The document provides an overview of statistical thermodynamics including: - Its historic background beginning with Bernoulli's work in the 18th century and contributions from Maxwell, Boltzmann, Gibbs, and others. - The key difference between classical thermodynamics and statistical thermodynamics is that the latter links microscopic properties to macroscopic behaviors. - Statistical thermodynamics is needed to explain thermodynamic parameters in terms of molecular properties and interactions since classical thermodynamics does not address this microscopic level. - Central topics in statistical thermodynamics include the partition function, degrees of freedom, heat capacity, and Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein statistics.

1. Presented by

2. MUHAMMAD ZAHID, MSC Final year (S-4)

3. Contents:
 Fundamental of statistical mechanics
I. Historic background
II. Classical thermodynamic vs. statistical thermodynamics
III. Why we need statistical thermodynamics
IV. Central topics in statistical thermodynamics
• Microstate
• Boltsman distribution law
• Partition function
• Thermodynamic equilibium
• Internal degree of freedom
• Heat capacity
• Nernst heat theorem
• Fluctuation
• Gibbse paradox
• Degenracy

4. Fundamental of statistical thermodynamics
Historic background
 In, 1738, Swiss physicist and mathematician Daniel.Bernoulli published Hydrodynamica.
Bernoulli posited the argument, that gases consist of great numbers of molecules moving in
all directions, their impact on a surface causes the gas pressure that we feel, and that what we
experience as heat is simply the kinetic energy of their motion.
 In 1959, James Clerk Maxwell formulated the Maxwell distribution of velocities amoung the
molecules of gas after reading Rudolf Clausius’s paper on diffusion of molecule. in
1864, Ludwig Boltzmann read the Maxwell’s paper and express distribution of energies
amoung the molecules.
𝑓 𝑣 𝑑𝑣 = 𝐶𝑒−2
1𝐵𝑚𝑣2
dv (maxvell velocity distribution)
𝐶 = (
𝛽𝑚
2𝜋
)1/2=proportionality constant, 𝛽 = 1/kT

5. 𝑝𝛼𝑒−𝜀/𝑘𝑇 (boltzman distribution of energy)
 Boltzmann known to be the "father" of statistical thermodynamics with his 1875
derivation of the relationship between entropy S and multiplicity Ω.
𝑆 = 𝑘𝑏 𝑙𝑛Ώ
Willard Gibbs delivered an address to the American Association for the Advancement
of Science, in 1884, in which he coined the word "statistical mechanics.
Classical thermodynamic vs. statistical
thermodynamics
 macroscopic approach to the study of thermodynamics
which does not require knowledge of the behavior of individual
particles is called classical thermodynamics.

6. Statistical thermodynamics -- link between microscopic properties of matter and its bulk
properties. Statistical thermodynamics is a theory that uses molecular properties to predict
the behavior of macroscopic quantities of compounds.
𝑆 = −𝑘𝑏 𝑖 𝑝𝑖 log𝑝𝑖 (Willard Gibbse, 1870: classical entropy)
where, pi =probability
𝑆 = 𝑘𝑏 𝑙𝑛Ώ (bolztman entropy, 1875: statistical entropy)
where, Ω = number of microstate
Why we need statistical mechanics in thermodynamic
 As an example, from a classical thermodynamics point of view we might ask what is it
about a thermodynamic system of gas molecules, such as ammonia NH3, that
determines the free energy characteristic of that compound? Classical thermodynamics
does not provide the answer.
 Actually, the free energy of the gas molecule could be spectroscopic data like bond
length, bond angle, rotation bond, flexibility of bond.

7. In other word they can not explain the thermodynamic parameter such as
temperature, pressure of constrituent of system.
 we need to bridge the gap between the microscopic realm of atoms and molecules and
the macroscopic realm of classical thermodynamics. From physics, statistical
mechanics provides such a bridge.
Aim of statistical thermodynamics
The goal of statistical thermodynamics is to understand and to interpret the measurable
macroscopic properties of materials in terms of the properties of their constituent
particles and the interactions between them. This is done by connecting thermodynamic
functions to quantum-mechanic equations.

8. Central topic in statistical thermodynamic
Partition function
The number of thermally accessible energy level at given temperature.
𝑄 =
𝑖
𝑔𝑖 𝑒−𝛽𝜀
Degree of freedom
In many scientific fields, the degrees of freedom of a system is the number of parameters of
the system that may vary independently. For example, a point in the plane has two degrees of
freedom for translation its two coordinates
Heat capacity
heat capacity or thermal capacity is a physical property of matter, defined as the amount
of heat to be supplied to a given mass of a material to produce a unit change in its temperature

9. Nernst heat theorem
The Nernst heat theorem says that as absolute zero is approached, the entropy change
ΔS for a chemical or physical transformation approaches 0. This can be expressed
mathematically as follows
Deneracy
Different state of quantum mechanical system with same value of energy.

10. Major Difference between
Maxvell-Boltzman, Fermic-
diract and Bose einstien
statistic

11. History
Bose-Einstien
 Developed by Bose in 1924
 The idea was later adopted and extended by Albert Einstein in
collaboration with Bose
Fermic-dirac
 F–D statistics was first published in 1926 by Enrico Fermi and Paul Dirac.
 F–D statistics was applied in 1926 by Ralph Fowler to describe the
collapse of a star to a white dwarf. In1927Arnold Sommerfeld applied it to
electrons in metals and developed the free electron model, and in1928
Ralph Howard Fowlerand Lothar Wolfgang Nordheim applied it to field
electron emission from metals

12. Particles and spin
Maxvell-boltzman statistic
 Sometimes called classical case
 Particle are distinguishable: non interecting ideal gas
Bose-Einstien statistic
 Quantum mechanical case
 Paricle are indistinguishable like photon, phonon etc. which are called boson
 Boson have integer spin i-e 0,1,2….
Fermic-dirac statistic
 Qauntum mechanical case
 Particle are indistinguishable like electron called Fermion
 Fermion have half-integer spin i-e ½, 3/2 etc.

13. Pualing excusion principle
 Bose-Einstien : does not obey the pualing exclusion principle
 Fermic-dirac :- obey the pualing exclusion principle
 Acorrding to PEP two or more Fermion (with half integer spin) cannot
occupy in same quantum state within quantum system at same time.
 In case of electron in atom it is impossible for two electron to have the
same values of the four quantum numbers: n, ℓ mℓ and ms, the spin
quantum number.
 So, when two electron reside in same orbital then both electron have
same value of n, l and m, therefore it must be different the spin quantum
number.

14. Formulas
Maxvell-Boltzman statistic
𝑁𝑖 =
𝑔𝑖
℮(ℰ𝑖−𝜇)/𝑘𝑇
gi =degeneracy of the enery level
Ԑi = ith enregy level
μ= chemical potential (change in the energy of the system
when one particle are added
Valid for Ni ˂˂1 and gi˃˃ Ni , any number can go into an energy
state.

15. Bose-Einstien statistics
𝒏𝜺 =
𝟏
℮𝜷(ℇ𝒊−𝝁) − 𝟏
Ԑi = 𝜇, n=0, the distribution is infinite
Ԑi ˂ 𝜇 then it does not make scence
The Bose distribution only make scence when Ԑi ˃ 𝜇