Dft (Density Functional Theory; The Integral Equation Of Adsorption - Micromeritics ASAP 2020 Confirm Operator's Manual

ASAP 2020 Confirm

DFT (Density Functional Theory)

The adsorption isotherm is known to convey a great deal of information about the energetic
heterogeneity and geometric topology of the sample under study. The data of physical adsorp-
tion have been used for many years as the basis for methods to characterize the surface area
and porosity of adsorbents. 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 material is therefore the convolution of an adsorp-
tion process with the distribution of one or more properties which affect that process. This
was first stated mathematically by Ross and Olivier
tion 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  
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 provide 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 mul-
tiple distribution functions requires more data than provided by a single adsorption isotherm.
C-40
    
= a d b d cq p 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,...
13 for the case of surface energy distribu-
 f a b c    
Appendix C
(1)
202-42811-01 - Mar 2011