Page 323 - Physical Chemistry

lev38627_ch10.qxd 3/14/08 1:07 PM Page 304

304
Chapter 10 Since x x 1, we have dx dx 0, and the last equation becomes
B
B
A
A
Nonideal Solutions
d ln g 1x >x 2 d ln g const. T, P (10.22)
B A B A
Integrating between states 1 and 2, and choosing Convention II, we get
ln g II,B,2 ln g II,B,1 2 x A d ln g II,A const. T, P (10.23)
1 1 x A
Let state 1 be pure solvent A. Then g 1 [Eq. (10.10)] and ln g 0. We plot
II,B,1 II,B,1
x /(1 x ) versus ln g . The area under the curve from x 1 to x x gives
A A II,A A A A,2
ln g . Even though the integrand x /(1 x ) → q as x → 1, the area under the
II,B,2 A A A
curve is finite; but the infinity makes it hard to accurately evaluate the integral in
(10.23) graphically. A convenient way to avoid this infinity is discussed in Prob. 10.34.
[Equation (10.23) is for a two-component solution. Surprisingly, if activity-coefficient
data for one component of a multicomponent solution are available over the full range
of compositions, one can find the activity coefficients of all the other components; see
Pitzer (1995) pp. 220, 250, 300.]
Some activity coefficients for aqueous solutions of sucrose at 25°C calculated
from vapor-pressure measurements and the Gibbs–Duhem equation are:
x(H O) 0.999 0.995 0.980 0.960 0.930 0.900
2
g (H O) 1.0000 0.9999 0.998 0.990 0.968 0.939
II 2
g (C H O ) 1.009 1.047 1.231 1.58 2.31 3.23
II 12 22 11
Note from (10.22) that g must increase when g decreases at constant T
II,sucrose II,H 2 O
and P. Because of the large size of a sucrose molecule (molecular weight 342) com-
pared with a water molecule, the mole-fraction values can mislead one into thinking
that a solution is more dilute than it actually is. For example, in an aqueous sucrose
solution with x(sucrose) 0.10, 62% of the atoms are in sucrose molecules, and the
solution is extremely concentrated. Even though only 1 molecule in 10 is sucrose, the
large size of sucrose molecules makes it highly likely for a given sucrose molecule to
be close to several other sucrose molecules, and g deviates greatly from 1.
II,sucrose
Aqueous sucrose solutions have g 1 and g 1. The same reasoning used
II,i II,A
for acetone–chloroform solutions shows that sucrose–H O interactions are more fa-
2
vorable than sucrose–sucrose interactions.
Other Methods for Finding Activity Coefficients
Some other phase-equilibrium properties that can be used to find activity coefficients
are freezing points of solutions (Sec. 12.3) and osmotic pressures of solutions
(Sec. 12.4). Activity coefficients of electrolytes in solution can be found from galvanic-
cell data (Sec. 13.9).
In industrial processes, liquid mixtures are often separated into their pure compo-
nents by distillation. The efficient design of distillation apparatus requires knowledge
of the partial vapor pressures of the mixture’s components, which in turn requires
knowledge of the activity coefficients in the mixture. Therefore, chemical engineers
have devised various methods of estimating activity coefficients. Group-contribution
methods express the activity coefficients as functions of the mole fractions and of pa-
rameters for interactions between the various chemical groups in the molecules of the
components of the solution. The parameter values were chosen to give good fits to
known activity coefficients. Such group-contribution methods (with names like ASOG
and UNIFAC) often work fairly well but sometimes give very inaccurate results.
Other approaches to estimating activity coefficients are considered in Probs. 10.13
to 10.19. A thorough discussion of activity-coefficient estimation methods in liquid
mixtures is given in Poling, Prausnitz, and O’Connell, chap. 8.