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Is there a relation of the boiling point of a gas with its van der Waals constants, $a$ & $b$?

I know if constant $a$ is greater, then the boiling point will also be greater, because of more attraction between the molecules, but does the boiling point depend on $b$ too? If two gases have the same $a$ but different $b$, can I tell which of the two gases will have a higher boiling point or nearly same boiling point?

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  • $\begingroup$ Boiling point is a fact of nature; van der Waals constants aren't. That said, there surely is a relation, albeit a complicated one (it involves the roots of a cubic equation and stuff). $\endgroup$ – Ivan Neretin Apr 1 '18 at 21:54
  • $\begingroup$ Without information on the liquid side of the phase transition, this is not answerable. $\endgroup$ – Jon Custer Apr 2 '18 at 13:47
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There will be a relationship, but as the comments on the question indicate, the relationship will not be simple. There are two complications:

  1. The van der Waals equation is an empirical model built to fit experimental pressure/volume/temperature data for real gasses.
  2. The boiling point of a liquid is not a single number but a line on the pressure/temperature phase diagram.

Supposed the van der Waals equation can predict phase behavior (source: 1) However, I made some graphs. The van der Waals constant data are from Wikipedia and the boiling point data looked up from the Wikipedia articles.

Constant a

The blue line is a cubic polynomial fit, while the grey shaded area is the error. With the exception of the metal outlier (mercury), this fit seems pretty good.

enter image description here

Constant b

The blue line is a logarithmic fit, while the grey shaded area is the error. Again, with the exception of the metal outlier (mercury), this fit seems pretty good.

enter image description here

Finally, I decided to plot the van der Waals constants against each other.

The van der Waals constants are fairly co-linear, so any clear relationship between one and boiling points implies a clear relationship between the other and boiling points.

enter image description here


Source:

  1. Terrell L. Hill, 2012 [1960], "An Introduction to Statistical Thermodynamics" [Dover Books on Physics], Chicago, IL, USA:R.R. Donnelly (Courier/Dover), ISBN 0486130908, referenced on the Wikipedia article, but I do not have access.
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