Multicomponent

adsorption equilibria

Ali Ahmadpour

Chemical Eng. Dept.

Ferdowsi University of Mashhad

2

Contents

Introduction

Langmuir Multicomponent Theory

The Basic Thermodynamic Theory

Ideal Adsorption Solution Theory (IAS)

Fast IAS Theory

Real Adsorption Solution Theory (RAST)

Potential Theory

Conclusions

3

Introduction

In most cases, adsorption systems usually involve

more than one component, therefore adsorption

equilibria involving competition between

molecules of different type is needed for

understanding the system as well as the design

purposes.

4

Langmuir Multicomponent

Theory

The extended Langmuir isotherm equation is the

simplest theory for describing multicomponent

equilibria.

This equation has assumed that each species maintains its own surface

area (i.e. the area covered by one species is not affected by the others).

5

Cont.

The selectivity of species "i" in relation to the

species "j" is defined as:

It is reminded that the extended Langmuir isotherm is derived assuming

the surface is ideal and the adsorption is localized, and one molecule can

adsorb onto one adsorption site.

6

Empirical Approaches Based on

the Langmuir Equation

When there are lateral interactions between adsorbed species, which are

generally different from the self interactions among molecules of the

same type, the following empirical equation is recommended:

The parameter ηi allows for interaction among molecules of the same

type, and ni allows for other effects of heterogeneity.

This equation is the generalization of the Langmuir-Freundlich (Sips)

isotherm.

7

The Basic Thermodynamic Theory

Recognizing the deficiency of the extended

Langmuir equation, despite its sound theoretical

footing on basic thermodynamics and kinetics

theories, other approaches have been proposed.

ideal adsorbed solution theory of Myers and Prausnitz,

real adsorption solution theory,

vacancy solution theory,

potential theory.

8

Ideal Adsorption Solution Theory

(IAS)

n°j : number of mole of pure component j per unit mass of the adsorbent;

Фj : surface potential of the pure component j;

xi, yi: mole fraction in the adsorbed phase and gas phase, respectively.

9

Cont.

If the pure component isotherm n°i(Pi) is known, the surface potential

of the pure component i can be obtained from the integral.

When IAS theory apply to the usual case of adsorption equilibria, we

should specify the total pressure and the mole fractions in the gas

phase and then determine the properties of the adsorbed phase which is

in equilibrium with the gas phase.

Then, when the gas phase is specified the total number of unknowns is:

10

Cont.

Once the total adsorbed amount is determined, the adsorbed phase

concentration of the component i is:

In theory of IAS, adsorption isotherm equation for pure components

can take any form which fits the data best.

Two isotherms commonly used are Toth and Unilan equations.

Toth equation is preferred from the computational point of view

because it usually gives faster convergence than the Unilan equation.

11

Cont.

The success of the calculations of IAS depends on how

well the single component data are fitted, especially in the

low pressure region as well as at high pressure region

where the pure hypothetical pressure lies.

An error in these regions, particularly in the low pressure

region, can cause a large error in the multicomponent

calculations.

12

Practical Considerations of The

IAS Equations

The IAS theory is recast in the form convenient for the

purpose of computation.

For a system containing N species, the IAS equations are:

P°i() is the hypothetical pressure of the pure component that gives the

same spreading pressure on the surface.

13

Cont.

The spreading pressure is the negative of the surface potential.

For a given total pressure (P) and the mole fraction in the gas

phase (yi), these equations provide a total of 2N+1 equations.

With this set of 2N+1 equations, there are 2N+1 unknowns.

N values of mole fractions in the adsorbed phase (xi),

one value of the spreading pressure π, and

N values of the hypothetical pressure of the pure component, P°j,

which gives the same spreading pressure as that of the mixture.

Solving the equations numerically will give a set of solution for

the adsorbed phase mole fractions and a solution for the

spreading pressure.

14

Cont.

For general case, the IAS equations must be

solved numerically and this can be done with

standard numerical tools, such as the Newton-

Raphson method for the solution of algebraic

equations and the quadrature method for the

evaluation of integral.

15

IAS Theory

The IAS theory provides a convenient means to

calculate the multicomponent equilibrium.

It has a number of attractive features:

The theory does not require any mixture data,

It is independent of the actual model of physical adsorption,

The form of adsorption isotherm equation for pure component

data is arbitrary (hence any equations that best fit the data can

be used),

Different solutes can have different forms of the isotherm

equation.

16

Cont.

Despite the versatility of the IAS theory, one disadvantage

is the evaluation of the reduced spreading pressure.

With the exception of few isotherm equations, the integral

equation for the spreading pressure is generally evaluated

numerically.

Even when the spreading pressure can be analytically

evaluated, the inverse problem of obtaining the pure

component "hypothetical“ pressure in terms of the reduced

spreading pressure is not always available analytically.

17

Fast IAS Theory

An isotherm, introduced by O'Brien and Myers (1984), is

obtained as a truncation to two terms of a series expansion of the

adsorption integral equation in terms of the central moments of

the adsorption energy distribution.

Using this equation which allows the computation of reduced

spreading pressure analytically is called the Fast IAS procedure.

Any isotherm that has explicit expression for the spreading

pressure can be used in the Fast IAS formalism.

18

Cont.

The difference between IAS and Fast IAS is that:

The Fast IAS involves the solution of N variables of pure

component pressure, bjPj0, while the solution for the spreading

pressure is sought in the IAS theory.

Once the spreading pressure is known in the IAS theory,

the hypothetical pure component pressures can be obtained

as the inverse of the integral equation.

Thus, solving directly for the hypothetical pressures of

pure component makes the computation using the Fast IAS

faster than the IAS theory.

19

Cont.

Although the Fast IAS is attractive to obtain the set of

solution for the hypothetical pressures, the applicability of

this theory is restricted to only a few equations which yield

analytical expressions for the reduced spreading pressure.

For other equations, the IAS theory has to be used instead.

20

Real Adsorption Solution

Theory (RAST)

The procedure used in IAS theory is simple and the

method of calculation is also straight forward.

Unfortunately, the method only works well when the

adsorption systems do not behave too far from ideality.

However for some nonideal cases (azeotropic), the IAS

theory is inadequate to describe the systems.

In such systems, the activity coefficient is used to account

for the nonideality of the adsorbed phase.

This procedure called as the real adsorption solution theory

(RAST).

21

Cont.

The activity coefficients can be either calculated from known

theoretical correlations or from binary equilibrium data.

They are a function of the composition of the adsorbed phase and

the spreading pressure. To calculate them, we need the binary

experimental data:

(a) the total pressure of the gas phase and its compositions

(b) the compositions of the adsorbed phase

(c) the spreading pressure of the mixture

Knowing these activity coefficients, they are then used in some

theoretical models, such as the Wilson equation to derive the binary

interaction parameters of that model.

22

Binary adsorption curves

G a s p h a se m o le fra c tio n , Y1

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Am

ou

nt

ad

so

rbe

d (

mm

ole

/g)

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0N u x it c a rb o n , 2 9 3 K

C2H

6

C H4

IH F L

IA S T

M P S D

E D

S ip s

G a s p h a se m o le fra c tio n , Y1

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Am

ou

nt

ad

so

rbe

d (

mm

ole

/g)

0 .0

0 .5

1 .0

1 .5

2 .0

2 .5

3 .0

A ja x ca rb o n , 2 7 3 K

C O2

C H4

IH F L

M P S D

E D

IA S T

S ip s

23

Binary adsorption curves

(X-Y diagrams)

M e th a n e in th e a d s o rb e d p h a s e , X1

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

Me

tha

ne

in

th

e g

as

ph

as

e,

Y1

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

C H4-C O

2

(a )

M e th a n e in th e a d s o rb e d p h a se , X1

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

(b )

C H4-C

2H

6

M e th a n e in th e a d s o rb e d p h a s e , X1

0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

C H4-C

3H

8

(c )

24

Potential Theory

The extension of the potential theory was studied by Doong

and Yang (1988) to multicomponent systems.

The approach is simple in concept, and it results in analytical

solution for the multicomponent adsorption isotherm.

The basic assumption of the model is that:

there is no lateral interaction between molecules of different types,

pure component isotherm data are described by the DA equation.

25

Conclusions

Theories for adsorption equilibria in multicomponent systems

are not as advanced as those for single component systems.

The slow progress in this area is due to a number of reasons:

(i) lack of extensive experimental data for multicomponent

systems,

(ii) solid surface is too complex to model adequately.

However, some good progress has been steadily achieved in

this area.