Determination of Fugacity of a Gas

We know that G = Gº + nRT lnf
for one mole of a gas-
G = Gº + RT lnf
Differentiating the above equation with respect to pressure at constant temperature and number of moles of various components, we get-
(δG/δP)_T = RT [δ(lnf)/δP]_T
Since, (δG/δP)_T = V
Thus, V = RT [δ(lnf)/δP]_T
or, V/RT = [δ(lnf)/δP]_T
Thus, at definite temperature, the above equation may be written as-
RT d(lnf = VdP --- Equation:1

Let 'α' be the quantity which defines the deviation from ideal behavior. Substituting for ideal gas (V = RT/P), the quantity 'α' is given by-
α = (RT/P) – V
Multiplying the above equation by dP, we get-
αdP = RT(dP/P) – VdP --- Equation:2

Now combining equation:1 and Equation:2 we get-
RT d(lnf = RT(dP/P) – αdP
d(lnf = d(lnP) – (α/RT)dp

Integrating the above equation between the pressure range 0 to P, we get-

Here α is an experimentally determined quantity measured at different pressures. Thus, according to the above equation, 'α' may be positive or negative. Thus, fugacity of a gas may be positive or negative.

From graph, it is clear that the value of is positive at low pressure and negative at very high pressure. Hence, according to Equation:3, fugacity of a gas would be less than pressure P at low pressure while higher than pressure P at high pressure.