Callen Thermodynamics 2.8-2 matter flow equilibrium

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1. Feb 16, 2015

It's me

1. The problem statement, all variables and given/known data
A two component gaseous system has a fundamental equation of the form
$$S=AU^{1/3} V^{1/3} N^{1/3} + \frac{BN_1N_2}{N}$$ where $$N=N_1+N_2$$
and A and B are positive constants. A closed cylinder of total volume 2V_0 is separated into two equal subvolumes by a rigid diathermal partition permeable only to the first component. One mole of the first component at a temperature T_l is introduced in the left-hand subvolume and a mixture of 1/2 mole of each component is introduced in the right-hand subvolume at a temperature T_r.

Find the equilibrium temperature T_e and the mole numbers in each subvolume when the system has come to equilibrium, assuming that $$T_r=2T_l=400K$$ and that $$37B^2=100A^3V_0$$. Neglect the heat capacity of the walls of the container.

2. Relevant equations
$$\frac{1}{T}=(\frac{\partial S}{\partial U})$$
$$\frac{P}{T}=(\frac{\partial S}{\partial V})$$
$$\frac{-\mu_1}{T}=(\frac{\partial S}{\partial N_1})$$

3. The attempt at a solution
I am trying to use the definitions of the equations of state and conservation of internal energy and chemical potential in order to find relationships that will allow me to solve for N_1l, N_1r, and T_e. This is how far I've gone:

$$\frac{1}{T}=(\frac{\partial S}{\partial U})=\frac{1}{3}AU^{-2/3}V^{1/3}N^{1/3}$$
$$\Rightarrow U=(\frac{ATV^{1/3}N^{1/3}}{3})^{3/2}$$
$$\frac{-\mu_{1l}}{T}=(\frac{\partial S}{\partial V})=\frac{1}{3}AU^{1/3}V^{1/3}(N_{1l}+N_{2l})^{-2/3}+\frac{(N_{1l}+N_{2l})BN_{2l}-BN_{1l}N_{2l}}{(N_{1l}+N_{2l})^2}=\frac{1}{3}AU^{1/3}V^{1/3}N_{1l}^{-2/3}$$
$$\frac{-\mu_{1r}}{T}=(\frac{\partial S}{\partial V})=\frac{1}{3}AU^{1/3}V^{1/3}(N_{1r}+N_{2r})^{-2/3}+\frac{(N_{1r}+N_{2r})BN_{2r}-BN_{1r}N_{2r}}{(N_{1r}+N_{2r})^2}$$
$$\mu_{1l}=\mu_{1r}$$

I really don't know how to proceed from here and how to use the relation given at the end of the problem. Any help would be greatly appreciated.

2. Feb 17, 2015

TSny

Hello, and welcome to PF!

So far, your equations look good to me. The last term on the right side of your next to last equation will simplify somewhat.

Looks like you're in for some fairly complicated algebra to solve for the equilibrium temperature $T_e$ and the final mole numbers $N_{1l}$ and $N_{1r}$. I think you will need a few more relations. As you noted, $\mu_{1l} = \mu_{1r}$ at equilibrium. What other quantity must be the same for the left and right compartments at equilbrium?

There are also a couple of conserved quantities that don't change during the approach to equilibrium. This should give you two more equations to work with.

3. Feb 17, 2015

It's me

At equilibrium $$U_l=U_r$$, right? And also $$N_{1l} + N_{1r}= 3/2$$ I haven't been able to use these relations though, I still get more variables than equations. Am I missing something?

4. Feb 17, 2015

TSny

No, the energy on the left does not need to equal the energy on the right. What can you say about the final total energy of the system?

This equation is correct.

Can you say anything about the temperatures of the left and right compartments at equilibrium? Does that give you another relation?

5. Feb 17, 2015

It's me

Oh right, I meant to say $$U_i=U_f=U_l+U_r$$. $$T_l=T_r=T_e$$

6. Feb 17, 2015

TSny

Yes. That looks good. Conservation of total energy should lead to a relation between Te, the initial temperatures, and the initial final number of moles on the left and right.

Last edited: Feb 18, 2015
7. Feb 6, 2016

Othin

Am I missing something? Doesn't the last equation contradict the imposition that $T_r=2T_l ?$

8. Feb 6, 2016

Othin

Wouldn't I need to know something about their heat capacity to find this relation?

9. Feb 6, 2016

TSny

In post #5, $T_r$ and $T_l$ refer to the final equilibrium temperatures of the left and right compartments rather than the initial temperatures.

10. Feb 6, 2016

TSny

No, I don't believe you do. There are 3 unknowns: $T_e$, and the final values of $N_{1l}$, and $N_{1r}$.

And you have three relations:
(1) $T_{l, final} = T_{r,final}$
(2) $\mu_{1l,final} = \mu_{1r,final}$
(3) Conservation of energy

11. Feb 7, 2016

Othin

Yes, after a little work I came to a system which is theoretically solvable, though far too big.

12. Feb 7, 2016

TSny

From post #1
Just for the record, there is a typographical error above where the first $(\frac{\partial S}{\partial V})$ should be $(\frac{\partial S}{\partial N_{1l}})$ and the second $(\frac{\partial S}{\partial V})$ should be $(\frac{\partial S}{\partial N_{1r}})$

13. Feb 7, 2016

TSny

I used Mathematica to get a numerical solution to one of the equations.