2.6.2 The Virial Expansion of van der Waals Equation

Van der Waals equation can be put into virial form by expanding it in a power series. We start by solving for p:

$$p = \frac{RT}{ V_{ \textrm{m}} -b} - \frac{ a}{V_{ \textrm{m}}^2}$$ (2.6.26)

and then expanding in powers of $1/V_{\textrm{m}}$. There's a trick involved.^2.20 If we write the ${1/(V_{ \textrm{m}} -b)}$ term as:

$$\frac{1}{ V_{ \textrm{m}} -b} = \frac{1}{V_{ \textrm{m}} } \left(\frac{1}{1 - b/V_{ \textrm{m}}}\right)$$ (2.6.27)

we can then use the series expansion

$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + ... \quad \quad \vert x\vert < 1$$ (2.6.28)

(where x has to be less than one). Of course, ${b/V_{ \textrm{m}}}$ is always less than 1,^2.21 so we can write:

$$\frac{1}{1 - b/V_{ \textrm{m}} } = 1 + \frac{b}{V_{ \textrm{m}}} ... ...{V_{ \textrm{m}}} \right)^2 + \left( \frac{b}{V_{ \textrm{m}}} \right)^3 + ...$$ (2.6.29)

Putting all the bits together gives:

$$p = \frac{RT}{V_{ \textrm{m}}} \left\{ 1 + \frac{b}{V_{ \textrm{m... ...{b}{V_{ \textrm{m}}^3} \right) + ... - \frac{a}{RTV_{ \textrm{m}}} \right\}$$

for a final expansion

$$pV_{ \textrm{m}} = RT \left\{ 1 + \left( b - \frac{a}{RT}\right) ... ...+ \frac{b^2}{V_{ \textrm{m}}^2} + \frac{b^3}{V_{ \textrm{m}}^3} + ... \right\}$$ (2.6.30)

Thus the van der Waals equation corresponds to a second virial coefficient of

$$B = b - \frac{a}{RT}$$ (2.6.31)

where, of course b is the van der Waals b, not a virial coefficient. This almost has the right sort of behavior for a second virial coefficient. As can be seen, B will be negative at low temperature and positive (b is always positive) at high temperatures.

The third virial coefficient is

$$C = b^2$$ (2.6.32)

and is constant, which is not really a good approximation of reality.