2.5.1 The van der Waals Equation

The most famous equation of state for real gases is the van der Waals equation. It looks tantalizingly like the ideal gas law:

$$\left( p + \frac{ a n^2 }{ V^2 } \right) \left( V - n b \right) = n R T$$ (2.5.5)

where a and b are constants having different values for each different gas.^2.10

Now how was this equation derived?

The short answer is that it was not derived. It was just thought up. Can you do that in science? Yes, if you are clever, and van der Waals was clever.

He seems to have reasoned as follows: the ideal gas law allows a volume V for each molecule in the gas to roam around in. That isn't actually true. A real gas molecule can roam anywhere except in the space taken up by other molecules. So the real volume available to a gas molecule isn't V, but is less than V by an amount that depends on the number of moles of gas n and the volume occupied by a mole of that gas b.^2.11 So van der Waals argued that in the ideal gas law V should be replace by V - nb, the actual volume available to the gas.

Repulsive forces between molecules account for the fact that a molecule has "volume"; you can't force two molecules too close together because the repulsive forces rise very rapidly as the distance between molecules becomes very small.

Molecules also have attractive forces. Van der Waals accounted for those by reasoning as follows: pressure is caused by impacts of molecules on the walls of the container holding a gas. A molecule moving toward a wall will hit it causing an observed pressure p. But, van der Waals argued, that isn't the real pressure in the gas. Gas molecules are slowed down before they hit the wall, so the pressure actually measured is too low.

Why are molecules slowed down in this way? In the body of the gas, attractive forces pull the molecule equally in all directions. In other words, they balance out. But as the molecule moves toward the wall, most of the molecules are behind it. The attractive forces are unbalanced, and act to slow the molecule down as it approaches the wall.^2.12

The slowdown will clearly depend on the number of molecules near the one hitting the wall. And that depends on what is called the number density of the molecules, the number per unit volume, n/V. The larger n/V, the more molecules are present to slow the one we are watching down.

But more than one molecule is hitting the wall. The pressure is due to the cumulative effects of all the molecules that hit the wall. And the number hitting the wall depends on the number near the wall. This is the number per unit volume (the number density) again. So the actual reduction in pressure should, van der Waals reasoned, be proportional to n/V times n/V. The proportionality factor is a (different for each different gas), and so a term $an^2/V^2$ has to be added to the measured pressure to bring it back to what it would be ideally if there were no attractive forces.

The result of these two changes is van der Waals' equation 2.5.1.

One thing more must be dealt with. The constant b isn't exactly Avogadro's number times the volume per molecule. That's because a molecule actually keeps other molecules out of a volume a good bit larger than just its own volume as the figure shows.

Figure 2.5: Excluded Volume

Molecule a, shown by a solid black line in Figure 2.5.1, keeps the centers of all other molecules (such as b) outside of the dotted circle. The dotted circle (really a sphere) has a radius twice that of molecule a and so has eight times the volume.

The molecules have been shown here as circles (really spheres) but in reality only monatomic gases have that shape. So the diagram is only suggestive and not the basis for an accurate calculation.

Of course, that isn't the entire story either, since three or more molecules could be very close. The best that can be said is that b is roughly eight times the volume of a mole of gas molecules. But only roughly.