Tag Archives: dimensions

Switching Units

(i) Homunculus sitting at a desk: "Continuously switching from atomic to SI to Plank units devastated my brain." -- The doorbell rings. (ii) "Fresh & Frozen Ltd. Are you the idiot who ordered 9.11E31 kg vanilla ice cream?" -- Homunculus: "Shit, I knew it."

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(c) M.Klammler 2010, Creative Commons Share Alike.

Dimensional Abuse

(i) Homunculus next to a heap of smoking dimensions presenting a report: "I have written this report in atomic units. That means: reduced Planck constant = electric constant = mass of the electron = 1"

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(c) M.Klammler 2010, Creative Commons Share Alike.

0.000000125 are Not a Coincidence

Recently, I had to use some ab initio results for a kinetic computation. I was interested in a transition dipole moment of the molecule but none was given. However, there were some “infrared intensities” given in — crazy enougth — km mol-1. I was rather confused about that because, by all convention, this is no unit of an intensity.

[Klippenstein:1996] gives a conversion factor that claims basically:

A = 1.25E-7 I(in km mol-1) ν2(in cm-1)

where A is the Einstein coefficient for spontaneous emission.

I generally do not trust such formulas. Furthermore I think they are an unnecessary attack on healthy physical thinking. But how to do better? A colleague found a tutorial of Dunbar where he presented essentially the same formula.

Finally [Neugebauer:2002] could give me deeper insight. The values given as “infrared intensities” seem to be what Neugebauer calls integrated absorption coefficients. Using the definitions given in [Neugebauer:2002] it is a simple and straight forward exercise to calculate the Einstein coefficient as

A = I 8 π c NA-1 (ν/c)2

In this formula I let I be the symbol for the integrated molar absorption coefficient for the sake of continuity.

If one investigates the numerical value of 8 π c NA-1 he finds it to be about 1.25115E-14. The difference of a factor of E7 compared with Klippenstein’s formula is explained by recalling that he is suggesting to use km and cm-2 instead of SI units.

References

[Klippenstein:1996] Klippenstein et. al. Ion–Molecule Radiative Association Kinetics. J. Chem. Phys., 104(12), 1996.

[Neugebauer:2002] Neugebauer et. al. Raman and IR Spectra for Buckminsterfullerene. J. Comp. Chem., 23(9), 2002.

Another Pitfall

I was never fully satisfied by the usage of dimensions like m6mol-2s-1 (or worse) for rate constants in chemical kinetics. I used to think there should be a way to express this by a dimension free coefficient and use mole fractions (xi) instead.

After many pages of paper written I have to admit that this is not possible. It is due to the fact that kinetics and thermodynamics are not fully interchangeable and the concept of molar fractions shows to be rather useless in kinetics. An obvious example is given by a container filled with two gases to react. The rate is – assuming constant temperature – determined only by the chance of two molecules colliding in a certain amount of time. This probability is not affected whatsoever by adding a third gas to the container. In terms of molecular fractions this couldn’t be expressed because increasing the overall amount of substance will decrease the molar fraction of the components. Hence, there is no way beside using concentrations even if they are not as “nice” as fractions could be…

A minute ago I mailed an article [1] to a professor of mine in which I discussed the topic in more detail. It is written in German but I’ll still attach it to this post. However, I just got aware that I didn’t mention the case of a second order reaction between molecules of the same substance in that work. I hope that it still might be useful to some of you somehow under certain kind of circumstances.

[1] Einheiten von Reaktionsgeschwindigkeitskonstanten