CHAPTER 5 THERMODYNAMIC POTENTIALS
5.8 CRITERIA FORSPONTANEITY 145
and pressure will be assumed to be practically uniform during the process, even if the pro-
cess is irreversible. For example, the volume might be changing at a finite rate but very
slowly, or there might be a spontaneous homogeneous reaction in a mixture of uniform
temperature and pressure.
The second law states that dSis equal to∂q=Tif the process is reversible, and is greater
than∂q=Tif the process is irreversible:
dS∂q=T (5.8.1)
(irrevrev, closed system)or
∂qTdS (5.8.2)
(irrevrev, closed system)Theinequalitiesin these relations refer to an irreversible process and theequalitiesto a
reversible process, as indicated by the notationirrevrev.
When we substitute∂qfrom Eq.5.8.2into the first law in the form dUD∂q pdVC
∂w^0 , where∂w^0 is nonexpansion work, we obtain the relation
dUTdS pdVC∂w^0 (5.8.3)
(irrevrev, closed system)We substitute this relation for dUinto the differentials of enthalpy, Helmholtz energy, and
Gibbs energy given by Eqs.5.3.4–5.3.6to obtain three more relations:
dHTdSCVdpC∂w^0 (5.8.4)
(irrevrev, closed system)dA SdT pdVC∂w^0 (5.8.5)
(irrevrev, closed system)dG SdTCVdpC∂w^0 (5.8.6)
(irrevrev, closed system)The last two of these relations provide valuable criteria for spontaneity under common
laboratory conditions. Equation5.8.5shows that during a spontaneous irreversible change
at constant temperature and volume, dAis less than∂w^0. If the only work is expansion
work (i.e.,∂w^0 is zero), the Helmholtz energy decreases during a spontaneous process at
constantTandVand has its minimum value when the system reaches an equilibrium state.
Equation5.8.6is especially useful. From it, we can conclude the following:
Reversible nonexpansion work at constantT andpis equal to the Gibbs energy
change. For example, if the system is a galvanic cell operated in the reversible limit
(Sec.3.8.3) at constantTandp, the electrical work is given by∂wel, revDdG. There
is an application of this relation in Sec.14.3.1.