Thermodynamics equilibrium with respect to matter flow

[kiyoshi7]

Homework Statement

I don`t know if the image will show so I`m also adding a link to the image of the problem. This problem is a modification my professor made to the one in the link below(*), he changed the rigid diathermic partition into a movable partition. I`m supposed to find the equilibrium temperature and volume. He also mentioned that he isn`t sure that it can be solved, he kind of changed it spontaneously. Also sorry for the math, I don`t know how to format it here in the forums.

https://drive.google.com/open?id=18F64i9f9BeFHGuygQj4hpIK5yQjYzZPl

* taken from: thermodynamics and introduction to thermalstatistics vol. 2, Herbert B. Callen.

Homework Equations

S= AU^1/3V^1/3N^1/3 + (BN₁N₂)/N
N = N₁+N₂
find equilibrium assuming the following
T_r = 2T_l = 400k
37B² = 100A³V₀

The Attempt at a Solution

I know how to solve it when the cylinder is separated by a rigid diathermic permeable partition, but I can`t figure out how deal with the movable partition in this problem. So I`ll describe the solution for the rigid partition.
first find the intensive parameters:
∂S/∂U = 1/T = (1/3)(AU^1/3V^1/3/N^2/3)
∂S/∂N₁ = -u₁/T
then rewrite ∂S/∂U as U in function of temperature:
U = T^3/2(A^3/2V^1/2N^1/2)/(3^3/2)
Total Energy:
U_t= [(A^3/2V^1/2)/(3^3/2)]( N_r^1/2 T_r^3/2 + N_l^1/2 T_l^3/2 )
I imagine that here I`d do the same as I did with N and T ie: (V^1/2_l N^1/2_l T^3/2_l + V^1/2_r N^1/2_r T^3/2_r), But I can't figure out how to solve it after this

 

[Chestermiller]

Homework Statement

I don`t know if the image will show so I`m also adding a link to the image of the problem. This problem is a modification my professor made to the one in the link below(*), he changed the rigid diathermic partition into a movable partition. I`m supposed to find the equilibrium temperature and volume. He also mentioned that he isn`t sure that it can be solved, he kind of changed it spontaneously. Also sorry for the math, I don`t know how to format it here in the forums.

https://drive.google.com/open?id=18F64i9f9BeFHGuygQj4hpIK5yQjYzZPl

* taken from: thermodynamics and introduction to thermalstatistics vol. 2, Herbert B. Callen.

Homework Equations

S= AU^1/3V^1/3N^1/3 + (BN₁N₂)/N
N = N₁+N₂
find equilibrium assuming the following
T_r = 2T_l = 400k
37B² = 100A³V₀

The Attempt at a Solution

I know how to solve it when the cylinder is separated by a rigid diathermic permeable partition, but I can`t figure out how deal with the movable partition in this problem. So I`ll describe the solution for the rigid partition.
first find the intensive parameters:
∂S/∂U = 1/T = (1/3)(AU^1/3V^1/3/N^2/3)
∂S/∂N₁ = -u₁/T
then rewrite ∂S/∂U as U in function of temperature:
U = T^3/2(A^3/2V^1/2N^1/2)/(3^3/2)
Total Energy:
U_t= [(A^3/2V^1/2)/(3^3/2)]( N_r^1/2 T_r^3/2 + N_l^1/2 T_l^3/2 )
I imagine that here I`d do the same as I did with N and T ie: (V^1/2_l N^1/2_l T^3/2_l + V^1/2_r N^1/2_r T^3/2_r), But I can't figure out how to solve it after this

It seems to me the constraints on this problem for the final state are:

The total volume is constant
The pressures in the two chambers are equal
The total internal energy is constant
The temperatures in the two chambers are equal
The chemical potentials of species 1 in the two chambers are equal. The chemical potential can be obtained by taking the partial derivative of U with respect to N1 at constant S and V.