The above derivation for the translations of a gas also holds for a polyatomic gas. Just make sure you use the total mass of the molecule. A monatomic gas has three degrees of freedom per molecule:

- movement in x-direction
- movement in y-direction
- movement in z-direction

A polyatomic gas has other levels that you can 'stuff' energy into. The molecule can rotate and vibrate and if enough energy is available you could also excite the electrons involved in the σ and π- bonds. In reasonable approximation the partition function of the molecule would become:

\[q_{tot} = q_{trans}q_{vib}q_{rot}q_{elect}\]

We will only scratch the surface of the additional degrees of freedom and their partition functions.

## Electronic

At room temperature the system is usually in its ground electronic state. This means that the **electronic partition function** q_{electronic} = 1 . Usually we do not have to worry about these degrees of freedom. If we do there are usually just a few levels to worry about. This includes their **degeneracy g**. If there is a single energy level at a certain energy g=1, if there are two g=2 etc. We must simply multiply the Boltzmann factor with this number.

If there is more than one level to worry about we could follow the same procedure as we did for the translational states:

- define the energy states and their degeneracies
- compose the partition function q for the molecule and Q for the gas
- use the (β or T) derivative of lnQ to determine <E>
- use the T derivative of U ≈ <E> to find the contribution to the heat capacity.