_{1}

^{*}

We show that most of the empirical or semi-empirical isotherms proposed to extend the Langmuir formula to sorption (adsorption, chimisorption and biosorption) on heterogeneous surfaces in the gaseous and liquid phase belong to the family and subfamily of the
*Burr _{XII}* cumulative distribution functions. As a consequence they obey relatively simple differential equations which describe birth and death phenomena resulting from mesoscopic and microscopic physicochemical processes. Using the probability theory, it is thus possible to give a physical meaning to their empirical coefficients, to calculate well defined quantities and to compare the results obtained from different isotherms. Another interesting consequence of this finding is that it is possible to relate the shape of the isotherm to the distribution of sorption energies which we have calculated for each isotherm. In particular, we show that the energy distribution corresponding to the Brouers-Sotolongo (

*BS*) isotherm [1] is the Gumbel extreme value distribution. We propose a generalized

*GBS*isotherm, calculate its relevant statistical properties and recover all the previous results by giving well defined values to its coefficients. Finally we show that the Langmuir, the Hill-Sips, the BS and GBS isotherms satisfy the maximum Bolzmann-Shannon entropy principle and therefore should be favoured.

Every year hundreds or more papers are devoted to the analysis of sorption (physical adsorption, chemi- and bio-sorption) of gas or solutions on a variety of substrates [

In this paper which is a contribution to that effort, we want to emphasize that since some of these isotherms appear to be genuine cumulative probability distributions, they should be favoured, formulated in the language of the theory of probability and might bring more quantitative and more structured information making advantage of their mathematical properties. The probability theory of complex systems has made considerable progress these last years and one can expect that its introduction in the field of sorption could be of great help.

A few years ago we published a paper [

As a consequence of this study we proposed an isotherm using a Weibull distribution known since as Brouers-Sotolongo (BS) isotherm which has been used among others in sorption on porous/nonporous surface interface [

The present paper is a extension of some of the ideas developed in our previous works. We will take advantage of the recent progress in the statistical theory of complex and deterministic chaotic systems. We will show that many of the isotherms used in the literature, especially in the treatment of water, form a subfamily of the Burr_{XII} distribution. This will lead us to propose a generalization (GBS) of the BS isotherm replacing the exponential in the Weibull function by a deformed exponential used now in the formulation of the nonextensive thermodynamics [

If we view the isotherm as a cumulative distribution function we can write the isotherms in the following forms:

In Equation (1),

In an heterogeneous system, as we increase the pressure or the concentration, the most active sites with the highest sorption energy are first occupied until complete saturation. With a change of variable, one can write

where

We will now demonstrate that if we choose for

In probability theory and statistical sciences, the

The

where

In previous papers [

The cumulative distribution functions belonging to the Burr family are solution of the general differential equation

where

In its discrete form it has been one of the first model of deterministic chaos [

For the

The

An other interesting feature of the

The

・ For

This is a Weibull distribution. The corresponding isotherm in the sorption literature is known as the Brouers- Sotolongo (BS) isotherm:

If

If moreover one puts

・ For

which is called in probability theory the loglogistic function. The corresponding isotherms are the Hill, the Langmuir-Freundlich and Sips isotherms

・ If both

the corresponding isotherm is the Langmuir isotherm.

As discussed in [

In the isotherms we have just reviewed, the exponent

This generalized BS isotherm has a unified character since it contains the Langmuir, the Freundlich-Langmuir, the Hill and the Sips isotherm and as we will see in the next section, the Generalized Freundlich-Langmuir and the Toth isotherms. The GBS isotherm can be written in a more compact form

We have used the definition of the deformed exponential function introduced in mathematics in the XIX century and appearing to day in the theory of many complex systems

When

This new isotherm has four parameters

A two exponents isotherm (GFL) generalizing the Freundlich-Langmuir (Hill, Sips) isotherm was proposed by Marczewski and Jaroniec [

The corresponding cdf function

has the characteristics of a cdf

It appears that

As

We have moreover:

These asymptotic behaviors which are supposed to be the same as the ones of

Starting from the GLF isotherm equation, one can recover some of the empirical isotherms: for

The Burr_{XII} cdf and pdf functions (Equations (4), (5)) can be written

If we make the change of variables

Therefore one has the relation

The relation between the Generalized Freundlich-Langmuir function and the Burr_{XII} function can be written using the previous results:

This allows the GLF isotherm and the Toth [

The others empirical isotherms [

We can now derive quite simply the shape of the sorption energy distribution giving rise to the various isotherms we have just derived.

As we already discussed in a previous publication, starting from the thermodynamic relation

and using the probability theory relation

it is possible to calculate the sorption energy distribution corresponding to each isotherm. As discussed later, this sorption energy

In that way we have obtained the following results:

・ For the proposed GBS. isotherm derived from the Burr_{XII} distribution function:

The other distributions can be obtained easily:

・ For

・ For

It is worth noticing that the BS. distribution has the form of the Gumbel [

The standard deviation of this function is well known

confirming the conclusions of reference [

The function

It is the symmetric of the Fisher-Tippett [

It is worth noticing that this last

To be complete we have calculated the energy distributions corresponding to the Freundlich-Langmuir isotherm

If

If

If

Some of the these distributions have been obtained earlier by various authors without reference to the pro- bability theory and using the Cerofolini condensation approximation method [

Before concluding this study it is worthwhile to point out that the distribution functions giving the Langmuir, the Hill-Sips-“Langmuir-Freundlich”, the Brouers-Sotolongo and Generalized Brouers-Sotolongo can be derived maximizing the Boltzmann-Shannon entropy measure:

using the Lagrange multipliers methods [

in the same spirit as the deformation of the exponential function (Equation (22)). One uses the following con- straints:

where

For

which are the well known Weibull constraints and

which are the loglogistic constraints. The fact that these isotherms correspond to the maximum entropy show that they are the best and less biased isotherms when the parameters a, b and

In this paper we have shown that a generalized isotherm having the analytical form of a

The statistical expressions given in the appendix allow a mathematically well defined characterization of the data. Extensions of the

Another important conclusion of this study is that the energy distributions giving rise to the

Two last remarks have to be made on the range of applicability of the results of this paper. One has to emphasize that it deals with one aspect of sorption i.e. the generalization of the Langmuir isotherm to highly heterogeneous surfaces and solid-liquid interfaces and in some cases of complex composition of sorbates and sorbent. It concerns in particular most works done in water and air decontamination research with pure or treated natural products.

The sorption of simple molecules on smooth surfaces and well defined rough surfaces [

The statistical quantities of all isotherms deriving from the unified

What we need to characterize statistically the experimental isotherms are the maximum sorbed quantity corresponding to the saturation

corresponding to the maximum sorption rate

We can determine the value of

For

The expression for the k-th moment is

where

From Equation (62), one can calculate the expectation value

The inverse of the

allows us to know the pressure (or the concentration) corresponding to a given percentage of the sorbed quantity.

One can then calculate the quantile

sorbed quantity ranging from 0 to 1. We have therefore using Equation (65):

These are the values of

What to do when

Starting from the expression of the kth moment (Equation (65)) and choosing the value

the expression (64) yields

This statistical quantity

The pdf

Calculating the limits

In the first case when

For the second case one gets

It is convenient to have a relation between

More results can be found in [

All these results can be obtained directly by performing the corresponding integrals.

The characterization of sorption using the values of