Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.2 SOLVENTCHEMICALPOTENTIALS FROMPHASEEQUILIBRIA 369

Table 12.1 Comparison of the Clausius–Clapeyron and van’t Hoff equations for vaporiza-

tion of a liquid.

Clausius–Clapeyron equation van’t Hoff equation

ÅvapHR

d ln.p=p/

d.1=T /

ÅvapHDR

d lnK

d.1=T /

Derivation assumesVm(g) Vm(l) and

ideal-gas behavior.

An exact relation.

ÅvapH is the difference of the molar en-

thalpies of the real gas and the liquid at the

saturation vapor pressure of the liquid.

ÅvapHis the difference of the molar en-

thalpies of the ideal gas and the liquid at

pressurep.

pis the saturation vapor pressure of the liq-

uid.

Kis equal toa(g)=a(l)D.f=p/=(l),

and is only approximately equal top=p.

Equations12.1.13and12.1.14are two forms of thevan’t Hoff equation. They allow us
to evaluate the standard molar reaction enthalpy of a reaction by a noncalorimetric method
from the temperature dependence of lnK. For example, we can plot lnKversus1=T; then
according to Eq.12.1.14, the slope of the curve at any value of1=Tis equal toÅrH=R
at the corresponding temperatureT.

A simple way to derive the equation for this last procedure is to substituteÅrGD

ÅrHTÅrSinÅrGDRTlnKand rearrange to

lnKD

ÅrH

R



1

T



C

ÅrS

R

(12.1.15)

Suppose we plot lnKversus1=T. In a small temperature interval in whichÅrHand

ÅrSare practically constant, the curve will appear linear. According to Eq.12.1.15,

the curve in this interval has a slope ofÅrH=R, and the tangent to a point on the

curve has its intercept at1=TD 0 equal toÅrS=R.

When we apply Eq.12.1.14to thevaporization processA(l)!A(g) of pure A, it resem-
bles the Clausius–Clapeyron equation for the same process (Eq.8.4.15on page 219 ). These
equations are not exactly equivalent, however, as the comparison in Table12.1shows.

12.2 Solvent Chemical Potentials from Phase Equilibria

Section9.6.3explained how we can evaluate the activity coefficient (^) m;Bof a nonelectrolyte
solute of a binary solution if we know the variation of the osmotic coefficient of the solution
from infinite dilution to the molality of interest. A similar procedure for the mean ionic
activity coefficient of an electrolyte solute was described in Sec.10.6.
The physical measurements needed to find the osmotic coefficientmof a binary solu-
tion must be directed to the calculation of the quantityAA, the difference between the
chemical potentials of the pure solvent and the solvent in the solution at the temperature and
pressure of interest. This difference is positive, because the presence of the solute reduces
the solvent’s chemical potential.