Surfactants (Soaps and Detergents) and Physicochemical Properties - Surface and Colloid Chemistry: Principles and Applications - page 57

where V α and V β are the actual volumes of each bulk phase, and p and p′ are the
respective pressures. Since the volume of the interfacial region was considered to be
negligible, V t = V α + V β . Further, if the surface is almost planar, then p α = p′ β , and
(p dV α + p β dV β ) = p dV t .
The changes in the internal energy for idealized phases α and β may similarly be
expressed as follows:

dE α = T dS α − p dV α + μ i dn i + … +μ i , dn iα (3.16)

and

dE β = T dS β − p dV β + μ i dn i + … +μ i,β dn i,β (3.17)

In the real system, the contribution due to change in the surface energy, γ dA, is
included as an additional work. Such a contribution is absent in the idealized system
containing only two bulk phases without the existence of any physical interface.
By subtracting Equations 3.16 and 3.17 from Equation 3.15, the following rela-
tionship is obtained:

d(E t − E α − E β ) = T d(S t − S α − S β )+ γ dA + μ 1 d(nt − n1 − n1) + … +

μ i d(n i,t − n i − n i β ) (3.18)

or

d E x = T dS x + γ dA + μ 1 dn 1 x + . . . + μ i dn ix (3.19)

This equation, on integration at constant T, γ, and μ i , etc., gives

E x = T S x + γ A + μ 1 n 1 ix + … + μ 1 n ix (3.20)

This relationship may be differentiated in general to give

dE x = T dS x + γ dA + Σ i (μ i dn i ) + Σ i n i dμ i + A dγ + S dT (3.21)

Combining Equations 3.21 and 3.20 gives

− A d γ = S x dT + Σ i n ix dμ i (3.22)

Let S s,x and Γ ix denote the surface excess entropy and moles of the i -th component per
surface area, respectively. This gives

S s,x = S x /A

Γ ix = n ix /A (3.23)

and