Page 145 - Physical Chemistry

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Chapter 4 replace the actual irreversible change by a reversible change and calculate dG for the
Material Equilibrium reversible change. We imagine using an anticatalyst to “freeze out” any chemical reac-
tions in the system. We then reversibly add dn moles of substance 1, dn moles of 2,
1
2
etc., and reversibly change T and P by dT and dP.
To add substance 1 to a system reversibly, we use a rigid membrane permeable to substance
1 only. If pure substance 1 is on one side of the membrane and the system is on the other
side, we can adjust the pressure of pure 1 so that there is no tendency for component 1 to
flow between system and surroundings. An infinitesimal change in the pressure of pure 1
then reversibly changes n in the system.
1
The total differential of (4.67) is

0G 0G 0G 0G
dG a b dT a b dP a b dn . . . a b dn k
1
0T 0P 0n 1 0n k
P,n i T,n i T,P,n j 1 T,P,n j k
(4.68)
where the following conventions are used: the subscript n on a partial derivative means
i
that all mole numbers are held constant; the subscript n j i on a partial derivative means
that all mole numbers except n are held fixed. For a reversible process where no
i
change in composition occurs, Eq. (4.36) reads
dG S dT V dP rev. proc., n fixed, P-V work only (4.69)
i
It follows from (4.69) that

0G 0G
a b S, a b V (4.70)
0T 0P
P,n i T,n i
where we added the subscripts n to emphasize the constant composition. Substitution
i
of (4.70) in (4.68) gives for dG in a reversible process in a one-phase system with only
P-V work:
k 0G
dG S dT V dP a a b dn i (4.71)
i 1 0n i T,P,n j i
Now suppose the state variables change because of an irreversible material
change. Since G is a state function, dG is independent of the process that connects
states (T, P, n , n , . . .) and (T dT, P dP, n dn , n dn , . . .). Therefore dG
1 2 1 1 2 2
for the irreversible change is the same as dG for a reversible change that connects these
two states. Hence Eq. (4.71) gives dG for the irreversible material change.
To save time in writing, we define the chemical potential m (mu eye) of substance
i
i in the one-phase system as
0G
m a b one-phase syst. (4.72)*
i
0n i T,P,n j i
where G is the Gibbs energy of the one-phase system. Equation (4.71) then becomes

dG S dT V dP a m dn one-phase syst. in therm. (4.73)*
i
i
i and mech. equilib., P-V work only
Equation (4.73) is the key equation of chemical thermodynamics. It applies to a
process in which the single-phase system is in thermal and mechanical equilibrium but
is not necessarily in material equilibrium. Thus (4.73) holds during an irreversible
chemical reaction and during transport of matter into or out of the system. Our previ-
ous equations were for closed systems, but we now have an equation applicable to
open systems.