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.

4

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.

6

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.

8

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.

10

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.

15

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.

16

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

21

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.

25

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.

26

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

28

Potential energy equations and

their characteristics

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A Species and Two Parallel

Lattice Planes

d-z

d+z

d

30

Cont.

31

Potential energy for two parallel

lattice planes

E

32

A Species and Two Parallel Slabs

33

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)

36

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