Page 62 - Adsorption Technology & Design, Elsevier (1998)

Fundamentals of adsorption equilibria 59

parameters introduced to account for non-ideality are the activity coef-
ficients ?,~. Hence a similar procedure of calculation as for the IAS model
may be adopted to obtain isotherms for non-ideal mixtures from single-
component data if the activity coefficients are known. Alternatively, the
method may be used to determine activity coefficients for the adsorbed
phase having calculated the spreading pressure from an equation such as
(3.40) and then employing equation (3.42).

3.4.5 Vacancy solution model
Suwanayuen and Danner (1980) formulated a theory of adsorption for single
components which was extended to a binary mixture. They also claimed that
the model is applicable to multicomponent mixtures. It was assumed that the
gas phase and the adsorbed phase are each composed of a hypothetical
solvent (termed the vacancy) and identifiable adsorbates. A vacancy, in this
theory, is considered to be a vacuum entity which occupies adsorption space
which can be filled by an adsorbate. Thus, adsorption equilibrium for a
mixture of n adsorbable components translates to an equilibrium between (n
+ 1) vacancy solutions (n adsorbate species and the vacancy). The chemical
potential of a component i in the gas phase is

[dig "- lld~ + RgT ln(r (3.43)

where r the fugacity coefficient accounting for any non-ideality, y~ is the
mole fraction of component i in the gas phase and P is the total pressure.
Application of classical thermodynamics to the adsorbed phase (considered
as the adsorbate in the potential field of an inert adsorbent) gives the Gibbs
function as
Gs = - irA + ns p~ (T, tr)
(3.44)
at constant temperature and pressure. The chemical potential/~ (T, tr) is a
function of temperature T and spreading pressure tr rather than temperature
and pressure as it would be in the gas phase. Differentiation of this equation
with respect to molar quantity ns at constant spreading pressure yields

/As(T, A) = -- ~mi +/.Is( T, ~) (3.45)
where ~u~(T, A) is the chemical potential as a function of temperature and
partial molal area. Classical solution thermodynamics gives the chemical
potential as a function of temperature and partial molal area and for a
component i is written
/zi~(T, Ai) =///~(T) +ng T ln(?,i~xi~)
(3.46)