Thermodynamics and Chemistry

CHAPTER 8 PHASE TRANSITIONS AND EQUILIBRIA OF PURE SUBSTANCES

8.2 PHASEDIAGRAMS OFPURESUBSTANCES 199

By the same reasoning, mechanical equilibrium involvesP 1 independent relations
among pressures, and transfer equilibrium involvesP 1 independent relations among
chemical potentials.
The total number of independent relations for equilibrium is3.P1/, which we subtract
from3P (the number of independent variables in the absence of equilibrium) to obtain
the number of independent variables in the equilibrium system: 3P3.P1/D 3.
Thus,an open single-substance system with any number of phases has at equilibrium three
independent variables. For example, in equilibrium states of a two-phase system we may
varyT,ní, andnìindependently, in which casepis a dependent variable; for a given value
ofT, the value ofpis the one that allows both phases to have the same chemical potential.

8.1.7 The Gibbs phase rule for a pure substance

The complete description of the state of a system must include the value of anextensive
variable of each phase (e.g., the volume, mass, or amount) in order to specify how much of
the phase is present. For an equilibrium system ofPphases with a total of 3 independent
variables, we may choose the remaining 3 Pvariables to beintensive. The number of these
intensive independent variables is called thenumber of degrees of freedomorvariance,
F, of the system:

FD 3 P (8.1.17)

(pure substance)

The application of the phase rule to multicomponent systems will be taken up in Sec.

13.1. Equation8.1.17is a special case, forCD 1 , of the more general Gibbs phase

ruleFDCPC 2.

We may interpret the varianceFin either of two ways:
 F is the number of intensive variables needed to describe an equilibrium state, in
addition to the amount of each phase;
 F is the maximum number of intensive properties that we may vary independently
while the phases remain in equilibrium.
A system with two degrees of freedom is calledbivariant, one with one degree of free-
dom isunivariant, and one with no degrees of freedom isinvariant. For a system of a
pure substance, these three cases correspond to one, two, and three phases respectively.
For instance, a system of liquid and gaseous H 2 O (and no other substances) is univariant
(F D 3 PD 3 2 D 1 ); we are able to independently vary only one intensive property,
such asT, while the liquid and gas remain in equilibrium.

8.2 Phase Diagrams of Pure Substances

Aphase diagramis a two-dimensional map showing which phase or phases are able to exist
in an equilibrium state under given conditions. This chapter describes pressure–volume and
pressure–temperature phase diagrams for a single substance, and Chap. 13 will describe
numerous types of phase diagrams for multicomponent systems.