Concise Physical Chemistry

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

128 THE PHASE RULE

p

T

.

vapor

liquid

solid

FIGURE 9.3 A single-component phase diagram. The unusual solid–liquid coexistence

curve for water is shown as a dotted line. The terminus (•) of the curve on the right is

thecritical point. The intersection of the three curves is thetriple point.

9.3 THE CLAUSIUS–CLAPEYRON EQUATION

TheClapeyron equationfor a phase transition is

dp

dT

=

transH

T
V

where
transHis the enthalpy change of the phase transition and
Vis the corre-

sponding volume change. It is customary to take both as molar quantities.

Clausius observed that the volume of a liquid in equilibrium with its vapor at

the boiling point is normally very much smaller than the volume of the vapor itself:

V(l)<<V(g). Thus, taking the special case of vaporization of a liquid where
V=

V(g)−V(l), he ignored the smaller volumeV(l) and substituted the larger volume

V(g) for
V. He then assumed that the vapor over the liquid is an ideal gas, which

permits the substitution

dp

dT

=

vapH

TV(g)

=

vapH

T

( p

RT

)

Rearranging, we obtain

dp

p

=

vapH

R

(

1

T^2

)

dT

and taking the indefinite integral,

∫

dp

p

=

vapH

R

∫

1

T^2

dT