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E
E
Chapter 10 so G can be found from the activity coefficients. Conversely, if G is known as a func-
Nonideal Solutions tion of solution composition, the activity coefficients can be calculated from G (see
E
Prob. 10.5).
Excess functions are found from mixing quantities. We have
id
id
E
id
G G G G G G* G* G G* 1G G*2
E id
G ¢ mix G ¢ mix G
The same argument holds for other excess properties, and (since mix H id 0 and
id
mix V 0)
E id E id
G ¢ mix G ¢ mix G , S ¢ mix S ¢ mix S ,
E E
H ¢ mix H, V ¢ V
mix
id
id
where G and S are given by (9.44) and (9.46).
mix mix
id
id
Figure 10.1 shows typical curves of G, G , G, G , and G versus com-
E
mix mix
position at constant T and P for solutions of two liquids B and C that show positive
deviations from ideality. In drawing the curves, it was arbitrarily assumed that G
m,C
0 and G 10 kJ/mol.
m,B
10.3 DETERMINATION OF ACTIVITIES
AND ACTIVITY COEFFICIENTS
The formalism of Sec. 10.1 leads nowhere unless we can determine activity coeffi-
cients. Once these are known, the chemical potentials m are known, since m m°
i i i
RT ln g x [Eq. (10.6)]. From the chemical potentials, the other thermodynamic prop-
i i
erties can be found.
Activity coefficients are usually found from data on phase equilibria, most com-
monly from vapor-pressure measurements. The condition for phase equilibrium be-
tween the solution and its vapor is that for each species i the chemical potential m in
i
v
the solution must equal the chemical potential m of i in the vapor phase. We shall as-
i
sume the vapor in equilibrium with the solution to be an ideal gas mixture. Departures
mix G/n from ideality in gases are ordinarily much smaller than are departures from ideal-
solution behavior in liquids. (See Sec. 10.10 for allowance for gas nonideality.) Since
v
m depends on the vapor partial pressure P and since m in solution depends on g ,
mix G /n i i i i
id
measurement of P allows the activity coefficient g to be found. The vapor partial
i i
pressure P allows us to probe the escaping tendency of i from the solution.
i
Convention I
Suppose we want a solution’s activities a and activity coefficients g for the
I,i I,i
Figure 10.1 Convention I choice of standard states. Recall that for an ideal solution we started from
l
l
m m° RT ln x and derived Raoult’s law P x P* (Sec. 9.6). For a real solution,
i
i
i
i
i i
E
Typical curves of G, G , and the activity replaces the mole fraction in m , and we have m m° RT ln a . Also,
i
I,i
i
I,i
mix G at 25°C for solutions of the Convention I standard states are the same as the ideal-solution standard states, so
two liquids that show positive
id
deviations from ideality. G and m° has the same meaning in these two expressions for m . Therefore, exactly the same
i
i
l
mix G are the corresponding steps that gave Raoult’s law P x P* in Sec. 9.6 will give for a nonideal solution
id
i
i
i
quantities for ideal solutions. Of
i
i
E
I,i
course, G 0. n is the total P a P* ideal vapor, P not very high (10.13)*
id
number of moles.
Thus a P /P*, where P is the partial vapor pressure of i above the solution and P*
I,i i i i i
is the vapor pressure of pure i at the temperature of the solution.
At a given temperature, P* is a constant, so (10.13) shows that the activity a of
i I,i
a substance in a solution is proportional to the vapor partial pressure P of the solution.
i
l
Therefore a plot of P versus x is, except for a change in scale, the same as a plot of
i i