Question

-Show that for the truncated virial equation, Z = 1 + BP/RT, the fugacity is given by f = e^Z-1 ⋅P. Further show that f ≈ Z⋅P when Z is near unity (make use of a series approximation). -Show that for a van der Waals gas, with Z near unity, the second virial coefficient is given by B = (b – a/RT).

-Use the derived relationship to calculate the fugacities and fugacity coefficients for hydrogen and for carbon dioxide gases at 50 bar and 298K.

Answer 1

The real gas equation can be written as:

(P + a/V²)(V - b) = RT

i.e P = RT/(V - b) - a/V²

i.e. PV/RT = V/(V - b) - a/VRT

i.e. Z = (1-b/V)^-1 - a/VRT (since (1-x)^-1 = 1 + x + x²/2 + .....)

i.e. Z = 1 + b/V + b²/V² + .... - aV/RT

i.e. Z = 1 + (b - a/RT) / V (since the higher coefficients can be neglected).

i.e. Z = 1 + BP/RT, where B = b - a/RT and PV = RT

Therefore, the second virial coeffiecient (B) = b - a/RT

According to the given data:

f = e^Z-1 ⋅P (since e^x = 1 + x + .....)

~ {1 + (Z-1)} .P

~ Z.P

Therefore, f ~ Z.P

If you know the Z value (i.e. nearly equal to fugacity coefficient), you can easily calculate the fugacity (f) using the above relation.