Physical Chemistry Third Edition

246 6 The Thermodynamics of Solutions

graph, and a temperature–composition diagram is created by passing a plane of con-

stant pressure though the three-dimensional graph. The intersection of these two planes

contains the common tie line of the two diagrams, as shown in the figure.

A constant-pressuredistillationprocess can be described using the temperature–

composition phase diagram, as depicted in Figure 6.5. Pointaon the diagram in this

figure represents the composition of a liquid solution that is being boiled in the still.

Pointb, at the other end of the tie line, represents the composition of the vapor at

equilibrium with this liquid. This vapor condenses at the temperature represented by

pointc. A simple still in which this process can be carried out is said to have one

theoretical plate. A still can be made to produce a greater separation of the components

by packing its column with glass beads or other objects. The liquid condenses on the

glass beads part way up the column and then evaporates again, making this part of

the column equivalent to one theoretical plate. For the process beginning at pointa,

a second evaporation at pointcleads to a vapor with the composition at pointd, and

this vapor can condense still further up the column, giving a liquid corresponding to

pointe. This process corresponds to two theoretical plates. Three theoretical plates lead

to a liquid with the composition at pointg. A still with a large number of theoretical

plates can lead to a condensate that is almost entirely made up of the more volatile

component. A “spinning-band” still has a rotating helical wire screen that wipes the

walls of the column and can provide several hundred theoretical plates.

Vapor

T/

K

400

390

380

370

360

350

340

0

a b

c d

e f

g

0.2

Mole fraction of benzene

0.4 0.6 0.8 10

Liquid

1st

theoretical

plate 2nd

theoretical

plate

3rd

theoretical

plate

Figure 6.5 Diagram Representing a

Constant-Pressure Distillation Pro-

cess. Each theoretical plate corres-

ponds to one step in the “staircase.”

Exercise 6.7

Estimate from Figure 6.5 the composition and boiling temperature of the condensate produced

from the liquid at pointaby a still with three theoretical plates.

A two-component solid–liquid phase diagram is similar to a liquid–vapor phase

diagram if the solids are miscible. Figure 6.6 shows the temperature–composition

phase diagram of silicon and germanium, which form a nearly ideal solid solution and

a nearly ideal liquid solution. The area below the lower curve represents one-phase solid

solutions, and the area above the upper curve represents one-phase liquid solutions. The

area between the curves is a tie-line region, with the two ends of a tie line representing

the compositions of the solid phase and the liquid phase at equilibrium with each other.

A continuation of the silicon–germanium diagram to higher temperatures leads to the

liquid–vapor transition region, giving a diagram with two areas of tie lines, as shown

in Figure 6.7.

1750

1700

1650

1600

1550

1500

T/K^1450

1400

1350

1300

1250

1200

1150

0 0.2

Mole fraction of Si

0.4 0.6 0.8 10

Figure 6.6 The Solid–Liquid Tem-

perature–Composition Phase Diag-

ram of Silicon and Germanium.Since

both the solid and liquid phases are

nearly ideal solutions, this diagram

resembles the liquid–vapor phase dia-

gram of an ideal liquid solution. From

C. D. Thurmond,J. Phys. Chem., 57 , 827

(1953).

Maximum Solubilities in Ideal Solutions

Benzene and naphthalene form a nearly ideal liquid solution. However, these two

substances are nearly insoluble in each other’s solid phase since they cannot both fit into

a single crystal lattice. Figure 6.8 schematically shows the vapor pressure and chemical

potential of naphthalene in a liquid solution with benzene near room temperature. The

standard state of naphthalene for the liquid solution is supercooled liquid naphthalene,

which has a higher chemical potential than solid naphthalene. There is a range of mole

fractions of naphthalene near unity in which the chemical potential of naphthalene in the

solution would exceed that of the solid naphthalene. The mole fraction of naphthalene at

which its chemical potential equals that of the solid represents themaximum solubility

of naphthalene in a solution with benzene. A solution with this composition is said