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N particles, Partition Function and finding U and Cv

  1. Mar 18, 2009 #1

    TFM

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

    Consider a system of N identical particles. Each particle has two energy levels: a ground
    state with energy 0, and an upper level with energy [tex] epsilon [/tex]. The upper level is four-fold degenerate (i.e., there are four excited states with the same energy [tex] epsilon [/tex]).

    (a) Write down the partition function for a single particle.

    (b) Find an expression for the internal energy of the system of N particles.

    (c) Calculate the heat capacity at constant volume of this system, and sketch a graph to
    show its temperature dependence.

    (d) Find an expression for the Helmholtz free energy of the system.

    (e) Find an expression for the entropy of the system, as a function of temperature. Verify
    that the entropy goes to zero in the limit T --> 0. What is the entropy in the limit
    T --> infinity? How many microstates are accessible in the high-temperature limit?

    2. Relevant equations

    [tex] z_1 = z_{int} =\sum{e^{E_{int}(s)}/k_BT} [/tex]


    3. The attempt at a solution

    Okay for, a), I have used:

    [tex] z_1 = z_{int} =\sum{e^{E_{int}(s)}/k_BT} [/tex]

    this has given me:

    [tex] 1+4(e^{\epsilon/k_BT}) [/tex]

    now b)

    I have used:

    [tex] z_{total} = \frac{1/N!}(1+4(e^{\epsilon/k_BT}))^N [/tex]

    and:

    [tex] U = \frac{\partial}{\partial \beta}ln z [/tex]

    [tex] \beta = \frac{1}{k_BT} [/tex]

    and I have found ln z to be:
    [tex] -2(ln N!) +N ln 4 + N - \beta \epsilon [/tex]

    thus U equal the beta derivative, Thus I have found :

    [tex] U = frac{\partial}{\partial \beta} = --\epsilon = \epsilon [\tex]

    however this doesn't fit into the next question, find Cv, which needs the formula:

    [tex] C_V = \frac{\partial U}{\partial T} [/tex] sice this would make Cv 0, thus meaning I can't plot a graph.

    Any ideas where I have gone wrong?

    Many Thanks,

    TFM
     
  2. jcsd
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