Thermodynamics and Chemistry

CHAPTER 5 THERMODYNAMIC POTENTIALS

5.8 CRITERIA FORSPONTANEITY 146

 During a spontaneous process at constantT andpin a closed system with expan-

sion work only, the Gibbs energy continuously decreases until the system reaches an

equilibrium state.

Ben-Amotz and Honig^4 developed a “rectification” procedure that simplifies the math-

ematical manipulation of inequalities. Following this procedure, we can write

dSD∂q=TC∂ (5.8.7)

where∂is anexcess entropy functionthat is positive for an irreversible change and

zero for a reversible change (∂  0 ). Solving for∂qgives the expression∂qD

TdST ∂that, when substituted in the first law expression dUD∂qpdVC∂w^0 ,

produces

dUDTdSpdVC∂w^0 T ∂ (5.8.8)

The equality of this equation is equivalent to the combined equality and inequality of

Eq.5.8.3. Then by substitution of this expression for dUinto Eqs.5.3.4–5.3.6, we

obtain equalities equivalent to Eqs.5.8.4–5.8.6, for example

dGDSdTCVdpC∂w^0 T ∂ (5.8.9)

Equation5.8.9tells us that during a process at constantTandp, with expansion work

only (∂w^0 D 0 ), dGhas the same sign asT ∂: negative for an irreversible change

and zero for a reversible change.

(^4) Refs. [ 11 ] and [ 81 ].