Physical Chemistry Third Edition

7.1 Gibbs Energy Changes and the Equilibrium Constant 305

reactionξby

nini(initial)+νiξ (definition ofξ) (7.1-4)

Equation (7.1-4) gives the same value ofξfor any choice of the substancei. The extent

of reaction has the dimensions of moles. Ifξchanges in value from 0 to 1 mol, we

say that 1 mol of reaction has occurred. If 1 mol of reaction occurs,νimoles ofihave

appeared ifiis a product, and|νi|moles ofihave disappeared ifiis a reactant. Think

of a stoichiometric coefficient as representing moles of substance per mole of reaction,

so that the stoichiometric coefficients are dimensionless.

For an infinitesimal extent of reaction,dξ,

dniνidξ (7.1-5)

Equation (7.1-3) now becomes, for constantTandP,

dG

∑c

i 1

μividξ

[c

∑

i 1

viμi

]

dξ (7.1-6)

where we have factored the common factordξout of the sum. For our reacting system

Gis a function ofT,P, andξso that

(

∂G

∂ξ

)

T,P



∑c

i 1

viμi (7.1-7)

The quantity (∂G/∂ξ)T,Pis therate of change of Gibbs energy per mole of reaction.

A spontaneous process at constantTandPcorresponds todG <0. If the forward

reaction is spontaneous,dξ >0 and

(

∂G

∂ξ

)

T,P

< 0

(forward reaction

spontaneous)

(7.1-8)

If the reverse reaction is spontaneous,dξ <0 and

(

∂G

∂ξ

)

T,P

> 0

(reverse reaction

spontaneous)

(7.1-9)

If the equilibrium state has been attained, there is no tendency for the reaction to

proceed, anddG0, so that

(

∂G

∂ξ

)

T,P



∑c

i 1

viμi 0 (equilibrium) (7.1-10)

eq



G

Figure 7.1 The Gibbs Energy of a

Reacting System as a Function of

the Progress Variable.

The situation is as represented in Figure 7.1, with a smooth minimum inGat the

equilibrium value ofξ. A system in any nonequilibrium state will spontaneously react

to approach the equilibrium state at the minimum in the curve representingGas a

function ofξ, beginning from either side of the minimum.