# Determination of Fugacity of a Gas

We know that G = Gº + nRT ln*f*

for one mole of a gas-

G = Gº + RT ln*f*

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(ln*f*)/δP]_{T}

Since, (δG/δP)_{T} = V

Thus, V = RT [δ(ln*f*)/δP]_{T}

or, V/RT = [δ(ln*f*)/δP]_{T}

Thus, at definite temperature, the above equation may be written as-

RT d(ln*f* = 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(ln*f* = RT(dP/P) – αdP

d(ln*f* = 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.