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Review of second-order models for adsorption systems

Journal of Hazardous Materials B136(2006)

681–689

Yuh-Shan Ho?

School of Public Health,Taipei Medical University,250Wu-Hsing Street,Taipei11014,Taiwan

Received24November2004;received in revised form2April2005;accepted29December2005

Available online7February2006

Abstract

Applications of second-order kinetic models to adsorption systems were reviewed.An overview of second-order kinetic expressions is described in this paper based on the solid adsorption capacity.An early empirical second-order equation was applied in the adsorption of gases onto a solid.

A similar second-order equation was applied to describe ion exchange reactions.In recent years,a pseudo-second-order rate expression has been widely applied to the adsorption of pollutants from aqueous solutions onto adsorbents.In addition,the earliest rate equation based on the solid adsorption capacity is also presented in detail.

Keywords:Kinetics;Second-order;Pseudo-second-order;Biosorption;Sorption

1.Introduction

Predicting the rate at which adsorption takes place for a given system is probably the most important factor in adsorption system design,with adsorbate residence time and the reac-tor dimensions controlled by the system’s kinetics.A number of adsorption processes for pollutants have been studied in an attempt to?nd a suitable explanation for the mechanisms and kinetics for sorting out environment solutions.In order to inves-tigate the mechanisms of adsorption,various kinetic models have been suggested.In recent years,adsorption mechanisms involving kinetics-based models have been reported.Numerous kinetic models have described the reaction order of adsorption systems based on solution concentration.These include?rst-order[1]and second-order[2]reversible ones,and?rst-order [3]and second-order[4]irreversible ones,pseudo-?rst-order [5]and pseudo-second-order ones[6]based on the solution concentration.On the other hand,reaction orders based on the capacity of the adsorbent have also been presented,such as Lagergren’s?rst-order equation[7],Zeldowitsch’s model[8], and Ho’s second-order expression[9–12].

This paper describes an earlier adsorption rate equation based on the solid capacity for a system of liquids and solids[7],the Elovich equation for adsorption of gases onto a solid and apply-

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E-mail address:ysho@https://www.wendangku.net/doc/af4656240.html,.tw.ing a second-order rate equation for gas/solid and solution/solid adsorption systems[8],a second-order rate expression for ion exchange reactions[13],and a pseudo-second-order expression [9].

2.Modeling

2.1.Second-order rate equation

A linear form of the typical second-order rate equation is

C t

=k2t+1

,(1)

where C t is the equilibrium concentration(mg/dm3),C0the ini-tial concentration(mg/dm3),t the time(min),and k2is the rate constant(dm3/mg min).

Early applied second-order rate equations in solid/liquid sys-tems described reactions between soil and soil minerals[14,15]. Others applied which the second-order rate equation included the adsorption of?uoride onto acid-treated spent bleaching earth [16];and the adsorption of water using the dealumination of HZSM-5zeolite by thermal treatment[17].Moreover,Varsh-ney et al.reported the kinetics of adsorption of the pesticide, phosphamidon,on beads of an antimony(V)phosphate cation exchanger during the?rst15min[18].

682Y.-S.Ho /Journal of Hazardous Materials B136(2006)681–689

https://www.wendangku.net/doc/af4656240.html,gergren’s equation

As early as 1898[7],Lagergren described liquid–solid phase adsorption systems,which consisted of the adsorption of oxalic acid and malonic acid onto https://www.wendangku.net/doc/af4656240.html,gergren’s ?rst-order rate equation is the earliest known one describing the adsorption rate based on the adsorption capacity.It is summarised as follows:d x

d t

=k (X ?x ),(2)where X and x (mg/g)are the adsorption capacities at equilibrium and at time t (min),respectively,and k is the rate constant of the ?rst-order adsorption (1/min).

Eq.(2)was integrated with the boundary conditions of t =0to t =t and x =0to x =x to yield ln X X ?x

=kt (3)

and

x =X (1?e ?kt ).

(4)

Eq.(3)may be rearranged to a linear form:

log(X ?x )=log(X )?k

2.303

t.(5)In order to distinguish kinetics equations based on concentra-tions of solution from adsorption capacities of solids,Lager-gren’s ?rst-order rate equation has been called pseudo-?rst-order [10,19–22].An early known application of Lagergren’s kinetics equation to adsorption was undertaken by Trivedi et al.[23]for the adsorption of cellulose triacetate from chloroform onto cal-cium silicate.During the last four decades,the kinetics equation has been widely applied to the adsorption of pollutants from aqueous solutions [24].2.3.Elovich’s equation

Elovich’s equation is another rate equation based on the adsorption capacity.In 1934[8],the kinetic law of chemisorp-tion was established though the work of Zeldowitsch.The rate of adsorption of carbon monoxide on manganese dioxide decreas-ing exponentially with an increase in the amount of gas adsorbed was described by Zeldowitsch [8].It has commonly been called the Elovich equation in the following years:

d q

d t

=a e ?αq ,

(6)where q is the quantity of gas adsorbed during the time t ,αthe initial adsorption rate,and a is the desorption constant during any one experiment.The integrated form of Eq.(6)can be written in the form

q = 2.3α log(t +t 0)? 2.3α log t 0

(7)with t 0=

1αa

.(8)With a correctly chosen t 0,the plot of q as a function of log(t +t 0)should yield a straight line with a slope of 2.3/α;Eq.(8)then gives a which obviously represents the initial rate of adsorption for q =0.The test thus involves one single disposable parameter,t 0,which is found by trial;if t 0is too small,the curve is con-vex,and if t 0is too large,it is concave to the axis of log(t +t 0)[25].This Elovich equation is commonly used to determine the kinetics of chemisorption of gases onto heterogeneous solids,and is quite restricted,as it only describes a limiting property ultimately reached by the kinetic curve [26].

To simplify Elovich’s equation,Chien and Clayton [27]assumed that a αt 1and by applying the boundary conditions of q =0at t =0and q =q at t =t ,then Eq.(6)becomes [28]:q =αln(aα)+αln(t ).

(9)

Thus,the constants can be obtained from the slope and the inter-cept of a straight line plot of q against ln(t ).Recently,Rudzinski and Panczyk [29]published an exhaustive analysis of existing rationalizations for the Elovich equation found in the literature for the kinetics of adsorption onto heterogeneous surfaces.In earlier years,numerous applications of Elovich’s equa-tion to the adsorption of gases onto solid systems were reported [30,31].During the last three decades,the equation has been widely used to describe the kinetics of adsorption of gases onto solids [29,32–35].The most frequently cited paper for the application of Elovich’s equation to adsorption systems was an alternative to Elovich’s equation for kinetics of adsorption of gases onto solids [33].An earlier application of the rate equa-tion of Elovich was the exchange of 32P between the goethite (?-FeOOH)crystal surface and the solution phase [36].The application of Elovich equation to the kinetics of phosphate release and adsorption in soils [27]is the most frequently cited paper on the adsorption in solution/solid systems.In addition,the Elovich equation has also been used to describe the adsorption of pollutants from aqueous solutions in recent years [19,37,38].2.4.Ritchie’s equation

In 1977[33],Ritchie reported a model for the adsorption of gaseous systems.Assumptions were made as follows:θis the fraction of surface sites which are occupied by an adsorbed gas,n the number of surface sites occupied by each molecule of the adsorbed gas,and αis the rate constant.Assuming that the rate of adsorption depends solely on the fraction of sites which are unoccupied at time t ,then d θ

d t

=α(1?θ)n .(10)

Eq.(10)integrates to 1

(1?θ)n ?1=(n ?1)αt +1

for n =1

(11)or

θ=1?e ?αt

for n =1.

(12)

Y.-S.Ho/Journal of Hazardous Materials B136(2006)681–689683 It is assumed that no site is occupied at t=0.When introducing

the amount of adsorption,q,at time t,Eq.(11)becomes

q n?1

∞

(q∞?q)n?1

=(n?1)αt+1(13)

and similarly Eq.(12)becomes

q=q∞(1?e?αt),(14)

where q∞is the amount of adsorption after an in?nite time.

In earlier years,Sobkowsk and Czerwi′n ski[39]presented a

rate equation for the reaction of carbon dioxide adsorption onto

a platinum electrode:

dθ

d t

=k(1?θ)n,(15)

whereθ=Γ/Γ∞denotes the surface coverage by the reaction

products,ΓandΓ∞the surface concentrations at time t and after

completion of the reaction,respectively,k the rate constant,and

n is the order of the reaction.

When n=1,

?ln(1?θ)=k1t.(16)

When n=2,

1?θ

=k2t.(17)

Sobkowsk and Czerwi′n ski[39]concluded that the?rst-order

is only for low surface concentrations of a solid,con?rmed by

using plots of?ln(1?θ)versus time as Eq.(16),and the second-

order is for higher concentrations of a solid,con?rmed by using

plots ofθ/(1?θ)versus time as Eq.(17).In addition,Trasatti and

Formaro reported that the plot of?ln(1?θ)versus time is not

linear for very long times,when the coverage reaches a station-

ary value for the adsorption of glycolaldehyde onto a platinum

electrode[40].In the case of the sorption of basic dyes from

aqueous solution onto sphagnum moss peat,Ho and McKay

[22]found that log(q e?q t)versus time was only applicable in

the early stage of the reaction.In the case of adsorption of gases

onto a solid surface,Sobkowsk and Czerwi′n ski reported that

the?rst-order rate equation could only be used for a low surface

concentration of gases adsorbed onto a solid surface,and the

second-order rate evaluation could be applied to higher concen-

trations[39].

Several adsorption results were examined using the Ritchie

equation[33].In the early years,the Elovich equation was

applied to describe gas and vapour adsorption systems,such

as the adsorption of carbon monoxide during the oxidation

of polyvinylidene chloride[41],the chemisorption of hydro-

gen onto graphon[42],the measuring of the kinetics of the

chemisorption of H2onto a MoS2+Al2O3catalyst[43],and the

adsorption of water vapour by Vycor?bre[44].These systems

did not?t the Elovich equation very well.Ritchie[33]examined

these results using Eq.(13)when n=2.Eq.(13)becomes

q∞(q∞?q)=αt+1.(18)

The value for q∞is obtained from the intercept at(1/t)=0on

a plot of(1/q)against(1/t).Ritchie found a good linear rela-

tionship between t and q∞/(q∞?q)for the results of Austin et

al.[41],Bansal et al.[42],Deitz and Turner[44],and Samuel

and Yeddanapalli[43].In recent years,the Ritchie equation

has also been applied to solution/solid adsorption systems,for

example,the adsorption of cadmium ions onto bone char[37],

and the adsorption of Cd(II)onto acid-treated jackfruit peel

[45].

2.5.Second-order rate expressions

In1984[13],Blanchard et al.presented the overall exchange

reaction of NH4+ions?xed in zeolite by divalent metallic ions

in the solution which can be written:

Z(2NH

+)+M2+→Z(M2+)+2NH4+,(19)

where Z(2NH

+)and Z(M2+)are the amounts of NH4+ion?xed in

the zeolite(meq/g),and M2+and NH4+are the concentrations

(meq/dm3).

The authors assumed that the metallic concentration varies

very slightly during the?rst hours,and the kinetic order is two

with respect to the number(n0?n)of available sites for the

exchange;thus,the differential equation can be written as

?d n

d t

=K[n0?n]2(20)

and integration gives

(n0?n)

?α=Kt,(21)

where n is the amount of M2+?xed or the amount of NH4+

released at each instant,n0the exchange capacity,and K is the

rate constant.

Considering the boundary condition n=0for t=0,it follows

thatα=1/n0.By plotting1/(n0?n)as a function of time,a

straight line must be obtained,the slope of which gives the rate

constant,K,and the intercept gives the exchange capacity.In

recent years,the Blanchard second-order expression has been

used to describe the kinetics of exchange processes between

sodium ions from zeolite A and cadmium,copper,and nickel

ions from solutions[46].

An expression of second-order rate based on solid capacity

has also been presented for the kinetics of adsorption of diva-

lent metal ions onto peat[9–12].Peat contains polar functional

groups such as aldehydes,ketones,acids,and phenolics.These

groups can be involved in chemical bonding and are respon-

sible for the cation exchange capacity of the peat.Thus,the

peat–copper reaction may be represented in two ways[47]:

2P?+Cu2+?CuP2(22)

and

2HP+Cu2+?CuP2+2H+,(23)

where P?and HP are polar sites on the peat surface.

684Y.-S.Ho/Journal of Hazardous Materials B136(2006)681–689 In an attempt to present the equation representing adsorption

of divalent metals onto sphagnum moss peat during agitation,

the assumption was made that the process may be second-order

and that chemisorption occurs involving valency forces through

sharing or the exchange of electrons between the peat and diva-

lent metal ions as covalent forces.The rate of the second-order

reaction may be dependent on the amount of divalent metal ions

on the surface of the peat,and the amount of divalent metal

ions adsorbed at equilibrium[9,12].The rate expression for the

adsorption described by Eqs.(24)and(25)is

d(P)t

d t

=k[(P)0?(P)t]2(24)

d(HP)t

d t

=k[(HP)0?(HP)t]2,(25)

where(P)t and(HP)t are the number of active sites occupied

on the peat at time,t,and(P)0and(HP)0are the number of

equilibrium sites available on the peat.

The driving force,(q e?q t),is proportional to the available

fraction of active sites.The kinetic rate equations can be rewrit-

ten as follows:

d q t

d t

=k(q e?q t)2,(26)

where k is the rate constant of adsorption(g/mg min),q e the

amount of divalent metal ions adsorbed at equilibrium(mg/g),

and q t is the amount of divalent metal ions on the surface of the

adsorbent at any time,t(mg/g).

Separating the variables in Eq.(26)gives

d q t

(q e?q t)2

=k d t(27)

and integrating this for the boundary conditions t=0to t=t and

q t=0to q t=q t,gives

q t=

q2e kt

1+q e kt

(28)

which is the integrated rate law for a second-order reaction.Eq.

(28)can be rearranged to obtain

q t=

+t q

(29)

which has a linear form of

t q t =1

kq2e

q e

t(30)

and

h=kq2e,(31) where h is the initial adsorption rate(mg/g min)as q t/t approaches0,and Eq.(29)can be rearranged to obtain

q t=

+t q

(32)

and

q t

q e

t.(33)

The rate of a reaction is de?ned as the change in concentration of

a reactant or product per unit time.Concentrations of products

do not appear in the rate law because the reaction rate is studied

under conditions where the reverse reactions do not contribute

to the overall rate.The reaction order and rate constant must

be determined by experiments.In order to distinguish the kinet-

ics equation based on the concentration of a solution from the

adsorption capacity of solids,this second-order rate equation

has been called a pseudo-second-order one[9].The pseudo-

second-order model constants can be determined experimentally

by plotting t/q t against t.Although there are many factors which

in?uence the adsorption capacity,including the initial adsorbate

concentration[12,48–51],the reaction temperature[10,12,50],

the solution pH value[52,53],the adsorbent particle size[48]and

dose[12,48,51],and the nature of the solute[12,54],a kinetic

model is concerned only with the effect of observable parame-

ters on the overall rate.The pseudo-second-order expression has

been successfully applied to the adsorption of metal ions,dyes,

herbicides,oils,and organic substances from aqueous solutions

(Table1).

Recently,a theoretical analysis of the pseudo-second-order

model was reported[139].The advantage of the Azizian deriva-

tion is that when the initial concentration of solute is low,then

the adsorption process obeys the pseudo-second-order model.

Conversely pseudo-?rst-order models can be applied to higher

initial concentrations.The rate constant of the pseudo-second-

order model is a complex function of the initial concentration of

the solute.

Table2shows a comparison of second-order rate equations

of Sobkowsk and Czerwi′n ski[39],Ritchie[33],Blanchard et al.

[13],and Ho[9].In earlier years,Sobkowsk and Czerwi′n ski used

the second-order rate equation based on the adsorption capacity

of a solid for higher concentrations of solids with the rate of

reaction of carbon dioxide adsorption onto a platinum electrode

[39].Ritchie presented a second-order empirical equation to test

the adsorption of gases onto solids[33].Blanchard et al.reported

a similar rate equation for the exchange reaction of NH4+ions

?xed in zeolite by divalent metallic ions in solution[13].Ho

described adsorption which included chemisorption and gave

a different idea of the second-order equation called a pseudo-

second-order rate expression[9].

In many cases,the equilibrium adsorption capacity is

unknown,and chemisorption tends to become immeasurably

slow and the amount adsorbed is still signi?cantly smaller than

the equilibrium amount[140].On the other hand,achieving equi-

librium takes a long time in some adsorption systems[141–143].

However,the pseudo-second-order equation has the following

advantages:it does not have the problem of assigning an effec-

tive adsorption capacity,i.e.,the adsorption capacity,the rate

constant of pseudo-second-order,and the initial adsorption rate

all can be determined from the equation without knowing any

parameter beforehand.

Y.-S.Ho/Journal of Hazardous Materials B136(2006)681–689685 Table1

Pseudo-second-order kinetic model of various related systems from the literature

Adsorbent Adsorbate References

2-Mercaptobenzimidazole

clay

Hg(II)[55]

Activated carbon2,4-Dichlorophenoxy-acetic

acid

[56]

Activated carbon Cd(II)[57]

Activated carbon Cd(II)[58]

Activated carbon Cd(II),Ni(II)[59]

Activated carbon Congo red[60]

Activated carbon Direct blue2B,Direct

green B

[61]

Activated carbon Hg(II)[62]

Activated carbon Hg(II)[63]

Activated carbon Methylene blue[64]

Activated carbon Paraquat dichloride[65]

Activated carbon Co(II)[66]

Activated carbon Pb(II)[67]

Activated carbon Pb(II)[68]

Activated carbon Pb(II),Hg(II),Cd(II),

Co(II)

[69]

Activated clay Basic red18,Acid blue9[70]

Aeromonas caviae Cr(VI)[71]

Alginate Ni(II)[72]

Anaerobic granular sludges Ni(II),Co(II)[73]

Aspergillus niger Acid blue29[74]

Aspergillus niger Basic blue9[75]

Aspergillus niger Congo red[76]

Aspergillus niger Pb(II),Cd(II),Cu(II),Ni(II)[77]

Azadirachta indica(Neem)

leaf

Congo red[78]

Azadirachta indica(Neem)

leaf

Pb(II)[79]

Baker’s yeast Cd(II)[80]

Banana stalk Musa

paradisiacal

Hg(II)[81]

Beech leaves Cd(II)[11]

Bentonite Acid red57,Acid blue294[82]

Bi2O3Cr(VI)[11]

Blast furnace slag,dust,

sludge,carbon slurry

Chlorophenols[83]

Bottom ash Cu(II),Pb(II)[11]

Calabrian pine bark Zn(II),Pb(II)[84]

Calcined alunite Phosphorus[85]

Calcined Mg–Al–CO3

hydrotalcite

Cr(VI)[86]

Cassava waste biomass Cu(II),Cd(II)[87]

Chitin Cd(II)[88]

Chitin,chitosan,Rhizopus

arrhizus

Cr(VI),Cu(II)[38]

Chitosan Cu(II)[89]

Chitosan Ni(II)[90]

Clinoptilolite Pb(II)[91]

Coconut coir pith2,4-Dichlorophenol[92]

Coconut coir pith Cr(VI)[93]

Coir Cu(II),Pb(II)[94]

Cypress leaves Pb(II)[11]

Date pits Methylene blue[95]

Date pits Phenol[96]

Diatomaceous clay Methylene blue[97]

Dolomite Phosphate[98]

Fly ash Congo red[99]

Fly ash Omega chrome red ME,

o-cresol,p-nitrophenol

[100]

Fly ash Victoria blue,OCL,PNP,

OCRME [11]

Table1(Continued)

Adsorbent Adsorbate References

Grafted silica Pb(II),Cu(II)[101]

Grape stalks Cr(VI)[102]

Iron oxide-coated sand As(V),As(III)[103]

Jordanian low-grade

phosphate

Pb(II)[104]

Juniper?ber Cd(II)[105]

Juniper?ber Phosphorus[106]

Mesoporous silicate Phosphate[107]

Mg–Al–CO3hydrotalcite Cr(VI)[108]

Microcystis Ni(II),Cr(VI)[109]

Microporous titanosilicate

ETS-10

Pb(II)[110] Mixed clay/carbon Acid blue9[111]

Mucor rouxii Pb(II),Cd(II),Ni(II),Zn(II)[112]

Myriophyllum spicatum Pb(II),Zn(II),Cd(II)[113]

Na-bentonite Oil[114]

Oil shale4-Nitrophenol[115]

Peat Basic blue69,Acid blue25[11]

Peat Basic green4,Basic violet

4,Basic blue24

[116] Peat Cu(II)[117]

Peat Cu(II)[11]

Peat Cu(II)[118]

Peat-resin particle Basic magenta,Basic

brilliant green

[119]

Perlite Cd(II)[120]

Perlite Methylene blue[121]

Pith Basic red22,Acid red114[122]

Reed leaves Cd(II)[11]

Rhizopus oligosporus Cu(II)[123]

Rhodotorula aurantiaca Pb(II)[124]

Sago Cu(II),Pb(II)[125]

Sawdust Cd(II),Pb(II)[126]

Sawdust Phenol[127]

Schizomeris leibleinii Pb(II)[128]

Sepiolite Pb(II)[129]

Spent grain Pb(II),Cd(II)[130]

Sphagnum moss peat Chrysoidine,Astrazon blue,

Astrazone blue

[22]

Sphagnum moss peat Cu(II),Ni(II)[131]

Sphagnum moss peat Cu(II),Ni(II),Pb(II)[12]

Sugar beet pulp Pb(II)[132]

Sugar beet pulp Pb(II),Cu(II),Zn(II),

Cd(II),Ni(II)

[133] Surfactant-modi?ed

clinoptilolite

Phosphate[134]

TNSAC Phosphate[11]

Tree fern Basic red13[135]

Tree fern Cd(II)[136]

Tree fern Cu(II)[49]

Tree fern Pb(II)[50]

Vermiculite Cd(II)[137]

Waste tyres,sawdust Cr(VI)[138]

Wollastonite Ni(II)[11]

Wood Basic blue69,Acid blue25[21]

686Y.-S.Ho/Journal of Hazardous Materials B136(2006)681–689 Table2

Comparison of second-order models

Author Year Linear form Plot

Sobkowsk and Czerwi′n ski1974θ

1?θ=k2tθ

1?θ

vs.t

Ritchie1977q∞

q∞?q =αt+1q∞

q∞?q

vs.t

Blanchard et al.19841

n0?n ?α=Kt1n

0?n

vs.t

Ho1995t q

t =1

k2q2e

+1q

t t q

vs.t

3.Conclusion

Adsorption rate equations have considered the adsorption capacities of solids since Lagergren’s?rst-order equation was presented.Several rate equations were reported with the same idea in the following years.In earlier years,Elovich’s equa-tion and Ritchie’s equation were applied to the adsorption of gases onto solid https://www.wendangku.net/doc/af4656240.html,ter,application of these equations to the adsorption of pollutants from aqueous solutions were investigated.A second-order rate equation was used to describe chemisorption for the adsorption of gases used to describe ion exchange reactions.The pseudo-second-order rate expression was used to describe chemisorption involving valency forces through the sharing or exchange of electrons between the adsor-bent and adsorbate as covalent forces,and ion exchange.In recent years,the pseudo-second-order rate expression has been widely applied to the adsorption of pollutants from aqueous solu-tions.The advantage of using this model is that there is no need to know the equilibrium capacity from the experiments,as it can be calculated from the model.In addition,the initial adsorption rate can also be obtained from the model.

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