Physical Chemistry Third Edition

4.1 Criteria for Spontaneous Processes and for Equilibrium: The Gibbs and Helmholtz Energies 155

The Equilibrium State of a Closed Simple System

For a system to be at macroscopic equilibrium, every process must have proceeded to the

state at which the appropriate criterion for spontaneity has been satisfied. For example,

when a closed simple system at constant temperature and pressure reaches equilibrium,

the Gibbs energy must have reached the minimum value possible at that pressure and

temperature. In Figure 4.1 the variablexschematically represents the extent to which a

chemical reaction or some other process has occurred. We assume that a state variable

such as the Gibbs energy is a differentiable function of its independent variables, so that

there is a smooth minimum as shown in Figure 4.1a, not a cusp as shown in Figure 4.1b.

The value ofxat the minimum in the curve corresponds to the equilibrium state for the

particular constant values ofPandTand conforms to

(

∂G

∂x

)

T,P

 0

(closed simple system at

macroscopic equilibrium

withTandPconstant)

(4.1-20)

If the temperature and pressure are constant, an infinitesimal change in a simple system

at equilibrium must obey

dG 0

(closed simple system at macroscopic

equilibrium withTandPconstant) (4.1-21)

For a closed simple system at constant temperature and volume, the value ofxat

equilibrium corresponds to a minimum inA:

(

∂A

∂x

)

T,P

 0

(closed simple system at

macroscopic equilibrium

withTandVconstant)

(4.1-22)

An infinitesimal change at equilibrium at constantTandVmust obey

dA 0

(closed simple system at macroscopic

equilibriumTandVconstant)

(4.1-23)

(a) (b)

NOT POSSIBLE

Equilibrium

value of x

G G

xx

Figure 4.1 The Gibbs Energy as a Function of the Extent of a Process.(a) As it actually is assumed to be (schematic). (b) As it is
assumed not to be (schematic).