Rovibration spectroscopy (Worksheet)

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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. The harmonic oscillator model connects to vibrational spectroscopy. This exercise aims to help you connect rotation-vibration spectroscopy.

The frequency of a transition in the harmonic oscillator is

\[ \tilde{G}(v) = h \tilde{\nu} (v+1/ 2)\]

The frequency of a transition in the rigid rotator is

\[\tilde{F}(J)= 2 \tilde{B} (J +1)\]

For the ¹²C¹⁶O molecule: \(\tilde{\nu} = 2169.8 \;cm^{−1}\) and \(\tilde{B}= 1.93\; cm^{−1}\).

Q1

On the plot below, draw the three lowest vibrational energy levels, that is,

!G0 ( ), !G 1 ( ), and !G (2).

Can you accurately show the position of the lowest rotational levels on the graph above? Draw a line indicating where the J=500 rotational level would appear on the graph.

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Q2

On the graph below, accurately show the lowest three rotational levels. Make sure to label the numbers on the y-axis of the graph.

Figure 13.1 from McQuarrie and Simon is shown below. What features in this figure are inaccurate?

In Figure 13.1 from McQuarrie and Simon, the ninth rotational level is depicted to lie above the first vibrational excited state, v=1. Is this accurate? If not, at what energy should this level lie? Why do McQuarrie and Simon include this figure in their textbook?