heterogeneous adsorption equilibria - ali...
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Heterogeneous
Adsorption Equilibria
Ali Ahmadpour
Chemical Eng. Dept.
Ferdowsi University of Mashhad
2
Contents
Introduction
Heterogeneous solids
Surface topography
Langmuir Approach
Energy Distribution Approach
Relationship between Slit Shape Micropore
and Adsorption Energy
Adsorption Isotherm for Slit Shape Pore
Micropore size distribution
3
Introduction
Adsorption in practical solids is a very complex
process because the solid structure is generally
complex and is not so well defined.
The complexity of the system is usually associated
with the heterogeneity between the solid and the
adsorbate.
In other words, heterogeneity is not a solid
characteristic alone but rather it is a characteristics
of the specific solid and adsorbate pair.
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Heterogeneous solids
The direct evidence of the solid heterogeneity is
the decrease of the isosteric heat of adsorption
versus loading.
A behavior of constant heat of adsorption versus
loading is not necessary to indicate that the solid is
homogeneous. This constant heat behavior could
be the result of the combination of the surface
heterogeneity and the interaction between
adsorbed molecules.
5
Cont.
One practical approach in dealing with the problem of
heterogeneity is to take some macroscopic thermodynamic
quantity and impose a statistical attribute (that is a
distribution function) on such quantity .
Once a local isotherm is chosen, the overall (observed)
isotherm can be obtained by averaging it over the
distribution of that thermodynamic quantity.
Many local isotherms have been used, and among them
the Langmuir equation is the most widely used.
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System heterogeneity
The contribution of solid toward heterogeneity is the geometrical
and energetical characteristics, such as the micropore size
distribution and the functional group distribution (they both give
rise to the overall energy distribution which characterizes the
interaction between the solid and the adsorbate molecule), while
the contribution of the adsorbate molecule is its size, shape and
conformation.
All these factors will affect the system heterogeneity, which is
macroscopically observed in the adsorption isotherm and dynamics.
Therefore, by measuring adsorption equilibrium, isosteric heat, and
dynamics, one could deduce some information about heterogeneity,
which is usually characterized by a so called apparent energy
distribution.
7
Cont.
The inverse problem of determining this energy
distribution would depend on the choice of the local
adsorption isotherm, the shape of the energy distribution,
and the topography of the surface (that is whether it is
patchwise or random) as the observed adsorption isotherm
is an integral of the local adsorption isotherm over the full
energy distribution.
The choice of the local isotherm depends on the nature of
the surface.
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Surface topography
The surface topography might also be an important factor in the calculation of the overall adsorption isotherm.
These surfaces of different energy can distribute between the two extremes. In one extreme, the solid is composed of patches,
wherein all sites of the same energy are grouped together, and there is no interaction between these patches i.e. patchwise topography.
The other extreme is the case where surfaces of different energy are randomly distributed.
Of course, real solids would have a topography which is somewhere between these two extremes.
9
Cont.
Schematic diagram of the surface topography composed of
patches of sites, of which each patch contains sites of the
same energy.
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Patchwise topography
Topography of adsorption sites is usually taken as the patchwise
model, whereby all sites having the same energy are grouped
into one patch and there is no interaction between patches.
The parameters used as the distributed variable can be one of the
following:
the interaction energy between the solid and the adsorbate molecule,
the micropore size,
the Henry constant, and
the free energy of adsorption.
Among these, the interaction energy is the commonly used as
the distributed variable.
11
Cont.
When the micropore size is used as the distributed
variable, a relationship between the interaction energy and
the micropore size has to be known, and this can be
determined from the potential energy theory, or if the local
isotherm used is the DR or DA equation, the relationship
between the characteristic energy and the micropore size
proposed by Dubinin and Stoeckli could be used.
12
Cont.
The topography is only important when the interaction between
adsorbed molecules is important.
Adsorption equations such as the Fowler-Guggenheim, the Hill-
deBoer, and the Nitta et al. equations are capable of describing the
adsorbate-adsorbate interaction.
The energy accounting for this interaction depends on the number of
neighboring adsorbed molecules.
In the patchwise topography, the average number of neighbor
molecules is proportional to the fractional loading of that particular
patch (i.e. local fractional loading), while in the random topography,
the average number of neighboring molecules is proportional to the
average fractional loading of the solid (i.e. the observed fractional
loading).
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Langmuir Approach
The simplest model describing the heterogeneity of solid surface is that
of Langmuir. He assumed that the surface contains several different
regions. Each region follows the usual Langmuir assumptions of one
molecule adsorbing onto one site, homogeneous surface and localized
adsorption.
The further assumptions are that there is no interaction between these
regions, i.e. they act independently, and within each region there is no
interaction between adsorbed molecules.
If there are N such regions, the adsorption equation is simply the
summation of all the individual Langmuir equations for each region,
that is:
14
Cont.
For low enough pressures, the equation reduces to the usual
Henry law relation:
If all the patches are very different in terms of energy, then the
overall Henry law constant is approximately equal to that of the
strongest patch, that is to say at low pressures almost all
adsorbed molecules are located in the patch of highest energy.
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Energy Distribution Approach
Adsorption of molecule in surfaces having constant energy
of interaction is very rare in practice as most solids are
very heterogeneous.
We can explain the degree of heterogeneity by assuming
that the energy of interaction between the surface and the
adsorbing molecule is governed by some distribution.
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Surface Topography
The observed adsorption equilibria, expressed as the
observed fractional loading obs (amount adsorbed at a
given pressure and temperature divided by the maximum
adsorption capacity), can be written in terms of a local
adsorption isotherm and an energy distribution:
: local isotherm (isotherm of a homogenous patch with interaction energy
between that patch and the adsorbing molecule of E),
P : gas pressure, T : temperature,
u : interaction energy between adsorbed molecules,
F(E) :energy distribution with F(E)dE being the fraction of surfaces having energy
between E and E+dE.
17
Cont.
If the local isotherm and the energy distribution are known, the
previous equation can be readily integrated to yield the overall
adsorption isotherm.
This is a direct problem, but the problem usually facing us is that very
often the local isotherm is not known and neither is the energy
distribution.
Facing with these two unknown functions, one must carry out
experiments (preferably at wider range of experimental conditions as
possible), and then solve equation as an inverse problem for the
unknown integrand.
Very often, we assume a form for the local isotherm and then solve the
inverse problem for the energy distribution.
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The Energy Distribution
The energy distribution for real solids is largely unknown
a priori, and therefore the usual and logical approach is to
assume a functional form for the energy distribution,
such as:
Uniform distribution
Exponential distribution
Gamma distribution
Shifted Gamma distribution
Normal distribution
log-normal distribution
Rayleigh distribution
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Relationship between Slit Shape
Micropore and Adsorption Energy
The energy distribution approach provides a useful
means to describe the adsorption isotherm of
heterogeneous solids.
But the fundamental question still remains in that how
does this energy distribution relates to the intrinsic
parameters of the system (solid + adsorbate).
In dealing with adsorption of some adsorbates in
microporous solids where the mechanism of adsorption is
resulted from the enhancement of the dispersive force,
we can relate this interaction energy with the intrinsic
parameter of the solid (the micropore size) and the
molecular properties of the adsorbate.
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Two Atoms or Molecules Interaction
(12-6 potential)
The basic equation of calculating the potential energy of
interaction between two atoms or molecules of the same
type "k" is the empirical Lennard-Jones 12-6 potential
equation.
r : distance between the nuclei of the two atoms or molecules,
kk : depth of the potential energy minimum,
kk : distance at which kk is zero (characteristic or collision diameter).
The minimum of the potential occurs (when F=d/dr=0) at 21/6 kk (= 1.122kk).
Repulsion Attraction
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Intermolecular forces
-4E-07
-2E-07
0
2E-07
4E-07
8 10 12 14 16 18 20 22 24
Series 1
Lennard-Jones potential function
r
kk
0
22
Potential energy of interaction
23
Cont.
For two atoms or molecules of different type (type 1 and
type 2), the relevant 12-6 potential energy equation is:
Where:
24
Cont.
F=0 when r= 1.122 12
F>0 when r < 1.122 12 implying repulsion,
F<0 when r > 1.122 12 suggesting attraction.
At r 3 12 the force becomes negligible.
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An Atom or Molecule and a Lattice
Plane (10-4 potential)
Where:
n : number of the interacting centers per unit area of the lattice plane.
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An Atom or Molecule and a Slab
(9-3 potential)
Where:
n' : number of interacting centers per unit volume of solid.
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Comparison
Comparison between 12-6,10-4, 9-3 potentials
versus the reduced distance
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Potential energy equations and
their characteristics
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A Species and Two Parallel
Lattice Planes
d-z
d+z
d
30
Cont.
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Potential energy for two parallel
lattice planes
E
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A Species and Two Parallel Slabs
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Potential energy for two parallel
slabs
34
Cont.
Four different half-widths and corresponding values of
the central potential energies.
When =0 : Situation is similar to the gas phase
35
Cont.
To calculate minimum potential energy, we
need to know:
Collision diameter (12)
Minimum energy at infinite spacing (*1SLP)
Minimum potential energy = Energy of interaction between the
molecule and the pore (E)
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Adsorption Isotherm for Slit
Shape Pore
The analysis of the last two cases are particularly useful for
the study of adsorption of nonpolar molecules in slit-
shaped micropore solids, such as activated carbon:
(i) a molecule and two parallel lattice planes
(ii) a molecule and two parallel slabs
Using the previous equations, an overall adsorption
isotherm can be obtained from the local adsorption
isotherm and a micropore size distribution.
37
Cont.
Let us denote the micropore size distribution (MPSD) as f(r) such that:
is the volume of the micropores having half width from rmin to r, where
Vμ is the micropore volume.
The minimum micropore half width rmin is defined as the minimum
micropore size accessible to the adsorbate, hence it is a function of the
adsorbate.
If the local adsorption in a micropore having a half width of r is
denoted as:
38
Cont.
where E is the interaction energy between the adsorbent and the
adsorbate (which is a function of pore half-width), then the overall
adsorption isotherm is taken in the following form:
where rmax is the maximum half width of the micropore region.
In writing this eq., we have assumed that the state of the adsorbate in
the micropore is liquid-like with vM being the liquid molar volume
(m3/mole). We could write eq. as follows:
39
Cont.
These integral can only be evaluated if the relationship
between the energy of interaction E and the pore half width
r is known.
This relationship is possible with the information for two
parallel lattice planes and two parallel slabs.
The depth of the potential minimum is the interaction
energy between the micropore and the adsorbate.
40
Cont.
If we have MPSD, then using the previous
eqn. we can calculate the amount adsorbed
If we have the amount adsorbed versus
pressure, then we can calculate MPSD. (In
most cases we have this situation)
41
Micropore Size-Induced Energy
Distribution
Knowing the relationship between the energy of interaction
versus the pore size, the energy distribution can be
obtained from the micropore size distribution by using the
following formula:
where F(E)dE is the fraction of the micropore volume
having energy of interaction between E and E+dE. Thus,
we have:
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Micropore size distribution