P21_045 - box, then the multiplicity is W B = N ! ( N/ 2)!...

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45. (a) Suppose there are n L molecules in the left third of the box, n C molecules in the center third, and n R molecules in the right third. There are N ! arrangements of the N molecules, but n L ! are simply rearrangements of the n L molecules in the right third, n C ! are rearrangements of the n C molecules in the center third, and n R ! are rearrangements of the n R molecules in the right third. These rearrangements do not produce a new conFguration. Thus, the multiplicity is W = N ! n L ! n C ! n R ! . (b) If half the molecules are in the right half of the box and the other half are in the left half of the
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Unformatted text preview: box, then the multiplicity is W B = N ! ( N/ 2)! ( N/ 2)! . If one-third of the molecules are in each third of the box, then the multiplicity is W A = N ! ( N/ 3)! ( N/ 3)! ( N/ 3)! . The ratio is W A W B = ( N/ 2)! ( N/ 2)! ( N/ 3)! ( N/ 3)! ( N/ 3)! . (c) or N = 100, W A W B = 50! 50! 33! 33! 34! = 4 . 16 10 16 ....
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