Page 16 - Partition & Adsorption of Organic Contaminants in Environmental Systems

CHEMICAL POTENTIALS IN MULTIPLE PHASES 7

two phases. For a system involving a change in the quantity of its components,
due, for example, to chemical reactions or transfer of mass to and out of the
system, the previous differential equations are adjusted to take into account
the changes in the moles (n 1, n 2, n 3, etc.) of individual components. By extend-
ing Eqs. (1.8) and (1.15), one now obtains

dE T dS P dV + ∂ ( E ∂ ) dn + ◊◊◊ + ∂ ( E ∂ ) dn k (1.19)
=
-
1
n k
n 1
,,
,,
VS n j VS n j
and
dG V dP SdT + ∂ ( G ∂ ) dn 1 + ◊◊◊ ( ∂ G ∂ ) dn k (1.20)
+
-
=
n 1
n k
,,
,,
TP n j TP n j
Thus, for a reversible process involving a change in individual-component
mass in a phase,
dE = T dS - P dV + Âm i dn i (1.21)
and
dG = V dP - S dT + Âm i dn i (1.22)
in which the chemical potential or the molar Gibbs function of component i
is defined as

m i =∂ ( E ∂ ) =∂ ( G ∂ ) (1.23)
n
n
, ,
,,
i VS n j i TP n j
Since the chemical potential of a substance is an intensive property, the dif-
ference in its values between regions in a phase or between phases of a system
determines the direction of mass transfer (from the one of higher potential to
the one of lower potential), just as the temperature gradient determines the
direction of heat flow. The usefulness of the chemical potential as a criterion
for equilibrium of a substance between phases is illustrated below.

1.6 CHEMICAL POTENTIALS IN MULTIPLE PHASES

Consider a closed system consisting of two separate phases,A and B, to which
an organic compound (solute) is added at constant temperature and pressure,
as shown in Figure 1.1. The solute i will then distribute itself between phases
A and B, to arrive eventually at some stable concentrations when the system
reaches the state of equilibrium. Here one may express the change in Gibbs
free energy of the entire system as

DG i = DG i + DG i,B or dG i = dG i + dG i,B (1.24)
,A
,A