Physics 9B Practice Final Problem #5 Again

Two volumes A and B of ideal gas with N_A and N_B molecules respectively are separated by a piston that is free to move while the total volume V = V_A + V_B is constant. Both gases are in contact with a heat reservoir at a fixed temperature T.
For each volumes of gas A and B, the multiplicity depends on the volume according to
ω_A = aV_A^N_A
ω_B = bV_B^N_B
with a and b independent of V_A and V_B . After the piston comes to equilibrium, what is the pressure in the gas? Express it in terms of k, V, N_A, N_B, T, a, and b. Do the problem using the given information and the ideas of statistical mechanics. There are at least a couple ways to do the problem. If you want to use the equation of state, you must derive it from the information given above and general results on the properties of entropy.
Question so kindly provided by Joe Kiskis

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This is a stat mech problem, so to find the equilibrium, we need to maximize entropy.

Again, maximize entropy.

Third time, for the readers at home: Maximize entropy. S = k ln(ω), so S is maximum when ∂S = 0.

• S_A = k ln (aV_A^N_A)
• S_A = k N_A ln(V_A) + k ln(a)
• Repeat this for S_B, but they both need to be in terms of one variable V, using the relation:
• V = V_A + V_B
• S_total = S(V_A) + S(V−V_A)
• ∂S⁄∂V_A = ∂S(V_A)⁄∂V_A + ∂S(V−V_A)⁄∂V_A
• Realize that since V is constant, it's derivative is 0 and the second part can be rewritten.
• ∂S⁄∂V_A = ∂S(V_A)⁄∂V_A − ∂S(V_B)⁄∂V_A
• ∂S⁄∂V_A = k N_A⁄V_A − k N_B⁄V_B = 0
  
• And to maximize entropy, this is all equal to 0.
• k N_A⁄V_A = k N_B⁄V_B
• At the same time, note that ∂S(V_A)⁄∂V_A = ∂S(V_B)⁄∂V_A
• This is important because ∂S⁄∂V|_U = P⁄T
• P_A⁄T = P_B⁄T
• Replug in V = V_A + V_B and solve for V_A
• V_A = V⋅N_A⁄[N_A + N_B]
• Now we have two equations for ∂S⁄∂V, set them equal:
• P_A⁄T = k N_A⁄V_A
• Plug in the solution for V_A and solve for P
• P = k N_A T⁄[V⋅N_A⁄(N_A+N_B)]
• P = k T (N_A + N_B)⁄V
• Holy crap! That's the answer!!!

Now we were able to come up with that answer from the very beginning, because it's essentially PV=nRT, replacing nR with Nk, which is fortunate since it gives you a goal. Unfortunately, like so many things in life, its not the destination, but the journey that matters.