2.5.5 Solving van der Waals Equation for the Volume

We've already noted that van der Waals equation is a cubic in the volume. That means that given the temperature, pressure, and number of moles, a cubic equation has to be solved. There are a number of fairly cumbersome ways to do this. A few are discussed below.

An Iterative Solution

A semi-cumbersome method involves iteration or, as it is sometimes called: successive approximations.

To use this first write van der Waals equation in the form:

$$V = \frac{1}{V^2} \left\{ n \left( \frac{RT}{p} + b \right) V^2 - \frac{n^2 a }{p} V + \frac{n^3 a b }{p} \right\}$$ (2.5.14)

All that need be done is to take an approximate value of V, (often usefully obtained from the ideal gas law), plug it into the right-hand side of Equation 2.5.10, and solve for a better value of V. This process can be made much more easy by precomputing three constants:

$$\alpha = n \left( \frac{RT}{p} + b \right), \quad \beta = \frac{n^2 a }{p}, \quad \gamma = \frac{n^3 a b }{p}$$ (2.5.15)

and then using:

$$V = \frac{ \alpha V^2 - \beta V + \gamma }{V^2}$$ (2.5.16)

Given modern calculators, the three constants can be computed once and stored.

This iterative process is really suited only for computing the volume on the gas branch of the van der Waals equation. It converges slowly, but with a bit of patience, gives the right answer.

Another Series Solution

To be added.