Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.7 GIBBSENERGY ANDREACTIONEQUILIBRIUM 347

Converting partial pressures to mole fractions withpiDyipandpi;0Dyi;0pgives

G./G.0/D

X

i

.nini;0/i(g)CRT

X

i

nilnyi

RT

X

i

ni;0lnyi;0CRT

X

i

.nini;0/ln

p

p

(11.7.17)

With the substitutionnini;0Di(Eq.11.2.11) in the first and last terms on the right
side of Eq.11.7.17, the result is

G./G.0/D

X

i

ii(g)CRT

X

i

nilnyi

RT

X

i

ni:0lnyi;0CRT

X

i

i

!

ln

p

p

(11.7.18)

The sum

P

ii



i(g) in the first term on the right side of Eq.11.7.18isÅrG

, the

standard molar reaction Gibbs energy. Making this substitution gives finally

G./G.0/DÅrGCRT

X

i

nilnyiRT

X

i

ni;0lnyi;0

CRT

X

i

i

!

ln

p

p

(11.7.19)

(ideal gas mixture)

There are four terms on the right side of Eq.11.7.19. The first term is the Gibbs en-
ergy change for the reaction of pure reactants to form pure products under standard-state
conditions, the second is a mixing term, the third term is constant, and the last term is an
adjustment ofGfrom the standard pressure to the pressure of the gas mixture. Note that
the first and last terms are proportional to the advancement and cannot be the cause of a
minimum in the curve of the plot ofGversus. It is themixing termRT

P

inilnyithat
is responsible for the observed minimum.^16 This term divided bynD

P

iniisÅG
id
m(mix),
the molar differential Gibbs energy of mixing to form an ideal mixture (see Eq.11.1.8on
page 304 ); the term is also equal tonTÅSmid(mix) (Eq.11.1.9), showing that the minimum
is entirely an entropy effect.
Now let us consider specifically the simple reaction

A.g/!B.g/

in an ideal gas mixture, for whichAis 1 andBisC 1. Let the initial state be one of pure
A:nB;0D 0. The initial mole fractions are thenyA;0D 1 andyB;0D 0. In this reaction, the
total amountnDnACnBis constant. Substituting these values in Eq.11.7.19gives^17

G./G.0/DÅrGCnRT .yAlnyACyBlnyB/ (11.7.20)

(^16) This term also causes the slope of the curve ofG./G.0/versusto be1andC1at the left and right
extremes of the curve.
(^17) Note that although lnyAapproaches1asyAapproaches zero, the productyAlnyAapproacheszeroin
this limit. This behavior can be proved with l’Hospital’s rule (see any calculus textbook).