Is Carbon Monoxide a Rigid Rotator?

Introduction

The excited states of the diatomic molecule carbon monoxide (CO)  can be observed by passing a beam of light through a gas cell containing CO. The absorption spectrum of the transmitted light is displayed in the figure below and can be found on the course website containing this laboratory.  The CO molecule can both vibrate and rotate so its energy spectrum should consist of two components.  If the rotational energy states of this diatomic molecule can be represented as those of a rigid rotator then the energy spectrum should be described by 

                                                 (Subscript[E, ℓ]) = ((ℏ^2)/(2ℐ)) ℓ(ℓ+1)

where ℐ is the moment of inertia of the rotating molecule and ℓ is the angular momentum quantum number. The vibrational part of the energy is

                                                (Subscript[E, n]) = (n + (1/2))(Subscript[ℏω, 0])

where (Subscript[ℏω, 0]) represents one vibrational quantum of energy. We found previously that the separation between vibrational states is (Subscript[ℏω, 0]) = 0.25 ± 0.05 eV by analyzing the energy-loss spectrum of an electron-CO colliding beam experiment.  For reasons that we will develop later the allowed transitions must satisfy the selection rule that  Δℓ =± 1.

Figure 1. Absorption spectrum for Carbon Monoxide.

Necessary Commands

The sub-sections here contain commands that you might find useful in analyzing the spectrum of the C-O molecule.

2.1 Enter raw data in a table.

There is listing of the energies of all the peaks in the spectrum in Figure 1 on the course website.

rawdata = { 0.72, 0.93, 1.13, 1.25}
ListPlot[rawdata, PlotStyle->PointSize[0.02],Frame->True,FrameLabel->{"x","y","Title"}]

2.2 Create a table of peak energy separations.

In the Analysis section we will develop a prediction for the energy spacing between adjacent peaks in the C-O spectrum. For now we want to measure those spacings. However, we must be careful. Notice on the C-O spectrum there are two 'families' of peaks. Create two tables of energy differences, one for each family.

deltaE1=Table[Abs[(rawdata[[n+1]]-rawdata[[n]])],{n,1,Length[rawdata]-1}]

2.3 Get means and standard deviations.

Now generate some statistics. The set of commands in this section join the energy-difference tables of the two families together and calculates the mean and standard deviation. The first command is required to merge the two sets of data for the two families of peaks. The other commands calculate the mean and standard deviation of the merged data set.

deltaE = Join[deltaE1,deltaE1]
Mean[deltaE]
StandardDeviation[deltaE]

Analysis

We hypothesize the transitions measured below are ones where the CO molecule goes from the initial state with quantum numbers (Subscript[n, 0]) = n and (Subscript[ℓ, 0]) = ℓ, to some new state where (Subscript[n, 1])= (Subscript[n, 0])+1 and (Subscript[ℓ, 1]) = ℓ+1 or where (Subscript[n, 1]) = (Subscript[n, 0])+1 and  (Subscript[ℓ, 1]) = ℓ-1.  Consider the questions below.

1. What would the photon absorption spectrum look like if the molecule did not rotate?  

2. For the two different molecular transitions mentioned above, what are the possible energies of the absorbed photon starting from the initial (Subscript[n, 0]) and (Subscript[ℓ, 0])? Include vibrational and rotational effects in your calculation. What pattern does this model predict for the absorption spectrum?  Hint: Try calculating the difference in energy between the initial and final states of the CO molecule for the two types of transitions. This difference in energy between initial and final states is the energy of the absorbed photon.

3. What does this model predict for the difference in energy between adjacent peaks in the photon absorption spectrum? In other words, what is the energy difference between two photons that were absorbed by two, different CO molecules and excited those molecules into adjacent rotational states? Is there qualitative agreement between this prediction and the observed data?  

4. How would you test the agreement between the theory and the experimental data? How good is the rigid rotator hypothesis at reproducing this data?   Be quantitative in your answer and consider the two types of transitions (ℓ->ℓ±1) separately. 

5. What value of the moment of inertia, ℐ, do you find for the CO  molecule? What equilibrium separation between the carbon and oxygen atoms does this correspond to? To answer this last question start with the moment of inertia in the center of mass (CM) of the CO molecule

                                                                           ℐ=(Subscript[m, C])((Subscript[r, C])^2) + (Subscript[m, O])((Subscript[r, O])^2)

where (Subscript[m, C]) and (Subscript[m, O]) are the carbon and oxygen masses (we have treated them as point particles) and  (OverVector[(Subscript[r, C])]) and  (OverVector[(Subscript[r, O])])  are the position vectors from the CM for carbon and oxygen respectively. Rewrite ℐ  in terms of the C-O separation  r using the definition of the CM (remember we have placed the origin at the CM) and (OverVector[r])= (OverVector[(Subscript[r, C])])  - (OverVector[(Subscript[r, O])]). Is your result reasonable? Why? 

6. If you look closely at the absorption spectrum of CO, it appears to consist of two components with a hole at the center between the two ‘lobes’. Can you explain this hole? Be quantitative. The results of part 2 of this analysis might be useful.