[Schrödinger]

Hi!

I understand that Microwave spec can be used to calculate bond lengths in molecules. When one records a spectrum of say, carbon monoxide, here's how one goes about calculating the inter-atomic distance:

1. Value of B is measured
2. I (moment of inertia) is calculated as $$ I_{CO} =  \frac {h}{8π^2 Bc} $$
3. I = μr², from which 'r', the interatomic distance is calculated

But every sample containing carbon atoms also contains a certain amount of ¹³C. Does this mean that the value of B (as measured), is a weighted average of sorts?

I ask this question, because as I read in a textbook (Colin and Banwell), the calculation has been carried out assuming the relative atomic weight of Carbon to be 12.000 amu, although the sample whose MW spectrum has been recorded is just plain CO. Nothing's mentioned about its isotopic purity.


Edit: Title changed

[Borek]

What is MW spectroscopy?

Millimeter wave?

What is the resolution - is it high enough to get separate signals for molecules of a different isotope composition?

[Schrödinger]

What is MW spectroscopy?
Millimeter wave?

Sorry about that. I've changed the subject accordingly - microwave spectroscopy.

What is the resolution - is it high enough to get separate signals for molecules of a different isotope composition?

I'm not entirely sure. This is just something mentioned in a textbook - says nothing about resolution. Although, I will check with another source and get back.

[Enthalpy]

This Wiki article (redirected from "microwave spectroscopy") shows distinct lines, to my surprise too:
http://en.wikipedia.org/wiki/Microwave_spectroscopy
The graphic's estimated 10MHz resolution demands a free flight time of 100ns, obtained from 1mbar vacuum - reasonable after all.

10MHz resolution around 10GHz would distinguish between ¹²C and ¹³C frequencies (and ¹⁸O as well), but only if their lines are intense enough, and this might be a limit, especially at low pressure. The lines of ¹³CO must be faint or unvisible, and the book just skips them, that's my explanation attempt.

In Wiki's article, I doubt much about this choice of words
"Those molecules that absorb the energy from this pulse are induced to rotate coherently in phase with the incident radiation."

[mjc123]

In the deep cold and low pressure of space, very narrow lines and high resolution are possible - see the spectrum at 3.1.1. here: http://mackenzie.chem.ox.ac.uk/teaching/Molecular%20Rotational%20Spectroscopy.pdf. Whether you can get that in lab conditions I don't know.
The individual isotopomers of CO have been measured to high precision: http://www.nist.gov/data/PDFfiles/jpcrd47.pdf. So non-averaged values are certainly available.

[Corribus]

@Schrodinger

Rovibrational spectra are more commonly measured for simple molecules to accomplish this (because you get more pieces of information for a single experiment) and it is very easy to measure all the rotational transitions, and even discriminate isotopomers, with basic instrumentation. HCl/DCl is a common undergraduate laboratory experiment.  Pure rotational spectra are not much different, though.

Note that discriminating isotopomers depends not only on spectral resolution but also the relative natural abundances of the isotopes involved.

Modern instruments have a spectral resolution on the order of ~0.001 cm^-1. The spectral bandwidth of a rotational transition is on the order of ~0.01 wavenumber. No problem there - you are more likely to be limited by spectral bandwidth overlapping than instrumental shortcomings.

Turning to CO:

Based on natural abundances (NA) of the various nuclei, only three permutations of CO really need be considered. ¹²C-¹⁶O (NA ~ 98.65%), ¹²C-¹⁸O (NA ~ 0.2%), and ¹³C-¹⁶O (NA ~ 1.1%). All remaining combinations have NA values below 0.05%.

The rotational constant B of ¹²C-¹⁶O can be easily estimated by calculating the moment of inertia from molecular weights and bond lengths to be ~ 8.7247 amu Ų. The analogous values for ¹²C-¹⁸O and ¹³C-¹⁷O are 9.1612 and 9.1260. From these values, the respective rotational constants are ~1.932 cm^-1, ~ 1.8404 cm^-1 and ~1.847 cm^-1, respectively. Take as an example the E(J=7) to E(J=8) rotational transition, which occurs at a value of 14B. For ¹²C-¹⁶O, ¹²C-¹⁸O, and ¹³C-¹⁶O the transition energies are therefore predicted to be ~27.050 cm^-1, ~25.762 cm^-1, ~25.858 cm^-1, respectively. Given the bandwidths of these transitions and even allowing for a much poorer instrument resolution, there should be no trouble distinguishing the intense transitions of ¹²C-¹⁶O from the other to isotopomers. However, 25.762 and 25.858 cm^-1 are <0.01 cm^-1 separated, on the order of the peak widths. Coupled with the fact that they are expected to be weak, owing to low abundance, it is predictable that it will be hard to resolve these peaks.

This is exactly what you see in the pure rotational spectrum of CO, shown below from Spectra of Atoms and Molecules by Bernath (Figure 6.15, p 172). Note the little peaks in between the big peaks that correspond to the overlapped transitions from low NA isotopomers. You might be able to convince yourself by their symmetry that they are the overlap of two peaks. But they are so weak that spectral deconvolution would be difficult. Resolving these peaks may be possible by making other adjustments to the experiment.

So, yes you can resolve isotopomers using even basic pure rotational spectroscopy (and rovibrational spectroscopy) provided their reduced masses are sufficiently different from each other.

[Enthalpy]

Nice explanations and figure, thanks Corribus!
How would microwaves compare with this infrared spectrum? Are the absorption peaks so intense?

Yes mjc123, and radioastronomers let distant objects speak in great detail. One fellow student (during the stone age...) made his master thesis on a part of a receiver at 300GHz - back then, 40GHz was the limit for ultra-specialists, no component existed commercially, and it would still be puzzling today.

[Corribus]

Nice explanations and figure, thanks Corribus!
How would microwaves compare with this infrared spectrum? Are the absorption peaks so intense?

This is a pure rotational spectrum, not an IR (vibrational) spectrum. (Yeah, it is identified as far-infrared, but that's just because it's where the rotational states of CO show up; the far infrared is adjacent to the microwave region).

[Schrödinger]

Quite a helpful discussion this has turned out to be! So let me summarize what I've understood so far regarding isotopomers, and contribute a bit to how to increase resolution.

In order to differentiate between isotopomers, we obviously need better resolution, which is obtained via:
1. Low pressure (which reduces collisional broadening)
2. Low temperature (which presumably forces the molecule to occupy lower energy levels, and hence prevents peak broadening due to this - especially when you have an instrument with moderately good resolution)
3. Reduction in Doppler broadening - This was something introduced to me by a colleague of mine who has worked with jet spectroscopy. To reduce broadening due to Doppler effect, the radiation source, the jet stream of the sample and the detector are maintained at right angles to each other. This way, apparently, all the incoming molecules 'observe' the same frequency incident on them

All of the above having been said, it is also necessary, as Corribus and Enthalpy pointed out, to take into consideration the natural abundance of the isotopes, so that you have respectable intensities.

[Corribus]

Low temperature reduces broadening for a number of reasons, one of which for a gas phase experiment is a reduction in Doppler broadening. I haven't heard of heard of the right angle solution. I suppose that would work, but most conventional rotational spectral experiments don't use a "jet stream" as far as I remember - the gas is in a sealed cuvette and molecular motion is random.