component. Also note that the bubble point line and the dew point line have
switched places.
Fractional distillations can also be illustrated using temperature-composition
phase diagrams. A solution of initial composition vaporizes into a vapor having
a different composition. If this vapor is cooled, it condenses into a liquid hav-
ing the same composition. This new liquid can establish an equilibrium with
another vapor having a more enriched composition, which condenses, and so
on. Figure 7.11 illustrates the stepwise process. Three theoretical plates are
shown explicitly.
Raoult’s law is one requirement for an ideal liquid solution. There are a few
other requirements for an ideal solution. When two pure components are
mixed, there should be no change in the total internal energy or enthalpy of
the components:(^) mixU 0 (7.25)
(^) mixH 0 (7.26)
If the solution is mixed under conditions of constant pressure (which is usu-
ally an applicable restriction), then equation 7.26 implies that
qmix 0
Mixing is usually a spontaneous process, which means that (^) mixSand (^) mixG
for the process must have the proper magnitudes. Indeed, in analogy to gas
mixtures, for ideal liquids they are
(^) mixGRT
i
xiln xi (7.27)
(^) mixSR
i
xiln xi (7.28)
for constant-temperature processes. Since xiis always less than 1, the loga-
rithms ofxiare always negative, so (^) mixGand (^) mixSwill always be nega-
tive and positive, respectively. Mixing is a spontaneous, entropy-driven
process. When one uses equations 7.27 and 7.28 and the units come out as
joules per mole, the “per mole” part refers to the moles of components in
the system. To calculate a total quantity, the amount per mole must be
multiplied by the number of moles in the system, as shown in the follow-
ing example.
Example 7.6
What are (^) mixH, (^) mixU, (^) mixG, and (^) mixSfor a system that mixes 1.00 mol
of toluene and 3.00 mol of benzene? Assume ideal behavior and 298 K.
Solution
By definition, (^) mixHand (^) mixUare exactly zero. The total number of moles
in our system is 4.00 mol, so for (^) mixG, we use x 1 0.250 and x 2 0.750.
Therefore,
(^) mixG8.314
mo
J
lK(298 K)(0.250 ln 0.250 0.750 ln 0.750)
(^) mixG1390 J/mol 4.00 mol 5560 J
178 CHAPTER 7 Equilibria in Multiple-Component Systems
T(BP 2 )
T(BP 1 )
Temperature
0.5
x 1 , y 1
0.0 1.0
Tie line
Figure 7.11 Fractional distillations can also
be represented using temperature-composition
phase diagrams. This diagram shows the same
process as Figure 7.8. Can you explain the differ-
ences between the two representations of the same
process?