Page 42 - Partition & Adsorption of Organic Contaminants in Environmental Systems

TEMPERATURE DEPENDENCE OF PARTITION COEFFICIENT 33

estimated by Raoult’s law. The more rigorous expression for K pw is obtained
by equating the activity of the solute in water [Eq. (2.1)] with that in the poly-
meric organic phase [Eq. (2.10)]. With a dilute-solution approximation [Eq.
(3.3)] for the solute in water, this gives
*
logK pw = logg w - logV + logV w - ( [ 1 - V V p )+ ] c . 2 303 - log(g w g * )
w
(3.12)
where V is the molar volume of the solute and V p is the molar volume of the
polymeric or macromolecular substance. Substituting Eq. (3.7) into (3.12) with
w = V
V * w, one obtains for liquid (or supercooled-liquid) solutes,
*
logK pw =- logS V - ( [ 1 - V V p )+ ] c . 2 303 - log(g w g w ) (3.13)
w
and, if /VV p 0,

+ )
logK pw =- logS V - (1 c . 2 303 - log(g w g * w ) (3.14)
w
It is often more convenient to express the concentration of the solute in a
polymeric or macromolecular phase on a mass-to-mass basis when the condi-
tion /VV p 0 suffices. The K pw in Eq. (3.14), expressed as a mass-to-mass con-
centration ratio, is therefore

*
logK pw =- logS V - log -(1 + ) c . 2 303 - log(g w g w ) (3.15)
r
w
where r is the density (g/mL) of the macromolecular organic phase and the
density of the water phase is assumed to be 1. The superiority of Eq. (3.14) or
(3.15) to (3.11) for the partition of liquid and supercooled-liquid solutes from
water into a macromolecular phase is illustrated later.

3.4 TEMPERATURE DEPENDENCE OF
PARTITION COEFFICIENT

Temperature affects the partition coefficient of an organic solute between two
separated solvent phases. Since the partition coefficient of a solute (K) is a
function of its relative solubilities in the two solvents, the dependence of K on
temperature reflects the net temperature effect of the solute solubilities with
the solvents.
One may start with Eq. (3.11) for a solute at dilution in a solvent–water
mixture. The solute is considered to have a limited solubility in water. Taking
a derivative of logK sw with T gives
* * *
dlog K sw dlog S w dlog V s dlogg s dlog (g w g w )
=- - - - (3.16)
dT dT dT dT dT
For simplicity, the second and fourth terms on the right of Eq. (3.16) may be
approximated as zero because the molar volume of a liquid (solvent) is not