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Liquid-gas and liquid-liquid interfaces 81

P P

A "i Region of

S Uarying

D J composition

(a) Ideal system (b) Real system

Figure 4.12 Representations of an interface between bulk phases a and /3

ently in terms of surface excess concentrations. If nf is the amount of

component i in the surface phase cr (Figure 4.12b) in excess of that

which would have been in cr had the bulk phases a and /8 extended to

a surface SS with unchanging composition, the surface excess

concentration of component / is given by

r* (4.15)

where A is the interfacial area. F, may be positive or negative, and its
magnitude clearly depends on the location of SS, which (as illustrated
in the following derivation) must be chosen somewhat arbitrarily.
The total thermodynamic energy of a system is given by the
expression

U=TS-pV + 2^

The corresponding expression for the thermodynamic energy of a

surface phase cr is

U" = yA (4.16)

(The pV" and yA terms have opposite signs, since pressure is an
expanding force and surface tension is a contracting force. A
superscript is not necessary for T, p and the chemical potential terms,
since these must have uniform values throughout for a heterogeneous
system to be in equilibrium.) Differentiating equation (4.16) generally

dU" = TdS" + S"dT - pAV" - V"dp + ydA + Ady

+ 2/Mnf + Snfdf*,. (4.17)