Rotational Partition Functions (Worksheet)

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The partition function for a single atom or molecule can be expressed

$$q(V ) = \sum_i e^{−\epsilon_i/k_BT} \label{Eq1}$$

where $\epsilon _i$ describes the energy of the atom or molecule (e.g., derived from solving the Schrödinger equation for a specific system). For systems that exhibit a degeneracy of eigenstates (for example in rotation or electronic spectroscopy), Equation $\ref{Eq1}$ is modified

$$q(V ) = \sum_i g_i e^{−\epsilon_i/k_BT} \label{Eq2}$$

where $g_i$ is the degeneracy of the i^th eigenstate.

Q1

We can model rotational motion as a rigid rotor quantum problem with the following expression for energies

$$\epsilon_J = \dfrac{h^2 J(J+1)}{2I}$$

where $I$ is the moment of inertia. For a diatomic molecule

$$I = mR_{eq}^2$$

The degeneracy is $g_J = 2J + 1$ for a rigid rotor.

Substitute your the energies and degenerates into Equation $\ref{Eq2}$ for the starting equation of the rotational molecular partition function.

Q2

While the spacing of the energy levels for a rigid rotor are appreciable bigger than for translation, they are are often quite small relative to thermal energy $K_BT$ at ambient temperature. This means that we can approximate the sum derived in Q1 with an integral, just like in the translation partition function. Write this integral and pay attention to the limits of integration.

Q3

Since the differential below can be expanded

$$d[J(J+1)] = d[J^2+J] = (2J+1)dJ$$

the integral derived in Q2 can be expressed in terms of $d[J(J+1)]$ instead of $dJ$. Write this integral.

Q4

The integral in Q3 is an exponential integral with a known solution (e.g., via an integration table). Use that solution to obtain the final form of the rotational partition constant (of a diatomic molecule)

$$q_{rot} = \dfrac{8 \pi I k_B}{h^2} T = \Theta_r T$$

where $\Theta_r$ is the rotational temperature for the molecule (akin to the thermal de Brogle wavelength in the translation partition function).

Q5

What is the rotational partition function of $B_2$ at 300 K? How does this compare to the diatomic molecule's corresponding translational partition function? How do you interpret this comparison (in words)?