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For a uniform mixture of ideal gases, I think that the concentration of a gas reported in parts per million by volume (ppmV) and micro-moles per mole (μmol/mol) would be the same.

When I see atmospheric CO2 discussed, I have seen both units used, depending on the source. See for example, the comments under this answer.

Since multi-atomic gases like N2, O2, H2O and CO2 are not ideal gases (even monatomic gases are not absolutely ideal) I would guess that for realistic atmospheric samples at different altitudes and times, the CO2 concentrations expressed in ppmV and μmol/mol would not be numerically identical. Further, due to the variations in concentrations of the other constituents they wouldn't even be strictly proportional.

Is there a handy, approximate conversion constant, or function, that is more accurate than unity? Is there a nearly-exact one?

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According to the US Carbon Dioxide Information Analysis Center :

Though in their data table they use the term "ppmv", they explain:

The CO2 mixing ratios are reported as micromoles per mole (µmol/mol = ppmv) of dry air in the World Meteorological Organization (WMO) X85 mole fraction scale, traceable to primary standards at the Scripps Institution of Oceanography (SIO)

This is further explained in The WMO Mole Fraction Scales for CO2 and other greenhouse gases, and uncertainty of the atmospheric measurements:

Abbreviations can be ppm, ppb, etc., whereas widely used abbreviations such as ppmv (the v is for volume) are wrong and should be abandoned. When reviewing papers, or when we are co-authors, we should convince first authors to abandon the ppmv notation. Volume has nothing to do with mole fraction.

Supposing you know the mole fraction, if you literally wanted to calculate the parts per million by volume, first you need to define what that would mean. Are you interested in the ratio of the volume of CO2 separately isolated relative to the whole volume? Or are you interested in the volume of CO2 separately isolated, relative to the sum of all the gases separately isolate? For either of these methods, you would need to choose a temperature and pressure, such as the actual temperature and pressure of a sample, or standard temperature and pressure.

Then you need an equation of state at least for the CO2, that is more real than the ideal gas law, such as van der Waals.

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    $\begingroup$ This is exactly what I was hoping for. You've resolved the quandary by explaining the ambiguity when trying to define partial volumes, and found a definitive reference to settle the matter. Thanks! $\endgroup$ – uhoh May 12 '17 at 13:01
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I think this is it:

ppm(mol)=ppm(v) x (density CO2/density total atmos) x (Molecular Weight total atmos/Molecular Weight CO2)​​​​.

Try some number crunching to see if that makes sense.

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  • $\begingroup$ Well I'm looking for something more subtle, like a 1% or even smaller correction factor. 1ppmV is very close to 1umol/mol. For ideal gas they would be the same. So it's really just the deviation from the ideal gas law that I'm trying to find. $\endgroup$ – uhoh Apr 22 '17 at 16:03
  • $\begingroup$ I think the equation you wrote is for ppm by mass, not by number (in moles). $\endgroup$ – uhoh Apr 22 '17 at 16:06
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From deal gas law, amount (mol) of gas is directly proportional to volume or pressure. Hence ratio micromol per mol is same as ppmv at atmospheric temperatures and pressures.

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    $\begingroup$ Thank you for your answer, but it doesn't really add anything of value beyond what is present in the accepted answer. Also, if you look at my question again, you'll see that it is about deviations from the ideal gas law; "Since multi-atomic gases like N2, O2, H2O and CO2 are not ideal gases..." as well as other effects. So while they are roughly the same, my question is about the small difference between the two. Welcome to stackexchange! If you like, you can take the tour. $\endgroup$ – uhoh Aug 16 '17 at 5:19
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    $\begingroup$ I agree, I actually misread the question! Thanks for clarifying. $\endgroup$ – Harish Aug 16 '17 at 11:36
  • $\begingroup$ It looks like some users have down voted your answer. I gave you an up vote "for effort" when I left my comment. Down votes are not personal in stackexchange, they apply to the post itself. It's probably a good idea to delete your answer to make sure more down votes aren't cast and to remove these from your reputation score. Eventually it will probably done in the review queue anyway, so doing it yourself first is ideal. But please consider posting again! Hint: asking questions is a little easier than posting answers when starting off on any SE site, until you get the hang of it. $\endgroup$ – uhoh Aug 16 '17 at 11:41

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