## Question: Rotation of diatomic molecules ([2] problem 3.6)

In our first look at the ideal gas we considered only the translational energy of the particles. But molecules can rotate, with kinetic energy. The rotation motion is quantized; and the energy levels of a diatomic molecule are of the form

where is any positive integer including zero: . The multiplicity of each rotation level is .

### a

Find the partition function for the rotational states of one molecule. Remember that is a sum over all states, not over all levels — this makes a difference.

### b

Evaluate approximately for , by converting the sum to an integral.

### c

Do the same for , by truncating the sum after the second term.

### d

Give expressions for the energy and the heat capacity , as functions of , in both limits. Observe that the rotational contribution to the heat capacity of a diatomic molecule approaches 1 (or, in conventional units, ) when .

### e

Sketch the behavior of and , showing the limiting behaviors for and .

## Answer

### a. Partition function

To understand the reference to multiplicity recall (section 4.13 [1]) that the rotational Hamiltonian was of the form

where the eigenvectors satisfied

\begin{subequations}

\end{subequations}

and , where is a positive integer. We see that is of the form

and our partition function is

We have no dependence on in the sum, and just have to sum terms like fig 1, and are able to sum over trivially, which is where the multiplicity comes from.

To get a feel for how many terms are significant in these sums, we refer to the plot of fig 2. We plot the partition function itself in, truncation at terms in fig 3.

### b. Evaluate partition function for large temperatures

If , so that , all our exponentials are close to unity. Employing an integral approximation of the partition function, we can somewhat miraculously integrate this directly

### c. Evaluate partition function for small temperatures

When , so that , all our exponentials are increasingly close to zero as increases. Dropping all the second and higher order terms we have

### d. Energy and heat capacity

In the large domain (small temperatures) we have

The specific heat in this domain is

For the small (large temperatures) case we have

The heat capacity in this large temperature region is

which is unity as described in the problem.

### e. Sketch

The energy and heat capacities are roughly sketched in fig 4.

It’s somewhat odd seeming that we have a zero point energy at zero temperature. Plotting the energy (truncating the sums to 30 terms) in fig 5, we don’t see such a zero point energy.

That plotted energy is as follows, computed without first dropping any terms of the partition function

To avoid the zero point energy, we have to use this and not the truncated partition function to do the integral approximation. Doing that calculation (which isn’t as convenient, so I cheated and used Mathematica). We obtain

This approximation, which has taken the sums to infinity, is plotted in fig 6.

From eq. 1.0.12, we can take one more derivative to calculate the exact specific heat

This is plotted to 30 terms in fig 7.

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

[2] C. Kittel and H. Kroemer. *Thermal physics*. WH Freeman, 1980.