# 1.The Statistical Basis of Thermodynamics 1.The Macroscopic & the Microscopic States 2.Contact between Statistics & Thermodynamics: Physical Significance.

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1.The Statistical Basis of Thermodynamics 1.The Macroscopic & the Microscopic States 2.Contact between Statistics & Thermodynamics: Physical Significance of the Number  (N,V,E) 3.Further Contact between Statistics & Thermodynamics 4.The Classical Ideal Gas 5.The Entropy of Mixing & the Gibbs Paradox 6.The “Correct” Enumeration of the Microstates

1.1.The Macroscopic & the Microscopic States System of N identical particles in volume V, with (Thermodynamic limit ) E.g., Non-interacting particles:  i = single particle energies n i = # of p’cles with energy  i A macrostate is specified by parameters ( N, V, E,... ). Postulate of equal a priori probabilities: All microstates satisfying the macrostate parameters are equally likely to occur. = # of all microstates that give rise to the macrostate (extensive) parameters N, V, E,.... Let

1.2.Contact between Statistics & Thermodynamics: Physical Significance of the Number  (N,V,E) Consider 2 systems A 1 & A 2 in thermal contact with each other, i.e., partition is fixed, impermeable but heat conducting. ( N j, V j & E (0) = E 1 + E 2 are fixed ) A 1 ( N 1, V 1, E 1 ) A 2 ( N 2, V 2, E 2 )  Equilibrium is achieved if E 1 ( with E 2 = E (0)  E 1 ) maximizes  (0) : (0) denotes properties of the composite system

  Let  2 systems are in thermal equilibrium if they have the same . Thermodynamics : Planck : Boltzmann :  k = Boltzmann constant  3 rd law  0 th law ( thermal eqm.)

1.3. Further Contact between Statistics & Thermodynamics For an impermeable but movable & heat conducting partition, N j, V (0) = V 1 +V 2 & E (0) = E 1 + E 2 are fixed. Equilibrium is achieved, i.e.,  (0) is maximized, if and i.e., both system have the same values of  & 1 st law:    chemical potential ~ mech. eqm.

For a permeable, movable & heat conducting partition, N (0) = N 1 + N 2, V (0) = V 1 +V 2 & E (0) = E 1 + E 2 are fixed. Equilibrium is achieved, i.e.,  (0) is maximized, if i.e., Both system have the same values of , , & 1 st law:  ~ chemical eqm.

Summary Connection between statistical mechanics & thermodynamics is Once  is written in terms of the independent thermodynamical variables, all other thermodynamic quantities can be obtained via the Maxwell relations.

U Internal Energy SV, X  P, Y T H Enthalpy G Gibbs free energy F Helmholtz free energy Mnemonics for the Maxwell Relations Good Physicists Have Studied Under Very Fine Teachers = U  (  P) V  Y X = F  (  P) V  Y X = H  TS = U  TS = U(V,S,X)

1.4.The Classical Ideal Gas Non-interacting, classical (  distinguishable), point particles:  Cf  const here means indep. of V.

Quantum (Obeying Schrodinger Eq) Free Particles Let these particles be confined within a cube of edge L. Dirichlet boundary conditions:   0 at walls ( where x,y,z = 0,L ). Neumann boundary conditions:  n   0 at walls. 1-particle energy :

i.e. Let (  * is a positive integer )  # of { n x, n y, n z } satisfying  # of { n ix, n iy, n iz } satisfying For N non-interacting particles  

 For reversible adiabatic processes, S & N are kept constant.     Valid for both classical & quantum statistics (adiabatic processes) 

Better behaved quantity is  ( N,E,V), defined as the # of lattice points with non-negative coordinates & lying within the volume bounded by the surface of a sphere, centered at the origin, and with radius Counting States:Distinguishable Particles State labels { n ix, n iy, n iz } form a lattice in the 3N-D n- space.   ( N,E,V) = # of lattice points with non-negative coordinates & lying on the surface of a sphere, centered at the origin, and with radius  fluctuates wildly even for small E changes unless N >>1.

As R  , the lattice points become a continuum. Better approximations: Number of points on the x-y, y-z, z-x planes is Since these points are shared by 2 neighboring sectors, the volume integral counts each as half a point. Dirichlet B.C. (exclude all n j = 0 points ) Neumann B.C. (include all n j = 0 points ) ( Density of states in n-space is 1. )

Volume of an n-D sphere of radius R is ( see App.C ) Volume of points with non-negative coordinates ( Take non-negative-half of every dimension ) 

Stirling’s formula: for n >>1   Let  (N,V,E) = # of states lying between E  ½  & E+ ½ .

  

Isothermal processes ( N, T = const ) :  Adiabatic processes ( N, S = const ) :   Alternatively,also leads to

1.5.The Entropy of Mixing & the Gibbs Paradox This S is not extensive, i.e., Mixing of 2 ideal gases 1 & 2 (at fixed T ) :  Thermal wavelength

Entropy of mixing of gases : Gibb’s paradox : For the mixing of different parts of the same gas in equilibrium ( N i / V i = N / V,  i = ), the formula still applies & we also have  S > 0, which is unacceptable. Irreversible process:  S > 0 is expected.

For the mixing of different parts of the same gas in eqm.,   Thus, Gibbs’ paradox is resolved using Gibbs’ recipe : Sackur-Tetrode eq. S is now extensive, i.e., or

Revised Formulae  extensive In general, relations derived using the previous definition of S are not modified if they do not involve explicit expression of S. intensive Gibbs’ recipe is cancelled by removing all terms in red.

1.6.The “Correct” Enumeration of the Microstates Elementary particles are all indistinguishable. In the distribution of N particles such that n i particles occupy the  i state, for distinguishable particles for indistinguishable particles In the classical (high T ) limit,  Gibbs’ recipe corresponds to

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