980 CHEMICAL ENGINEERING

of adsorption has been put forward which satisfactorily explains all systems. Fortunatelyfor the engineer, what is needed is an accurate representation of equilibrium, the theoreticalminutiae of which is not of concern. For this reason, some of the earliest theories ofadsorption are still the most useful, even though the assumptions on which they werebased were seen in later years to be not entirely valid. Most theories have been developedfor gassolid systems because the gaseous state is better understood than the liquid.Statistical theories are being developed which should apply equally well to gassolid andliquidsolid equilibria, though these are not yet at a stage when they can be applied easilyand confidently to the design of equipment.

The capacity of an adsorbent for a particular adsorbate involves the interaction of threeproperties the concentration C of the adsorbate in the fluid phase, the concentration Csof the adsorbate in the solid phase and the temperature T of the system. If one of theseproperties is kept constant, the other two may be graphed to represent the equilibrium.The commonest practice is to keep the temperature constant and to plot C against Csto give an adsorption isotherm. When Cs is kept constant, the plot of C against T isknown as an adsorption isostere. In gassolid systems, it is often convenient to expressC as a pressure of adsorbate. Keeping the pressure constant and plotting Cs against Tgives adsorption isobars. The three plots are shown for the ammonia-charcoal system inFigure 17.4 which is taken from the work of BRUNAUER(10).

17.3.1. Single component adsorption

Most early theories were concerned with adsorption from the gas phase. Sufficient wasknown about the behaviour of ideal gases for relatively simple mechanisms to be postu-lated, and for equations relating concentrations in gaseous and adsorbed phases to beproposed. At very low concentrations the molecules adsorbed are widely spaced over theadsorbent surface so that one molecule has no influence on another. For these limitingconditions it is reasonable to assume that the concentration in one phase is proportionalto the concentration in the other, that is:

Cs = KaC (17.1)This expression is analogous to Henrys Law for gasliquid systems even to the extentthat the proportionality constant obeys the vant Hoff equation and Ka = K0 eH/RTwhere H is the enthalpy change per mole of adsorbate as it transfers from gaseousto adsorbed phase. At constant temperature, equation 17.1 becomes the simplest form ofadsorption isotherm. Unfortunately, few systems are so simple.

17.3.2. The Langmuir isotherm

At higher gas phase concentrations, the number of molecules absorbed soon increases tothe point at which further adsorption is hindered by lack of space on the adsorbent surface.The rate of adsorption then becomes proportional to the empty surface available, as wellas to the fluid concentration. At the same time as molecules are adsorbing, other molecules

ADSORPTION 981

Figure 17.4. Equilibrium data for the adsorption of ammonia on charcoal(10)

(a) Adsorption isotherm (b) Adsorption isobar (c) Adsorption isostere

982 CHEMICAL ENGINEERING

will be desorbing if they have sufficient activation energy. At a fixed temperature, the rateof desorption will be proportional to the surface area occupied by adsorbate. When therates of adsorption and desorption are equal, a dynamic equilibrium exists. For adsorptionwhich is confined to a mono-molecular layer, the equilibrium may be written as:

k0a0C = k0(1 a1)C = k1a1or:

a1 = B0C1 + B0C (17.2)

where: a0 is the fraction of empty surface,a1 is the fraction of surface occupied by a monolayer of adsorbed molecules,B0 = k0/k1,k0 is the velocity constant for adsorption on to empty surface, andk1 is the velocity constant for desorption from a monolayer.

Equation 17.2 has been developed for adsorption from the gas phase. It is convenient toalso express it in terms of partial pressures, which gives:

Cs

Csm= B1P

1 + B1P (17.3)

where: Cs is the concentration of the adsorbed phase,Csm is the concentration of the adsorbed phase when the monolayer is complete,B1 = B0/RT ,andP is the partial pressure of adsorbate in the gas phase

.

Equations 17.2 and 17.3 have the form of the LANGMUIR(11) equation, developed in 1916,which describes the adsorption of gases on to plane surfaces of glass, mica and platinum.A number of assumptions is implicit in this development. As well as being limited tomonolayer adsorption, the Langmuir equation assumes that:

(a) these are no interactions between adjacent molecules on the surface.(b) the energy of adsorption is the same all over the surface.(c) molecules adsorb at fixed sites and do not migrate over the surface.

When B1P 1, equation 17.3 reverts to the form of Henrys Law, as given inequation 17.1. Equation 17.3 can be rewritten in linear form to give:

P

Cs= PCsm

+ 1B1Csm

(17.4)

so that a plot of P/Cs against P will be a straight line when applied to a system thatbehaves in accordance with the Langmuir isotherm.

It has been shown experimentally, however, that many systems do not follow thisisotherm and efforts continue to find an improved equation.

ADSORPTION 983

17.3.3. The BET isotherm

In 1938, BRUNAUER, EMMETT and TELLER(12) and EMMETT and DE WITT(13) developed whatis now known as the BET theory. As in the case in Langmuirs isotherm, the theory isbased on the concept of an adsorbed molecule which is not free to move over the surface,and which exerts no lateral forces on adjacent molecules of adsorbate. The BET theorydoes, however, allow different numbers of adsorbed layers to build up on different partsof the surface, although it assumes that the net amount of surface which is empty orwhich is associated with a monolayer, bilayer and so on is constant for any particularequilibrium condition. Monolayers are created by adsorption on to empty surface and bydesorption from bilayers. Monolayers are lost both through desorption and through theadsorption of additional layers. The rate of adsorption is proportional to the frequencywith which molecules strike the surface and the area of that surface. From the kinetictheory of gases, the frequency is proportional to the pressure of the molecules and hence:

The rate of adsorption on to empty surface = k0a0P , andthe rate of desorption from a monolayer = k1a1

Desorption is an activated process. If E1 is the excess energy required for one mole in themonolayer to overcome the surface forces, the proportion of molecules possessing suchenergy is eE1/RT . Hence the rate of desorption from a monolayer may be written as:

A1 eE1/RT a1

where A1 is the frequency factor for monolayer desorption.The dynamic equilibrium of the monolayer is given by:

k0a0P + A2 eE2/RT a2 = k1a1P +A1 eE1/RT a1 (17.5)where A2 is the frequency factor for description from a bilayer, thus creating a monolayer.

Applying similar arguments to the empty surface, then:

k0a0P = A1 eE1/RT a1 (17.6)From equations 17.5 and 17.6:

k1a1P = A2 eE2/RT a2a1 = k0

A1eE1/RT a0P = 0a0 (17.7)

and: a2 = k1A2

eE2/RT a1P = a1 (17.8)

The BET theory assumes that the reasoning used for one or two layers of molecules maybe extended to n layers. It argues that energies of activation after the first layer are allequal to the latent heat of condensation, so that:

E2 = E3 = E4 = = En = M

984 CHEMICAL ENGINEERING

Hence it may be assumed that is constant for layers after the first and:

ai = i1a1 = B2ia0where B2 = 0/, and ai is the fraction of the surface area containing i layers of adsorbate.

Since a0, a1, . . . are fractional areas, their summation over n layers will be unity and:

1 = a0 +ni=1

ai

= a0 +ni=1

B2ia0 (17.9)

The total volume of adsorbate associated with unit area of surface is given by:

vs = v1sni=1

iai = v1sni=1

iB2ia0 (17.10)

where v1s is the volume of adsorbate in a unit area of each layer.Since v1s does not change with n, a geometrically plane surface is implied. Strictly,

equation 17.10 is not applicable to highly convex or concave surfaces. Equations 17.9and 17.10 may be combined to give:

vs

v1s=

ni=1

iB2ia0

a0 +ni=1

B2ia0

(17.11)

The numerator of equation 17.11 may be written as:

B2a0d

d

(ni=1

i

)= B2a0 d

d

{(1 n1

)

}

and the denominator as:

a0

[1 + B2

(1 n1

)]

Substituting these values into equation 17.11 and rearranging gives:

vs

v1s= B2

1 [1 (n+ 1)n + nn+1]

[1 + (B2 1) B2n+1] (17.12)

On a flat unrestricted surface, there is no theoretical limit to the number of layers thatcan build up. When n = , equation 17.12 becomes:

vs

v1s= B2(1 )(1 + B2) (17.13)

ADSORPTION 985

When the pressure of the adsorbate in the gas phase is increased to the saturated vapourpressure, condensation occurs on the solid surface and vs/v1s approaches infinity. Inequation 17.13, this condition corresponds to putting = 1. It may be noted that putting = 1/(1 B2) is not helpful.

Hence from equation 17.8:

1 = k1A2

eM/RT P 0

where: M is the molar latent heat andP 0 is the saturated vapour pressure.

Hence, from equation 17.8, = P/P 0.Equation 17.12 may be rewritten for unit mass of adsorbent instead of unit surface.

This is known as the limited form of the BET equation which is:

Vs

V 1s= B2 P

P 0

[1 (n+ 1)(P/P 0)n + n(P/P 0)n+1](1 P/P 0)[1 + (B2 1)(P/P 0) B2(P/P 0)n+1] (17.14)

where V 1s is the volume of adsorbate contained in a monolayer spread over the surfacearea present in unit mass of adsorbent.

When n = 1, adsorption is confined to a monolayer and equation 17.14 reduces to theLangmuir equation.

When n = , (P/P 0)n approaches zero and equation 17.13 may be rearranged in aconvenient linear form to give:

P/P0

V (1 P/P0) =1

V 1B2+ B2 1

V 1B2

(P

P 0

)(17.15)

where V and V 1 are the equivalent gas phase volumes of Vs and V 1s .If a plot of the left-hand term against P/P 0 is linear, the experimental data may be

said to fit the infinite form of the BET equation. From the slope and the intercept, V 1

and B2 may be calculated.The advantage of equation 17.14 is that it may be fitted to all known shapes of

adsorption isotherm. In 1938, a classification of isotherms was proposed which consistedof the five shapes shown in Figure 17.5 which is taken from the work of BRUNAUERet al.(14). Only gassolid systems provide examples of all the shapes, and not all occurfrequently. It is not possible to predict the shape of an isotherm for a given system,although it has been observed that some shapes are often associated with a particularadsorbent or adsorbate properties. Charcoal, with pores just a few molecules in diameter,almost always gives a Type I isotherm. A non-porous solid is likely to give a Type IIisotherm. If the cohesive forces between adsorbate molecules are greater than the adhesiveforces between adsorbate and adsorbent, a Type V isotherm is likely to be obtained for aporous adsorbent and a Type III isotherm for a non-porous one.

In some systems, three stages of adsorption may be discerned. In the activated alumina-air-water vapour system at normal temperature, the isotherm is found to be of Type IV.This consists of two regions which are concave to the gas concentration axis separatedby a region which is convex. The concave region that occurs at low gas concentrationsis usually associated with the formation of a single layer of adsorbate molecules over the

986 CHEMICAL ENGINEERING

Figure 17.5. Classification of isotherms into five types of BRUNAEUR, DEMING, DEMING and TELLER(14)

surface. The convex portion corresponds to the build-up of additional layers, whilst theother concave region is the result of condensation of adsorbate in the pores so calledcapillary condensation as discussed earlier in this Section.

At low gas concentrations, whilst the monolayer is still incomplete, the absorbedmolecules are relatively immobile. In the multilayer region, the adsorbed molecules behavemore like a liquid film. The amount of capillary condensation that occurs depends on thepore sizes and their distribution, as well as on the concentration in the gas phase.

When n = 1, equation 17.14 represents a Type I isotherm.When n = , equation 17.14 represents a Type II, and the rarer Type III isotherm by

choosing a suitable value for B2. As B2 is increased, the point of inflexion or kneeof Type II becomes more prominent. This corresponds to an increasing tendency for themonolayer to become complete before a second layer starts. In the extreme case of anadsorbent whose surface is very uniform from an energy point of view, the adsorbatebuilds up in well-defined layers. This gives rise to a stepped isotherm, in which eachstep corresponds to another layer. When B2 is less than 2, there is no point of inflexionand Type III isotherms are obtained. The condition 1 > B2 > 0 often corresponds to atendency for molecules to adsorb in clusters rather than in complete layers.

The success of the BET equation in representing experimental data should not beregarded as a measure of the accuracy of the model on which it is based. Its capabilityof modelling the mobile multilayers of a Type IV isotherm is entirely fortuitous because,in the derivation of the equation, it is assumed that adsorbed molecules are immobile.

Example 17.1

Spherical particles of 15 nm diameter and density 2290 kg/m3 are pressed together to form a pellet.The following equilibrium data were obtained for the sorption of nitrogen at 77 K. Obtain estimates

ADSORPTION 987

of the surface area of the pellet from the adsorption isotherm and compare the estimates with thegeometric surface. The density of liquid nitrogen at 77 K is 808 kg/m3.

P/P 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9m3 liq N2 106/kg solid 66.7 75.2 83.9 93.4 108.4 130.0 150.2 202.0 348.0where P is the pressure of the sorbate and P 0 is its vapour pressure at 77 K.

Use the following data:

density of liquid nitrogen = 808 kg/m3area occupied by one adsorbed molecule of nitrogen = 0.162 nm2

Avogadro Number = 6.02 1026 molecules/kmol

Solution

For 1 m3 of pellet with a voidage , then:

Number of particles = (1 )/(/6)(15 109)3

Surface area per unit volume = (1 )(15 109)2/(/6)(15 109)3

= 6(1 )/(15 109) m2/m3

1 m3 of pellet contains 2290 (1 ) kg solid and hence:

specific surface = 6(1 )/[(15 109(1 )2290]= 1.747 105 m2/kg

(a) Using the BET isotherm

(P/P 0)/[V (1 P/P 0)] = 1/V B + (B 1)(P/P 0)/V B (equation 17.15)

where V and V are the liquid volumes of adsorbed nitrogen.From the adsorption data given:

(P/P 0) 0.1 0.2 0.3 0.4 0.5 0.6V (m3 liquid N2/kg solid 106) 66.7 75.2 83.9 93.4 108.4 130.0((P/P 0)/V ) 106 1500 2660 3576 4283 4613 4615(P/P 0)/[V (1 P/P 0)] 1666 3333 5109 7138 9226 11538

Plotting (P/P 0)/[V (1 P/P 0)] against (P/P 0), as shown in Figure 17.6, then:

intercept, 1/V B = 300, and slope, (B 1)/V B = 13,902

988 CHEMICAL ENGINEERING

00

2000

4000

6000

8000

10000

12000

0.2 0.4

P/P00.6 0.8

V(1

P

/P0)

P/P

0

orV

P/P

0

(BET isotherm)

(Langmuir isotherm)

Figure 17.6. Adsorption isotherms for Example 17.1.

from which:B = (13,902/300)+ 1 = 47.34

and:

V = 1/(300 47.34) = 70.4 106 m3/kg.The total surface area = [(70.4 106 808 6.2 1026 0.162 1018)]/28

= 2.040 105 m2/kg .

ADSORPTION 989

(b) Using the Langmuir form of the isotherm:

Assuming this applies at low concentrations then, expressing pressure as the ratio P/P 0, and theamount adsorbed as a volume of liquid adsorbate, equation 17.4 becomes:

(P/P 0)/V = (P/P 0)/V + 1/(B2V )Thus, a plot of (P/P 0)/V against (P/P 0) will have a slope of (1/V ).

Thus, from Figure 17.6:

1/V = 13,902and: V = 71.9 106m3/kgwhich agrees with the value from the BET isotherm.

It may be noted that areas calculated from the isotherm are some 20 per cent greater than thegeometric surface, probably due to the existence of some internal surface within the particles.

17.3.4. The Gibbs isotherm

An entirely different approach to equilibrium adsorption is to assume that adsorbed layersbehave like liquid films, and that the adsorbed molecules are free to move over the surface.It is then possible to apply the equations of classical thermodynamics. The properties whichdetermine the free energy of the film are pressure and temperature, the number of moleculescontained and the area available to the film. The Gibbs free energy G may be written as:

G = F(P, T , ns,As) (17.16)Hence:

dG =(G

P

)T ,ns ,As

; dP +(G

T

)P,ns,As

; dT +(G

ns

)T ,P,As

; dns +(G

As

)T ,P,ns

dAs

(17.17)At constant temperature and pressure this becomes:

dG =(G

ns

)dns +

(G

As

)dAs (17.18)

= s dns % dAs (17.19)

where: s is the free energy per mole or chemical potential of the film, and% is defined as a two-dimensional or spreading pressure.

The total Gibbs free energy may be written as:

G = sns %As (17.20)so that: dG = s dns + ns ds % dAs As d% (17.21)A comparison of equations 17.19 and 17.21 shows that:

d% = nsAs

ds