Physical Chemistry Third Edition

5.6 Surfaces in Multicomponent Systems 231

(b)

z(position)

z 1 z 0 z 2

(a)

z(position)

Concentration

Density

Figure 5.19 A Typical Density Profile Through a Surface Region (Schematic).

(a) A one-component system. (b) A two-component system.

dividing plane, so that no volume is ascribed to the surface. The concentration in the

homogeneous portion of each phase (the “bulk” portion) is extrapolated up to the surface

plane as shown in Figure 5.19a for a single component. The amount of substance in

each phase is assigned to be the amount that would occur if the concentration obeyed

this extrapolation. The shaded area to the right of the surface plane in Figure 5.19a

represents the amount of substance that is present but is not ascribed to phase II by

this convention. The shaded area to the left of the surface plane represents the amount

of substance that is ascribed to phase I by the convention, but which is actually not

present. For a one-component system it is possible to place the plane in the interfacial

region so that the two shaded areas are equal in size and no amount of the substance is

ascribed to the surface.

In a multicomponent system, this cancellation can generally be achieved for only one

substance. Figure 5.19b shows a schematic concentration profile for a two-component

system. There is no placement of the plane that will produce equal areas for both

substances. We choose to place the plane so that the cancellation occurs for the solvent.

The same extrapolation is carried out for each other substance (a solute) as was carried

out for a one-component system, and we denote the amount of substanceithus assigned

to phase I byn

(I)

i and the amount assigned to phase II byn

(II)

i. We define thesurface

excessof substance numberi, denoted byn(iσ):

n

(σ)

i ni−n

(I)

i −n

(II)

i (definition) (5.6-1)

whereniis the total amount of substance numberi(a solute). For a solute that accu-

mulates at the surface, the surface excess is positive, and for a substance that avoids

the surface, the surface excess is negative.

The thermodynamic variables of each phase are defined as though the phase were

uniform up to the dividing plane. They obey all of the equations of thermodynamics

without surface contributions. For phase I,

dG(I)−S(I)dT+V(I)dP+

∑c

i 1

μidn(I)i (5.6-2)