Page 299 - Physical Chemistry

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Chapter 9 Let P* be the vapor pressure of pure liquid i at temperature T. For equilibrium be-
i
v
l
Solutions tween pure liquid i and its vapor, we have m* (T, P*) m* (T, P*) or [Eq. (6.4)]
i
i
i
i
l v
m* 1T, P*2 m° 1T 2 RT ln 1P*>P°2 (9.49)
i
i
i
i
Subtraction of (9.49) from (9.48) gives
l l l
m* 1T, P2 m* 1T, P*2 RT ln x RT ln 1P >P*2 (9.50)
i
i
i
i
i
i
For liquids, m* (which equals G* ) varies very slowly with pressure (Sec. 4.4), so it
m,i
i
l
l
is an excellent approximation to take m* (T, P) m* (T, P*) (unless the pressure is
i
i
i
l
very high). Equation (9.50) then simplifies to RT ln x RT ln (P /P*). If ln a ln b,
i
i
i
l
then a b. Therefore x P /P* and
i
i
i
l
P x P* ideal soln., ideal vapor, P not very high (9.51)*
i
i
i
In Raoult’s law (9.51), P is the partial pressure of substance i in the vapor in
i
l
equilibrium with an ideal liquid solution at temperature T, x is the mole fraction of i
i
in the ideal solution, and P* is the vapor pressure of pure liquid i at the same temper-
i
l
ature T as the solution. Note that as x in (9.51) goes to 1, P goes to P*, as it should.
i
i
i
l
l
As x increases, both the chemical potential m (Fig. 9.15) and the partial vapor pres-
i
i
sure P increase. Recall that m is a measure of the escaping tendency of i from a phase.
i
i
v
Since P x P [Eq. (1.23)], Raoult’s law can be written as
i
i
v
l
x P x P* (9.52)
i
i
i
where P is the (total) vapor pressure of the ideal solution.
The vapor pressure P in equilibrium with an ideal solution is the sum of the par-
tial pressures. For a two-component solution, Raoult’s law gives
l
l
l
l
P P P x P* x P* x P * 11 x 2P* (9.53)
B
C
B
C
B
B
B
B
C
C
P 1P* P * 2x P * (9.54)
l
B
C
B
C
At fixed temperature, P* and P* are constants, and the two-component ideal-solution
C
B
l
l
vapor pressure P varies linearly with x . For x 0, we have pure C, and P P*. For
B
C
B
l
x 1, the solution is pure B, and P P*. Figure 9.18a shows the Raoult’s law partial
B
B
pressures P and P [Eq. (9.51)] and the total vapor pressure P of an ideal solution as
B
C
a function of composition at fixed T. A nearly ideal solution such as benzene– toluene
shows a vapor-pressure curve that conforms closely to Fig. 9.18a. Figure 9.18b plots
v
l
x versus x in an ideal two-component solution for the three cases P* 3P*, P*
C
B
B
B
B
P*, and P* P*/3. Note that the vapor is richer than the liquid in the more volatile
C
B
C
x y B
Figure 9.18
(a) Partial pressures P and P C
B
and (total) vapor pressure P
P P above an ideal solution
B
C
as a function of composition at
fixed T. (b) Vapor-phase mole
l
fraction of B versus x for an ideal
B
solution of B C plotted for three
different ratios P*/P* of the pure- x B l x B l
C
B
component vapor pressures.