Adsorption Equilibria

Ali Ahmadpour

Chemical Eng. Dept.

Ferdowsi University of Mashhad

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Contents

Introduction

Adsorption isotherm models

Langmuir isotherm

Volmer isotherm

Fowler-Guggenheim isotherm

Hill-deBoer isotherm

Concluding remarks

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Introduction

Adsorption equilibria is the most important information in

understanding an adsorption process.

No matter how many components are present in the system,

the adsorption equilibria of pure components indicate how

much those components can be accommodated by a solid

adsorbent.

This information can be used in the study of adsorption

kinetics of a single component, adsorption equilibria of

multicomponent systems, and then adsorption kinetics of

multicomponent systems.

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Adsorption Process

Adsorbent

AdsorbateAdsorptive

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The Isotherm

The amount of gas adsorbed is a function of

The strength of interaction between gas and solid (intrinsic)

Temperature (fixed)

Pressure (controlled variable)… expressed as relative pressure P/Po

T)(C, fq

T)(P, fV

V : cm3 STP/g ; mole/g ; mole/cm3

q : mole/g ; mole/cm3

Gas-Solid system:

Liquid-Solid system:

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Adsorption isotherm models

• Some of the famous models describing the adsorptionequilibria are:

• Basic model: Langmuir (1918)

• Extended model: BET (1938)

• Empirical model: Freundlich (1932)

• There are various types of theoretical, empirical andsemi-empirical models for equilibrium of adsorption.

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Different types of adsorption

isotherms

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Langmuir adsorption isotherm

(Type I)

bp

1+bp

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Type II and IV isotherms

Similar to type II

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Type III and V isotherms

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The adsorption theory

The Langmuir theory (1918) is the most basic theory in

adsorption. Langmuir described the kinetic behavior of the

adsorption process.

This theory allows us to understand the monolayer surface

adsorption on an ideal surface.

He postulated that at equilibrium, the rate of arrival of

adsorptive (adsorption) and the rate of evaporation of

adsorbate (desorption) were equal.

Furthermore, the heat of adsorption was taken to be constant

and unchanging with the degree of coverage, θ.

I. Langmuir, J. Amer. Chem. Soc., 40, 1368 (1918)

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• Graduated as a metallurgical engineer from the

School of Mines at Columbia University in 1903.

• 1903-1906 M.A. and Ph.D. in 1906 from

Göttingen.

• 1906-1909 Instructor in Chemistry at Stevens

Institute of Technology, Hoboken, New Jersey.

• 1909 –1950 General Electric Company at

Schenectady where he eventually became

Associate Director.

• 1913 :Invented the gas filled, coiled tungsten

filament incandescent lamp.

Irving Langmuir (1881-1957)

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• 1919 to 1921, his interest turned to an examination of atomic theory, and

he published his "concentric theory of atomic structure". In it he

proposed that all atoms try to complete an outer electron shell of eight

electrons.

• 1927 Coined the use of the term "plasma" for an ionized gas.

• 1935-1937 With Katherine Blodgett studied thin films.

• 1948-1953 With Vincent Schaefer discovered that the introduction of dry

ice and iodide into a sufficiently moist cloud of low temperature could

induce precipitation.

• 1932 The Nobel Prize in Chemistry "for his discoveries and

investigations in surface chemistry"

Cont.

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• Langmuir theory describe that the rate of accumulation of

molecules at the flat surface is zero at equilibrium.

• The assumptions of the Langmuir model are:

Homogeneous surface (all adsorption sites energetically identical).

Monolayer adsorption (Adsorption on surface is localized, i.e. adsorbed

atoms or molecules are adsorbed at definite, localized sites).

No interaction between adsorbed molecules (Each site can

accommodate only one molecule or atom).

Langmuir Theory

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Cont.

Rate of adsorption (the striking rate at the surface)× sticking coefficient (α: the

accommodation coefficient) = Rate of desorption from the surface

The rate of striking the surface (mole / time × area), from the kinetic theory of gas is:

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Cont.

The rate of adsorption (mole adsorbed/ surface area × time) is:

The rate of desorption, corresponds to fully covered surface (kd ) × fractional

coverage, is:

Ed = Q : activation energy for desorption (is equal to the heat of adsorption for

physically sorbed species)

kd : rate constant for desorption at infinite temperature

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Cont.

The average residence time of adsorption is defined as:

The higher Ed , the longer

is the time for adsorption

Physisorption: a = 10-13 to 10-9 sec

Chemisorption: a= 10-6 to 109 sec

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Famous form of Langmuir

isotherm in the gas phase

“b” is affinity constant and a measure of adsorbate-adsorbent attraction forces.

bp

bp

1

)/exp(2

)/exp(TRQb

TMRk

TRQ

k

kb

g

gd

g

d

a

lim bpp 0

lim = 1p

1bpor

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Behavior of Langmuir model

The Langmuir isotherm reduces to the Henry law isotherm when the pressure is

very low (bP << 1).

When pressure is sufficiently high, the amount adsorbed reaches the saturation

capacity or monolayer coverage (1).

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Useful form of Langmuir isotherm

for gas phase

P b(T )1

P b(T )CC

μs

RT

QbTb exp

Cμ : amount adsorbed (mole per unit mass or volume).

Cμs : maximum adsorbed concentration corresponding to a

complete monolayer coverage.

μ denote the

adsorbed phase

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Linear form of equation

P

1

bC

1

C

1

C

1

μsμs

Plotting 1/Cμ against 1/P gives straight line with the slope 1/bCμs

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Isosteric Heat of Adsorption

• Isosteric heat is the ratio of the small change in the adsorbate enthalpy

to the small change in the amount adsorbed.

• The knowledge of isosteric heat is essential in the study of adsorption

kinetics.

• The isosteric heat may or may not vary with loading. It is calculated

from the van't Hoff equation:

: thermal expansion coefficient

of the saturation concentration

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Cont.

• The negativity of the enthalpy change indicates that the

adsorption process is an exothermic process.

• For the isosteric heat to take a finite value at high coverage

(that is 1), the parameter must be zero.

• Therefore, the saturation capacity is independent of

temperature, and as a result the heat of adsorption is constant,

independent of loading and temperature (very ideal system).

• Prove that for the Langmuir isotherm:

dEQH

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emme

Cbq/q/q/ 111

m

VCCq

e

e

)(0

qe : amount adsorbed at equilibrium in the adsorbed phase (mg/g)

qm : Langmuir constant related to maximum adsorption capacity (mg/g)

C0 : initial concentration in the aqueous solution (mg/L)

Ce : equilibrium concentration in the aqueous solution (mg/L)

b : Langmuir constant related to energy of adsorption (L/mg)

V : volume of the solution (L)

m : sorbent dose in the mixture (g)

Form of Langmuir isotherm in

Liquid phase

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Isotherms based on the Gibbs

thermodynamic approachIf adsorbed phase is treated as a 2D surface, we have the following Gibbs equation for pure component systems:

At equilibrium, the chemical potential of the adsorbed phase is equal

to that of the gas phase, which is assumed to be ideal, i.e.

Substituting μ into the eqn. gives the following Gibbs isotherm:

This is the fundamental equation relating gas pressure, spreading

pressure (Π) and amount adsorbed.

03 nd μVdP:D in

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Linear isotherm

For an ideal surface at infinite dilution, the EOS relating the spreading pressure and the number of mole on the surface is:

nRTPV:D 3nRTA:D2

Integrating this equation at constant T gives:

At equilibrium the spreading pressure in the

adsorbed phase is linearly proportional to the

pressure in the gas phase.

To relate the amount adsorbed in the adsorbed

phase in terms of the gas phase pressure, we

use EOS to finally get:

Where:

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Volmer isotherm

If consider the case where allow for the finite size of adsorbed molecules, the EOS for a surface takes the following form:

Ao : the minimum area

occupied by n molecules

The Gibbs equation can be written in terms of the area per unit

molecule as follows:

σ is the area per unit molecule of adsorbate

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Cont.

we have:

σ0 is the minimum area per

unit molecule of adsorbate

where:

Volmer

Isotherm

Volmer isotherm is a fundamental equation describing

the adsorption on surfaces where the mobility of

adsorbed molecules is allowed, but no interaction is

allowed among the adsorbed molecules.

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Cont.

Rearranging the volmer equation:

the Volmer equation is similar to the Langmuir isotherm with the

apparent affinity as:

b is constant in the Langmuir mechanism,

bapp decreases with loading in the Volmer mechanism.

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Fowler-Guggenheim isotherm

From statistical thermodynamics, the simplest equation allowing for

the lateral (adsorbate-adsorbate) interaction is obtained:

z : coordination no. (usually 4 or 6 depending on the packing of molecules)

w : the interaction energy between adsorbed molecules (w>0 means

attraction between adsorbed species and w<0 means repulsion)

w 500-1000 cal/mole

where:

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Cont.

Plots of the fractional loading versus bP for the FG equation

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Hill-deBoer isotherm

When EOS of the adsorbate allows for the co-volume term and the

attractive force term, the following van der Waals equation can be used:

Here mobile adsorption and lateral interaction among adsorbed

molecules are considered.

Then, the isotherm equation obtained is:

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Isotherms derived from the Gibbs equation