Multicomponent Equilibrium Thermodynamics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 72

γ i with
γ i ,theso-called infinitedilutionactivitycoefficient. Theconditionofinfinitedilution
is the limit as x i →

For sparingly soluble organic compounds in water, it is possible to equate

0. For a large number of sparingly soluble organics in water,
this condition is satisfied even at their saturation solubility. Hence the two activity
coefficientsareindistinguishable.Themolefractionatsaturationformostcompounds
of environmental significance lies between 10 − 6 and 10 − 3 . The mole fraction of 10 − 6
is usually taken as the practical limit of infinite dilution . For those compounds for
which the solubility values are very large, γ i .

Using the Gibbs-Helmholtz relation obtained in Chapter 2 and replacing the Gibbs
free energy change in that expression by the excess molar Gibbs free energy of dis-
solution of liquid i , we can derive the following fundamental relationship for activity
coefficient:

γ i is significantly different from

γ i h i d ln

d T =− RT 2 ,

(3.62)

where h i is called the excess molar enthalpy of solution for component i . If the excess
molar enthalpy of dissolution is constant over a small range of temperature, we should
get a linear relationship

h i

γ i = ln

RT +

c , (3.63)

where c is a constant. Since no phase change is involved for liquid solutes, the excess
enthalpyisidenticaltotheenthalpychangeforsolution.Experimentalevidenceforthe
effectoftemperatureonthesolubilityofseveralorganicliquidsinwatershowsthatthe
decrease in activity coefficient (increase in solubility) over a 20 ◦ rise in temperature
is in the range 1-1.2 (Tse, Orbey, and Sandler, 1992). This is shown in Figure 3.7.
For some of the compounds the value of activity coefficient decreases slightly with
increase in temperature, particularly if the excess molar enthalpy is near zero or
changes sign within the narrow range of temperature. In conclusion, we may state that
within the narrow range of temperatures encountered in the environment, the activity
coefficients of liquid solutes in water do not change appreciably with temperature.
For compounds that are solids or gases at the temperature of dissolution in water,
the total enthalpy of solution will include an additional term resulting from a phase
change (sub-cooled liquids for solid solutes and super-heated fluids for gases) that
will have to be added to the excess enthalpy of solution. For most solids, the additional
term for phase change (enthalpy of melting) will dominate and hence the effect of
temperature on activity coefficient will become significant.The result for naphthalene
(a solid at room temperature) is shown in Figure 3.7. The mole fraction solubility of
naphthalene at 281 K was 2.52

10 − 6 .
Thus, in environmental engineering the effects of temperature on activity coefficients
for solids in water cannot be ignored.

10 − 6 whereas at 305 K it increased to 5.09 × ×

3.4.3 C ORRELATIONS WITH H YDROPHOBICITY
Organic compounds of low aqueous solubility have generally large activity coeffi-
cients. In order to explain the large activity coefficients, we need a knowledge of the
structure of water in both the liquid and solid states. The following discussion focuses