Page 32 - Partition & Adsorption of Organic Contaminants in Environmental Systems

MOLAR HEAT OF SOLUTION 23

stance in all phases must be the same. Therefore, for solute i at equilibrium
between the vapor and solution phases,

(
(
m i sol) = m i vap) (2.24)
where m i(sol) is the chemical potential of solute i in solution and m i(vap) is the
chemical potential in the vapor phase. By Eq. (1.35), one arrives at

o
(
m i vap) = m i + RT ln i P (2.25)
where m° i is the chemical potential of the vapor i at 1 atm and T.
If we choose Raoult’s law to express the dissolved solute activity and
assume that the solute is a liquid at T, then P i = P° i x ig i, where P° i is the vapor
pressure of pure liquid (or supercooled liquid) i. Thus, for liquid or super-
cooled-liquid solutes at temperature T, Eq. (2.25) can be written as

o
(
m i vap) = m i + RT ln o i P + RT ln i x g i (2.26)
or
*
(
(
m i vap) = m i liq + ) RT ln i x g i (2.27)
where m* i (liq) is recognized as the chemical potential of pure liquid i at
T. Hence, the chemical potential of solute i in solution can be related to its
concentration as
*
m i = m i (sol ) = m i ( )+ RT ln i x g i (2.28)
liq
or alternatively,
* * o
liq
ln
liq
m i = m i ( )+ RT ln i a = m i ( )+ RT ( i P P ) (2.29)
i
In light of the fact that the supercooled-liquid state of a solid substance is
metastable, it is also desirable to express the chemical potential of the dis-
solved solid with the pure solid as the reference state. The relation between
the chemical potentials of the solid, m* i (sld), and its supercooled liquid, m* i (liq),
at temperature T is given as
*
*
)
s
m i ( ) = m i (sld + RT ln o i P ( i P ) (2.30)
liq
s
In Eq. (2.30), since P° i > P i one finds that m* i (liq) >m* i (sld), as the supercooled-
liquid state is unstable. Here m* i (liq) -m* i (sld) =DG i (fus) is called the molar free
energy of fusion of the solid at T. Substitution of Eq. (2.30) into (2.28) gives
an alternative expression for a dissolved solid solute as
* o s * s
)(
)
)
ln
m i = m i (sld + RT ln i x ( [ g i P i P )] = m i (sld + RT ( i P P ) (2.31)
i
i