Multicomponent Equilibrium Thermodynamics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 83

Note that if the free energy were 19.3 kJ/mol instead of the calculated value of
23 kJ/mol, we would have predicted the actual experimental value of the activity coef-
ficient. The above calculation shows how sensitive the estimates are to the free energy
of solution due to its exponential dependence. The free energy is a function of both
surface energy and molecular area, and hence both of these contribute to uncertainty
in the free energy calculation. The dilemma arises as to what is the exact value of sur-
face energy that should be used, since at small radius of curvature the surface energy
is dependent on curvature. The concept of surface tension does not hold for a single
molecule; however, the concept of interfacial energy remains valid even for an isolated
molecule (Israelchvili, 1992). Sinanoglu (1981) showed that clusters of gaseous car-
bon tetrachloride composed of even as small as seven molecules show surface energies
similar to those of a planar macroscopic surface.
If we use the correlation given in Table 3.13, the solubility in water for benzene is
C i = 0.0467 mol/dm 3 , which gives a γ i = 1.2 × 10 3 .

E XAMPLE 3.15 A QUEOUS S OLUBILITY FROM M OLECULAR P ARAMETERS

Calculate the aqueous solubility of naphthalene from molecular area.
For naphthalene A m is 156.76Å 2 . Hence log C i =− 0.0282 ( 156.76 ) + 1.42 −
0.0095 ( 80.3 ) =− 3.76. C i = 0.00017 mol/L.

E XAMPLE 3.16 C ALCULATIONS OF A QUEOUS S OLUBILITY FROM K ow
(a) Naphthalene: Melting point = 80.3 ◦ C (a solid at room temperature), log K ow =
3.25. Hence from Table 3.13 we have − log C i = ( 0.880 )( 3.25 ) + ( 0.01 )( 80.3 ) +
0.012 = 3.675. Hence C i = 2.11 × 10 − 4 mol/dm 3

= 27 mg/dm 3 . The experimental
value is 30 mg/dm 3 . Hence the error in the estimation is − 10%.
(b) 1,2,4-Trichlorobenzene: Melting point = 17 ◦ C (a liquid at room temperature),
log K ow = 4.1. Hence from Table 3.13 we have − log C i = ( 0.987 )( 4.1 ) + 0.718 =
4.76. Hence C ∗ t
i

= 3.1 mg/dm 3 . The experimental value is
31 mg/dm 3 . The error in the estimation is − 90%. Such large errors are not unusual and
hence care should be taken in accepting the estimated solubilities at their face values.

= 1.72 × 10 − 5 mol/dm 3

3.4.5 T HEORETICAL AND S EMI -E MPIRICAL A PPROACHES TO A QUEOUS

S OLUBILITY P REDICTION

3.4.5.1 First-Generation Group Contribution Methods

Langmuir (1925) suggested that each group in a molecule interacts separately with
the solvent molecules around it, and hence the overall interaction can be determined