Solutions4

Solutions4 - PHYS 333 Winter 2009 Assignment 4 1. Entropy...

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Unformatted text preview: PHYS 333 Winter 2009 Assignment 4 1. Entropy of Mixing. Calculate the increase in entropy when two different ideal gases diffuse into each other. The two gases are originally in separate containers, but with the same pressure and temperature. There are N A molecules of gas A, and N B molecules of gas B, with N = N A + N B , occupying volumes V A and V B respectively. After they have mixed, their volume is V = V A + V B . Do the calculation in three independent ways, and show, each time, that the answer is S = k B N A ln [ N/N A ] + k B N B ln [ N/N B ] (a) Take the Sackur-Tetrode expression for the entropy, and calculate the change directly. Since the temperature will not change, the respective internal energies U A and U B will not change either. Thus, since N A and N B do not change (!), the only variables are the volumes. Thus, for A, the change in entropy will be N A k B ln V/V A , and for B will be N B k B ln V/V B . However, since the pressures and temperatures were originally equal, the equation of state gives P/k B T = N A /V A = N B /V B = N/V , so that the total change of entropy is S = k B N A ln [ N/N A ] + k B N B ln [ N/N B ] as required. (b) Schroeder, 2.38. Without using the S-T expression, show that the entropy change corre- sponding to mixing any two substances (think of two Einstein solids, for example) must be k B ln [ N ! /N A ! N B !]. Show that this is the same expression. N objects can be arranged in N ! ways. But if N A of them are identical, and of type A, with N B of type B, the total number of distinguishable ways is N ! /N A ! N B ! . Originally, with the N A objects and the N B objects separated out, there is only one way to arrange them! Thus, the change in entropy is S = k B ln [ N ! /N A ! N B !]- ln (1) which, using Stirlings approximation, is S = k B [( N A + N B ) ln N- N A ln N A- N B ln N B giving the same result as before. 1 (c) Use a thermodynamic argument, as follows: imagine that the two gases, already mixed, at volume V , are separated by a process involving two semi-permeable membranes....
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Solutions4 - PHYS 333 Winter 2009 Assignment 4 1. Entropy...

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