Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.1 EFFECTS OFTEMPERATURE 367

For a pure substance in a closed system, Eq.12.1.3when multiplied by the amountn

becomes 

@ .G=T /

@T



p

D

H

T^2

(12.1.4)

This is theGibbs–Helmholtz equation.

12.1.2 Variation ofi=Twith temperature

If we make the substitutioniDiCRTlnaiin Eq.12.1.3and rearrange, we obtain

d.i=T /

dT

D

Hi

T^2

R



@lnai

@T



p;fnig

(12.1.5)

Becausei=T is a function only ofT, its derivative with respect toTis itself a function
only ofT. We can therefore use any convenient combination of pressure and composition
in the expression on the right side of Eq.12.1.5in order to evaluate d.i=T /=dTat a given
temperature.
If speciesiis a constituent of a gas mixture, we take a constant pressure of the gas that
is low enough for the gas to behave ideally. Under these conditionsHiis the standard molar
enthalpyHi(Eq.9.3.7). In the expression for activity,ai(g)Di(g)ipi=p(Table9.5),
the pressure factori(g) is constant whenpis constant, the fugacity coefficientifor the
ideal gas is unity, andpi=pDyiis constant at constantfnig, so that the partial derivative
å@lnai(g)=@T çp;fnigis zero.
For componentiof a condensed-phase mixture, we take a constant pressure equal to
the standard pressurep, and a mixture composition in the limit given by Eqs.9.5.20–
9.5.24in which the activity coefficient is unity.Hiis then the standard molar enthalpyHi,
and the activity is given by an expression in Table9.5with the pressure factor and activity
coefficient set equal to 1:aiDxi,aADxA,ax;BDxB,ac;BDcB=c, oram;BDmB=m.^2 With
the exception ofac;B, these activities are constant asTchanges at constantpandfnig.
Thus for a gas-phase species, or a species with a standard state based on mole fraction
or molality,å@lnai(g)=@T çp;fnigis zero and Eq.12.1.5becomes

d.i=T /

dT

D

Hi

T^2

(12.1.6)

(standard state not based
on concentration)
Equation12.1.6, as the conditions of validity indicate, does not apply to a solute stan-
dard state based on concentration, except as an approximation. The reason is the volume
change that accompanies an isobaric temperature change. We can treat this case by consid-
ering the following behavior of ln.cB=c/:

@ln.cB=c/
@T



p;fnig

D

1

cB



@cB

@T



p;fnig

D

1

nB=V



@.nB=V /

@T



p;fnig

DV



@.1=V /

@T



p;fnig

D

1

V



@V

@T



p;fnig

D (12.1.7)

(^2) If solute B is an electrolyte,am;Bis given instead by Eq.10.3.10; likeam;Bfor a nonelectrolyte, it is constant
asTchanges at constantpandfnig.