Page 25 - Adsorbents fundamentals and applications

10 FUNDAMENTAL FACTORS FOR DESIGNING ADSORBENT

Field (of an ion) and point dipole:
qµ cos θ
φ Fµ =−Fµ cos θ =− (2.7)
2
r (4π ∈ 0 )
Field gradient (F) and linear point quadrupole:
˙
2
1 ˙ Qq(3cos θ − 1)
φ ˙ FQ = QF =− 3 (2.8)
2 4r (4π ∈ 0 )
where A and B are constants, α = polarizability, F = electric field, q =
electronic charge of ion on surface, ∈ 0 = permittivity of a vacuum, µ =
permanent dipole moment, θ = angle between the direction of the field or field
gradient and the axis of the dipole or linear quadrupole, Q = linear quadrupole
moment (+ or −). The important parameter, r, is the distance between the centers
of the interacting pair. It can be shown that the field-quadrupole interaction is
always zero for all θ.
The dispersion and repulsion interactions form the Lennard–Jones potential
(Barrer, 1978; Masel, 1996; Razmus and Hall, 1991; Gregg and Sing, 1982;
Steele, 1974; Adamson and Gast, 1997; Rigby, et al., 1986), with an equilibrium
distance (r 0 ) at which point φ D + φ R = 0. This distance is taken as the mean
of the van der Waals radii of the interacting pair. Once the attractive, disper-
sion constant, A, is known, B is readily obtained by setting dφ/dr = 0at r 0 .
6
Hence, B = Ar 0 /2. Interestingly, at r 0 , φ D =−2φ R . The most commonly used
expression for calculating A is the Kirkwood–M¨ uller formula:

2
6mc α i α j
A = (2.9)
(α i /χ i ) + (α j /χ j )
where m is the mass of electron, c is the speed of light, χ is the magnetic
susceptibility, and i and j refer to the two interacting atoms or molecules. For
, the maximum potentials are obtained when the dipole or quadrupole
φ Fu and φ ˙ FQ
is arranged linearly with the charge on the surface.
The dispersion potential, Eq. 2.4, was derived by F. London in 1930, starting
from Eq. 2.6, and summarized by Adamson and Gast, 1997. The repulsion term,
Eq. 2.5, was not rigorously derived. Equation 2.6 can be obtained from µ = αF,
where µ is the induced dipole moment and α is, by definition, the polarizability.
The derivation of Eqs. 2.7 and 2.8 is straightforward.

2.2. HEAT OF ADSORPTION

In 2.1, we summarized the different contributions to the potential energy for the
interactions between an adsorbate molecule (or atom) and an atom on the solid
surface. Pairwise additivity is generally assumed when calculating the interaction