# 6A: Rotational Spectroscopy (Worksheet)

• • Contributed by Nancy Levinger
• Professor and University Distinguished Teaching Scholar (Chemistry) at Colorado State University

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

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)?