# 6A: Rotational Spectroscopy (Worksheet)

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Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

*The rigid rotator model is used to interpret rotational spectra of diatomic molecules. This groupwork exercise aims to help you connect the rigid rotator model to rotational spectroscopy.*

## Q1: Absolute Energies

The energy for the rigid rotator is given by \(E_J=\dfrac{\hbar^2}{2I}J(J+1)\). What is \(J\) in this expression? What values can \(J\) have?

What is the lowest value for \(J\)? What is the energy, \(E_{J}\), for this level?

What is the next lowest value for \(J\)? What is the energy for this level?

What is the zero point energy for a rigid rotor?

## Q2: Energy Differences

In the laboratory, we cannot measure absolute energy levels. Instead, we measure transitions between energy levels. What is the general expression for the energy associated with a transition from rotational from a particular energy level, \(E_J\), to the next higher energy level, \(E_{J+1}\)?

It is much more common to refer to the frequency of a rotational spectroscopic transition that to use energy. If the energy of a transition is given by \(\Delta E=h\nu\), modify your general expression for the energy associated with a rotational transition that you wrote above with an expression that only refers to the frequency.

Usually we use the **rotational constant** \(B=\dfrac{h}{8\pi^2 I}\) to describe rotational transitions and spectroscopy. With \(I= \mu r^2\) as the moment of inertia. Rewrite your expression for the frequency of a rotational spectroscopic transition using \(B\).

Generally, spectral transitions for rotations are only allowed when \(\Delta J=\pm 1\). Use the diagram below to draw where you would expect to see spectral lines in a rotational spectrum of a generic diatomic molecule. Label your axes with *units *and *values*.

How do the rotational spectra for HCl and HBr to differ? How do spectra of \(\ce{HCl}\) and \(\ce{DCl}\) differ?

## Q3: Degeneracy of Eigenstates

The eigenstates of a rigid rotor are **degenerate **(i.e., multiple eigenstates correspond to the same energy) with a set of \(2J + 1\) for each energy of \(E_J\). How many different transitions from one specific eigenstate to another coexist in the lowest energy line in a microwave spectrum (i.e., the \(J=0 \rightarrow J=1\) line) if there were no **selection rules** limiting the options.

What about the \(J=1 \rightarrow J=2\) line (again assuming no selection rules pertain)?