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- 1. Some notes on thermodynamics 1. Introduction to thermodynamics 2. Relevance for atmosphere 3. Global perspectives 4. Heat machines & GeophysicsFrom Wikipedia, the free encyclopediaThermodynamics (from the Greek thermos meaning heat and dynamis meaning power)is a branch of physics that studies the effects of changes in temperature, pressure, andvolume on physical systems at the macroscopic scale by analyzing the collective motionof their particles using statistics. Roughly, heat means "energy in transit" and dynamicsrelates to "movement"; thus, in essence thermodynamics studies the movement of energyand how energy instills movement. Historically, thermodynamics developed out of theneed to increase the efficiency of early steam engines.The starting point for most thermodynamic considerations are the laws ofthermodynamics, which postulate that energy can be exchanged between physicalsystems as heat or work. They also postulate the existence of a quantity named entropy,which can be defined for any system. In thermodynamics, interactions between largeensembles of objects are studied and categorized. Central to this are the concepts ofsystem and surroundings. A system is composed of particles, whose average motionsdefine its properties, which in turn are related to one another through equations of state.Properties can be combined to express internal energy and thermodynamic potentials areuseful for determining conditions for equilibrium and spontaneous processes.With these tools, thermodynamics describes how systems respond to changes in theirsurroundings. This can be applied to a wide variety of topics in science and engineering,such as engines, phase transitions, chemical reactions, transport phenomena, and evenblack holes. The results of thermodynamics are essential for other fields of physics andfor chemistry, chemical engineering, cell biology, biomedical engineering, and materialsscience to name a few. Quotes • "Thermodynamics is the only physical theory of universal content which, within the framework of the applicability of its basic concepts, I am convinced will never be overthrown." — Albert Einstein • "The law that entropy always increases - the Second Law of Thermodynamics - holds, I think, the supreme position among the laws of physics. If someone points out to you that your pet theory of the universe is in disagreement with Maxwells equations - then so much the worse for Maxwells equations. If it is found to be contradicted by observation - well, 1.1
- 2. these experimentalists do bungle things from time to time. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation." — Sir Arthur Eddington• “Isn’t thermodynamics considered a fine intellectual structure, bequeathed by past decades, whose every subtlety only experts in the art of handling Hamiltonians would be able to appreciate?” Pierre Perrot, author: “A to Z Dictionary of Thermodynamics”• "Thermodynamics is a funny subject. The first time you go through it, you dont understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you dont understand it, but by that time you are so used to it, it doesnt bother you any more." — Arnold Sommerfeld 1.2
- 3. 1. Introduction to (atmospheric) thermodynamics 1.1 On the atmosphere The atmosphere is an ever changing fluent layer that circumvents our earth. Theatmosphere constantly exchange heat, forces, and mass between the solid earth surfaceand the outer space. The atmosphere consists of a number of different gases and materials. Many of theseplay an important role for setting the structure of the atmosphere. The atmosphere mainly 1.3
- 4. consists of nitrogen gas, N2, Oxygen, O2, Argon, Ar, etc. To a smaller extent there is alsocarbon dioxide, CO2, Methane, CH4. These latter gases are strongly affected by biologicalactivity, and by burning of fossil fuels and organic material. Accordingly, theconcentration of these gases varies significantly with time. Notably, they also play animportant role for the radiation budget of the earth although they are not importantdirectly for the thermodynamical properties of the atmosphere – which is the subject ofthis course. A material of special interest for this course is water, which exist in its all three phasesin the natural environment. Furthermore, the transition between water vapor andwater/ice is very important for many atmospheric processes and describing thesetransition properties will be one of the main tasks in this course. 1.1.2 Specific for meteorology There are few things which may be considered to be specific for the atmosphere andhow we deal with it from a thermodynamic viewpoint. 1. Mixture of gases, including water vapor. 2. Pressure decreases with height. 3. Is forced thermally at the ground and some heights (by radiation). Exchange with ground (and space). 4. Is in constant movement (is not in equilibrium). Thermodynamically the following point receive most attention 1. Dry air (torr luft) 2. Water in its three phases (aggregationstillstånd). Ice, water, gas (is, vatten, ånga). 3. Mixture of dry air and moisture (luft+ånga=fuktig luft) 4. A mixture of moist air and water droplets and/or ice crystals.From Wikipedia, the free encyclopediaIn the physical sciences, atmospheric thermodynamics is the study of heat and energytransformations in the earth’s atmospheric system. Following the fundamental laws ofclassical thermodynamics, atmospheric thermodynamics studies such phenomenon asproperties of moist air, formation of clouds, atmospheric convection, boundary layermeteorology, and vertical stabilities in the atmosphere. Atmospheric thermodynamicdiagrams are used as tools in the forecasting of storm development. Atmosphericthermodynamics forms a basis for cloud microphysics and convection parameterizationsin numerical weather models, and is used in many climate considerations, includingconvective-equilibrium climate models. 1.4
- 5. 1.1.3 This course We will analyze some thermodynamic properties of the atmosphere highlighting theinfluence of water – and its phase transitions – on the atmosphere. We will also considerthe main force balances of the atmosphere in this course and the vertical acceleration ofair parcels – this is referred to as “statik”. 1.2 Basic thermodynamic stuff Thermodynamic principles are firmly rooted in experiments. Thermodynamics is ascience of measurable quantities rather than invisible constructors. It seeks noexplanation at a level below what we can observe directly with our coarse senses andmeasuring instruments. Thus thermodynamics treats phenomena on a macroscopic scale.Accessible to us and our instruments, as opposed to a microscopic scale, the atomic andmolecular scale. Thus, thermodynamics do not rely on the existence of invisible atomsand molecules – they provide a way to explain processes in a different way but thethermodynamic theory would essentially remain unchanged if we discover that theprevailing theory of atoms and molecules were wrong. In some regards thermodynamicsis phenomenological in the way that it does not try to explain underlying causes in detail,rather it deals the description and classification of phenomenon. 1.2.1 There are some important concepts in thermodynamics.From Wikipedia, the free encyclopediaAn important concept in thermodynamics is the“system”. A system is the region of the universe understudy. A system is separated from the remainder of theuniverse by a boundary which may be imaginary or not,but which by convention delimits a finite volume. Thepossible exchanges of work, heat, or matter between thesystem and the surroundings take place across thisboundary. There are five dominant classes of systems: Isolated Systems – matter and energy may not cross the boundary. Adiabatic Systems – heat may not cross the boundary. Diathermic Systems - heat may cross boundary. Closed Systems – matter may not cross the boundary. Open Systems – heat, work, and matter may cross the boundary.For isolated systems, as time goes by, internal differences in the system tend to even out;pressures and temperatures tend to equalize, as do density differences. A system in whichall equalizing processes have gone practically to completion, is considered to be in a stateof thermodynamic equilibrium. 1.5
- 6. In thermodynamic equilibrium, a systems properties are, by definition, unchanging intime. Systems in equilibrium are much simpler and easier to understand than systemswhich are not in equilibrium. Often, when analyzing a thermodynamic process, it can beassumed that each intermediate state in the process is at equilibrium. This will alsoconsiderably simplify the situation. Thermodynamic processes which develop so slowlyas to allow each intermediate step to be an equilibrium state are said to be reversibleprocesses. 1. System: A system is a well defined volume that we intend to study. a. Closed system: Is not in contact with the outside world?? i. There are no sharp and impermeable boundaries in nature. All such boundaries are purely mathematic. b. Open system: Interacts with the outside world, for instance can receive energy. 2. Equilibrium: This is an important concept in thermodynamics. The postulate of local equilibrium is just that, a postulate, not a law. More than a postulate it is a hope that nature is kind to us. a. Internal equilibrium (inre jämvikt.). A system that does to exchange any properties with the surrounding is in equilibrium. b. External equilibrium (yttre jämvikt). A system is in equilibrium with an external contact. c. Systems can be described relatively well even if there is not a true equilibrium. Requires that a system is relatively close to equilibrium. i. Thus we will only consider small infinitesimal disturbances. ii. Packets that we will frequently refer to has scales molecules<<package<<system. 3. Temperature: Can only be defined as something that is in equilibrium with an external system. Temperature requires equilibrium to be defined and it is a complicated variable to define and measure. There exist very many different meteorological properties defined as temperature. Potential temperature, virtual temperature, potential virtual temperature, dew point, etc. 4. Macro variables. a. Mass (m or ∆ m); The total mass of the system is a key variable of the system b. Volume (V or ∆ V); In meteorology the volume is often expressed in a different form namely as i. Density, (ρ =M/V or ρ =∆M/∆V); Density is a commonly used variable in dynamical meteorology. 1.6
- 7. ii. Specific volume (α =∆V/∆M). Is used in the thermodynamic analysis of meteorology. Notably ρα=1. In principle we can always use α instead of V in the thermodynamics relation provided that we change appropriate constants. c. Pressure (p); Pressure implies the presence of a force perpendicular to an area A which equals ∆F i. p= . ∆A ii. Unit is Pa=1N/m2. Often used is mbar=102Pa=1hPa. d. Temperature (T or θ ); Is relatively complicated. Requires equilibrium to be defined in a proper way. Uses Kelvin scale (K) or Celsius (oC). 0oC=273.15 K. i. There are a large number of temperature definitions in meteorology.5. Micro variables: Molecular speed, number of molecules etc. Not really considered in thermodynamics, belongs to statistical mechanics.6. Equation of state (tillståndsekvationen). An important pile stone of the thermodynamics is that there exist some well described relations between the physical quantities P, V, and T. (in meteorology, p, α , T). a. pV = nR *T : n is number of moles, R*(=8.3144 J mol-1 K-1) universal gas constant. i. Alternative form pv = RT , v is molar volume (volume V per mole n. Avogadros number v=22.414 m3/mol for p0=1 atm, T0=273 K).) ii. Meteorology: pα = RT , or p = ρRT . R=R*/M (M=molmassa) is the specific gas constant (specifika eller inviduella gaskonstanten). a iii. p + 2 ( v − b ) = nR T . Van der Waals equation. a, b are * v constants appropriate for each gas. b. Daltons law: A mixture of ideal gases will behave as an ideal gas. In a gas containing more than one component, each component add to the total pressure (for an ideal gas). i. p k V = n k R *T , thus Volume and Temperature are “global” quantities. * T T n k Rk ii. Total pressure p = ∑ p k = ∑ nk R = V n∑ n * k k V k k 1.7
- 8. iii. For air: pα = Rd T , Rd=287.0 J/K. Specific molmass Md=R*/ Rd=28.97 kg/mol. iv. Air composition is relatively constant up to 100 km, thereafter it decreases with height. Figure text: Mean molecular weight versus height for U.S. Standard Atmosphere. c. An important concept is that the relation between state variables can be written F(p, α, T)=0. i. If two variables are known, the third can be calculated. We may write p= p (α,T), α = α (p, T), T = T(p, α). The direct implication is that all functions will varies independently on two of the states variables only, that is U= U1(α,T), U2 (p, T), U3(p, α). Notably, U1, U2, U3 are three completely different functions. However to avoid naming an incomprehensive number of functions one usually skip the index on the function. 1.3 Laws of thermodynamicsFrom Wikipedia, the free encyclopediaIn thermodynamics, there are four laws of very general validity, and as such they do notdepend on the details of the interactions or the systems being studied. Hence, they can beapplied to systems about which one knows nothing other than the balance of energy andmatter transfer. Examples of this include Einsteins prediction of spontaneous emissionaround the turn of the 20th century and current research into the thermodynamics of blackholes.The four laws are: 1.8
- 9. Zeroth law of thermodynamics, stating that thermodynamic equilibrium is anequivalence relation. If two thermodynamic systems are in thermal equilibrium with a third, they are also in thermal equilibrium with each other. First law of thermodynamics, about the conservation of energy The increase in the energy of a closed system is equal to the amount of energy added to the system by heating, minus the amount lost in the form of work done by the system on its surroundings. Second law of thermodynamics, about entropy The total entropy of any isolated thermodynamic system tends to increase over time, approaching a maximum value. Third law of thermodynamics, about absolute zero temperature As a system asymptotically approaches absolute zero of temperature all processes virtually cease and the entropy of the system asymptotically approaches a minimum value. My view • 0’th law: Two systems that are in thermal equilibrium with a third are in thermal equilibrium with each other. • 1’st law: Energy is conserved. For instance dv d v2 dz v ⋅ m = mge z ⇒ m = mg ⇒ o dt dt 2 dt , d ( K + P ) = 0 ⇒ K + P = const dt Energy is conserved in a mechanistic system, can change between the kinetic energy of the point mass (energy of motion) and the potential energy (energy of position). For system with many particles, it can be shown that we also consider changes in kinetic and potential energy in the way the mass centre moves (the kinetic energy of a fictious body with mass equal the entire system of point masses and moving with its centre of mass velocity) and in the way the particles move randomly or disorganized, this thus also represent a kinetic energy but may represent internal kinetic energy (or temperature) Ordered mean motions are easily transferred to internal random motions. Consider a balloon filled with water. If we drop it will have a certain mean motion before it strikes the floor. However, after hitting the floor the mean motion becomes zero and must 1.9
- 10. have been transferred to disorganized random motion with zero net mass transfer. Conservation of energy implies that internal motion must have increased. In other words mean motion (external) have been transformed into disorganized motion (heat). Thermodynamics is a way to describe these processes with giving an exact description of the entire procedure. o If a closed system is caused to change from an initial state to a final state by adiabatic means only, then the work done on the system is same far all adiabatic paths connecting the two states. o If there is an exchange of properties between the system and the outside word, the work will depend on the exact pathway. o Internal energy. dU = Q − W ; Q is heating rate, W working rate. (these are dt equal zero for a closed system). dα • In general the work is W = p which implies that we dt dU dα often write =Q− p . dt dt • It should be noted that we require that ∆x ∆t << v s , where vs is the speed of molecules (i.e., roughly the speed of sound) for the system to be in a reasonable quasi-stationary state. This feature is always valid for the atmosphere and is actually well fulfilled in a normal engine. dU = dQ + dW • We need to distinguish between exact differentials (denoted d) which refers to state variables, and inexact differentials (denoted d ) that refers to external forcing parameters. • 2’nd law: Entropy always increases. (Whatever than means) The best way to understand energy and entropy – indeed, all concepts, scientific orotherwise – is to use them in as many contexts as possible, proceeding from the familiarto the unfamiliar.From Wikipedia, the free encyclopedia 1.10
- 11. Thermodynamic processesA thermodynamic process may be defined as the energetic evolution of athermodynamic system proceeding from an initial state to a final state. Typically, eachthermodynamic process is distinguished from other processes, in energetic character,according to what parameters, as temperature, pressure, or volume, etc., are held fixed.Furthermore, it is useful to group these processes into pairs, in which each variable heldconstant is one member of a conjugate pair. The six most common thermodynamicprocesses are shown below: 1. An isobaric process occurs at constant pressure. 2. An isochoric process, or isometric/isovolumetric process, occurs at constant volume. 3. An isothermal process occurs at a constant temperature. 4. An isentropic process occurs at a constant entropy. 5. An isenthalpic process occurs at a constant enthalpy. 6. An adiabatic process occurs without loss or gain of heat. Thermodynamic potentialsAs can be derived from the energy balance equation on a thermodynamic system thereexist energetic quantities called thermodynamic potentials, being the quantitative measureof the stored energy in the system. The four most well known potentials are: Internal energy Helmholtz free energy Enthalpy Gibbs free energyPotentials are used to measure energy changes in systems as they evolve from an initialstate to a final state. The potential used depends on the constraints of the system, such asconstant temperature or pressure. Internal energy is the internal energy of the system,enthalpy is the internal energy of the system plus the energy related to pressure-volumework, and Helmholtz and Gibbs free energy are the energies available in a system to douseful work when the temperature and volume or the pressure and temperature are fixed,respectively. 1.11
- 12. 1.4 Changes due to heating Lets us assume that we heat a certain volume of gas. The response may be consideredfrom the response in time or as ordinary differentials. Personally I do think that it iseasier to consider changes in time than in differentials. Let us write dU dα Q= +p dt dt The internal energy U for a simple closed system such as a gas may be considered sfunction of the two independent variables, temperature T and volume V, while the thirdvariable p is related to these two by the ideal gas law. 1.4.1 Constant volume Applying the chain rule simply provides dU ∂U dT ∂U dα = + dt ∂T dt ∂α dt Notably ∂U ∂α has the dimension of pressure andis sometimes called the internal pressure (for an idealgas ∂U ∂α =0, Joules law and has been confirmed tobe small in experiments using real gases). Continuing ∂U ∂α = 0 for an ideal gas (pdV=0 for ∂U dT ∂U dα dα Q= + +p above experiment) ∂T dt ∂α dt dt ∂U dT ∂U dα = + + p ∂T dt ∂α dt The first partial derivative on the right hand side of this equation appears withsufficient frequency that is has acquired a name, heat capacity (at constant volume) ∂U Cv = ∂T Its relevance is clear from the following equation appearing for constant volume. dT Q = Cv , V=const. dt Notably we also find that ∂C v ∂ ∂U ∂ ∂U = = ≈0 ∂V ∂V ∂T ∂T ∂VThis relation holds for an ideal gas where interactions between molecules are negligible.Cv depends on the total mass of the system and sometimes it is convenient to deal withthe specific heat capacity instead, thus 1.12
- 13. cv Cv = . m 1.4.2 Constant pressure To find how the system respond to heating under constant pressure we need to find anexpression which has dp/dt. Starting as before dU dα Q= +p dt dt dα d T R dT RT dp using that pα = RT we can write =R = − 2 dt dt p p dt p dt dU R dT RT dp dU dT RT dp Q= + p p dt − p 2 dt = dt + R dt − p dt dt dU dT dT RT dp = +R − dT dt dt p dt under constant pressure we find dT dT Q = ( Cv + R ) = Cp dt dt . C p = Cv + R We thus find that the heat capacity (or resistance to become warm when heated) islarger under constant pressure than under constant volume. To show that cp greater thancv makes physical sense, consider an ideal gas confined to a cylinder fitted with the usualfrictionless nut tightly fitted piston. Fix the piston in place and heat the gas for a certainamount of time. The temperature of the gas rises in this constant volume process. Now letthe piston move freely so that gas pressure is constant. Heat the gas for the same amountof time a before. Again the temperature increases, but in this process the piston rises; thuswork is done by the gas and consequently its internal energy doesn’t increase as much asbefore. This implies that the temperature increase isn’t as great. Stated another way, cp isgreater than cv, which is consistent with what we derived. 1.13
- 14. For liquids and solid material the compressibility (implying that pdα≈0) is essentiallyzero and thus C p ≈ Cv . This can be shown but I am too lazy 1.4.3 Constant pressure 2 The derivations for the temperature response to heating under constant volume andconstant pressure took different paths. However, let us redo the calculation in anotherway, it is convenient to introduce enthalpy, defined as H = U + pα dU dαThe first law Q = +p can now be written dt dt dH dp Q= −α dt dt the chain rule gives dH ∂H dT ∂H dp = + . dt ∂T dt ∂p dt We thus have ∂H dT ∂H dP dP Q= + −α ∂T dt ∂P dt dt ∂H dT ∂H dp = + −α ∂T dt ∂V dt 1.14
- 15. For constant pressure we have that the heat capacity (at constant volume) is ∂H Cp = ∂T Its relevance is clear from the following equation appearing for constant volume. dT Q = CP , p=const. dt Using the definition of entalphy (and conidering the independent variables to be T andp) H = U + pα H (T , p ) = U (T , α ) + pα ∂H dT ∂H dp ∂U dT ∂U dα dp dα + = + +α +p ∂T dt ∂P dt ∂T dt ∂α dt dt dt dα ∂α dp ∂α dT α = α ( p, T ) ⇒ = + dt ∂p dt ∂T dt ∂H ∂U ∂U ∂α ∂α dT ∂H ∂U ∂α dα dp − − −p + ∂P − V ∂α ∂p − p dp dt = 0 ∂T ∂T ∂α ∂T ∂T dt thus ∂U ∂α ∂α C p = Cv + +p ∂α ∂T ∂T ∂U , ideal gas = 0 (Joules law). ∂α ∂α C p = Cv + p = Cv + R ∂T Note that the intermolecular forces are negligible in an ideal law. Joules law can beshowed in experiments where a chamber is split in two parts. The left part has a certainpressure and the right volume has no pressure. If the wall is removed gas will go from leftto the right side. There is no exchange of heat, no work done and thus the internal energymust remain constant. ∂U ∂α ∂α Joules law and the expression C p = C v + +p will be considered in later ∂α ∂T ∂Tsections. However, to pave the way forward we need to introduce the concept entropy,which is done in section 3. From Wikipedia, the free encyclopediaIn thermodynamics, the quantity enthalpy, symbolized by H, also called heat content, isthe sum of the internal energy of a thermodynamic system plus the energy associated 1.15
- 16. with work done by the system on the atmosphere which is the product of the pressuretimes the volume. The term enthalpy is composed of the prefix en-, meaning to "putinto", plus the Greek suffix -thalpein, meaning "to heat".Enthalpy is a quantifiable state function, and the total enthalpy of a system cannot bemeasured directly; the enthalpy change of a system is measured instead. A possibleinterpretation of enthalpy is as follows. Imagine we are to create the system out ofnothing, then, in addition to supplying the internal energy U for the system, we need todo work to push the atmosphere away in order to make room for the system. Assumingthe environment is at some constant pressure P, this mechanical work required is just PVwhere V is the volume of the system. Therefore, colloquially, enthalpy is the total amountof energy one needs to provide to create the system and then place it in the atmosphere.Conversely, if the system is annihilated, the energy extracted is not just U, but also thework done by the atmosphere as it collapses to fill the space previously occupied by thesystem, which is PV.Enthalpy is a thermodynamic potential, and is useful particularly for nearly-constantpressure processes, where any energy input to the system must go into internal energy orthe mechanical work of expanding the system. For systems at constant pressure, thechange in enthalpy is the heat received by the system plus the non-mechanical work thathas been done. In other words, when considering change in enthalpy, one can ignore thecompression/expansion mechanical work. Therefore, for a simple system, with a constantnumber of particles, the difference in enthalpy is the maximum amount of thermal energyderivable from a thermodynamic process in which the pressure is held constant. 1.16

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