Why is there a spike in the heat capacity of a diatomic gas, at around the rotational temperature of the molecule?

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Solution 1

The bump can be observed through an explicit calculation.

If $\it l$ is the angular momentum quantum number of a molecule, then the rotational energy levels are

$$ \varepsilon_\text{rot} = \frac{\hbar^2}{2I}\it l(\it l + 1) = \frac{k\theta_\text{r}}{2} \it l(\it l + 1) \,, \quad \textrm{ where }\quad \theta_\text{r} \equiv \frac{\hbar^2}{Ik} \,,$$

and $I$ is the moment of inertia of the molecule.

Since each $\it l$ is $(2\it l + 1)$-fold degenerate, the partition function over each rotational mode reads

$$ Z_\text{rot} = \sum_{l=0}^\infty (2\it l+1)\exp\left( - \frac{\theta_\text{r}}{2T} \it l(\it l + 1) \right) = \begin{cases} 1 + 3 e^{-\theta_r/T} + 5e^{-3\theta_r/T} + \mathcal O\left( e^{-6\theta_r/T} \right) &\text{for } T \ll \theta_r \,, \\ 2\frac{T}{\theta_r} + \frac13 + \frac1{30}\frac{\theta_r}{T} + \mathcal O\left( \left( \frac{\theta_r}{T} \right)^2 \right) &\text{for } T \gg \theta_r \,. \end{cases} $$

Using this, we can calculate the contribution to the internal energy per rotational degree of freedom.

$$ E_\text{rot} = NkT^2\frac{\partial}{\partial T}\ Z_\text{rot} = \begin{cases} 3Nk\ \theta_r\left( e^{-\theta_r/T} - 3e^{-2\theta_r/T} + \dots \right) &\text{for } T \ll \theta_r \,, \\ NkT\left(1 - \frac{\theta_r}{6T} - \frac{1}{180}\left(\frac{\theta_r}{T}\right)^2 + \dots \right) &\text{for } T \gg \theta_r \,. \end{cases} $$

Therefore, the contribution to heat capacity at constant volume by each rotational mode is $$ C_V^\text{rot} = \frac{\partial }{\partial T}E_\text{rot} = Nk \begin{cases} 3 \left(\frac{\theta_r}T\right)^2 e^{-\theta_r/T} \left( 1 - 6 e^{-\theta_r/T} + \dots \right) &\text{for } T \ll \theta_r \,, \\ 1 + \frac{1}{180}\left(\frac{\theta_r}{T}\right)^2 + \dots &\text{for } T \gg \theta_r \,. \end{cases} $$

The above function has a maximum of $1.1\ Nk$ at about the temperature $0.81\ \theta_r/2$. As the temperature is increased way above $\theta_r/2$, it settles down to $1\ Nk$ and we recover the classical flat result.

Solution 2

You asked for a physically intuitive explanation, so here we go. For the full derivation, see Nanashi No Gombe's answer.

I believe this "bump" phenomenon is related to the Skottky anomaly, which is explained nicely for a two state system in this wikipedia article: https://en.wikipedia.org/wiki/Schottky_anomaly. I will begin by explaining the "bump" phenomenon for a two-state system and then generalise to a rotational system.

Here is the heat capacity for a two state system:

Heat capacity for two state system

Here are the energy levels for a two state system:

Energy levels for a two state system

For $k_BT << \Delta$, only the ground state is occupied, and increasing $T$ slightly isn't going to change this, hence $C \rightarrow 0$

For $k_BT >> \Delta$, both states are equally occupied, and increasing $T$ slightly won't make much difference to the average energy, hence $C \rightarrow 0$

In between these two extremes, increasing T will have a dramatic effect on the average energy, since it is now possible to excite transitions from the lower energy state. This is the cause of the large bump in the heat capacity.

Now, let's consider a rotational system. Here are the energy levels of a rotational system:

enter image description here

Since $E_2 - E_1 > E_1 - E_0$, the system looks somewhat like a two state system at low temperature. Therefore, you also get this "bump" behaviour in the heat capacity at low temperature.

Images are from wikipedia and "Concepts in thermal physics" by Blundell.

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JRF
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Updated on July 11, 2021

Comments

  • JRF
    JRF over 2 years

    While studying for my Statistical thermodynamics test, I encountered this graph Graph

    Source: https://www.physicsforums.com/threads/variation-of-specific-heat-with-temperature.399514/

    I know this isn't the best graph you'll ever see, but the "bump" was present on several other graphs as well. So just to clarify my question once more; I'm looking for a physically intuitive explanation of the "bump", which occurs at temperatures, where rotational degrees of freedom become relevant.

    Thank you!

    • dmckee --- ex-moderator kitten
      dmckee --- ex-moderator kitten over 6 years
      I've never seen a figure depicting such a bump before, but then I've seen only a few plots of data and a lot of schematics that were presumably draw to show the plateaus. Do you have an example of a data plot showing such a bump. I can think of a possible reason for such a bump to appear in a practical experiment that does not correspond to some fundamental physics but to difficulties actually running the experiment, but if that was the case I'd expect a similar bump at the transition to including vibrational modes.
    • Admin
      Admin over 6 years
      Where did you get the graph, and a credit/ attribution would do no harm ;)
    • JRF
      JRF over 6 years
      @dmcke We've drawn a graph like that in class as well, but our professor said that he won't go into details about it. There was no bump at the transition to vibrational modes however. Moreover, I think the bump has a theoretical background, as can be seen here: link in the "Heat capacity at low temperature" section.
    • dmckee --- ex-moderator kitten
      dmckee --- ex-moderator kitten over 6 years
      This figure rkt.chem.ox.ac.uk/tutorials/statmech/hydrogen.jpg suggests two things. That it is a real phenomena and that it is related to spin degrees of freedom (which would be why it is associated with the rotational turn-on but not with the vibrational turn-on). No time to follow up now, but it promises to be a very interesting question indeed.
  • Chris
    Chris about 5 years
    That's for $C_V$. The plot is depicting $C_P$. For an ideal gas, $C_P=C_V+R$.