Page 53 - Adsorption Technology & Design, Elsevier (1998)

50 Fundamentals of adsorption equilibria

If an equation of state relating tr, A and T is specified then substitution
into equation (3.27) followed by integration leads directly to a relation
between 7r and p. The simplest equation of state is zrA = nRgT, analogous to
the perfect gas law, and it is a simple matter to show that this equation of
state substituted into the differential form of the Gibbs isotherm leads to
Henry's law (q.v. Section 3.3.2). If, on the other hand, (A -a) replaces A in
the equation of state (therefore allowing for the total extent, a, of area
which the adsorbate molecules occupy), the result of substitution in
equation (3.27) is Volmer's isotherm
bp = [0/(1 - O)][exp (0/1-0)]
(3.29)
provided the integration constant is defined so that Henry's law (0 = bp) is
obeyed as the pressure tends to zero and 0 is assumed to equal a/A.

3.3.7 Statistical thermodynamic model

Although it is beyond the scope of this chapter to discuss details of the
statistical thermodynamic approach to the formulation of isotherms, it is
nevertheless important to point out that there has been some success
(Ruthven 1984, Ruthven and Wong 1985) in applying the theory to the
adsorption of gases in regular cage-like structures of zeolites. The
well-defined cages with well-defined interconnecting channels can be
considered as independent subsystems so that interaction between mole-
cules adsorbed in neighbouring cages can be ignored. Each cage will only
adsorb a limited number of adsorbate molecules. The so-called grand
partition function is then formulated which describes how the potential
energy of molecules within the cage is distributed. For this purpose a
configurational integral is written which is the integral of the exponential of
the negative value of potential energies over the position of each and every
molecule within the cage. A Lennard-Jones potential is employed as the
potential energy function. Ruthven (1971) and Barrer (1978) assumed that
the sorbate behaves as a van der Waals gas and thus accounted for the
reduction in free volume caused by the presence of the adsorbate molecules.
The configurational integral was thus greatly simplified and results in an
expression for the number of sorbate molecules per cage. As the number of
cages is known from the structure of the zeolite, the result can be expressed
as an isotherm which takes the form
Kp + A2(Kp) 2 +... A,,(Kp)"/(n- 1)!
g - (3.30)
1 + Kp + A2(Kp)2/2!... An(Kp)n/n!
where g is the average number of molecules per zeolite cage, K is Henry's