# Callen problems 2.8-2

1. Jul 26, 2010

### daudaudaudau

I'm reading Callen's thermodynamics book in my vacation. I cant solve this problem.

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 close 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, at a temperature $T_r$, is introduced into the right-hand subvolume.
Find the equilibrium temperature $T_e$ and the mole numbers in each subvolume when the system has come to equilibrium, assuming that $T_l=2T_r=400K$ and that $37B^2=100A^3V_0$.

Answer: $N_{1l}=0.9$

3. The attempt at a solution

First I use the initial conditions to solve for the initial values of $U_1$ and $U_2$. Then I write up the equations
$$\frac{\partial S_1}{\partial U_1}=\frac{\partial S_2}{\partial U_2}$$
$$\frac{\partial S_1}{\partial N_1}=\frac{\partial S_2}{\partial N_2}$$

and use the conditions $U_1+U_2=U$ and the condition on the mole numbers. I think this procedure is correct, but I cannot solve the equations that it produces. Even Mathematica cannot solve them...