Page 148 - Physical Chemistry

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The chemical potential of substance i in the phase is a state function that depends on Section 4.7
the temperature, pressure, and composition of the phase. Since m is the ratio of infini- Phase Equilibrium
i
tesimal changes in two extensive properties, it is an intensive property. From m
i
10G>0n 2 , the chemical potential of substance i gives the rate of change of the
i T,P,n j i
Gibbs energy G of the phase with respect to the moles of i added at constant T, P, and
other mole numbers. The state function m was introduced into thermodynamics by
i
Gibbs.
Because chemical potentials are intensive properties, we can use mole fractions
instead of moles to express the composition dependence of m. For a several-phase
system, the chemical potential of substance i in phase a is
a
a
a
a
a
a
m m 1T , P , x , x , . . .2 (4.85)
i
i
1
2
a
Note that, even if substance i is absent from phase a (n 0), its chemical poten-
i
a
tial m in phase a is still defined. There is always the possibility of introducing sub-
i
a
stance i into the phase. When dn moles of i is introduced at constant T, P, and n , the
i
j i
a
a
a
a
Gibbs energy of the phase changes by dG and m is given by dG /dn .
i
i
The simplest possible system is a single phase of pure substance i, for example,
solid copper or liquid water. Let G (T, P) be the molar Gibbs energy of pure i at the
m,i
temperature and pressure of the system. By definition, G m,i G/n , so the Gibbs en-
i
ergy of the pure, one-phase system is G n G m,i (T, P). Partial differentiation of this
i
equation gives
m 10G>0n 2 G one-phase pure substance (4.86)*
i T,P
i
m,i
For a pure substance, m is the molar Gibbs free energy. However, m in a one-phase
i
i
mixture need not equal G of pure i.
m
4.7 PHASE EQUILIBRIUM
The two kinds of material equilibrium are phase equilibrium and reaction equilibrium
(Sec. 4.1). A phase equilibrium involves the same chemical species present in different
phases [for example, C H O (s) ∆ C H O (aq)]. A reaction equilibrium involves
6
6
12
6
12
6
different chemical species, which may or may not be present in the same phase [for ex-
ample, CaCO (s) ∆ CaO(s) CO (g) and N (g) 3H (g) ∆ 2NH (g)]. Phase equi-
2
2
3
3
2
librium is considered in this section, reaction equilibrium in the next.
The condition for material equilibrium in a closed system with P-V work only is
a
a
given by Eq. (4.83) as m dn 0, which holds for any possible infinitesimal
i
a
i
i
a
change in the mole numbers n . Consider a several-phase system that is in equilibrium,
i
and suppose that dn moles of substance j were to flow from phase b (beta) to phase
j
d (delta) (Fig. 4.6). For this process, Eq. (4.83) becomes
b
d
d
b
m dn m dn 0 (4.87)
j
j
j
j
d
b
b
d
From Fig. 4.6, we have dn dn and dn dn . Therefore m dn m dn j
j
j
j
j
j
j
j
0, and Phase b
b
d
1m m 2 dn 0 dn j
j
j
j
d
b
Since dn 0, we must have m m 0, or
j j j
b
d
m m phase equilib. in closed syst., P-V work only (4.88)* Phase d
j
j
For a closed system with P-V work only in thermal and mechanical equilibrium, the
phase equilibrium condition is that the chemical potential of a given substance is the
same in every phase of the system. Figure 4.6
Now suppose the closed system (which is in thermal and mechanical equilibrium dn moles of substance j flows
j
and is capable of P-V work only) has not yet reached phase equilibrium. Let dn moles from phase b to phase d.
j