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Homework Help: Callen problems 2.8-2

  1. Jul 26, 2010 #1
    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
    [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...
     
  2. jcsd
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