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# hw3 - PHYS 333 Winter 2009 Assignment 3 Due 5:00 pm Friday...

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PHYS 333 Winter 2009 Assignment 3 Due: 5:00 pm Friday January 23 Most of these problems are from Schroeder’s book. 1. Schroeder, problems 2.18/22/30 (a) Use Stirling’s approximation to show that the multiplicity of an Einstein solid, for any large values of N and q , is approximately Ω( N, q ) ' ± q + N q ² q ± q + N N ² N p 2 πq ( q + N ) /N The square root in the denominator is merely large, and can often be neglected. However, it what follows, it is needed. (b) 2.22. Consider two Einstein solids, each with N oscillators, in thermal contact with each other. Suppose that the total number of energy units in the combined system is exactly 2 N . How many diﬀerent macrostates (that is, possible values for the total energy in the ﬁrst solid) are there for this combined system? (c) Use the result of part (a) (2.18) to ﬁnd an approximate expression for the total number of microstates for the combined system. (Hint: treat the combined system as a single Einstein solid. Do not throw away factors of “large” numbers, since you will eventually be dividing two “very large” numbers that are nearly equal.) Ans: 2 4 N / 8 πN . (d) The most likely macrostate for this system is, of course, the one in which the energy is shared equally between the two solids. Use the result of part (a) (2.18) to ﬁnd an approximate expression for the multiplicity of this macrostate. Ans:

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hw3 - PHYS 333 Winter 2009 Assignment 3 Due 5:00 pm Friday...

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