7.3.3.P10
The entropy for a weakly interacting gas of N molecules at a particular temperature T and pressure p is given by S = k_{B} ln W, where W is the number of microstates possible for that particular temperature and pressure.
When a gas at a given temperature freely expands from a volume V to a volume 2V, the number of microstates in equilibrium after the expansion (W_{2}) is 2N times larger than it was before the expansion (W_{1}), where N is the number of gas molecules: W_{2} = 2NW_{1}.
Now let’s consider the mixing scenario shown in the figure. Assume we have one mole of each of two different weakly interacting gases. At the beginning one mole of blue molecules (the bigger dots) is on the left side, and one mole of red molecules (the smaller dots) is on the right side of a valve. The volume on each side of the valve is V.


A. If W_{0} is the total number of microstates for 1 mole of gas confined to a volume V, what is the initial entropy for the whole system?
B. How does the total number of microstates after the mixing, W_{mixed}, relate to the total number of microstates in the separated state, W_{separated}? (Express your answer in terms of the variables given in the problem.)
C. What is the total entropy change ΔS_{mixing} for the process?
D. What is the total free energy change ΔG_{mixing} for the process, assuming it happens at room temperature?
E. Provide a qualitative explanation for the sign of ΔG_{mixing}.
Ben Dreyfus 3/23/16
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