Page 298 - Physical Chemistry

lev38627_ch09.qxd 3/14/08 1:31 PM Page 279

279
Section 9.6
Thermodynamic
T mix S/n Properties of Ideal Solutions

Figure 9.16
mix H/n Mixing quantities for a two-
component ideal solution as a
function of composition at 25°C.

mix G/n

From G H T S and (9.44) and (9.46), we find
mix mix mix
¢ mix H 0 ideal soln., const. T, P (9.47)

There is no heat of mixing on formation of an ideal solution at constant T and P.
From H U P V at constant P and T and Eqs. (9.45) and (9.47),
mix mix mix
we have U 0 for forming an ideal solution at constant T and P, as expected from
mix
the molecular picture.
Figure 9.16 plots G/n, H/n, and T S/n for an ideal two-component
mix mix mix
solution against the B mole fraction x at 25°C, where n n n .
B B C
Vapor Pressure
If the applied pressure on an ideal liquid solution is reduced until the solution begins
to vaporize, we obtain a two-phase system of solution in equilibrium with its vapor.
As we shall see, the mole fractions in the vapor phase will generally differ from those
v
v
v
in the liquid phase. Let x , x , . . . , x , . . . be the mole fractions in the vapor phase in
1 2 i
equilibrium at temperature T with an ideal liquid solution whose mole fractions are
l
l
l
x , x , ..., x , . . . (Fig. 9.17). The vapor pressure is P and equals the sum of the partial
1 2 i
v
pressures of the gases: P P P P , where P x P [Eq. (1.23)].
1 2 i i i
The system’s pressure equals the vapor pressure P. We now derive the vapor-pressure
equation for an ideal solution.
The condition for phase equilibrium between the ideal solution and its vapor is
v
l
v
l
m m [Eq. (4.88)] for each substance i, where m and m are the chemical potentials
i i i i x , x , . . . , x , . . .
y
y
y
of i in the liquid solution and in the vapor, respectively. We shall assume that the vapor 1 2 i
is an ideal gas mixture, which is a pretty good assumption at the low or moderate pres-
v
sures at which solutions are usually studied. In an ideal gas mixture, m m°
v
i i
v
RT ln (P /P°) [Eq. (6.4)], where m° is the chemical potential of pure ideal gas i at T
i i
l
l
l
and P° 1 bar, and P is the partial pressure of i in the vapor in equilibrium with the x , x , . . . , x , . . .
i 1 2 i
l
v
l
l
solution. Substitution of this expression for m and of m m* RT ln x [Eq. (9.42)]
i i i i
l
l
v
for the ideal solution m into the equilibrium condition m m gives
i i i Figure 9.17
l
v
m m
i
i
An ideal solution in equilibrium
l
l
v
m* 1T, P2 RT ln x m° 1T2 RT ln 1P >P°2 (9.48) with its vapor (y).
i i i i