Physical Chemistry Third Edition

5.6 Surfaces in Multicomponent Systems 233

Equating Eq. (5.6-4) and Eq. (5.6-10), canceling equal terms, and dividing byA,we

obtain a surface version of the Gibbs–Duhem equation:

0 

S(σ)

A

dT+dγ+

∑c

i 1

Γ(iσ)dμi (5.6-11)

The surface excess per unit area,Γ

(σ)

i , is defined by

Γ(iσ)

n(iσ)

A

(definition) (5.6-12)

and is called thesurface concentration. If the temperature is constant,

dγ−

∑c

i 1

Γ

(σ)

i dμi (constant temperature) (5.6-13)

This equation can be interpreted as follows: If adding substancei(which raisesμi)

decreases the surface tension, thenΓ

(σ)

i is positive and substanceiaccumulates at the

interface. A substance that significantly lowers the surface tension and accumulates

at the surface is called asurfactant. If raising the chemical potential of substancei

increases the surface tension, thenΓ(iσ)is negative, and substanceiavoids the interface.

We will see in Chapter 6 that for a solute in a dilute solution we can write to an

adequate approximation

μiμ◦i+RTln(ci/c◦) (5.6-14)

whereμ◦i is a constant at constantTandP, whereciis the molar concentration of

substanceiexpressed in mol L−^1 , and wherec◦is defined to equal exactly 1 mol L−^1.

For a two-component solution, we position the surface dividing the phases so thatΓ 1 ,

the surface concentration of the solvent, is zero. It can be shown that

Γ 2 −

1

RT

(

∂γ

∂ln(c 2 )

)

T,P

(5.6-15)

and

Γ 2 −

c 2

RT

(

∂γ

∂c 2

)

T,P

(5.6-16)

where the solute is called substance number 2.

Exercise 5.17

Show that Eq. (5.6-15) and Eq. (5.6-16) follow from Eqs. (5.6-13) and (5.6-14).

There are many kinds of systems with interfacial effects that are of practical interest.

Such systems have a large surface area per unit mass and either consist of very small

particles or have an extremely irregular surface. Surface effects can dominate in deter-

mining the behavior of such systems.Colloidsare suspensions of small solid particles

in a liquid medium.Aerosolsare suspensions of fine solid or liquid particles in a gas

and are important in atmospheric chemistry and physics. A number of solids with large