Page 150 - Thermodynamics of Biochemical Reactions

148 Chapter 8 Phase Equilibrium in Aqueous Systems

H 8.5 TRANSFORMED GIBBS ENERGY OF A TWO-PHASE
SYSTEM WITH A CHEMICAL REACTION AND A
MEMBRANE PERMEABLE BY A SINGLE ION

The equilibrium relations of the preceding section were derived on the assumption
that the charge transferred Q can be held constant, but that is not really practical
from an experimental point of view. It is better to consider the potential difference
between the phases to be a natural variable. That is accomplished by use of the
Legendre transform (Alberty, 1995c; Alberty, Barthel, Cohen, Ewing, Goldberg,
and Wilhelm, 2001)
G’ = G - $aQ (8.5-1)

that defines the transformed Gibbs energy G‘. Since
dG‘ = dG - 4adQ - Qd&a (8.5-2)

substituting equation 8.4-5 with $a = 0 yields
dC’ = - SdT+ VdP + pAzdncAa + pAOdncAa + pcdncc ~ Qdd, (8.5-3)

To learn more about the derivatives of the transformed Gibbs energy, the
chemical potentials of species are replaced by use of equation 8.3-8 to obtain

dG’ = - SdT+ VdP + (pi + RTln uAa)dnCAl + (pi + RTlnii,, + FzA‘bg)ducAg
+ (,u~RTIna,,)dn,, - Qd4, (8.5-4)
Thus

= pi + RTlna,, = piz (8.5-5)
1.P
kE) .Jl, .IIL( ,&
where pa, is the transformed chemical potential of A in the r phase. This
corresponds with writing equation 8.3-8 as
(8.5-6)
pi = p; + FZi4i
Thus pi is equal to the chemical potential of i in a phase where = 0.
Equation 8.5-3 indicates that the number of natural variables for the system
is 6, D = 6. Thus the number D of natural variables is the same for G and G‘, as
expected, since the Legendre transform interchanges conjugate variables. The
criterion for equilibrium is dC‘ < 0 at constant rP,ncA,, ticAa. ticc, and The
Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the
number of independent intensive variables is not changed. Equation 8.5-3 yields
the same membrane equations (8.4-13 and 8.4-14) derived in the preceding
section.
The integration of equation 8.5-3 at constant values of the intensive variables
yields
(8.5-7)
” = !‘Azr7cAz + ~A/J”CA[I + kJ7K
which agrees with equations 8.4-7 and 8.5-1.

H 8.6 EFFECTS OF ELECTRIC POTENTIALS ON MOLAR
PROPERTIES OF IONS

The fundamental equation for G for a two-phase system with a potential
difference can be written

dG = - SdT+ VdP + 1 ,u,,dnin + 1 p,Bdn,/, + d)/,dQ (8.6- 1 )