Thermodynamics and Chemistry

CHAPTER 13 THE PHASE RULE AND PHASE DIAGRAMS

13.1 THEGIBBSPHASERULE FORMULTICOMPONENTSYSTEMS 422

We may need toremovea component from a phase to achieve the final composition.

Note that it is not necessary to consider additional relations for electroneutrality or

initial conditions; they are implicit in the definitions of the components. For instance,

since each component is a substance of zero electric charge, the electrical neutrality of

the phase is assured.

We conclude that, regardless of the kind of system, the expression forF based on
components is given byFD 2 CCP. By comparing this expression andF D 2 Cs
rP, we see that the number of components is related to the number of species by

CDsr (13.1.5)

13.1.4 Examples

The five examples below illustrate various aspects of using the phase rule.

Example 1: liquid water

For a single phase of pure water,P equals 1. If we treat the water as the single species
H 2 O,sis 1 andris 0. The phase rule then predicts two degrees of freedom:

FD 2 CsrP

D 2 C 1  0  1 D 2 (13.1.6)

SinceFis the number of intensive variables that can be varied independently, we could for
instance varyTandpindependently, orTand, or any other pair of independent intensive
variables.
Next let us take into account the proton transfer equilibrium

2 H 2 O.l/ïH 3 OC.aq/COH.aq/

and consider the system to contain the three species H 2 O, H 3 OC, and OH. Then for the
species approach to the phase rule, we havesD 3. We can write two independent relations:
1.for reaction equilibrium,2H 2 OCH 3 OCCOHD 0 ;
2.for electroneutrality,mH 3 OCDmOH.

Thus, we have two relations involving intensive variables only. Nowsis 3,ris 2,Pis 1,
and the number of degrees of freedom is given by

FD 2 CsrPD 2 (13.1.7)

which is the same value ofFas before.
If we consider water to contain additional cation species (e.g., H 5 O 2 C), each such
species would add 1 tosand 1 tor, butF would remain equal to 2. Thus, no matter
how complicated are the equilibria that actually exist in liquid water, the number of degrees
of freedom remains 2.
Applying the components approach to water is simple. All species that may exist in
pure water are formed, in whatever proportions actually exist, from the single substance
H 2 O. Thus, there is only one component:C D 1. The component version of the phase
rule,FD 2 CCP, gives the same result as the species version:FD 2.