Page 92 - Introduction to Colloid and Surface Chemistry

82 Liquid-gas and liquid-liquid interfaces

From the first and second laws of thermodynamics,

d£7 = TdS - pdV + 2/M«/

or, for a surface phase,

a ff 4
dU = YdS" - pdV + ydA + 2/i,d«f ( -!8)
Subtracting equation (4.18) from equation (4.17),

S'dT - V^dp + Ady + S/ifd/t,- = 0

Therefore, at constant temperature and pressure
(4.19)

For a simple two-component solution (i.e. consisting of a solvent and
a single solute) equation (4.19) becomes

As explained above, surface excess concentrations are defined
relative to an arbitrarily chosen dividing surface. A convenient (and
seemingly realistic) choice of location of this surface for a binary
solution is that at which the surface excess concentration of the
solvent (F A) is zero. The above expression then simplifies to

dy = ~F Bd/x B
Since chemical potential changes are related to relative activities by

MB == MB ~^~ •**T In # B

then dfjL B = RT d In a B

Therefore T B = — = -^-. —L (4 20)
RT dlna B RT da B
or, for dilute solutions,
rB= (4 21)
~*F*!~" "

which is the form in which the Gibbs equation is usually quoted.