Page 41 - Adsorption Technology & Design, Elsevier (1998)

38 Fundamentals of adsorption equilibria

3.3 THEORIES OF ADSORPTION EQUILIBRIA

A variety of different isotherm equations have been proposed, some of
which have a theoretical foundation and some being of a more empirical
nature. Many of these equations are valid over small relative pressure ranges
but do not fit experimental data when tested over the full range of relative
pressures. Only those which are commonly used for the description of the
physical adsorption of gases or vapours onto the surface of porous
adsorbents will be outlined. Some of these theories, as shown later, can be
extended to describe the simultaneous adsorption of two or more com-
ponents.

3.3.1 The Langmuir isotherm
This isotherm describes adsorbate-adsorbent systems in which the extent of
adsorbate coverage is limited to one molecular layer at or before a relative
pressure of unity is reached. Although the isotherm, proposed originally by
Langmuir (1918), is more usually appropriate for the description of
chemisorption (when an ionic or covalent chemical bond is formed between
adsorbent and adsorbate), the equation is nevertheless obeyed at mod-
erately low coverages by a number of systems and can, moreover, be readily
extended to describe the behaviour of binary adsorbate systems. The
isotherm was formulated on the basis of a dynamic equilibrium between the
adsorbed phase and the gaseous or vapour phase. It was argued that the rate
at which adsorbate gas molecules strike a surface of an adsorbent is
proportional to the product of the partial pressure p of the gas and the
fraction (1 -0) of surface remaining uncovered by adsorbate and therefore
available as adsorption sites. Langmuir further supposed that the rate of
desorption from the surface is directly proportional to the fractional surface
coverage 0 and that the rates of adsorption and desorption are equal at
equilibrium. Thus

kap (1 - O) = kaO (3.5)
where ka and kd are the respective rate constants for adsorption and
desorption, respectively. The more usual form of the equation is written
(3.6)
0 = q/qm = bp/(1 + bp)
where b is ka/kd and qm is the quantity q of adsorbate adsorbed in a single
monolayer. The ratio q/qm can be measured and expressed in different ways.
For the present we will choose to represent the ratio by the number of moles
of a component adsorbed compared with the number of moles of that