Physical Chemistry Third Edition

1.4 The Coexistence of Phases and the Critical Point 33

Tt

(^0) Tc
Solid–liquid
tie line
Liquid–gas
tie line
Triple point
tie lines
Solid–gas
tie line
Vm
T
P
Figure 1.8 Surface Giving Pressure as a Function of Molar Volume and Temperature
Showing All Three Phases (Schematic).
are three sets of tie lines, corresponding to the three curves in Figure 1.4. At the triple
point, all three tie lines come together in a single tie line connecting three phases. As
shown in Figures 1.7 and 1.8, the pressure of a one-phase system of one substance
is a function of only two intensive variables,TandVm. Any intensive variable in a
one-component fluid system is also a function of two intensive variables. Theintensive
stateof an equilibrium system is the state of the system so far as only intensive variables
are concerned, and is specified by two independent intensive variables if an equilibrium
system contains a single substance and a single fluid phase. The size of the system is not
specified. For a one-phase fluid (liquid or gas) system ofcsubstances,c+1 intensive
variables specify the intensive state of the system.
The Law of Corresponding States
The van der Waals equation predicts that the value of the compression factor at the
critical point is equal to 0.375 for all substances. There is even a greater degree of
generality, expressed by an empirical law called thelaw of corresponding states:^6 All
substances obey the same equation of state in terms of reduced variables.The reduced
variables are dimensionless variables defined as follows: Thereduced volumeis the
ratio of the molar volume to the critical molar volume:
Vr
Vm
Vmc

(1.4-12)

Thereduced pressureis the ratio of the pressure to the critical pressure:

Pr

P

Pc

(1.4-13)

(^6) Hirschfelder, Curtiss, and Bird,op. cit., p. 235 [see Table 2.1].