© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

ADSORPTION:

Energy distribution function

(Used in General Integral Equation of Adsorption)

GL energy distribution | energy dispersion | Global Heterogeneity

En. distr.: Calculation ( Stieltjes transform | CA | CA examples | Practical appl.)

GIEA for: ( single gas | single solute | binary liquid | multicomponent ) adsorption

GIEA and En.Distr. ( Differential, χ(E) | Integral, F(E) | Normalization | Links )

Energy Dispersion, σ | Henry behavior | Surface Topography

Model pictures ( χ(E) vs. F(E) | χ(E) & θ | F(E), E(F) & θ )

Page under reconstruction

**General Integral Equation of Adsorption** (GIEA) is a general formulation of adsorption isotherm involving **energetic heterogeneity** of the adsorption system (i.e. adsorbate-surface).

**CAUTION!**

One has to be always cautious while describing energy distribution function as the property of adsorbent only. Though for homogeneous adsorbents it may be true, it may change not only with the change of such adsorbate properties like functional groups, but also with changing molecular size and shape and surface topography (see also below). Moreover, in adsorption of mixtures the obtained energy distribution function may have altered - and quite different - shape and width depending on energy correlations for molecule pairs on adsorption sites.

Differential Energy Distribution Function χ(E)

Global (overall) isotherm of adsorption θ_{t} is obtained by averaging of **local adsorption coverage θ _{t}** (adsorption energy dependent) corresponding to adsorption on various surface sites. This local isotherm equation describes adsorption on energetically homogeneous surface, however the weight factor is a non-negative

For

Integral (cumulative) Energy Distribution Function F(E)

The differential energy distribution χ(E) function is defined as the probability density function of adsorption energy. The General Integral Equation of Adsorption may be easily reformulated in a way allowing for use of **integral (cumulative) energy distribution function F(E)** instead of differential function χ(E).

For **homogeneous surface** (Langmuir isotherm) this function becomes a simple stepwise function (Heaviside step function) equal to zero for energies less than charactersitic adsorption energy and equal to 1 for energies greater than or equal to this energy value (alternative definition).

Normalization of Energy Distribution Function

The values of F(E) are always between 0 and 1, whereas the integral of χ(E) over energy E is **normalized** to 1. Such normalization approach simplifies formulation of generally applied equation, however, if needed such normalization conditions (integral = 1 and 0 to 1 range, correspondingly) may be redefined. If the General Integral Equation of Adsorption is reformulated by replacing relative coverages (θ) by actual adsorbed amounts, then in the above normalization conditions we must replace dimensionless 1 (corresponding to the monolayer capacity) by maximum adsorption capacity (in the monolayer!), a_{m} expressed in e.g. [mmol/g].

Dispersion of Adsorption Energy

Besides energy distribution general shape and symmetry an important parameter of energy distribution function is its **energy dispersion** (or - in general meaning - distribution width). This factor decides whether energetic heterogeneity effects will be visible in isotherm's behavior. For energy distributions of low energy dispersion the behavior of isotherm at moderate coverages will resemble Langmuir isotherm. If energy dispersion is high, the sensitivity of adsorption to the changes of pressure (or concentration) at moderate coverages will be much weaker than in the case of Langmuir isotherm (will resemble the middle section of Generalized Freundlich, Langmuir-Freundlich, Tóth etc. isotherm)

Henry behavior

If the distribution has finite width (i.e. has minimum and maximum energy) or it behaves in a special way at low and high adsorption energies the corresponding isotherm should display a so-called Henry and Langmuir behavior for low and high coverages, respectively.

Surface site **topography** and lateral interactions - local isotherm:

On **random topography** surfaces the interaction factor* depends on the mean field of adsorbate molecules which is surface-averaged anyway, i.e. it depends on the local average adsorbate density which is the same as average density of adsorbate molecules over the whole surface (that is: θ_{l,avg} = θ_{t}) and thus does not depend on which particular site the molecule is sitting (i.e. does not depend on local adsorption energy E).
However, for **patchwise topography**, the density of adsorbate molecules is characteristic - and different (depending on the local adsorbate density, i.e. local coverage θ_{l}) - for each of the patches characterized with different adsorption energy E.

* (Interacton factor describes how the presence of other adsorbate molecules affects adsorption forces by attractive Van der Waals forces, by specicic forrces like associative or hydrogen bonding, or by electrostatic forces - see expressions for Kiselev and Fowler-Guggengeim in gas and solute adsorption)

Global isotherm and surface topography

If for a given type of heterogeneity and without lateral interactions present the isotherm equation θ_{t}(c) (or for gas adsorption: θ_{t}(p)) has analytical form, it will be analytical with lateral interactions and **random site topography**, too. New isotherm θ_{t,int}(c) (or θ_{t,int}(p)) will have indentical form as the original equation if we replace concentration (or pressure for gas adsorption) by a suitable term c' = c f(θ_{t}) (or p' = p f(θ_{t})).
For **patchwise topography** with lateral interactions the obtained isotherms are generally non-analytical.

Model pictures

Comparison of differential and integral (cumulative) energy distribution functions
**Comparison of χ(E) and F(E)**

Differential energy distribution function χ(E) (blue lines; area under curve is 1) and corresponding integral distribution F(E) (red/orange lines; F_{min}=0, F_{max}=1) for LF isotherm (m=n= 0.9 and 0.5), E=ε/RT - reduced energy.

**Comparison of χ(z) and F(z)**

Differential energy distribution function χ(z) (blue lines; area under curve is 1) and corresponding integral distribution F(z) (red/orange lines; F_{min}=0, F_{max}=1) for LF isotherm (m=n= 0.9 and 0.5), z(F)=E(F)-E_{o} - relative energy (E=ε/RT - reduced energy).

Differential energy distribution χ(E) and surface coverage, local θ_{l} and global θ_{l}

Differential energy distribution functions χ(E) (blue lines; area under curve is 1) and products of local coverage and energy distribution function θ_{l}(p,E)χ(E) (red/orange lines; area under curve is equal to global/overall coverage θ_{t}(p)) for LF isotherm (m=n= 0.9 and 0.5, fixed pressure, Kp=1).

Integral (cumulative) energy distribution F(E) and surface coverage, local θ

**Influence of heterogeneity**

Influence of heterogeneity on energy distribution E(F) (red/orange lines; function inverse to integral (cumulative) energy distribution) and local coverage θ_{l}(p,E(F)) (blue lines; areas under curves are equal to global/overall coverages, θ(p)) for LF energy distribution.

**Influence of pressure (low heterogeneity)**

Influence of pressure on local coverage θ_{l}(p,E(F)) (blue lines; areas under curves are equal to global/overall coverages, θ(p)) for LF energy distribution (energy distribution E(F) - orange line; function inverse to integral (cumulative) energy distribution) (low heterogeneity).

**Influence of pressure (high heterogeneity)**

Influence of pressure on local coverage θ_{l}(p,E(F)) (blue lines; areas under curves are equal to global/overall coverages, θ(p)) for LF energy distribution (energy distribution E(F) - orange line; function inverse to integral (cumulative) energy distribution) (high heterogeneity).

More on the energy distribution functions in:

- General Integral Equation of Adsorption (and its special cases for gas and vapor, solute and binary liquid adsorption), see also (
*in preparation*) GIEA for multicomponent systems and its applications in prediction of mutlicomponent adsorption - Energy dispersion - definitions and calculation.
**Energy distributions**, χ(E), for the Generalized Langmuir isotherm (with its special cases: LF, GF, Tóth) and some other isotherms: Gauss and Rudzinski) - formulas and model graphs.**Alternative formulations of energy distributions**, F(z) and z(F), for the Generalized Langmuir isotherm (with its special cases: LF, GF, Tóth) and some other isotherms: Gauss, Rudzinski, Square aka UNILAN, k-centered and cut Freundlich) - formulas and relations, and series of model graphs.

Energy Dispersion, σ | Henry behavior | Surface Topography

Model pictures ( χ(E) vs. F(E) | χ(E) & θ | F(E), E(F) & θ )

GIEA for: ( single gas | single solute | binary liquid | multicomponent ) adsorption

GL energy distribution | energy dispersion | Global Heterogeneity

En. distr.: Calculation ( Stieltjes transform | CA | CA examples | Practical appl.)

isotherm equations | GL isotherm

Top

My papers

Search for papers

Main page

Send a message to *Adam.Marczewski AT@AT umcs.lublin.pl*