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Confused in stat. mech.-thermo., inconsistant result

  1. Oct 8, 2014 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    Hello guys,
    Long problem but I'll drastically shorten it. I've a diatomic ideal gas of N molecules (the 2 atoms are distinct). I must calculate the internal energy and the specific heat for high temperatures.
    I've got the solution but if I solve it my way in the last step, I get a different result.

    2. Relevant equations
    Partition function of one molecule: ##Z_1\approx \frac{T}{\omicron}##. This expression is to be found in the solution for this exercise.
    Thus, for N molecules, i.e. the whole system: ##Z_N\approx \left ( \frac{T}{\omicron} \right ) ^N## (according to me). These approximations are valid for ##T>>\omicron##.
    The solution states that from ##Z_1##, we can calculate the mean energy per molecule as ##\overline{\varepsilon}=-\frac{\partial Z_1 }{\partial \beta}=kT##.
    3. The attempt at a solution
    My idea was to get the Helmholz free energy and then, from it, getting the internal energy.
    I got ##A(\beta,N) \approx -\frac{N}{\beta} \ln \left ( \frac{1}{k\beta \omicron} \right )##.
    Since ##U=\left ( \frac{\partial A}{\partial \beta} \right ) _{V,N}##, I got ##U \approx\frac{N}{\beta ^2}[1-\ln (k\beta \omicron)]##. As you can see:
    1)My result for U is not extensive (so I'm fried).
    2)Dividing my expression for U by N, I don't reach the expression for the average energy per molecule the solution provides.

    I don't understand what I did wrong.
     
  2. jcsd
  3. Oct 10, 2014 #2

    fluidistic

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    Gold Member

    Nevermind guys, I found out the mistake. I had to multiply A by beta before taking the derivative with respect to beta, in order to get the internal energy.
     
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