Physical Chemistry Third Edition

218 5 Phase Equilibrium

24

22

20

18

16

14

12

C 10

/J gs

–1

K

–1

8

6

4

2

–1.5 –0.5 0 –4 –2 0 2 4 6 –20 –10 0 10 20 30

(T–Tl)/K (T–Tl) / mK (T–Tl)/mK

0.5 1.5

Figure 5.13 The Heat Capacity of Helium Near the Lambda Transition. The heat

capacity appears to become infinite, as in a first-order phase transition, but it rises smoothly

instead of showing a spike at one point as does a first-order phase transition.

order. The heat capacity rises smoothly toward infinity instead of rising abruptly as

in a first-order transition. A plot of the heat capacity versus the temperature resem-

bles the Greek letter lambda as shown in Figure 5.13, and the transition is called

alambda transition. The order-disorder transition in beta brass is also a lambda

transition.

The Critical Point of a Liquid–Vapor Transition

The tangents to the two curves in Figure 5.6 represent the molar volumes. In the case

of a liquid–vapor transition the two molar volumes become more and more nearly

equal to each other as the temperature is increased toward the critical temperature. The

tangents to the curves approach each other more and more closely until there is only one

curve at the critical temperature. The discontinuity in the graph of the molar volume

in Figure 5.7 gradually shrinks to zero at the critical temperature. However, there is

a vertical tangent at the critical pressure as schematically shown in Figure 5.14. The

isothermal compressibility is infinite, but it does not suddenly jump to an infinite value

at one point as it does below the critical temperature. It rises smoothly (and steeply)

toward an infinite value at the critical point.

At the critical point, the two curves representingSmin Figure 5.8 also merge into a

single curve. The molar entropy of the liquid and the molar entropy of the gas phase

approach each other. The discontinuity in the curve of Figure 5.8 shrinks to zero, but

there is a vertical tangent at the critical point. The heat capacity rises smoothly and

steeply toward an infinite value. Its behavior is similar to that of the heat capacity at a

lambda transition.

C

P

T

Figure 5.11 The Constant-Pres-

sure Heat Capacity as a Function

of Temperature at a Second-Order

Phase Transition (Schematic).

kT

P

Figure 5.12 The Isothermal Com-
pressibility as a Function of Pres-
sure at a Second-Order Phase
Transition (Schematic).

Vm

Pc

Vmc

P

Figure 5.14 The Molar Volume Near

a Liquid–Vapor Critical Point.