7.3.3.P9
The entropy for a particular macrostate of a weakly interacting gas is given by S = kB lnW, where W is the number of microstates that correspond to that particular macrostate of the gas.
When a thermally isolated gas freely expands from a volume V to a volume 2V, the number of microstates for the equilibrium state after the expansion (W2) is 2N times larger than the it was before the expansion (W1), where N is the number of gas molecules: W2 = 2NW1.
Now let’s consider the mixing scenario shown in the figure. Assume we have one mole each of two different weakly interacting gases on either side of a valve. Both gases have the same temperature.
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A. If W0 is the total number of microstates for 1 mole of gas confined to a volume V, what is the initial number of microstates for the system shown in the picture? Explain why you think so.
a. Winitial = W0
b. Winitial = 2W0
c. Winitial = W02
d. Winitial = ½ W0
B. How does the total number of microstates after the mixing, Wfinal, compare to the total number of microstates in the separated state, Winitial?
C. What is the total entropy change ∆Smixing for the process? Explain your reasoning.
D. What is the total free energy change ∆Gmixing for the process, assuming it happens at room tem-perature? Explain your reasoning.
E. Provide a qualitative explanation for the sign of ∆Gmixing.
Ben Geller and Joe Redish 3/1/13
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