Measuring Bond Lengths

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How do we know the bondlength for CO, for example, is 112.83 pm? One technique among many is the application of "rotational spectroscopy." In this method, electromagnetic radiation from the microwave region of the spectrum, is absorbed by a molecule causing the molecule to rotate faster. The wavelength or wavenumber or frequency of the microwave photon absorbed is related to the difference in energy of the molecule before and after absorbtion (as you know). In this case, the energy is rotational energy.

The classical energy of a rotation body depends on how the mass is distributed about the center of rotation. CO would be treated as a barbell shape as pictured below. The "bond" itself would have zero mass, the mass of the carbon atom would be 12.0000 (if the carbon were ¹²C) and that of the oxygen would be 15.9949 (if it were ¹⁶O) mass units. We can abbreviate these masses now as m₁₂ and m₁₆.

Treating the rotational motion rigorously using wave mechanics (done in most junior level physics and chemistry courses) gives some interesting results. First, the energy of rotation is quantized. The quantum number for allowed rotations is given the symbol J and can have values 0, 1, 2, ... The rotational energies then turn out to be

in which R is the bondlength between the mass centers (nuclei) and B is a collection of constants for the molecule in question. We can call B the rotational constant for CO now. Not only is the energy quantized, but the full wave mechanical treatment says there are restrictions on the absorption of photons. One of these restrictions is that if the molecule has a zero dipole moment, the photon won't be absorbed and the molecule will remain in its original rotational state. The other restriction is that a photon will be absorbed only if it increases J by 1.

Carbon monoxide does have a dipole moment. Hence transitions of the type J --> J+1 can occur if the photon exactly provides the missing energy. The absorptions that can take place are indicated by the vertical arrows below on the left. Unlike atoms' and molecules' electrons which are usually in the ground state when the species is in a room temperature environment, the rotational states are so close together that room temperature provides enough energy to populate excited rotational states to begin with.

Since the energy at which each "line" is measured is given by EJ'-EJ, the shortest line or lowest energy transition occurs for J=0 --> J=1; E(photon absorbed) = [1(1+1)B]-[0(0+1)B] = 2B. The next transition occurs at [2(2+1)B]-[1(1+1)B]= 4B. You should be able to convince yourself that the five transitions shown (there are actually more, of course) occur at energies 2B, 4B, 6B, 8B, and 10B. If we were to graph a schematic picture of the expected absorption spectrum, it would look like the one below where the top black line represents no absorption or 100% transmission.

The location of the various absorptions you can now see are evenly spaced They're all 2B apart. This means that measuring the difference between the absorption lines will allow you to determine the rotational constant B and hence the bondlength. Below is shown the actual absorption spectrum for carbon monoxide. The absorption lines are measured as "wavenumbers" in cm^-1, where wavenumber...you do recall!....is the reciprocal of the photon's wavelength or the frequency divided by c. The spacing between the lines can be determined very precisely, is seen to be equal, and actually corresponds to 3.8626 cm^-1.

The rotational constant determined from the spectrum evaluates to be B=3.836 X 10-23 J. From the various constants in B, this leads to a bondlength in CO of 112.8 pm.