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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

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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

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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

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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.)

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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.

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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.

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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.

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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)

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

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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 :

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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

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

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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.

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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. )

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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 )

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

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

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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

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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.

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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

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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.

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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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