CHM 5414 Lecture Notes
8 October 1996

Chance scatters the billiard balls randomly on a break unless you're a hustler. Nature's no hustler. She makes Entropy the old-fashion way; She rolls die.

We've already derived the most instructive expression for the entropy, S, from statistical mechanics; it was

S = - R X_i ln(X_i),

the classical molar Entropy of Mixing. Except in our derivation, X_i wasn't the mole fraction of a fluid (or even an isotope) but the probability of occupancy of an available state! All Entropy is the Entropy of Mixing.

But since we've elevated the partition function, z, to become the touchstone of thermodynamics, we ought to know how S relates to it. And we start with Boltzmann:

S = k ln(W) where, of course, W=N!/_i! (Remember that _i = N)
S = k { N ln(N) - N - _i ln(_i) + _i} (using Stirling's approximation)
S = k { N ln(N) - _i ln[ (N/z) exp(-_i) ] } (Maxwell-Boltzmann distribution)
S = k { N ln(N) - (_i)ln(N/z) + _ii }
S = k { N ln(N) - N ln(N) + N ln(z) + E }
S = k ln(z^N) + kE

S = k ln(Z) + E/T

Although we've not shown it, that same last line results if Z=z^N/N! for indistinguishable molecules. (See Nash's pink pages.)

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While that expression for S is simplicity itself, the Helmholtz Free Energy, A, takes the cake. You recall that A = E - TS is the physicists' preference for equilibrium calculations (at fixed V,T) while G = H - TS is the chemists' favorite for equilibrium at fixed P,T. (If that escaped you, fret not; it's coming again in a couple of lectures.)

So if we do the arithmetic, we find

A = - kT ln(Z).

While that couldn't be much simpler, it affords us instant access to statistical mechanical pressure. If that sounds unlikely, you've forgotten your multivariable calculus. Permit me to refresh it . . .

dE = dq_rev - PdV
dE = TdS - PdV (the fundamental expression of thermodynamics)
dA = dE - d(TS) = dE - TdS - SdT
dA = TdS - TdS - SdT - PdV
dA = - SdT - PdV, but
dA = [A/T]_V dT + [A/V]_T dV (see why A wants V,T fixed?)

By inspection then, P = - [A/V]_T or

P = + kT [ ln(Z)/V]_T, the statistical mechanical pressure! (Plug in Z_trans, and the Ideal Gas expression drops out!)

Now all we must do is develop Z from molecular modes, and we're home free. Next time.

Chris Parr
University of Texas at Dallas
Programs in Chemistry, Room BE3.506
P.O. Box 830688  M/S BE2.6 (for snailmail)
Richardson, TX  75083-0688

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