Page 147 - Physical Chemistry

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Chapter 4 entropies and volumes of the phases equal the total entropy S of the system and the
Material Equilibrium total volume V of the system, and (4.79) becomes

a
a
dG S dT V dP a a m dn syst. in mech. and therm. (4.81)*
i
i
a i equilib., P-V work only
Equation (4.81) is the extension of (4.78) to a several-phase system. Don’t be intimi-
dated by the double sum in (4.81). It simply tells us to add up m dn for each species
in each phase of the system. For example, for a system consisting of a liquid phase l
and a vapor phase y, each of which contains only water (w) and acetone (ac), we have
l
l
a
a
y
l
y
l
y
y
l
m dn m dn m dn m dn m dn , where m is the chemical
i
i
ac
w
a
w
ac
w
ac
ac
w
i
w
potential of water in the liquid phase.
Material Equilibrium
We now derive the condition for material equilibrium, including both phase equilib-
rium and reaction equilibrium. Consider a closed system in mechanical and thermal
equilibrium and held at constant T and P as it proceeds to material equilibrium. We
showed in Sec. 4.3 that, during an irreversible chemical reaction or interphase trans-
port of matter in a closed system at constant T and P, the Gibbs function G is de-
creasing (dG 0). At equilibrium, G has reached a minimum, and dG 0 for any
infinitesimal change at constant T and P [Eq. (4.19)]. At constant T and P, dT 0
dP, and from (4.81) the equilibrium condition dG 0 becomes
a a m dn 0 material equilib., closed syst., (4.82)
a
a
i
i
a i P-V work only, const. T, P
Not only is the material-equilibrium condition (4.82) valid for equilibrium reached
under conditions of constant T and P, but it holds no matter how the closed system
reaches equilibrium. To show this, consider an infinitesimal reversible process in a
closed system with P-V work only. Equation (4.81) applies. Also, Eq. (4.36), which
reads dG SdT VdP, applies. Subtraction of dG SdT VdP from (4.81)
gives
a a m dn 0 rev. proc., closed syst., P-V work only (4.83)
a
a
i
i
a i
Equation (4.83) must hold for any reversible process in a closed system with P-V work
only. An infinitesimal process in a system that is in equilibrium is a reversible process
(since it connects an equilibrium state with one infinitesimally close to equilibrium).
Hence (4.83) must hold for any infinitesimal change in a system that has reached ma-
terial equilibrium. Therefore (4.83) holds for any closed system in material equilib-
rium. If the system reaches material equilibrium under conditions of constant T and P,
then G is minimized at equilibrium. If equilibrium is reached under conditions of con-
stant T and V, then A is minimized at equilibrium. If equilibrium is reached under other
conditions, then neither A nor G is necessarily minimized at equilibrium, but in all
cases, Eq. (4.83) holds at equilibrium. Equation (4.83) is the desired general condition
for material equilibrium. This equation will take on simpler forms when we apply it to
phase and reaction equilibrium in the following sections.
Chemical Potentials
The chemical potential m of substance i in a one-phase system is m 10G>0n 2
i i i T,P,n j i
[Eq. (4.72)]. Since G is a function of T, P, n , n ,..., its partial derivative
1 2

G/
n m is also a function of these variables:
i i
m m 1T, P, n , n , . . .2 one-phase syst. (4.84)
i
2
1
i