2.5.2 The Ideal Limit

The first thing to check is what happens to van der Waals' equation when the volume gets very large. Does equation 2.5.1 reduce to the ideal gas equation?

It does. It does because in the quantity

$$p + \frac{n^2 a}{V^2}$$

for any value of p there is a (large) value of V that makes the second term as small as we wish. So the entire quantity reduces to p in that limit.

The other quantity, V - nb can be written as:

$$V \left( 1 - \frac{nb}{V} \right)$$

and again, one can always chose a (large) value of V that will make nb/V negligible compared to 1.

So in the limit of large volume, we do indeed recover the ideal gas equation.

We also recover the ideal gas equation if p is made very small. This can be done in two ways. One way to reduce the pressure is to increase the volume. That case has already been covered. The other is to decrease the number of moles per unit volume. This quantity is the number density n/V. Making the number density small has exactly the same effect as making V large, so again we have covered this case.

Last, note that if a and b are made small (actually zero), the ideal gas equation is immediately recovered.

This gives us some confidence that at least in a general way, the van der Waals equation behaves in a reasonable manner.