Page 63 - Adsorption Technology & Design, Elsevier (1998)

60 Fundamentals of adsorption equilibria

where (~is Xis) is the product of mole fraction Xis and the activity coefficient
~'is of component i in the adsorbed phase solution.
Substitution of ]-/is (T~/~i) into equation (3.45) for a component i in the
vacancy solution thus yields

~is(T~ ~) - ~0 s(T) + RgT In (~tisXis) + ~/~i (3.47)

Equilibrium between the two vacancy solutions in the gas and adsorbed
phases demands that pig - lais and so, from equations (3.43) and (3.47),
fiyiP/(Ti~xis)- exp {(/l~ - ll~ exp (rrAi/RgT) (3.48)

Equation (3.48) relates the mole fraction yi of component i in the gas phase
to the mole fraction xis in the adsorbed phase. The latter is based on the total
number of moles of the adsorbed vacancy solution. To interpret experimen-
tal data an expression is required relating the experimentally measured mole
fraction of component i in the adsorbed phase, x~, to the corresponding
vacancy solution quantity xi. For components i = 1, 2 the equation is x~s -
XieO where 0 is the ratio of the total number of moles nTs in the vacancy
solution to the total number of moles nre adsorbed as measured experiment-
ally. The mole fraction of vacancies in the vacancy solution is then Xvs = (1 -
0). Substitution of xis in equation (3.48) gives
= e0
r (Tisxi ) exp {(p~176 exp (rrfi~i/RgT) (3.49)
All of the terms on the right-hand side of equation (3.49) may be obtained
from pure component adsorption data. As the equation is valid for the whole
range of concentrations, it is therefore also applicable at infinite dilution
when r 7i, ~'is and xie are unity and the spreading pressure is zero. For such
conditions

exp {(/l~ - I~~ - P/O (3.50)

and so may be found from single-component data at low coverages where
the isotherm is linear. The partial molal area A may also be computed by
means of the integrated form of the Gibbs adsorption isotherm given by
equation (3.28). Fugacity and activity coefficients, however, are found by
somewhat tedious fitting of experimental data to an idealized isotherm such
as that of Langmuir. Multiplication of the Langmuir equation by a function
containing one or more parameters, only found by regression, then
correlates equation (3.49) with single-component data and, after a number
of iterations, gives values for r and 7i. The VSM theory, although based on
fundamental principles, is thus semi-empirical.