# Anomaly in van der Waals constants of noble gases

In van der Waals equation for a real gas, the equation is

$$\left(P + a\frac{n^2}{V^2}\right)(V - nb) = nRT$$

where $$P$$ is the pressure of the real gas, $$n$$ is the number of moles of the gas, $$R$$ is the ideal gas constant, $$T$$ is the temperature, $$a$$ is a constant to correct for the intermolecular forces of attraction, and $$b$$ is a constant to correct for the size of the molecules.

It follows that gases with large intermolecular forces have large $$a$$ values and gases with large molecular size have large $$b$$ values.

Below is a table of the $$a$$ and $$b$$ values of the ideal gases:

$$\begin{array}{ccc} \hline \text{gas} & a & b \\ \hline \ce{He} & 0.0341 & 0.0237 \\ \ce{Ne} & 0.211 & 0.0171 \\ \ce{Ar} & 1.35 & 0.0322 \\ \ce{Kr} & 2.32 & 0.0398 \\ \ce{Xe} & 4.19 & 0.0511 \\ \hline \end{array}$$

Why is the $$b$$ value for helium larger than the $$b$$ value for neon even though helium's atomic radius is smaller than that of neon? Does it maybe have to do with helium's small size, and thus higher electron density surrounding the nucleus, creating more repulsions with other helium atoms?

• What's the source? Also did you check other sources? Nov 28 '19 at 22:52
• This table is invalid, it lacks the temperature/pressure range in which the equation was fitted for each gas. If they used the same range for all of them, that would definitively make the results incomparable. $a$ and $b$ are not universal constants for each gas, and generally vdW is not a good model for any gas over a wider p/T range.
– Karl
Nov 28 '19 at 22:53
• @Mithoron from the Wikipedia page: en.wikipedia.org/wiki/Van_der_Waals_constants_(data_page) Nov 28 '19 at 23:33
• @Karl Why do you say the table is invalid when almost all of the chemical data pages for van der waals constants give only "a" and "b", as exemplified here: engineeringtoolbox.com/… and here: en.wikipedia.org/wiki/Van_der_Waals_constants_(data_page) and here: www2.ucdsb.on.ca/tiss/stretton/database/… Nov 28 '19 at 23:34
• I know, its terrible. a and b are just empiric parameters, valid for a limited p/T range. If you make the range too big, you get a bad fit and nothing makes sense, if you make it small enough, you still cannot compare between different gases. Giving vdW parameters without a T range is simply nonsense.
– Karl
Nov 29 '19 at 16:12

I will venture a guess, and hopefully a rational explanation. As discussed in the comments, the van der Waals equation parameter can be derived from mathematical considerations at the critical point and can be calculated as $$\displaystyle b = \frac{RT_\mathrm{crit}}{8P_\mathrm{crit}}$$.
$$\begin{array}{cr} \text{Gas} & T_\mathrm{crit}/\pu{K} \\ \hline \text{He} & 5.2 \\ \text{Ne} & 44.5 \\ \text{Ar} & 150.7 \end{array}$$