Two gases are initially separated by a valve, and at time zero they are allowed

Question

Two gases are initially separated by a valve, and at time zero they are allowed to mix. Assume each gas contains ten molecules. Using multiplicity, calculate the entropy for the following scenarios (assume in each case that the final number of molecules on either side of the valve is maintained at 10):

20 percent of gas molecules exchange

Half of the gas molecules exchange

All gas molecules exchange

Repeat A-C, but instead assume that each gas contains one hundred molecules. How does this affect the spontaneity of gas exchange?

Explanation / Answer

Multiplicity = W

Entropy = S = k lnW

k = boltzman constant = 1.38 × 10-23 m2 kg s-2 K-1

a) w= 0.2 * 20 = 4

S= k ln(w) = 1.91x10-23 J/K

b) w= 0.5* 20 = 10

S= k ln(w) = 3.17x10-23 J/K

c) w= 20

S= k ln(w) = 4.12x10-23 J/K

now when each gas 100 molecules ,

A) w= 0.2 * 200 = 40

S= k ln(w) = 5.09x10-23 J/K

B) w= 0.5 * 200 = 100

S= k ln(w) = 6.355x10-23 J/K

C) w= 200

S= k ln(w) = 7.311x10-23 J/K

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