BioPHYSICAL chemistry

CHAPTER 5 EQUILIBRIA AND REACTIONS INVOLVING PROTONS 97

Expressing the equilibrium constant in terms of the partial pressures:

(db5.6)

allows the standard Gibbs energy difference to be written in terms of the equilibrium
constant:

(ΔG)rec = 0 =(ΔG)°rec +RTlnKeq (db5.7)

y=exthen lny=x

For a more general situation, the chemical potential of the ith molecule can be related to
what is termed the acti 9 ity,ai, according to:

μi=μ^0 i+RTlnai (db5.8)

The activity is a measure of the concentration of a molecule. For an ideal solution, the activity
is equal to the mole fraction. For a nonideal solution, the activity of the ith molecule is pro-
portional to the mole fraction, xi, and the acti 9 ity coefficient, γ, according to:

ai=γixi (db5.9)

For the cases under consideration, solutions are considered to be ideal with γ=1. For the
reaction shown (eqn 5.3), the Gibbs energy of reaction can then be written as:

(ΔG)rec=dμD+cμC−bμB+aμA (db5.10)

Substituting the expression for activity (eqn db5.8) yields:

(ΔG)rec=d(μD^0 +RTlnaD) +c(μ^0 C+RTlnaC) −b(μ^0 B+RTlnaB) +a(μ^0 A+RTlnaA)

(ΔG)rec=(dμ^0 D+cμ^0 C−bμ^0 B−aμ^0 A) +RT(dlnaD+clnaC−blnaB−alnaA) (db5.11)

The standard terms can be collected and the terms depending on the activities can be
rewritten:

(ΔG)°rec=(dμ^0 D+cμ^0 C−bμ^0 B−aμ^0 A) (db5.12)

RT(dlnaD+clnaC−blnaB−alnaA) =RT (db5.13)

aa

aa

cd

ln ab

CD

AB

Keeq

G

RT

()rec

=

− °Δ

K

P

eq P

= B

A