Dft (Density Functional Theory; The Integral Equation Of Adsorption - Micromeritics ASAP 2460 Operator's Manual

Appendix C

DFT (Density Functional Theory)

The adsorption isotherm is known to convey a great deal of information about the energetic heteroge-
neity and geometric topology of the sample under study. The data of physical adsorption have been
used for many years as the basis for methods to characterize the surface area and porosity of adsor-
bents. Real solid surfaces rarely approach ideal uniformity of structure. It is accepted that in general,
the surface of even a nonporous material presents areas of greater or lesser attraction for adsorbed
molecules.
This energetic heterogeneity greatly affects the shape of the adsorption isotherm with the result that
simple theories such as the Langmuir and BET formulas can, at best, give only approximate estimates
of surface area. Porous solids virtually are never characterized by a single pore dimension, but instead
exhibit a more or less wide distribution of sizes. The observed adsorption isotherm for a typical mate-
rial is therefore the convolution of an adsorption process with the distribution of one or more
properties which affect that process. This was first stated mathematically by Ross and Olivier
case of surface energy distribution and has become known as the integral equation of adsorption.

The Integral Equation of Adsorption

In a general form for a single component adsorptive, the integral equation of adsorption can be written
as:

Q p  
= a d b d cq p a b c
where
Q(p) =
a,b,c,... =
f(a,b,c,...) =
q(p,a,b,c,...)=
Equation (1), a Fredholm integral of the first kind, is a member of a class of problems known as ill-
posed, in that there are an infinite number of functional combinations inside the integral that will pro-
vide solutions. Even when the kernel function is known, experimental error in the data can make
solving for even a single distribution function a difficult task. Solving for multiple distribution func-
tions requires more data than provided by a single adsorption isotherm.
C-34
     f a b c
d
the total quantity adsorbed per unit weight at pressure p,
a set of distributed properties,
the distribution function of the properties, and
the kernel function describing the adsorption isotherm on unit surface of
material with fixed properties a,b,c,...
ASAP 2460 Operator's Manual
   
12
for the
(1)
246-42800-01 - Aug 2013