Page 146 - Physical Chemistry

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Let us obtain the equation for dU that corresponds to (4.73). From G U PV Section 4.6
TS, we have dU dG PdV VdP TdS SdT. The use of (4.73) gives Chemical Potentials
and Material Equilibrium
dU T dS P dV a m dn i (4.74)
i
i
This equation may be compared with dU TdS PdV for a reversible process in a
closed system.
From H U PV and A U TS, together with (4.74), we can obtain expres-
sions for dH and dA for irreversible chemical changes. Collecting together the expres-
sions for dU, dH, dA, and dG, we have

dU T dS P dV a m dn i
i
i (4.75)*
dH T dS V dP a m dn i one-phase syst.
i
i in mech. and therm. (4.76)
w
dA S dT P dV a m dn i equilib., P-V work only
i
i (4.77)
dG S dT V dP a m dn i
i
i (4.78)*
These equations are the extensions of the Gibbs equations (4.33) to (4.36) to processes
involving exchange of matter with the surroundings or irreversible composition
changes. The extra terms m dn in (4.75) to (4.78) allow for the effect of the com-
i
i
i
position changes on the state functions U, H, A, and G. Equations (4.75) to (4.78) are
also called the Gibbs equations.
Equations (4.75) to (4.78) are for a one-phase system. Suppose the system has sev-
eral phases. Just as the letter i in (4.78) is a general index denoting any one of the
chemical species present in the system, let a (alpha) be a general index denoting any
a
one of the phases of the system. Let G be the Gibbs energy of phase a, and let G be
the Gibbs energy of the entire system. The state function G U PV TS is exten-
sive. Therefore we add the Gibbs energy of each phase to get G of the multiphase
a
a
system: G G . If the system has three phases, then G has three terms. The
a
a
relation d(u y) du dy shows that the differential of a sum is the sum of the dif-
a
a
ferentials. Therefore, dG d( G ) dG . The one-phase Gibbs equation (4.78)
a
a
written for phase a reads
a
a
a
a
dG S dT V dP a m dn a i
i
i
a
Substitution of this equation into dG dG gives
a
a
a
a
dG a S dT a V dP a a m dn a i (4.79)
i
a a a i
a
a
a
where S and V are the entropy and volume of phase a, m is the chemical potential
i
a
of chemical species i in phase a, and n is the number of moles of i in phase a.
i
Equation (4.72) written for phase a reads
0G a
a
m a a b (4.80)*
i
0n i T,P,n j i
a
(We have taken T of each phase to be the same and P of each phase to be the same. This
will be true for a system in mechanical and thermal equilibrium provided no rigid or
adiabatic walls separate the phases.) Since S and V are extensive, the sums over the