- #1
fluidistic
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Homework Statement
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.
Homework 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##.
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.