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The Crystal and Gas Partition Functions and the Indistinguishability of Molecules Edward J. O'Reilly University of North Dakota, Grand Forks, ND 58202 Conventionally, the partition function for a perfect crystal containing N molecules in a volume V, at a temperature T is written as where q, is the molecular partition function. For ararefied gas, in order to obtain agreement with the experimental values of entropy eqn. (1) must be divided by the factor N!';" This anomaly has been explained: 1) by invoking the concept of the indistinguishability of molecules of a perfect gas, the factor N! being the number of permutations of N indistin- guishable molecules; 2) by invoking the concept of the dis- tinguishability of molecules in a crystal, for while the mole- cules are per se indistinguishable, they lie on distinguishable lattice sites. An alternative derivation is possible, which is internally consistent in treating both the perfect crystal and perfect gas in the same fashion. Consider the above crystal to be divided into N cells of volume u, = V$N. These cells are distin- guishable, since they may be identified hy the coordinates of, say their centers, relative to some lahoratory-fixed origin. (The switch to cells, from lattice points is pure convenience, and of no fundamental significance). A randomly mixed crystal of N molecules of type-n, and H molecules of type-h, occupying ( N + H) equal volumed cells has a partition function The numerator (N + H)! is the number of ways (N + H) molecules call be arranged among L = (N + H ) distinguish- able cells. The divisor, N!H! takes into account the indistin- guishability of the n-type molecules among themselves, and the h-type molecules among themselves. In the limit that H - O., eqn. (3) reduces t o Q, = N!qnN/N! (4) This result is mathematically identical to eqn. (I), hut totally different in significance, since it admits of the indistinguish- ability of the molecules in the crystal through the divisor N! Having once introduced this indistinguishability concept for the crystal molecules, how does one then explain the par- tition function for a perfect gas in this fashion? Consider the h-type molecules to he in fact holes or vacancies with partition function qi, = 1. The partition function for the N type-n gas molecules in a cell of volume V, is For a real eas to a ~ ~ r o x i m a t e a nerfect gas. the pas volume V- . . approximation the logarithm of the iombinational factorln Eqn. (3) becomes From the relation (N +'HI = Vg/um,the first term on the right hand side of eqn. (6) becomes ln(V,lN~,)~. The second term may be written as ln(r + 1IH, x = NIH, and expanded by Taylor's theorem to become ln eN. The cornhinational factor in eqn. (3) is then eqn. (3) becomes for a perfect gas which gives rise to an entropy in agreement with experiment. Comparing eqn. (7) and eqn. (21, the factor (KV,) is the mo- lecular partition function q, for a gaseous molecule in volume V,. The factor (eIN)Nis the factor (N!)-'. We have shown that the appropriate partition functions for a perfect crystal and a perfect gas are both ohtained from the same consistent ideas: the number of ways of distributing N indistinguishable molecules among distinguishable cells. ' Moore, W. J., "Physical Chemisby;' 3rd Ed., Prentice-Hall Inc., New York. 1963. o. 627. );lc~ua;e, D. A,, 'Statistical Thermodynamics.'' Harper and Raw, New York, 1973, p. 70. 216 Journal of Chemical Education
