not show themselves until nearly approached. For some time my progress was much obstructed; but at last I came across a track which led towards the river, and which might, as I hoped, bring me to a ford. I found, in fact, when I came to the riverā€™s side that the track reappeared upon the opposite bank, plainly showing that the stream had been fordable at this place. Now, however, in consequence of the late rains the river was quite impracticable for baggage-horses. A body of waters about equal to the Thames at Eton, but confined to a narrower channel, poured down in a current so swift and heavy, that the idea of passing with laden baggage-horses was utterly forbidden. I could have swum across myself, and I might, perhaps, have succeeded in swimming a horse over; but this would have been useless, because in such case I must have abandoned not only my baggage, but all my attendants, for none of them were able to swim, and without that resource it would have been madness for them to rely upon the swimming of their beasts across such a powerful stream. I still hoped, however,
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    that there mightbe a chance of passing the river at the point of its actual junction with the Dead Sea, and I therefore went on in that direction. Night came upon us whilst labouring across gullies and sandy mounds, and we were obliged to come to a standstill quite suddenly upon the very edge of a precipitous descent. Every step towards the Dead Sea had brought us into a country more and more dreary; and this sandhill, which we were forced to choose for our resting-place, was dismal enough. A few slender blades of grass, which here and there singly pierced the sand, mocked bitterly the hunger of our jaded beasts, and with our small remaining fragment of goatā€™s-milk rock by way of supper, we were not much better off than our horses. We wanted, too, the great requisite of a cheery bivouac- fire. Moreover, the spot on which we had been so suddenly brought to a standstill was relatively high and unsheltered, and the night wind blew swiftly and cold. The next morning I reached the dĆ©bouchure of the Jordan, where I had hoped to find a bar of sand that might render its passage possible. The river, however, rolled its eddying waters fast down to the ā€œseaā€ in a strong, deep stream that shut out all hope of crossing. It now seemed necessary either to construct a raft of some kind, or else to retrace my steps and remount the banks of the Jordan. I had once happened to give some attention to the subject of military bridgesā€”a branch of military science which includes the construction of rafts and contrivances of the like sortā€”and I should have been very proud indeed if I could have carried my party and my baggage across by dint of any idea gathered from Sir Howard Douglas or Robinson Crusoe. But we were all faint and languid from want of food, and besides there were no materials. Higher up the river there were bushes and river plants, but nothing like timber; and the cord with which my baggage was tied to the pack-saddles amounted altogether to a very small quantity, not nearly enough to haul any sort of craft across the stream.
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    And now itwas, if I remember rightly, that Dthemetri submitted to me a plan for putting to death the Nazarene, whose misguidance had been the cause of our difficulties. There was something fascinating in this suggestion, for the slaying of the guide was of course easy enough, and would look like an act of what politicians call ā€œvigour.ā€ If it were only to become known to my friends in England that I had calmly killed a fellow-creature for taking me out of my way, I might remain perfectly quiet and tranquil for all the rest of my days, quite free from the danger of being considered ā€œslowā€; I might ever after live on upon my reputation, like ā€œsingle-speech Hamiltonā€ in the last century, or ā€œsingle sinā€”ā€ in this, without being obliged to take the trouble of doing any more harm in the world. This was a great temptation to an indolent person, but the motive was not strengthened by any sincere feeling of anger with the Nazarene. Whilst the question of his life and death was debated he was riding in front of our party, and there was something in the anxious writhing of his supple limbs that seemed to express a sense of his false position, and struck me as highly comic. I had no crotchet at that time against the punishment of death, but I was unused to blood, and the proposed victim looked so thoroughly capable of enjoying life (if he could only get to the other side of the river), that I thought it would be hard for him to die merely in order to give me a character for energy. Acting on the result of these considerations, and reserving to myself a free and unfettered discretion to have the poor villain shot at any future moment, I magnanimously decided that for the present he should live, and not die. I bathed in the Dead Sea. The ground covered by the water sloped so gradually, that I was not only forced to ā€œsneak in,ā€ but to walk through the water nearly a quarter of a mile before I could get out of my depth. When at last I was able to attempt to dive, the salts held in solution made my eyes smart so sharply, that the pain which I thus suffered, together with the weakness occasioned by want of food, made me giddy and faint for some moments, but I soon grew better. I knew beforehand the impossibility of sinking in this buoyant
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    water, but Iwas surprised to find that I could not swim at my accustomed pace; my legs and feet were lifted so high and dry out of the lake, that my stroke was baffled, and I found myself kicking against the thin air instead of the dense fluid upon which I was swimming. The water is perfectly bright and clear; its taste detestable. After finishing my attempts at swimming and diving, I took some time in regaining the shore, and before I began to dress I found that the sun had already evaporated the water which clung to me, and that my skin was thickly encrusted with salts.
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    CHAPTER XIV THE BLACKTENTS My steps were reluctantly turned towards the north. I had ridden some way, and still it seemed that all life was fenced and barred out from the desolate ground over which I was journeying. On the west there flowed the impassable Jordan, on the east stood an endless range of barren mountains, and on the south lay that desert sea that knew not the plashing of an oar; greatly therefore was I surprised when suddenly there broke upon my ear the long, ludicrous, persevering bray of a donkey. I was riding at this time some few hundred yards ahead of all my party except the Nazarene (who by a wise instinct kept closer to me than to Dthemetri), and I instantly went forward in the direction of the sound, for I fancied that where there were donkeys, there too most surely would be men. The ground on all sides of me seemed thoroughly void and lifeless, but at last I got down into a hollow, and presently a sudden turn brought me within thirty yards of an Arab encampment. The low black tents which I had so long lusted to see were right before me, and they were all teeming with live Arabsā€”men, women, and children. I wished to have let my party behind know where I was, but I recollected that they would be able to trace me by the prints of my horseā€™s hoofs in the sand; and having to do with Asiatics, I felt the danger of the slightest movement which might be looked upon as a sign of irresolution. Therefore, without looking behind me, without looking to the right or to the left, I rode straight up towards the foremost tent. Before this was strewed a semi-circular fence of dead boughs, through which there was an opening opposite to the front
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    of the tent.As I advanced, some twenty or thirty of the most uncouth-looking fellows imaginable came forward to meet me. In their appearance they showed nothing of the Bedouin blood; they were of many colours, from dingy brown to jet black, and some of these last had much of the negro look about them. They were tall, powerful fellows, but awfully ugly. They wore nothing but the Arab shirts, confined at the waist by leathern belts. I advanced to the gap left in the fence, and at once alighted from my horse. The chief greeted me after his fashion by alternately touching first my hand and then his own forehead, as if he were conveying the virtue of the touch like a spark of electricity. Presently I found myself seated upon a sheepskin, which was spread for me under the sacred shade of Arabian canvas. The tent was of a long, narrow, oblong form, and contained a quantity of men, women, and children so closely huddled together, that there was scarcely one of them who was not in actual contact with his neighbour. The moment I had taken my seat the chief repeated his salutations in the most enthusiastic manner, and then the people having gathered densely about me, got hold of my unresisting hand and passed it round like a claret jug for the benefit of everybody. The women soon brought me a wooden bowl full of buttermilk, and welcome indeed came the gift to my hungry and thirsty soul. After some time my party, as I had expected, came up, and when poor Dthemetri saw me on my sheepskin, ā€œthe life and soulā€ of this ragamuffin party, he was so astounded, that he even failed to check his cry of horror; he plainly thought that now, at last, the Lord had delivered me (interpreter and all) into the hands of the lowest Philistines. Mysseri carried a tobacco-pouch slung at his belt, and as soon as its contents were known the whole population of the tent began begging like spaniels for bits of the beloved weed. I concluded from the abject manner of these people that they could not possibly be thoroughbred Bedouins, and I saw, too, that they must be in the very last stage of misery, for poor indeed is the man in these climes
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    who cannot commanda pipeful of tobacco. I began to think that I had fallen amongst thorough savages, and it seemed likely enough that they would gain their very first knowledge of civilisation by ravishing and studying the contents of my dearest portmanteaus, but still my impression was that they would hardly venture upon such an attempt. I observed, indeed, that they did not offer me the bread and salt which I had understood to be the pledges of peace amongst wandering tribes, but I fancied that they refrained from this act of hospitality, not in consequence of any hostile determination, but in order that the notion of robbing me might remain for the present an ā€œopen question.ā€ I afterwards found that the poor fellows had no bread to offer. They were literally ā€œout at grass.ā€ It is true that they had a scanty supply of milk from goats, but they were living almost entirely upon certain grass stems, which were just in season at that time of the year. These, if not highly nourishing, are pleasant enough to the taste, and their acid juices come gratefully to thirsty lips.
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    CHAPTER XV PASSAGE OFTHE JORDAN And now Dthemetri began to enter into a negotiation with my hosts for a passage over the river. I never interfered with my worthy dragoman upon these occasions, because from my entire ignorance of the Arabic I should have been quite unable to exercise any real control over his words, and it would have been silly to break the stream of his eloquence to no purpose. I have reason to fear, however, that he lied transcendently, and especially in representing me as the bosom friend of Ibrahim Pasha. The mention of that name produced immense agitation and excitement, and the Sheik explained to Dthemetri the grounds of the infinite respect which he and his tribe entertained for the Pasha. A few weeks before Ibrahim had craftily sent a body of troops across the Jordan. The force went warily round to the foot of the mountains on the east, so as to cut off the retreat of this tribe, and then surrounded them as they lay encamped in the vale; their camels, and indeed all their possessions worth taking, were carried off by the soldiery, and moreover the then Sheik, together with every tenth man of the tribe, was brought out and shot. You would think that this conduct on the part of the Pasha might not procure for his ā€œfriendā€ a very gracious reception amongst the people whom he had thus despoiled and decimated; but the Asiatic seems to be animated with a feeling of profound respect, almost bordering upon affection, for all who have done him any bold and violent wrong; and there is always, too, so much of vague and undefined apprehension mixed up with his really well-
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    founded alarms, thatI can see no limit to the yielding and bending of his mind when it is wrought upon by the idea of power. After some discussion the Arabs agreed, as I thought, to conduct me to a ford, and we moved on towards the river, followed by seventeen of the most able-bodied of the tribe, under the guidance of several grey-bearded elders, and Sheik Ali Djoubran at the head of the whole detachment. Upon leaving the encampment a sort of ceremony was performed, for the purpose, it seemed, of ensuring, if possible, a happy result for the undertaking. There was an uplifting of arms, and a repeating of words that sounded like formulƦ, but there were no prostrations, and I did not understand that the ceremony was of a religious character. The tented Arabs are looked upon as very bad Mahometans. [149] We arrived upon the banks of the riverā€”not at a ford, but at a deep and rapid part of the stream, and I now understood that it was the plan of these men, if they helped me at all, to transport me across the river by some species of raft. But a reaction had taken place in the opinions of many, and a violent dispute arose upon a motion which seemed to have been made by some honourable member with a view to robbery. The fellows all gathered together in circle, at a little distance from my party, and there disputed with great vehemence and fury for nearly two hours. I canā€™t give a correct report of the debate, for it was held in a barbarous dialect of the Arabic unknown to my dragoman. I recollect I sincerely felt at the time that the arguments in favour of robbing me must have been almost unanswerable, and I gave great credit to the speakers on my side for the ingenuity and sophistry which they must have shown in maintaining the fight so well. During the discussion I remained lying in front of my baggage, which had all been taken from the pack-saddles and placed upon the ground. I was so languid from want of food, that I had scarcely animation enough to feel as deeply interested as you would suppose in the result of the discussion. I thought, however, that the
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    pleasantest toys toplay with during this interval were my pistols, and now and then, when I listlessly visited my loaded barrels with the swivel ramrods, or drew a sweet, musical click from my English firelocks, it seemed to me that I exercised a slight and gentle influence on the debate. Thanks to Ibrahim Pashaā€™s terrible visitation the men of the tribe were wholly unarmed, and my advantage in this respect might have counterbalanced in some measure the superiority of numbers. Mysseri (not interpreting in Arabic) had no duty to perform, and he seemed to be faint and listless as myself. Shereef looked perfectly resigned to any fate. But Dthemetri (faithful terrier!) was bristling with zeal and watchfulness. He could not understand the debate, which indeed was carried on at a distance too great to be easily heard, even if the language had been familiar; but he was always on the alert, and now and then conferring with men who had straggled out of the assembly. At last he found an opportunity of making a proposal, which at once produced immense sensation; he offered, on my behalf, that if the tribe should bear themselves loyally towards me, and take my party and my baggage in safety to the other bank of the river, I should give them a teskeri, or written certificate of their good conduct, which might avail them hereafter in the hour of their direst need. This proposal was received and instantly accepted by all the men of the tribe there present with the utmost enthusiasm. I was to give the men, too, a baksheish, that is, a present of money, which is usually made upon the conclusion of any sort of treaty; but although the people of the tribe were so miserably poor, they seemed to look upon the pecuniary part of the arrangement as a matter quite trivial in comparison with the teskeri. Indeed the sum which Dthemetri promised them was extremely small, and not the slightest attempt was made to extort any further reward. The council now broke up, and most of the men rushed madly towards me, and overwhelmed me with vehement gratulations; they
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    caressed my bootswith much affection, and my hands were severely kissed. The Arabs now went to work in right earnest to effect the passage of the river. They had brought with them a great number of the skins which they use for carrying water in the desert; these they filled with air, and fastened several of them to small boughs which they cut from the banks of the river. In this way they constructed a raft not more than about four or five feet square, but rendered buoyant by the inflated skins which supported it. On this a portion of my baggage was placed, and was firmly tied to it by the cords used on my pack-saddles. The little raft with its weighty cargo was then gently lifted into the water, and I had the satisfaction to see that it floated well. Twelve of the Arabs now stripped, and tied inflated skins to their loins; six of the men went down into the river, got in front of the little raft, and pulled it off a few feet from the bank. The other six then dashed into the stream with loud shouts, and swam along after the raft, pushing it from behind. Off went the craft in capital style at first, for the stream was easy on the eastern side; but I saw that the tug was to come, for the main torrent swept round in a bend near the western bank of the river. The old men, with their long grey grisly beards, stood shouting and cheering, praying and commanding. At length the raft entered upon the difficult part of its course; the whirling stream seized and twisted it about, and then bore it rapidly downwards; the swimmers flagged, and seemed to be beaten in the struggle. But now the old men on the bank, with their rigid arms uplifted straight, sent forth a cry and a shout that tore the wide air into tatters, and then to make their urging yet more strong they shrieked out the dreadful syllables, ā€œā€™Brahim Pasha!ā€ The swimmers, one moment before so blown and so weary, found lungs to answer the cry, and shouting back the name of their great destroyer, they dashed on through the torrent, and bore the raft in safety to the western bank.
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    Afterwards the swimmersreturned with the raft, and attached to it the rest of my baggage. I took my seat upon the top of the cargo, and the raft thus laden passed the river in the same way, and with the same struggle as before. The skins, however, not being perfectly air-tight, had lost a great part of their buoyancy, so that I, as well as the luggage that passed on this last voyage, got wet in the waters of Jordan. The raft could not be trusted for another trip, and the rest of my party passed the river in a different and (for them) much safer way. Inflated skins were fastened to their loins, and thus supported, they were tugged across by Arabs swimming on either side of them. The horses and mules were thrown into the water and forced to swim over. The poor beasts had a hard struggle for their lives in that swift stream; and I thought that one of the horses would have been drowned, for he was too weak to gain a footing on the western bank, and the stream bore him down. At last, however, he swam back to the side from which he had come. Before dark all had passed the river except this one horse and old Shereef. He, poor fellow, was shivering on the eastern bank, for his dread of the passage was so great, that he delayed it as long as he could, and at last it became so dark that he was obliged to wait till the morning. I lay that night on the banks of the river, and at a little distance from me the Arabs kindled a fire, round which they sat in a circle. They were made most savagely happy by the tobacco with which I supplied them, and they soon determined that the whole night should be one smoking festival. The poor fellows had only a cracked bowl, without any tube at all, but this morsel of a pipe they handed round from one to the other, allowing to each a fixed number of whiffs. In that way they passed the whole night. The next morning old Shereef was brought across. It was a strange sight to see this solemn old Mussulman, with his shaven head and his sacred beard, sprawling and puffing upon the surface of the water. When at last he reached the bank the people told him that by his baptism in Jordan he had surely become a mere Christian. Poor
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    Shereef!ā€”the holy man!the descendant of the Prophet!ā€”he was sadly hurt by the taunt, and the more so as he seemed to feel that there was some foundation for it, and that he really might have absorbed some Christian errors. When all was ready for departure I wrote the teskeri in French and delivered it to Sheik Ali Djoubran, together with the promised baksheish; he was exceedingly grateful, and I parted in a very friendly way from this ragged tribe. In two or three hours I gained Rihah, a village said to occupy the site of ancient Jericho. There was one building there which I observed with some emotion, for although it may not have been actually standing in the days of Jericho, it contained at this day a most interesting collection ofā€”modern loaves. Some hours after sunset I reached the convent of Santo Saba, and there remained for the night.
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    CHAPTER XVI TERRA SANTA Theenthusiasm that had glowed, or seemed to glow, within me for one blessed moment when I knelt by the shrine of the Virgin at Nazareth, was not rekindled at Jerusalem. In the stead of the solemn gloom and the deep stillness that of right belonged to the Holy City, there was the hum and the bustle of active life. It was the ā€œheight of the season.ā€ The Easter ceremonies drew near. The pilgrims were flocking in from all quarters; and although their objects were partly at least of a religious character, yet their ā€œarrivalsā€ brought as much stir and liveliness to the city as if they had come up to marry their daughters. The votaries who every year crowd to the Holy Sepulchre are chiefly of the Greek and Armenian Churches. They are not drawn into Palestine by a mere sentimental longing to stand upon the ground trodden by our Saviour, but rather they perform the pilgrimage as a plain duty strongly inculcated by their religion. A very great proportion of those who belong to the Greek Church contrive at some time or other in the course of their lives to achieve the enterprise. Many in their infancy and childhood are brought to the holy sites by their parents, but those who have not had this advantage will often make it the main object of their lives to save money enough for this holy undertaking. The pilgrims begin to arrive in Palestine some weeks before the Easter festival of the Greek Church. They come from Egypt, from all parts of Syria, from Armenia and Asia Minor, from Stamboul, from
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    Roumelia, from theprovinces of the Danube, and from all the Russias. Most of these people bring with them some articles of merchandise, but I myself believe (notwithstanding the common taunt against pilgrims) that they do this rather as a mode of paying the expenses of their journey, than from a spirit of mercenary speculation. They generally travel in families, for the women are of course more ardent than their husbands in undertaking these pious enterprises, and they take care to bring with them all their children, however young; for the efficacy of the rites does not depend upon the age of the votary, so that people whose careful mothers have obtained for them the benefit of the pilgrimage in early life, are saved from the expense and trouble of undertaking the journey at a later age. The superior veneration so often excited by objects that are distant and unknown shows not perhaps the wrongheadedness of a man, but rather the transcendent power of his imagination. However this may be, and whether it is by mere obstinacy that they poke their way through intervening distance, or whether they come by the winged strength of fancy, quite certainly the pilgrims who flock to Palestine from the most remote homes are the people most eager in the enterprise, and in number too they bear a very high proportion to the whole mass. The great bulk of the pilgrims make their way by sea to the port of Jaffa. A number of families charter a vessel amongst them, all bringing their own provisions, which are of the simplest and cheapest kind. On board every vessel thus freighted there is, I believe, a priest, who helps the people in their religious exercises, and tries (and fails) to maintain something like order and harmony. The vessels employed in this service are usually Greek brigs or brigantines and schooners, and the number of passengers stowed in them is almost always horribly excessive. The voyages are sadly protracted, not only by the land-seeking, storm-flying habits of the Greek seamen, but also by their endless schemes and speculations, which are for ever tempting them to touch at the nearest port. The voyage, too, must be made in winter, in order that Jerusalem may be reached some weeks before the Greek Easter, and thus by the time
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    they attain tothe holy shrines the pilgrims have really and truly undergone a very respectable quantity of suffering. I once saw one of these pious cargoes put ashore on the coast of Cyprus, where they had touched for the purpose of visiting (not Paphos, but) some Christian sanctuary; I never saw (no, never even in the most horridly stuffy ballroom) such a discomfortable collection of human beings. Long huddled together in a pitching and rolling prison, fed on beans, exposed to some real danger and to terrors without end, they had been tumbled about for many wintry weeks in the chopping seas of the Mediterranean. As soon as they landed they stood upon the beach and chanted a hymn of thanks; the chant was morne and doleful, but really the poor people were looking so miserable that one could not fairly expect from them any lively outpouring of gratitude. When the pilgrims have landed at Jaffa they hire camels, horses, mules, or donkeys, and make their way as well as they can to the Holy City. The space fronting the Church of the Holy Sepulchre soon becomes a kind of bazaar, or rather, perhaps, reminds you of an English fair. On this spot the pilgrims display their merchandise, and there too the trading residents of the place offer their goods for sale. I have never, I think, seen elsewhere in Asia so much commercial animation as upon this square of ground by the church door; the ā€œmoney-changersā€ seemed to be almost as brisk and lively as if they had been within the temple. When I entered the church I found a babel of worshippers. Greek, Roman, and Armenian priests were performing their different rites in various nooks and corners, and crowds of disciples were rushing about in all directions, some laughing and talking, some begging, but most of them going round in a regular and methodical way to kiss the sanctified spots, and speak the appointed syllables, and lay down the accustomed coin. If this kissing of the shrines had seemed as though it were done at the bidding of enthusiasm, or of any poor sentiment even feebly approaching to it, the sight would have been less odd to English eyes; but as it was, I stared to see
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    grown men thussteadily and carefully embracing the sticks and the stones, not from love or from zeal (else God forbid that I should have stared!), but from a calm sense of duty; they seemed to be not ā€œworking out,ā€ but transacting the great business of salvation. Dthemetri, however, who generally came with me when I went out, in order to do duty as interpreter, really had in him some enthusiasm. He was a zealous and almost fanatical member of the Greek Church, and had long since performed the pilgrimage, so now great indeed was the pride and delight with which he guided me from one holy spot to another. Every now and then, when he came to an unoccupied shrine, he fell down on his knees and performed devotion; he was almost distracted by the temptations that surrounded him; there were so many stones absolutely requiring to be kissed, that he rushed about happily puzzled and sweetly teased, like ā€œJack among the maidens.ā€ A Protestant, familiar with the Holy Scriptures, but ignorant of tradition and the geography of modern Jerusalem, finds himself a good deal ā€œmazedā€ when he first looks for the sacred sites. The Holy Sepulchre is not in a field without the walls, but in the midst, and in the best part of the town, under the roof of the great church which I have been talking about. It is a handsome tomb of oblong form, partly subterranean and partly above ground, and closed in on all sides except the one by which it is entered. You descend into the interior by a few steps, and there find an altar with burning tapers. This is the spot which is held in greater sanctity than any other at Jerusalem. When you have seen enough of it you feel perhaps weary of the busy crowd, and inclined for a gallop; you ask your dragoman whether there will be time before sunset to procure horses and take a ride to Mount Calvary. Mount Calvary, signor?ā€” eccolo! it is upstairsā€”on the first floor. In effect you ascend, if I remember rightly, just thirteen steps, and then you are shown the now golden sockets in which the crosses of our Lord and the two thieves were fixed. All this is startling, but the truth is, that the city having gathered round the Sepulchre, which is the main point of
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    interest, has creptnorthward, and thus in great measure are occasioned the many geographical surprises that puzzle the ā€œBible Christian.ā€ The Church of the Holy Sepulchre comprises very compendiously almost all the spots associated with the closing career of our Lord. Just there, on your right, He stood and wept; by the pillar, on your left, He was scourged; on the spot, just before you, He was crowned with the crown of thorns; up there He was crucified, and down here He was buried. A locality is assigned to every, the minutest, event connected with the recorded history of our Saviour; even the spot where the cock crew when Peter denied his Master is ascertained, and surrounded by the walls of an Armenian convent. Many Protestants are wont to treat these traditions contemptuously, and those who distinguish themselves from their brethren by the appellation of ā€œBible Christiansā€ are almost fierce in their denunciation of these supposed errors. It is admitted, I believe, by everybody that the formal sanctification of these spots was the act of the Empress Helena, the mother of Constantine, but I think it is fair to suppose that she was guided by a careful regard to the then prevailing traditions. Now the nature of the ground upon which Jerusalem stands is such, that the localities belonging to the events there enacted might have been more easily, and permanently, ascertained by tradition than those of any city that I know of. Jerusalem, whether ancient or modern, was built upon and surrounded by sharp, salient rocks intersected by deep ravines. Up to the time of the siege Mount Calvary of course must have been well enough known to the people of Jerusalem; the destruction of the mere buildings could not have obliterated from any manā€™s memory the names of those steep rocks and narrow ravines in the midst of which the city had stood. It seems to me, therefore, highly probable that in fixing the site of Calvary the Empress was rightly guided. Recollect, too, that the voice of tradition at Jerusalem is quite unanimous, and that Romans, Greeks, Armenians, and Jews, all hating each other sincerely, concur in assigning the same
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    localities to theevents told in the Gospel. I concede, however, that the attempt of the Empress to ascertain the sites of the minor events cannot be safely relied upon. With respect, for instance, to the certainty of the spot where the cock crew, I am far from being convinced. Supposing that the Empress acted arbitrarily in fixing the holy sites, it would seem that she followed the Gospel of St. John, and that the geography sanctioned by her can be more easily reconciled with that history than with the accounts of the other Evangelists. The authority exercised by the Mussulman Government in relation to the holy sites is in one view somewhat humbling to the Christians, for it is almost as an arbitrator between the contending sects (this always, of course, for the sake of pecuniary advantage) that the Mussulman lends his contemptuous aid; he not only grants, but enforces toleration. All persons, of whatever religion, are allowed to go as they will into every part of the Church of the Holy Sepulchre, but in order to prevent indecent contests, and also from motives arising out of money payments, the Turkish Government assigns the peculiar care of each sacred spot to one of the ecclesiastic bodies. Since this guardianship carries with it the receipt of the coins which the pilgrims leave upon the shrines, it is strenuously fought for by all the rival Churches, and the artifices of intrigue are busily exerted at Stamboul in order to procure the issue or revocation of the firmans by which the coveted privilege is granted. In this strife the Greek Church has of late years signally triumphed, and the most famous of the shrines are committed to the care of their priesthood. They possess the golden socket in which stood the cross of our Lord, whilst the Latins are obliged to content themselves with the apertures in which were inserted the crosses of the two thieves. They are naturally discontented with that poor privilege, and sorrowfully look back to the days of their former gloryā€”the days when Napoleon was Emperor, and Sebastiani ambassador at the Porte. It seems that the ā€œcitizenā€ sultan, old Louis Philippe, has done very little indeed for Holy Church in Palestine.
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