Monday, December 8, 2008

Physics 9B Practice Final Problem #5 Again

Two volumes A and B of ideal gas with NA and NB molecules respectively are separated by a piston that is free to move while the total volume V = VA + VB 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
ωA = aVANA
ωB = bVBNB
with a and b independent of VA and VB . After the piston comes to equilibrium, what is the pressure in the gas? Express it in terms of k, V, NA, NB, 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

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.
  • SA = k ln (aVANA)
  • SA = k NA ln(VA) + k ln(a)
    • Repeat this for SB, but they both need to be in terms of one variable V, using the relation:
    • V = VA + VB
  • Stotal = S(VA) + S(V−VA)
  • ∂S⁄∂VA = ∂S(VA)⁄∂VA + ∂S(V−VA)⁄∂VA
    • Realize that since V is constant, it's derivative is 0 and the second part can be rewritten.
  • ∂S⁄∂VA = ∂S(VA)⁄∂VA − ∂S(VB)⁄∂VA
  • ∂S⁄∂VA = k NA⁄VA − k NB⁄VB = 0
    • And to maximize entropy, this is all equal to 0.
  • k NA⁄VA = k NB⁄VB
    • At the same time, note that ∂S(VA)⁄∂VA = ∂S(VB)⁄∂VA
      • This is important because ∂S⁄∂V|U = P⁄T
      • PA⁄T = PB⁄T
    • Replug in V = VA + VB and solve for VA
  • VA = V⋅NA⁄[NA + NB]
    • Now we have two equations for ∂S⁄∂V, set them equal:
  • PA⁄T = k NA⁄VA
    • Plug in the solution for VA and solve for P
  • P = k NA T⁄[V⋅NA⁄(NA+NB)]
  • P = k T (NA + NB)⁄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.

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