# Surprising values of intermolecular forces of ethane, oxygen and xenon

I know that gases with stronger intermolecular forces should have a larger value of the van der Waals constant $a$. So I looked up a table on van der Waals constants and tried to reason about the relative values of that constant. Though I was able to explain most of the values, there was one thing I found absurd. Here:

$$\begin{array}{cc} \hline \text{Substance} & a~(\pu{atm {dm}^6 mol^{-2}}) \\ \hline \text{Oxygen} & 1.364 \\ \text{Xenon} & 4.137 \\ \text{Ammonia} & 4.225 \\ \text{Ethane} & 5.507 \\ \hline \end{array}$$

My intution says that dioxygen, which has a slightly higher molar mass (32 vs. 30) than ethane, but significantly less molecular size or volume, should have a only slightly lower value of $a$ than that of ethane, but here we find a significant difference! Another example is xenon; its molar mass (131) is much larger than ethane (30), but its value of $a$ is smaller. Does it have to do with the dominance of molecular size over mass? Or atomicity or atomic dipoles? I suspect that it's probably the large and branched structure of ethane that's causing this.

Why does the molecular size/volume seem to play such a dominating role in the value of the constant $a$? I thought it was primarily influenced by molecular mass.

Van der Waals forces are weak and depend on polarizability. There is a correspondence between the boiling points of oxygen (−183 °C), xenon (−108 °C), and ethane (−88.5 °C) and their constants (1.364, 4.137, 5.507). These gases are polarizable to varying extents, but not polar with charge separation. The size of the molecule and its mass will affect polarizability - perhaps a way to look at this is to say electron density in the outer regions of the molecule will determine how easily dipole-dipole interactions can evolve.

But ammonia stands out with a boiling point (−33 °C) which is too high to be explained by van der Waals forces alone. Of all the gases listed, ammonia is the only one with significant hydrogen bonding capability. Perhaps if $\ce{H2O}$ were included, it would be an even greater outlier.

• You seem to suggest that vdW forces are independent of molecular size. Am I correct? Mar 3, 2018 at 2:57
• van der Waals forces can be calculated from ionization potentials and polarizabilities. They vary approximately as 1/r^6, where r is the separation between two molecules (the space between them, not the distance between centers). Polarizability is related to how "soft" the molecule is, which is not directly related to size, but all other things being equal, a larger molecule would probably be more polarizable. Mar 3, 2018 at 4:42
• Ok, so you are suggesting here that the molecular size has a greater effect than the molecular mass of a molecule? Mar 3, 2018 at 5:18
• @Guarang Tandon I think you have the right idea. It is more about size than mass. A lot of times as a simplification, van der Waal forces are described as being related to mass, but that's just because for the simplest case of single atoms or small molecules, size is closely correlated with mass.
– Tyberius
Mar 3, 2018 at 5:29
• so let me just wind up the entire discussion- Van der waal forces actually do not depend on molar mass (atleast to a significant extent) but depend greatly on molecular size, since molar mass is generally proportional to molecular size we generally say that vanderwaal force increases with increasing molecular mass but that is not true, molecular size is the only dominating factor. If I am right please add this to your answer as this line alone would answer my question. Mar 3, 2018 at 6:23