Question

Find the Gibbs free energy for G(T,P) for an ideal gas with constant c_v.

Answer 1

Consider a system in contact with a resrvior at temprature T. For two equlibrium states at the same temprature T

deltaS(system)+deltaS(reservoir)>/0

deltaS-Q/T>/0

where Q is the amount of heat gained by the system.

or Q</TdeltaS - Equation 1

By the 1st law of thermodynamics,

Q= U2-U1+W

U2-U1+W</T(S2-S1)

W</(U1-TS1)-(U2-TS2)

W</F1-F2 - equation 2

where, F= U-TS called Helmholtz free energy.

Thus,the Helmholtz free energy decrease in an upper limit to the maximum work that may be done in the system.

We can consider W as a sum of pdV work and non - pdV work which is denoted by Wpv & Wnon-pv respectively.

At constant volume Cv

Wnon-pV</F1-F2 - equation 3

(F1-F2) sets an upper limit to the non-pdV work. if it is omly pdv work, then F1-F2>/0 or F1>/F2

Helmholtz free energy always decreased or remains constant in a system constant temorature and constant volume.

At constant pressure

Wnon-pV+p(V2-V1)</F1-F2

Wnon-pv</(U1-TS1+pV1)-(U2-TS2+pV2)

Wnon-pV</G1-G2 - equqtion 4

where, G= U-TS+pV is called the Gibbs free energy.

For an isolated system,the entropy always increases or remains constant