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Unformatted text preview: PHYS 333 Winter 2009 Assignment 3 Most of these problems are from Schroeder’s book. 1. (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 q q + N N 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. The starting point is equation (2.9) on page 55, namely Ω( N, q ) = ( q + N 1)! q !( N 1)! Write this as Ω( N, q ) = ( q + N )! q ! N ! N q + N and then use Stirling’s approximation for each of the three factorials, and cancel various common factors: Ω( N, q ) ' ( q + N ) q + N e ( q + N ) p 2 π ( q + N ) q q e q √ 2 πqN N e N √ 2 πN N q + N = ( q + N ) q + N q q N N s N 2 πq ( q + N ) which is the required result. (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? The number of energy units in A can be 0, 1, etc., up to 2N, so that the total number of macrostates is 2N+1. (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 . 1 Using the result of part (a), but substituting 2N for both N and q, we obtain Ω total = 2 2 N 2 2 N s 2 N 2 π · 2 N (2 N + 2 N ) which is the required result. (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: 2 4 N / (4 πN ) . The most probable macrostate has q = N for both A and B. For A alone, the number of microstates is therefore Ω A = 2 N 2 N s N 2 π · N ( N + N ) = 2 2 N √ 4 πN But the number of microstates for B has this same value, so Ω mostlikely = Ω A · Ω A = 2 4 N 4 πN (e) You can get a rough idea of the “sharpness” of the multiplicity function by comparing your answers to parts (c) and (d). Part (c) tells you the height of the peak, while part (d) tells you the total area under the entire graph.(d) tells you the total area under the entire graph....
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 Winter '09
 Harris
 Physics, Thermodynamics, Energy, Entropy, Black hole, ln Tf /Ti, kb ln tf

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