I'm reading Callen's thermodynamics book in my vacation. I cant solve this problem.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

A two-component gaseous system has a fundamental equation of the form

[tex]

S=AU^{1/3}V^{1/3}N^{1/3}+\frac{BN_1N_2}{N}

[/tex]

where [itex]N=N_1+N_2[/itex] and A and B are positive constants. A close cylinder of total volume [itex]2V_0[/itex] 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 [itex]T_l[/itex], is introduced in the left-hand subvolume, and a mixture of 1/2 mole of each component, at a temperature [itex]T_r[/itex], is introduced into the right-hand subvolume.

Find the equilibrium temperature [itex]T_e[/itex] and the mole numbers in each subvolume when the system has come to equilibrium, assuming that [itex]T_l=2T_r=400K[/itex] and that [itex]37B^2=100A^3V_0[/itex].

Answer: [itex]N_{1l}=0.9[/itex]

3. The attempt at a solution

First I use the initial conditions to solve for the initial values of [itex]U_1[/itex] and [itex]U_2[/itex]. Then I write up the equations

[tex]

\frac{\partial S_1}{\partial U_1}=\frac{\partial S_2}{\partial U_2}

[/tex]

[tex]

\frac{\partial S_1}{\partial N_1}=\frac{\partial S_2}{\partial N_2}

[/tex]

and use the conditions [itex]U_1+U_2=U[/itex] 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...

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Callen problems 2.8-2

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**