Thermodynamics and Chemistry

CHAPTER 8 PHASE TRANSITIONS AND EQUILIBRIA OF PURE SUBSTANCES

8.4 COEXISTENCECURVES 219

Then, using the mathematical identities d.p=p/=.p=p/ Dd ln.p=p/and dT=T^2 D
d.1=T /, we can write Eq.8.4.12in three alternative forms:

d ln.p=p/

dT



ÅtrsH

RT^2

(8.4.13)

(pure substance,

vaporization or sublimation)

d ln.p=p/

ÅtrsH

R

d.1=T / (8.4.14)

(pure substance,

vaporization or sublimation)

d ln.p=p/

d.1=T /



ÅtrsH

R

(8.4.15)

(pure substance,

vaporization or sublimation)

Equation8.4.15shows that the curve of a plot of ln.p=p/versus1=T(wherepis the
vapor pressure of a pure liquid or solid) has a slope at each temperature equal, usually to
a high degree of accuracy, toÅvapH=RorÅsubH=Rat that temperature. This kind of
plot provides an alternative to calorimetry for evaluating molar enthalpies of vaporization
and sublimation.

If we use the recommended standard pressure of 1 bar, the ratiop=pappearing in

these equations becomesp=bar. That is,p=pis simply the numerical value ofp

whenpis expressed in bars. For the purpose of using Eq.8.4.15to evaluateÅtrsH, we

can replacepby any convenient value. Thus, the curves of plots of ln.p=bar/versus

1=T, ln.p=Pa/versus1=T, and ln.p=Torr/versus1=T using the same temperature

and pressure data all have the same slope (but different intercepts) and yield the same

value ofÅtrsH.

If we assumeÅvapHorÅsubHis essentially constant in a temperature range, we may
integrate Eq.8.4.14from an initial to a final state along the coexistence curve to obtain

ln

p 2

p 1



ÅtrsH

R



1

T 2

1

T 1



(8.4.16)

(pure substance,

vaporization or sublimation)

Equation8.4.16allows us to estimate any one of the quantitiesp 1 ,p 2 ,T 1 ,T 2 , orÅtrsH,
given values of the other four.