2.6.1 Theory

Since equations of state cannot be derived from thermodynamics but must be determined by other means, all sorts of methods for obtaining them have been tried.

One of the most imaginative and theoretically useful^2.18 methods is that of the virial expansion.

The basic idea is simple. We know that one of the standard properties of a gas can be calculated from the other three. We take that property to be the pressure times the molar volume and write:

$$p V_{\textrm{m}} = f( p, T )$$ (2.6.17)

where I've written V_m for the molar volume, the actual volume divided by the number of moles. As a result, there's no n on the right-hand side of the equation.

Now, assuming that f(p,T) is continuous, it has a Taylor series expansion in p:^2.19 If I can abbreviate f(p,T) as f(p), the Taylor series can be written as:

$$f(p)  =   f(0) + f'(0)p + \frac{1}{2!} f''(0)p^2 + \frac{1}{3!}f'''(0) p^3 + ...$$ (2.6.18)

where prime marks indicate differentiation, so that $f''(0)$ means the second derivative of p evaluated at p = 0.

Thus what we have is:

$$p V_{\textrm{m}} = f(0) + f'(0)p + \frac{1}{2!} f''(0)p^2 + \frac{1}{3!}f'''(0) p^3 + ...$$ (2.6.19)

where the terms in f are also functions of T.

Since we don't know $f(p)$, we certainly can't find the derivatives. So we can rewrite equation 2.6.3

$$p V_{\textrm{m}} = \alpha + \beta p + \gamma p^2 + \delta p^3 + ...$$ (2.6.20)

Now, of what use is this formula? Well, for one thing we know that all gases become ideal as the pressure goes to zero. So:

$$\lim_{p \rightarrow 0} p V_{\textrm{m}} = RT = f(0)$$ (2.6.21)

and we've evaluated one coefficient already!

Sadly, the others can't be evaluated this way. To make equation 2.6.4 look a little like the start of the ideal gas equation we factor an RT out of it to get:

$$p V_{\textrm{m}} = RT \left( 1 + b(T) p +  c(T) p^2 + ... \right)$$ (2.6.22)

The term $b(T)$ is called the second virial coefficient. It can be (and has been) evaluated experimentally. It is a function of temperature and is different for each different gas.

Figure 2.8: Second Virial Coefficient for Argon

The term $c(T)$ is, logically, called the third virial coefficient. It also depends on temperature. It is much harder to evaluate experimentally, but this has been done in a number of cases.

There is another way to do a virial expansion. Instead of expanding in powers of p, we can instead expand in powers of $1/V_{\textrm{m}}$. The steps are the same, and we can again evaluate the first term if we take the limit as $1/V_{\textrm{m}}$ goes to zero. The result now is:

$$p V_{\textrm{m}} = R T \left( 1 + \frac{B}{V_{\textrm{m}}} + \frac{C}{V_{\textrm{m}^2}} + ... \right)$$ (2.6.23)

where, confusingly B and C are also called the second and third virial coefficients, respectively. And they too are functions of temperature and have been evaluated experimentally.

Of course the coefficients b and B must be related, as are the coefficients c and C. The relationships are:

$$B = R T b \quad\quad\quad C = R T b B + R^2 T^2 c$$ (2.6.24)

and

$$b = \frac{B}{RT} \quad\quad\quad c = \frac{C}{R^2 T^2} - B^2$$ (2.6.25)