Relationship between Free Energy and Equilibrium Constant

2. Understanding Corrosion
2.5.1 Relationship between 'Free Energy' and 'Equilibrium Constant'

The contribution made by one mole of any constituent, A, to the total free energy, G, of the mixture is G_A, which may be represented by

G_A = G_A^o RT ln(a_A)

where a_A = activity of the substance, T = absolute temperature, and R = gas constant.

From the Gibbs-Helmholtz equation:

G = H - TS

Substituting H from the First Law of Thermodynamics

G = U + PV - TS

Differentiating:

dG = dV + PdV + VdP - TdS - SdT

But,

dV = q_rev - PdV and q_rev = TdS

So,

dG = TdS - PdV + PdV + VdP - TdS - SdT

i.e., dG = VdP - SdT

At constant pressure, this equation becomes:

and at constant temperature:

dG = VdP

However, for ideal gas equation for 1 mole of reaction is

pV = RT

V = RT/p

Integration between the limits gives

If G'_A is taken to refer for standard conditions, it becomes G_A^o.

(1)

If activities are concerned instead of pressure, then

Applying DG for the reaction: aA + bB cC + dD gives

Expressing G_C, G_D, G_A and G_B as shown in (1) above, we get

So, for 1 mole forward reaction,

(2)

and at equilibrium, DG = 0

Equation (2) may be written as

The equations linking DG and K are referred to as VanHoff's Isotherms.