What Does the First Entropy Equation for a Permeable Membrane Indicate?

[LCSphysicist]

Homework Statement

All below

Relevant Equations

All below

Actually i am trying to see what the first equation to the entropy means, maybe N1 remets to the part 1 (the left suppose) of the system? (or the molecules type 1?)

I am not sure about the equations i will do below, probably it will be wrong, anyway.

∂S/∂U1 = 1/T1 = 3NR/2U1
Okay, it will give us U1+U2 = U, this will give us the initial energy.

Now i am not sure what to do now.
Derive with respect to N1? How to separate N1(2) + N2(2)?

 

[Chestermiller]

The trick to this problem is to first show that the starting equation describes an ideal gas mixture of two mono-atomic gases, in terms of the internal energy, the ideal gas law, and the chemical potentials of the two species.

 

[TSny]

Actually i am trying to see what the first equation to the entropy means, maybe N1 remets to the part 1 (the left suppose) of the system? (or the molecules type 1?)

Callen's notation can be confusing here. When "1" and "2" are used as subscripts, they refer to the type of molecule. When "1" and "2" appear in parentheses in superscripts, they refer to the left or right sides of the container. So, $N_2^{(1)}$ is the number of moles of type-2 molecules in the left side of the container.

∂S/∂U1 = 1/T1 = 3NR/2U1
Okay, it will give us U1+U2 = U, this will give us the initial energy.

Yes

Now i am not sure what to do now.
Derive with respect to N1?

Yes, $\large \frac{\partial S^{(1)}}{\partial N_1^{(1)}}$ will give you $-\large \frac{\mu_1^{(1)}}{T^{(1)}}$ for the left side of the container. Similarly for the right side.

How to separate N1(2) + N2(2)?

I'm not sure what you are asking here. Note that $N_2^{(1)}$ and $N_2^{(2)}$ are constants.

When taking the derivative of $S^{(1)}$ with respect to $N_1^{(1)}$, keep in mind that $N^{(1)} = N_1^{(1)} + N_2^{(1)}$.

 

[rude man]

Too bad the two compartments are initially at different temperatures because otherwise this would be a traditional osmosis problem.