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Given an ensemble of $N$ diatomic molecules, we know that the rotational partition function is given by $$Z_r = z_r^N$$ where $$z_r = \sum_{l} (2l+1)e^{-Kl(l+1)}$$ where $K = \beta \hbar ^2 /2I$.

I want to apply this to calculate the $C_v$ for an ensemble of $H_2$ (hydrogen) molecules in the low $T$ limit and the high $T$ limit for a certain quantum mechanical simulation. Before I run the simulation, I want to have an idea of how the results should look like.

For a general canonical ensemble, $C_v = \left( \partial E / \partial T \right)_V$, where $E = -(\partial Z / \partial \beta$). How would I evaluate the $C_v$ for a system of one mole of para-$H$ (singlet state) and one mole of ortho-$H$ (triplet state) in the high temperature limit and low temperature limit? How would I find the partition function for an HD molecule?

After doing some research, I see that $$Z_{para}= \sum_{l \,\text{even}} (2l+1)e^{-\beta \hbar ^2 l(l+1)/2I}$$ and $$Z_{ortho}= \sum_{l \,\text{even}} (2l+1)e^{-\beta \hbar ^2 l(l+1)/2I}$$

I know that because of the different multiplicities of these two states at equilibrium, the rotational partition function at low $T$ looks like

$$z_r = \frac{1}{4} \sum_{l \,\text{even}} (2l+1)e^{-\beta \hbar ^2 l(l+1)/2I} + \frac{3}{4} \sum_{l \, \text{odd}} (2l+1)e^{-\beta \hbar ^2 l(l+1)/2I} $$

After seeing this document, I can see how $C_v$ looks like for ortho, para-$H_2$, and their equilibrium. How do you find the partition function - and then $C_v$ for $HD$?

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  • $\begingroup$ Shouldn't the expression be $\langle E\rangle=-(\partial \ln Z/\partial \beta)$? $\endgroup$ – Anyon Sep 23 at 16:17
  • $\begingroup$ @Anyon, yes, I can make the change $\endgroup$ – megamence Sep 23 at 16:18
  • $\begingroup$ you can't simulate a mole of anything, using anything at the quantum or atomistic level. you can only simulate, at the quantum level, perhaps hundreds of atoms. But, that will take a very long time. Further, is not that expression for an ideal gas - which by definition is a single molecule. You cannot calculate the heat capacity of a molecules that interact accurately with the ideal gas assumption. Further, if you condition it such that they do not interact, and behave ideally... why not just simulate a single molecule? $\endgroup$ – Charlie Crown Sep 24 at 20:01
  • $\begingroup$ The real definition of heat capacity for bulk phases involved the fluctuation in energy. i.e., you will need to calculate ensemble averages involving $<E^2>$ and $<E>^2$. See any statistical mechanics reference textbook $\endgroup$ – Charlie Crown Sep 24 at 20:04