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Free energy of mixing in isothermal free expansion

Page history last edited by Joe Redish 6 years, 1 month ago

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 2times 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.

 

 

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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