Robert G. Keil
Un~vers~ty of Dayton Dayton, Ohro 45409 and Leonard K. Nash Horvard U n ~ v e r s ~ t y Combridge, Massochusetts 02138
I
The Most Probable Distribution in Statistical Thermodynamics
The first presentationof statistical thermodynamics to students of chemistry most often turns on a discussion of the microcanonical ensemble. The equilibrium properties of a macroscopic assembly are then derived not by averaging over all possible configurations but, rather, by considering only that one configuration in which the number of associated microstates assumes its maximum value. This policy is ordinarily justified only by (a) a purely verbal argument that it "works"; or (b) reference t o results obtained in more advanced treatises; or (c) demonstration for the special case in which only two quantum states are involved, e.g., spins "up" or "down"; or (d) demonstration for t,he special case of quant,um states with uniform energy-spacing. Examination of the most used texts for junior-year courses in physical chemistry indicates that options (a) and (b) are overwhelmingly those most frequently chosen. Yet some substantive analysis of the issue can easily be given in a manner that has proved accessible even to bright freshmen. This analysis demands no. explicit use of the calculus, which is involved implicitly only to the (minimal) extent required for derivation of Stirling's approximation and the further approximation that, for small x, In (1
+
z) _NX.
Consider an isolated macroscopic assembly of any one species of dist,inguishable units, with any kind of energy-spacing between their successive quantum states. Let us focus on two configurations of this assembly: the "predominant" configuration for which the number of associated microstates assumes its maximum value Wm,, and so,me other very slightly shifted configuration for which the number of microstates is W. I n the predominant configuration W,. = N!/ns,! (1) where the population (13of the nth quantum state is given by the Boltzmann distribution law The value of W for the shifted configuration is also calculablefromeqn. (1)if for the populationnumbers one substitutes the values characteristic of the shifted configuration. In the passage from the predominant configuration to the shifted configuration, let a. symbolize the resultant fractional change in the number of units present in the nth quantum level. That is, using primes to distinguish the population numbers in the shifted configuration, we define or. as
For any level in which the population increases, a will be positive; for any level in which the population decreases, a will be negative. And, since the shifted configuration has been assumed only minutely different from the predominant configuration, in all cases / a (
