Determination of Fugacity by Virial Equation

The fugacity of a real gas can also be determined with the help of the virial equation of state. This method is particularly useful at low and moderate pressures, where the virial equation gives a satisfactory representation of the behaviour of real gases.

For one mole of a gas, the virial equation may be written in terms of pressure as

$$
Z=1+B’P+C’P^2+D’P^3+\cdots
$$

where \(B’\), \(C’\), \(D’\), etc. are the pressure-form virial coefficients and may depend upon temperature.

The fugacity coefficient of a real gas is related to the compressibility factor by

$$
\ln\phi=\int_0^P\frac{Z-1}{P}\,dP
$$

For the virial equation,

$$
Z-1=B’P+C’P^2+D’P^3+\cdots
$$

Therefore,

$$
\frac{Z-1}{P}
=
B’+C’P+D’P^2+\cdots
$$

Substitution in the fugacity expression gives

$$
\ln\phi
=
\int_0^P
\left(B’+C’P+D’P^2+\cdots\right)dP
$$

At constant temperature, the virial coefficients are constants with respect to pressure. Hence, integration gives

$$
\ln\phi
=
B’P+\frac{C’P^2}{2}
+\frac{D’P^3}{3}+\cdots
$$

Thus,

$$
\boxed{
\ln\phi
=
B’P+\frac{C’P^2}{2}
+\frac{D’P^3}{3}+\cdots
}
$$

Since the fugacity coefficient is defined by

$$
\phi=\frac{f}{P},
$$

we have

$$
\ln\left(\frac{f}{P}\right)
=
B’P+\frac{C’P^2}{2}
+\frac{D’P^3}{3}+\cdots
$$

Taking the exponential of both sides,

$$
\boxed{
f
=
P\exp\left(
B’P+\frac{C’P^2}{2}
+\frac{D’P^3}{3}+\cdots
\right)
}
$$

At Low Pressure

At sufficiently low pressure, the higher-order terms become very small and may be neglected. The virial equation can then be approximated by

$$
Z=1+B’P
$$

Thus,

$$
Z-1=B’P
$$

and therefore

$$
\frac{Z-1}{P}=B’.
$$

The fugacity coefficient then becomes

$$
\ln\phi=\int_0^P B’\,dP
$$

giving

$$
\boxed{\ln\phi=B’P}
$$

or

$$
\boxed{\phi=e^{B’P}}
$$

Since \(f=\phi P\),

$$
\boxed{f=Pe^{B’P}}
$$

For very low pressures, \(B’P\) is small. Using the approximation \(e^x\approx1+x\) for small \(x\),

$$
f\approx P(1+B’P)
$$

Thus, at sufficiently low pressure, \(f\) approaches \(P\), as expected for a gas approaching ideal behaviour.

Relation with the Virial Coefficient in Volume Form

The virial equation is also commonly expressed in terms of molar volume as

$$
Z=1+\frac{B}{V}+\frac{C}{V^2}+\cdots
$$

where \(B\), \(C\), etc. are the virial coefficients in the volume form.

At low pressure, the pressure-form and volume-form coefficients are related approximately by

$$
B’= \frac{B}{RT}.
$$

Therefore, when only the second virial coefficient is significant,

$$
\boxed{
\ln\phi=\frac{BP}{RT}
}
$$

and hence

$$
\boxed{
f=P\exp\left(\frac{BP}{RT}\right)
}
$$

The sign of the second virial coefficient determines the direction of deviation from ideal behaviour. A negative \(B\) generally indicates predominance of attractive interactions and gives \(fP\).