Concise Physical Chemistry

c09 JWBS043-Rogers September 13, 2010 11:26 Printer Name: Yet to Come

134 THE PHASE RULE

9.5 THE GIBBS PHASE RULE

Of theCmole fractionsXi(i= 1 , 2 , 3 ,...,C) in a many-component one-phase mix-

ture,C−1 are independent because once you knowC−1 of them, you can get the

remaining one from

∑

i

Xi=1. The “last” concentration variable is not independent.

If there arePcoexisting phases, there areC−1 independent concentration variables

in each phase for a total ofP(C−1). However, allPphases must be at the same

chemical potential for equilibrium to exist. This makesP−1 equations connecting

each component in each phase,^5 and there areCcomponents for a total ofC(P−1)

equations. The number of degrees of freedom is equal to the total number of inde-

pendent variablesP(C−1) in the system minus the number of equationsC(P−1)

connecting them. In the most general case of many components that may break up

into many phases, we have

f=P(C−1)−C(P−1)=PC−P−CP+C=C−P

Adding the two remaining degrees of freedom for the system, which might bepand

T, we have the celebrated Gibbs phase rule

f=C−P+ 2

A problem of potentially great complexity has been reduced to simple terms. The

number of degrees of freedomfof the system is equal to the number of components

minus the number of phases plus 2.

9.6 TWO-COMPONENT PHASE DIAGRAMS

Two-component phase diagrams can be complicated. Without the phase rule, they

would seem to have a bewildering array of unrelated behaviors. Many two-component

phase diagrams, however, can be broken down into combinations of three simple

diagrams that we shall label I, II, and III here. Two-component phase diagrams are

characterized by one more degree of freedom than pure one-component diagrams

because of the added composition variable

f=C−P+ 2 = 2 −P+ 2 = 4 −P

We can express composition in many ways, but the mole fraction of a two-component

system is most convenient for our purposes.

XB=

nB

nA+nB

(^5) For example,μA=μB=μCconnects three variables but has only two equal signs, hence only two
equations.