Question

Derive equation,

SR(T, P, N) H GR NR NRT

Answer 1

This equation provides a method to compute energy and entropy changes for real gases using the residual property.

The specific residual property is M^R= M (T,P) - M^ig (T,P)

where M(T,P) is the specific property of a real gas at a given Temperature(T) and pressure (P)

M^ig (T,P) is the value of the same property if the gas were to behave ideally at the same temperature and pressure

Hence the residual Gibbs energy can be written as G^R = G – G^ig

We know that the Gibb's free energy is given by G= H-TS.

Partially differentiating both sides of this equation and replacing H by it's definition we get dG = VdP – SdT

The dimensionless form of the above equation is d(G/RT) = V/RT dP – H/RT² dT

In case of an idea gas the same equation becomes d(G^ig/RT) = V^ig/RT dP – H^ig/RT² dT

and for a real gas d(G^R/RT) = V^R/RT dP – H^R/RT² dT

The residual properties for the above equations at constant temperature and pressure are

V^R /RT = [δ(G^R /RT) / δ P]_T and H^R /RT = -T [δ(G^R /RT) / δ T]_P respectively.

By G= H-TS we have

For an ideal gas G^ig = H^ig – TS^ig , by the difference we get G^R = H^R – TS^R .

Therefore the residual entropy is given by S^R /R = H^R/RT – G^R /RT

In the expanded form in the terms of the dependent variables and including N we can write the same as follows:

S^R (T,P, {N_i}) /NR =( H^R – G^R )/ NRT