Applications of Equilibrium Thermodynamics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 156

following two acid-base surface exchange reactions can be written as

OH 2

H + ≡ M − ≡ M − OH +

(4.60)

O _ H + ≡ M − OH ≡ M − +

with equilibrium constants given by

H + ] O − ][ H + ]

K 1 = [≡

M − OH ][

, K 2 = [≡

M −

, (4.61)

OH 2 ]

[≡ − ]

[≡ M − M OH

The OH bonds in bulk water have an intrinsic activity derived solely from the disso-
ciation of water molecules, whereas the activity of the surface OH groups involved
in the above surface acid equilibrium constants will introduce the additional work
(energy) required to bring an H + ion from the bulk to the surface, which is given by
the electrostatic potential of the surface. Thus,

exp F

,

ψ
kT

K int r
1

K 1 =

(4.62)

exp F

.

ψ
kT

K int r
2

K 2 =

The intrinsic terms quantify the extent of H + exchange when the surface is uncharged.
With increasing pH, the surface dissociation increases since more of the surface H +
ions are consumed, and hence increasingly it becomes difficult to remove hydrogen
ions from the surface. The above equations represent this effect mathematically.

E XAMPLE 4.18 I NTRINSIC AND C ONDITIONAL E QUILIBRIUM C ONSTANTS

Derive the equations for the conditional equilibrium constants in terms of the intrinsic
constants given above.
For the two ionization reactions of concern, the conditional equilibrium constants are K 1
and K 2 as given by the equations earlier. The activity coefficients of surface-adsorbed
species are assumed to be equal. The equilibrium constants are conditional since they
depend on the surface ionization, which further depends on the pH.The total free energy
of sorption is composed of two terms: Δ G tot = Δ G intr + Δ zF ψ 0 , where the second
term is the “intrinsic” free energy term. The third term represents the “electrostatic”
term that gives the electrical work required to move ions through an interfacial potential
gradient. Δ z is the change in charge of the surface species upon sorption. Since Δ G 0 =

− RT ln K , we can obtain

K intr = K cond exp Δ Fz ψ 0
RT

.

(4.63)

Hence, we derive the equation for K 1 and K 2 .