# How are the Van der Waals constants a and b related to each other?

I know that the constant $$a$$ represents the attractive forces between the molecules while the constant $$b$$ represents the size/repulsion between them. Does that mean that $$a$$ and $$b$$ are "inversely proportional" or are they independent of each other? If we have two gases and we can compare their sizes and attractive natures, can we compare $$a$$ and $$b$$ for the two gases? If $$a$$ is greater for gas 1 will $$b$$ always be lesser for gas 1 as compared to gas 2?

• They are independent. Dec 19 '19 at 13:02
• I love how Ivan Neretin says so much with just few words. May 14 '21 at 7:56

Yes, they are certainly correlated as shown in the following plot based on data from the Wikipedia Van der Waals constants data page

The correlation should not be entirely surprising: the covolume b is a measure of the size of the gas molecules, whereas a is a measure of the strength of intermolecular interactions which is expected to depend through polarizability and dispersion interactions on the size of the molecules (molecular volume).

For a given gas parameters a and b tend to increase together. The shown correlation is empirical but since both a and b depend on molecular size and apparently in a very similar functional way, the result is a linear correlation.

Following the hint from Karl, here is a version of the plot with log(a) along the abscissa, and with superimposed labels.

• Thank you! I expected them to be decreasing functions of each other but it instead seems like a line with positive slope. I still have two questions: 1) Why are they related through an increasing function and 2) Does this mean that if a is greater for gas1 b will also be greater for gas1? Dec 19 '19 at 17:13
• Nice graph! If you label the dots and plot log(a) vs b, you notice a few notorious weirdos who stick out: H2, He, Ne, mercury and water.
– Karl
Dec 19 '19 at 18:03
• H2 and He are actually nicely on the line, while water and mercury stick out, which is somewhat tbe. I wonder whats wrong with neon: i.stack.imgur.com/B9s6W.png
– Karl
Dec 19 '19 at 20:00
• Neon is decidedly odd. It just wants to be left alone :-) Dec 19 '19 at 23:25
• @Karl: The dataset from Wikipedia gives neon a lower $b$ than helium, which seems a priori surprising (and apparent I'm not the only one surprised by it). I haven't checked the literature, but I wouldn't be too surprised if the value for neon was simply inaccurate (and/or poorly constrained by available data, or a result of some behavior that does not actually fit the van der Waals equation well). Dec 20 '19 at 13:58

Just for curiosity, I also plotted a vs b using the same date set used by Buck Thorn. What I want to know is what is the correlation shown in Buck Thorn's plot and what are the two outliers clearly in the plot:

As I marked in my plot, they are mercury and idobenzene. As Karl pointed out in one of his comments, mercury is the biggest outlier. Other than these two, other compounds have shown fairly good linear relationship with $$R^2 = 0.9308$$.

Again, based on few comments, I picked just the noble gasses and hydrogen to see how they would be doing since they are closed to ideal gas behavior. First, again following Karl's suggestion, I checked with $$\log a$$ versus $$b$$. The best fit ($$R^2 = 0.999$$) is the $$2^\circ$$-degree polynomial curve, but best fit come with the lost of hydrogen and helium (see the left top plot). Both values for helium and hydrogen are way off (not shown here). However, when I plot all $$a$$ and $$b$$ data of noble gasses and hydrogen (without log values), it gives a nice linear relationship ($$R^2 = 0.9571$$). Still, there is one outlier: Neon. As Buck Thorn nicely put it, neon just wants to be left alone!

So I eliminated the data for neon and re-plot the remaining data and it came out even a better fit linearly (compatible with the whole data set):

• Should not them be rather noble gases (like nobleman ) than Nobel gases ? :-) It makes an impression they may be Nobel prize laureates. :-) Dec 20 '19 at 8:29
• I always seems to make that mistake! :-) Will make the corrections. Dec 20 '19 at 8:35
• You make a quadratic fit in the half logarithmised data in your upper left plot. Im afraid that is highly unphysical. Also: 5 datapoints, four of which correlate very well, three parameters. Dont need much luck to get a good fit there.
– Karl
Dec 20 '19 at 17:05
• @Karl: Well, tha's what happens when a non-physical chemist try to be a one.:-) Nonetheless, I can't resist it's such a good relationship ($R^2=0.999$). Dec 20 '19 at 17:26

As Ivan writes $$a$$ and $$b$$ are independent, but it does not mean that they are not correlated; however, so are a child's shoe size and reading ability which clearly does not mean that buying a child larger shoes will increase its reading ability. A correlation does not imply cause and effect.

Plots of the compression factor with reduced pressure show that many gasses follow a general trend when using the van der Waals equation. The Critical temperature is defined as $$T_\mathrm c=8a/(27R b)$$ ($$R$$ is gas constant) which shows the connection between $$a$$ and $$b$$ for any gas but the values of $$T_\mathrm c$$ vary between gases. A plot of $$a$$ vs $$b$$ is thus effectively a plot of (scaled) $$T_\mathrm c$$ values. The other critical constants are $$V_\mathrm{cm}=3b$$ (molar volume) and $$P_\mathrm c=a/(27b^2)$$. Look for the Law of Corresponding States in a good phys. chem. texbook, such as Engel & Reid for more detail.

• So the relation between A and B was a false alarm? Dec 20 '19 at 17:12
• @aditya_stack No. The critical pressure and temperature both contain a and b. That is a strong hint that these two parameters are not independent at all.
– Karl
Dec 20 '19 at 17:25
• Of course you can make a hen and egg problem out of this: Are Pc and Tc independent? And they are of course not. After all, we don´t have statistical data here, but physical parameters of basic matter.
– Karl
Dec 20 '19 at 17:38
• @Karl, if $a$ and $b$ are fundamentally related then what is the physical basis for this, in other words, what is the derivation of $a$ and $b$ from first principles that unveils their relationship? Dec 21 '19 at 12:50