Physical Chemistry Third Edition

5.2 The Gibbs Phase Rule 203

equilibrium intensive state of a one-phase simple system requiresc+1 independent

variables, all of which must be intensive. A convenient set of independent variables

to specify the equilibrium intensive state of a one-phase simple system consists ofT,

P, andc−1 mole fractions. Onlyc−1 mole fractions can be independent variables,

since the mole fractions automatically obey the relation

∑c

i 1

xi 1 (5.2-1)

If all of the mole fractions but one are specified, the last one is determined.

Counting the Number of Phases and Components

in a System

We denote the number of phases in a system byp. In counting phases, we count

only regions that are different in their intensive properties from other regions. For

example, liquid water and crushed ice make up a two-phase system, just like a system

of liquid water and a single ice cube. Thecomponentsof a system are substances

whose amounts can be varied independently. The number of components is equal to

the number of chemical species present minus the number of relations that constrain

the amounts of the species. There are three principal types of constraints: (1) relations

due to chemical equilibrium; (2) a relation due to a requirement of electrical neutrality

(which we always assume to exist); and (3) relations due to the way the system was

prepared (such as a specification that two substances are in their stoichiometric ratio).

The number of components is also equal to the number of substances from which

the system could be prepared, given the conditions imposed on the system. A mixture

of gaseous hydrogen, oxygen, and water vapor can remain unreacted for a very long

time at room temperature if no catalyst is present. We can treat this metastable mixture

as if no reaction were possible and say that there are three components. If a platinum

catalyst is introduced into the system, a chemical equilibrium is rapidly established,

reducing the number of components to two (not counting the catalyst). In the presence

of the catalyst the amount of water vapor is determined by the amounts of hydrogen

and oxygen and the nature of the chemical equilibrium. If the additional constraint is

added that the hydrogen and oxygen are in the stoichiometric ratio of 2 moles to 1, then

the system has only one component. In this case the system could be produced from

water vapor in the presence of the catalyst.

EXAMPLE 5.1

Determine the number of components in:

a.An aqueous solution containing Na+,Cl−, and Br−.

b.An aqueous solution containing Na+,K+,Li+,Cl−, and Br−.

c.A gaseous system containing NO 2 and N 2 O 4 at chemical equilibrium with each other.

d.An aqueous solution containing Ca^2 +ions and Cl−ions.

Solution

a.There are three components. The system can be produced from three pure substances:

water, NaCl, and NaBr. The number can also be determined by counting up water and the

three ions and subtracting unity for the condition of electrical neutrality.