Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.7 GIBBSENERGY ANDREACTIONEQUILIBRIUM 346

tial ofAin a closed system with a chemical reaction:

dADSdTpdVC

X

i

ii

!

d (11.7.11)

We identify the coefficient of the last term on the right as a partial derivative:

X

i

iiD



@A

@



T;V

(11.7.12)

This equation shows that as the reaction proceeds spontaneously at constantTandV, it
reaches reaction equilibrium at the point where.@A=@/T;V is zero. This is simply an-
other way to express the criterion for spontaneity stated on page 145 : If the only work is
expansion work, the Helmholtz energy of a closed system decreases during a spontaneous
process at constantTandV and has its minimum value when the system attains an equilib-
rium state.

11.7.6 Reaction in an ideal gas mixture

Let us look in detail at the source of the minimum inGfor the case of a reaction occurring in
an ideal gas mixture in a closed system at constantTandp. During this process the system
has only one independent variable, which it is convenient to choose as the advancement.
The additivity rule (Eq.9.2.25) for the Gibbs energy is

GD

X

i

nii (11.7.13)

where bothniandidepend on. Thus,Gis a complicated function of.
For the chemical potential of each substance, we writeiDi(g)CRTln.pi=p/
(Eq.9.3.5), wherepiis the partial pressure ofiin the mixture. Substitution in Eq.11.7.13
gives, for the Gibbs energy at any value of,

G./D

X

i

ni



i(g)CRTln

pi

p



(11.7.14)

AtD 0 , the amounts and partial pressures have their initial valuesni;0andpi;0:

G.0/D

X

i

ni;0



i(g)CRTln

pi;0

p



(11.7.15)

The difference between these two expressions is

G./G.0/D

X

i

.nini;0/i(g)

CRT

X

i

niln

pi

p

RT

X

i

ni;0ln

pi;0

p

(11.7.16)