Multicomponent Equilibrium Thermodynamics - Elements of Environmental Engineering: Thermodynamics and Kinetics - page 91

by the respective surface activities to get a Γ /a o . Thus, the analogy of the definition

of fugacity of surface states of molecules with those in the bulk is evident. For the
adsorbate in equilibrium with the liquid phase, we obtain

Γ

θ i γ i f Γ 0

l

i f l 0

= x i γ . (3.75)

i i

For ideal dilute solutions, the activity coefficients are unity. Hence we have

f l 0
i
f Γ 0
i

θ i
x i =

K Γ l . = (3.76)

Further for dilute solutions, x i = C i V l , we have upon rearranging

Γ i = K

l C i .

(3.77)

Γ

This is called the linear (Henry's) adsorption equation with the linear adsorption
constant (units of length) given by K Γ l .
In a number of environmental systems, where dilute solutions are considered, the
above equation is used to represent adsorption from the liquid phase on both liquid
and solid surfaces. If a solid-gas interface is considered where adsorption occurs from
the gas phase, the concentration (

max

i ) in the above equation may be replaced by
the corresponding partial pressure. If we define a surface layer thickness,

Γ

δ (unit of

max
i

length), and express both ) , we can

express the linear adsorption constant as a dimensionless value, K ads . However, since
the definition of an interface thickness is difficult, the above approach is less useful.
It is conventional, both in soil chemistry and in environmental engineering, to
express adsorbed-phase concentrations on the solid surface in terms of amount
adsorbed per mass of the solid. This means that the surface concentrations

Γ i and Γ as concentration units ( C

Γ = Γ i /

δ

Γ i and

m

Γ

i , which are in moles per unit area of the solid, are converted to moles per gram of
solid, which is the product W ads = Γ i a m , where a m is the surface area per unit mass
of the solid. This definition then changes the units of the linear adsorption constant,
designated K ads , which is expressed in volume per unit mass of solid.

W ads = K ads C i . (3.78)

The linear adsorption constant is not applicable for many situations for a variety
of reasons. At high concentrations of molecules on the surface, the assumption
of no adsorbate-adsorbate interactions fails. Lateral interactions between adsorbed
molecules necessitate that we assume a limited space-filling model for the adsorbed
phase. This can be easily achieved by assuming that the expression for surface cov-
erage be written as ( − θ i ) to account for the removal of an equivalent amount of

solvent from the interface to accommodate the adsorbed solute i . In other words, the
adsorption of a solute i on the surface can be considered as an exchange process h the
solute and solvent molecules. Equating fugacity as above, for the liquid-gas interface
with the liquid chemical potential, we obtain

θ i / 1

f l 0
i
f Γ 0
i

θ i

K
Γ

− θ i =

x i · = x i ·

w .

(3.79)

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