Hence,as

(2.57)

(2.58)

(2.59)

(2.60)

Now,

(2.61)

Now μ is defined as chemical potential.

Gibbs-Duhem Equation

Let us consider a homogeneous phase of a multi-component system for which

(2.62)

If the phase is enlarged in size, U, S and V will increase in size, while T, p and μ will remain the same. Thus,

(2.63)

Let the system be enlarged to k times the original size.

∴ ΔU = kU - U = (k-1)U

(2.64a)

ΔS = kS - S = (k-1)S

(2.64b)

ΔV = kV - V = (k-1)V

(2.64c)

Δn_i = kn_i = n_i = (k-1)n_i

(2.64d)

Substituting Eqs. (2.64a-d) in Eq. (2.63), we get

(2.65)

(2.66)

(2.67)

(2.68)