Time Dependence of Adsorption at Solid Liquid Interfaces*

CROATICA CHEMICA ACTA CCACAA 60 (3) 457-475 (1987)

CCA-1741YU ISSN 0011-1643

UDC 541.183.13Conferenee Paper (Invited)

Time Dependence of Adsorption at Solid Liquid Interfaces*Ivica Ružić

Center for Marine Research Zagreb, »Ruđer Bošković" Institute, Zagreb, Croatia,Yugoslavia

Received January 28, 1987

In this paper the time dependence of adsorption at solid-Iiquidinterface is discussed for both the mass-transport and intrinsickinetic control of the adsorption process. It has been shown thatthe time dependence of adsorption of the mass-transport control'of the adsorption process is mostly dependent on the strength ofadsorption (natur al diffusion) and sametimes on the mass-transportcoefficient (dispersion coefficient in the case of eddies), whichappear in natural environments). For the intrinsic kinetic controlof adsorption several rate laws, are reviewed. Methods for theinterpretation of adsorption experiments available in the literatureare discussed. The possibility of distinguishing the mass-transportand kinetic control is mentioned. Simple models of a two-stepadsorption process are described assuming the first step to be avery fast attachment of adsorbate from the homogeneous phase tothe interface. It is concluded that for a correct interpretation ofadsorption experiments the impression that adsorption is underequilibrium conditions should always be supported by the rightshape of the adsorption isotherm. Otherwise additional experi-ments should be performed in a wider range of adsorption times.

INTRODUCTION

Adsorption kinetics has been long recognized as a subject of specialimportance from the academic, as well as from practical points of view.Adsorption kinetics certainly plays an important role, for example, in cata-lysis', detergency", drug administration", fate of pollutants in the environment+,flotationš-", and the uptake of nutrients by biota, Although most of the researchon adsorption processes at different types of interfaces has been concernedwith adsorption equilibrium, a substantial number of results on kinetics ofadsorption has been accumulated so far7-27•

There are some difficulties in studying adsorption kinetics, as mentionedin Iiterature'<>'. The first one is the paucity of theories and models availablefor comparison with experiments. Then there are also experimental difficultiesencountered in the measurements of the rate of adsorption. For exampleBenson and Ellis'" noted that the time dependence of adsorption of water

* Based on an invited lecture presented at the 7th »Ruđer Bošković" Institute'sInternational Summer Conference on the Chemistry of Solid/Liquid Interfaces, RedIsland - Rovinj, Croatia, Yugoslavia, June 25-July 3, 1986.

458 I. Ru2IC

on lyophilised proteins appeared due to the time variation of temperaturecaused by the heat evolved on adsorption on subsecond cooling of the sample.Mycels-" reported time dependent adsorption as an artifcat due to the slowadsorption of highly surface-active impurities.

Usually it is assumed that the relaxation time for the first attachmentof the adsorbate molecules on the interface is of the order of magnitude ofthe time necessary for the formation of the interface itself'", between 10 and100 f1. sec. Therefore, the first attachment of the adsorbate molecules to theinterface very often occurs under steady state conditions. Assuming theadsorbtion kinetics at the interface to be a first order process which is coupledwith mass transport of the adsorbate molecules from the bulk of homogeneousphase to the interface, such a steady state can be described using the followingrelation:

(1)

where Co and C* are the volume concentrations of the adsorbate near theinterface and in the bulk of homogeneous phase, while kA and kT are the rateconstants of the adsorption process at the interface and the mass transportrate constant, respectively. The rate of adsorption at the interface kA Co canthen be described by elimination Co from eqn. (1) in the following way:

(2)

The over all rate constant is k = kA kT/(kA + kT) or after some rearrangement:

lik = lIkA + l/kT (3)

Since the first attachment of the adsorbate to the interface is supposed tobe very fast, very often kA » kT should be true and consequently the overallrate will be often controlled by mass transport as a rate determining stepk ~ kT.30-48 The same reasoning is valid for any heterogeneous process atthe interface with the reactant dissolved in the homogeneous phase. However,the first attachment of the adsorbate to the interface could be followed bya whole series of different heterogeneous processes of very different relaxationtimes. Some of these processes may under certain conditions be slower thanthe mass transoprt itself. In such cases the equilibrium between the bulkof homogeneous phase and the volume concentration near the interface canbe established at the time scales of the relaxation time of these secondaryheterogeneous processes. Then, the time dependence of adsorption can becontrolled by the so called intrinsic adsorption kinetics. Such secondary hete-rogeneous processes can be: reorientation of adsorbed molecules, surfacediffusion, surface complexation, formation of new phases through an inter-action of adsorbed molecules, degradation of larger molecules producingadsorbates of lower molecular weight and different adsorption characteristcsetc.40-52• In addition, a whole series of homogeneous reactions can take placein some cases. The kinetics of these homogeneous reactions can be inducedwithin the mass transport layer by heterogeneous processes at the interface.

In this paper different models for time dependence of adsorption at solidliquid interfaces reported so far in literature will be discussed. Differentmethods for the determination of adsorption parameters will be also discussed.

ADSORPTION TIME DEPENDENCE 459

STATIONARY MASS TRANSPORT

Let us assume that a very fast adsorption process is taking place on aquiet interface with a certain radius of surface curvature ro. The adsorbateis present in the bulk of the homogeneous phase in concentration C* and thediffusion coefficient of the adsorbate is D. The mass transport can be describedby the following equation:

dC/dt = (D/q2) d (q2dC/dr)/dr (4)

where r is the space coordinate, t is time and q is the geometry parameter(q = O for planar, q = 1 cylindrical and q = 2 spherical interface). Eqn. (4)is subjected to the following initial, limiting and boundary conditions:

t = O, r;;;' ro, C = C*

t> O, r ~ 00, c ~ C*

r = ro' D (dC/dr) = dT/dt

where r is the surface concentration of the adsorbed molecules at the inter-face. Consequently, the rate of adsorption drtdt represents the surf'ace boun-dary condition to the mass transport eqn. (4). By solving eqn. (4) the expresionfor concentration profile near the interface can be obtained:

whereC = C* (I-I) (5)

1 ~ drI = ("v D/C*) S (dC/dr) r =ro ~ (r,ro,t - r) d. = -- ~ (r,r o,t - r) d.C* vD dr

and the function g depends on the geometry of the interface. Stationary masstransport is characterized by time independent concentration of adsorbatenear the interface:

where

(6)

Applying Laplace transformations to eqn. (6) the following result can beobtained if C r=ro is assumed to be time independent (steady state):

L{(vDjC*) (dC/drr=ro} = Io/sL{~o}

where 1Is is the Laplace transform of t. If the Laplace transformation of thefunction go is available, then the rate of adsorption can be obtained in thefollowing way:

(7)

Analytical expression for the go function is available only for planar (q = Oor ro ~ 00) and spherical (q = 2) interfaces:

dr/dt = lo ;,;D C* (IN nt + V D/ro)

460wherefrom

1. RUZIĆ

(8)

Here lo = 1- CofC*, is a constant with respect to time. From eqn. (8) describingthe surface concentration one can derive an expression for the surface cove-rage @ = Tir m (I'm being the surface concentration of the complete monolayer)in terms of two dimensionless parameters Gm = rm/c* .y Dt and S = .yDtlr;(known as sphericity):

(9)

As it has been shown by numerical calculations'" the time necessary to attainthe equilibrium between the bulk of the homogeneous phase and the volumenear the interface can be obtained from the following condition:

Gm < 0.1 or t> 100rm2/(C*2D) if S < 1

In cases where S is comparable to unity the mass transport is more efficientand the adsorption equilibrium can be attained faster. Usually rm/.yD is notvery different for different adsorbates and its value is between Ju" to 10-7 molsec'/2/cm3. Therefore, the condition necessary to attain the adsorption equilib-rium can be also written in the following way:

C* v' t > 10-7 to 10-6 mol sec'/2/cm~

In Table I the times necessary for attaining the adsorption equilibrium arepresented for different bulk concentrations of the adsorbate according to theabove mentioned condition. It is obvious that at bulk concentrations of about0.1 mol/l the relaxation time for the first attachment of the adsorbate to theinterface is of the order of time necessary to attain the equilibrium with thebulk of the homogeneous phase. Therefore, this is a case of coupling betweenthe time scales of the mass transport and the first attachment of the adsorbateto the interface.

However, in the case of lower ionic strengths the times necessary to attainthe adsorption equilibrium by natural diffusion could be significantly higherthan those suggested in Table 1. In such conditions the potential profile nearthe interface (rf;) represents an energy barrier to diffusion, as suggested byKimizuka et al. (54-56). In the case of sodium dodecyl sulphate, tridecyl sul-phate and hexadecyl sulfate these authors reported the energy barrier in therange between 0.5 to 2.5 kcal/mol at 28°C. As a consequence, the Gm values

TABLE I

Conditions Necessary for Attaining the Adsorption Equilibrium with the Bulk of theHomogeneous Phase by Ntiturai Diffusion for S < 1 and High Ionic Strengths

C* in mol/l 10-9

1-100[.Lsec

10-1000msec

1·-100min

10-1000days*

300-3000years*

* in reality influenced by natural convection and the adsorption equilibrium is the nattained foster.

I

ADSORPTION TIME DEPENDENCE 461decrease between one and two orders of magnitude and for the same factorthe time necessary to attain the adsorption equilibrium should be increased.For times of adsorption long er than several minutes, in addition to naturaldiffusion, one should also take into account the convective contributions tomass transport. Therefore, for bulk concentrations smaller than 10-6 mol/I thetime necessary to attain the adsorption equilibrium will be significantly shorterthan those predicted in Table I. In a natural environment, mass transport iscontrolled by different scales of eddies'" and therefore the establishment ofadsorption equilibrium will be more efficient, as shown in Table II.

TABLE II

The Longest Times Necessaru to .4ttain Adsorption Equilibrium in Ntiturai AquaticSystems

Concentration levels in mol/IAquatic systems

10-3 10-6 10-9

Pore water in sediments 100 1 3000msec day years

Ground water of alluvial aquifers 1 15 30msec min years

Surface waters a 10 10msec sec

Symbol »a« refers to the intrinsic kinetics of the first attachment to the interface.

INTRINSIC ADSORPTION KINETICS

From the steady state condition for adsorption equilibrium one can CDTl-

clude that mass transport will control time dependence of adsorption forstrongly adsorbable substances. In tis case measurable surface coverages willbe attained at concentrations which are of the order of magnitude of C' = liK(K being the adsorption equilibrium constant). It has been sugaested (50) thatthe condition to attain the adsorption equilibrium with the bulk of homo-geneous phase can be also described through a dimensionless quantity whic'icontains K instead of C*:

t>100Tm2K2fD

However, the time dependence of adsorption process at time scales of mi-nutes, hours and even days have been observed also for adsorbates undergoingrelatively weak adsorption (for example in cases of some aliphatic alcoholsand fatty acids as well as macromolecules). Measurable surface coverages canbe obtained in such cases only at relatively high bulk concentrations of ad-sorbates. Under such conditions, mass transport cannot explain the timedependence of adsorption, even when only natural diffusion can take place.Therefore, for such cases there must be an energy of barrier to adsorptionprocess responsible for time dependence of the adsorption process. If the firstattachment to the interface is very fast, then such intrinsic adsorption kineticsshould be a result of some kind of secondary inter action (reaction) with thesurface.

462 1. RUZIC

The first experiments of the intrinsic adsorption kinetics reported in thefirst half of this century suggest the following conclusionsš"?":

i) The rate of adsorption in always proportional to the concentration of theadsorbate in the bulk of the homogeneous phase: Va ex k.C.

ii) Sometimes the adsorption rate is also proportional to the free surface anddesorption rate to the surface coverage: Vo ex (1- 8) and Vd ex 8.

iii) Few examples are known of the rate of adsorption proportional to thehigher power of the free surface: Va ex (1- 8)2 or va ex (1- 8).3

On the basis of this knowledge the simplest first order kinetic model for theadsorption process can be obtained:

dG/dt = koC - ki9 (10)

This classical rate equation has the known analytical solution:

e = (k,C!kd) (1- e-k,,') = KC (1- e-kdt) (11)

The adsorption equilibrium according to this model e = KC eorresponds tothe so called Henry (or ideal) adsorption isotherm. Combining this adsorptionisotherm with the thermodynamic Gibbs relation ayialn C = - RTr (where yis the surface tension) one can easily derive the ideal equation of state foradsorption: 11: = RTr (which is equivalent of the equation of state for an idealgas (here 11: is the surface pressure yo - y, and Yo is the surface tension forr = 0).11,63

According to Langmuir hypothesis on the nature of the adsorbed layer,the above simplest rate law can be extended to the following more generalform58,59:

(12)

This Langmuirian rate law has the analytical pseudo-first order solution (forcases where the bulk concentration of adsorbate is not significantly affectedby the adsorption processes):

(13)

The adsorption equilibrium according to this model e = KC/(l + KC) cor-responds to the well known Langmuirian adsorption isotherm. This model hasbeen widely used for the interpretation of experimental results by manyauthors.13,64-68 On the basis of the Langrnuir type of adsorption isotherm aspecial method has been developed for the determination of the adsorptionequilibrium constant. According to this method, the plotting of the inversevalues of 8 vs C should produce a straight line of slope equal to liK in thecase of adsorption equilibrrum":

1/f) = 1 + lIKC

This method has been widely used in literature. However, it has been shownrecently+? that this method is not selective enough for distinguishing theadsorption equilibrium from the adsorption kinetics. Therefore, in the caseof the very slow time dependence of adsorption, the adsorption equilibriumcan be significantly underestimated using this method. Therefore, another


method has been proposed'", based on plotting B/(1- B) vs. C in alog-logdiagram. In the case of adsorption equilibrium this plot has the form of astraight line of unit slope:

log [el(l - el] = logC + logK

The adsorption equilibrium constant can be determined from the intersectionof this straight line with the abscissa. The form of this plot deviates from astraight line in the case of time dependence of adsorption as shown in Figure l.In some cases both the equilibrium constant and the rate constant can bedetermined from this plotš''.

91-9

//

////

//

//

'/V~

C"=l/K ///" /\,,<9 /

"OJ /-",'-: /,<YJ /

10

10

c

KC

Figure 1. Plot of el(l- e) vs. C (KC for theoretical predictions) in alog-log diagramfor the case of Langrnuirian adsorption, and two different values for kdt parameter.Equilibrium condition (which corresponds to kdt--+ oo) has the form of a straight line

which is at the same time an asymptote to the plot at higher surface coverages.

As mentioned above, the rate of adsorption is not always proportional tothe free surface. Roginsky and Zeldovich were able to fit some of the expe-rimental results using the exponential type of the adsorption rate law:

(14)

known as the Elovich eqn.69-72• This rate law has also an analyticale = (Ila) In (1+ aKaCt)

Delahay and Mohilner10,23,63 extended this kind ofadditional term for desorption rate:

solution:(15)

rate law by including an

(16)

No analytical solution for this rate law is available and, therefore, numericalmethods should be used to obtain the solution. The corresponding adsorption

464 1. RUZIC

equilibrium KC = ebe (where b = al + (2) is known as the Temkin adsorptionisotherm. A more general model of adsorption kinetics can be described bythe following rate equation:

(17)

The corresponding adsorption equilibrium KC = 0ebe represents the so calledvirial adsorption isothermv-". The meaning of coefficient b can be deducedfrom the corresponding equation of state for adsorption:

n + ni = RT where :iri = - RTb r2/2 rm

Bere Tri represents the internal surface pressure due to the lateral interactionbetween adsorbed molecules, which is equivalent to the internal pressure inthe Van der Waals equation of state for real gas. The combination of thisand the Langmuirian rate law is the so called Frumkin-Fowler-Guggenheimmodel of adsorption (FFG) which is, together with the Langmuirian model,most widely used rate law so faro The characteristics of different rate lawsare summarized in Table III.

TABLE III

Characteristics oj Different J\lIocleLs oj the Adsarption Process

Eqn. of stateAdsorption

Rate law of adsorption Ref.equilibrium

n = RT rm6KC= fJ

d6/dt = kaC - kd6Henry 24, 25

1 KC = fJ/(1-6)d6/dt = kaC (1 - 6) - kd6cl = RT rm . ln ( 1_ e) Langrnuir 26, 58, 59

Jr = RT rmb62/2KC = eh9

d6/dt = kaC-a,e- kden,eTemkin 10, 19, 22

n = RT rm. (6 +b62/2)KC = 6ebe

d6/dt = kaCe-a,e- kd6ea,eVirial

-r 1 KC = 6ebel d6/dt = kaC (1- 6) e-<t,e_Jr RT Im· [ln ( 1_ 6 ) + /(1-6) - kd6ea,e 51, 73, 84

+ b62/2] FFG

In the next section of this paper the FFG type of adsorption model will bediscussed in more.

FFG AND SIMILAR MODELS OF ADSORPTION

Combining the Langmuirian and virial models of adsorption, i. e. includingthe internal surface pressure into the Langmuirian type of equation of state,one can derive the FFG type of adsorption isotherm KC = 0ebej(1 - 0). Thecorresponding rate law can be written in the following way50,f,1,73:

(18)


This type of adsorption rate law has been used extensively for the interpre-tation of adsorption of electroactive species at the electrode-electrolyte inter-face10,20,24 :

(19)

where i is the electric current, a is the electro chemical transfer coefficient,A is the electro de surface, S = nF (E - Eo)/RT (n is the number of electronsinvolved in electrode reaction, F is the Faraday constant, E the electrodepotential), and Eo, ks are the standard potential and standard rate constantrespectively. The FFG type of adsorption rate law has also been successfullyused for the interpretation of adsorption of electroinactive species." The FFGtype of rate law can be rewritten in terms of parameters of adsorption equi-librium in the following way:

deldt = kd [KC (1 - 8) e-,\bĐ - ee(1- ,\)bĐ] (20)

where A in an additional kinetic parameters. The rate Jaw (20) cannot be solvedanalytically and some numerical methods, such as Runge-Kuta, shouldbe usedto obtain theoretical predictions. Comparison of experimental results with inthe FFG type of rate alw requires determination of two additional parametersto those of the Langmuirian type of rate law. These are the FFG type ofinreaction coefficient b and the corresponding kinetic par ameter A. For thecase of adsorption equilibrium Damaskin and Dyatkina'" pro posed the use ofthe plot of log C (1- e)/eC0~112vs. e. In such a case, this semilogariihmeticplot should have the form of a straight line of the slope equal to b log e.However, in the case of time dependence of adsorption, a whole range of sur-face coverages are not in adsorption equilibrium and the use of this kind ofplot can be a significant source of errors. It has been also suggested to usea similar semilogarithmic plot of log elc (1- e) vs. e, which can be describedfor the FFG type of adsorption by the following equaticnšv-":

log [elC (1 - e)] = log K - b log e

In the absence of time dependence of adsorption, on the basis of this plot onecan obtain the values of both the adsorption equilibrium constant K and theFFG type of inter action coefficient b. If the time dependence of adsorptioncannot be neglected, all adsorption parameters cannot be easily obtained fromthe experimental results. However, it has been shown that this plot is not verysensitive to the value of A parameter and that it can be assumed that A = 0.5,i. e. a2 =- al = b/2 is a reasonable approxirnation-". The rest of adsorptionparameters, K, kdt and b can be extracted from the experiments by compa-rison with theoretical plots for different times of adsorption. For the samepurpose one can use the same log-log plot as for the Langmuirian type of ad-sorption (el(l - e) vs. C). The separation between the asymptotes at low andhigh coverages for this plot will dependon the FFG type of interactioncoefficient, as shown in Figure 2. Adsorption parameters can be determinedfrom this plot by comparing the separation of these asymptotes(CoICoo) fordifferent times of adsorption, as showri in Figure 3. In the case of high valuesfor the FFG inter action coefficient b the separation between asymptotes can be

466 1. RUZrC

<D,i,,II,I /,,/

----------,(// ,

//

//

//

//

//

/

b =

10 C"= lIK

--CD

10 KC

Figure 2. Plot of 8/(1- 8) vs. C (KC for theoretical predictions) in alog-log dia-gram for the case of the FFG type of adsorption, and different values for the FFG

interaction coefficient »b«. Other parameters are kdt = 0.5 and ). = 0.5.

0.1 10

Figure 3. Plot of separation between asymptotes at low (Co) and high (Coo) surfacecoverages for the FFG type of adsorption and different values of the FFG inter-

action coefficient »b«. Adsorption parameters are the same as in Figure 2.


arbitrary constant. In this way one can increase the accuracy of determinationof the separation between the assymptotes'".

The determination of adsorption parameters and their reliability in pre-dicting the time dependence of adsorption depends strongly on the type ofadsorption isotherm chosen for the interpretation of experimental resu1ts.Therefore, it would be useful to develop reliable diagnostic criteria for theselection of an appropriate type of adsorption isotherm. For this purposeanother plot has been proposed where log elc is plotted vs. log (1 - e)50:

log (eIC) = logK + log (1- 6) - b6log e

In the case of the Langmuirian type of adsorption this plot should havethe form of a straight line, of unit slope while in the case of the FFG type ofadsorption a straight line can be obtained only at higher coverages. For verynegative values of the inter action coefficient b (attraction of adsorbed molecu-les) this plot is characterized by distrinct maximum at moderate surface cove-rages, as shown in Figure 4. This plot is also very selective in distinguishingthe time dependence of adsorption from the adsorption equilibrium. The inter-section with the ordinata at zero surface coverage gives the value for theadsorption equilibrium constant. From the comparison of experiments andtheoretical plots for different times of adsorption one can extract the valuefor the adsorption rate constant as well.

Long age Sweringen and Dickinson'" introduced the concept of the shiftfrom the adsorption equilibrium e = (CO - C)/(CO - COO) where Co and C"<>represent adsorbate concentrations at the beginning of the experiment and

10r-----------------------------------~

1 ~KC=l

U:><:.•....CD

0.1

00

0.001 1-80.01 0.1

Figure 4. Theoretical plots of 61KC vs. 1- 6 in alog-log diagram for the FFGtype of adsorption and different kdt values. Other parameters are b = 4 and). = 0.5.

l

4.68 I. RUZIC

in the equilibrium, respectively. Using this concept Fava and Eyring propossdthe following rate equation'":

de/dt = 2k (1- 13) sinh [b (1- el (21)

where they assumed the adsorption and desorption rate constants to he equal.This unnecessary assumption has been avoided by Lindstrom and Haque whoproposed the following modified version of the adsorption rate law'":

de/dt = ka (1- 13) (1- 13/2) e-Le - kd [(1 - 13/2) eb(2 - e) - 132 elJe/2] (22)

These rate laws are based on similar principles to the rate law which cor-responds to the FFG adsorption isotherm. However, in rate equations (21) and(22) the information about the adsorption equilibrium is lost and, therefore,they are not convenient for the interpretation of experimental results evenif a reasonably good fitting can be attained.

Theory of adsorption of metal ions at hydrous oxide surfaces is based onthe model of adsorption involving surface complexationtš". However, thismodel of adsorption cannot be conceptually directly connected .with any par-ticular type of adsorption isotherm. In fact, it can equally well accommodateLangmuirian, and the FFG type of behaviour. In comparison with the surfacecomplexation model the adsorption isotherm represents a formal mass lawequation (which can be mathematically similar for adsorption and for anyhomogeneous chemical reaction, such as complex formation in solution). Thesurface complexation model refers to the nature of the binding of adsorbedmolecules at the interface and not the type of mass law equation, it would,therefore, be incorrect to argue whether one should use Langmuirian, FFG arthe so called surface complexation model, but rather whether the energy arbinding or surface structure supports the idea on the surface complexation ornot. Even if we assume surface complexation model of adsorption, this modelcould be implemented using one of the known mass law equations (Langmui-rian 01' FFG type, for example). Recently Wherli and Stumm80,Sl suggested thatadsorption of oxide-water interface could be interpreted using the FFG typeof adsorption isotherm, interpreting at the same time the adsorption processby the surface complexation model. Therefore, one should use one of the abovementioned rate laws of adsorption even in the case where there is evidencethat the inter action between the adsorbate and the interface is manifestedthrough the formation of surface complexes. The rate law should be chosenon the basis of experimental results. The surface complexation model intro-duces the possibility of different energies of binding at different sites at theinterface and in this way a kind of mixed adsorption: isotherm'" could beobtained.

THE INFLUENCE OF MASS TRANSPORT ON ADSORPTION KINETICS

Diffusion: controlled adsorption kinetics has been discussed by Ward andTorday." They presented an integral equation for plan ar and quiet interfacesimilar to eqn. (6): Generally, one can write:

(23)


and in the case of the planar and spherical interface:

(1 V D ) Dt ( V nDt ) --T = C* V D S lo (t - r) ------==- + -- dr = C* 2 - '1 + --- - V DJ Co.! =.l

V rct ro n \ 2ro

(

1 VD'-(t-rl -= +--) dr

V nr ro(24l

This integral eqn. can be combined with a given adsorption isotherm and inthis way the solution for the pure mass transport control of adsorption canbe obtained. For Henry (ideal) adsorption isotherm such a solution shouldread:

r = C* VD[K TTr/V D + A ea2t erfc a vt+ B eb2t erfc b V tj

whereA = (1- VD!arol!(b - al

B = (1-- V D/brol/(a-- b)

a + b = vDIKT m

a : b = DIKrmro

For planar interface this solution reduces to the expression reported earlierby Delahay and Trahtenberg'":

T = KC* Tm [1 - eDtlK2 r",' erfc VDtlKT m

Analogous numerical solutions have been obtained for the Langmuir ianš'<š",as well as the FFG type of isotherms'".

Combining the eqn. (24) and a rate law for adsorption one can derive asolution for cases of coupling between the intrinsic adsorption kinetics andmass transport. In the case of the plan ar interface and linear rate law ofadsorption (eqn. 10) the following solution can be obtained24,25:

b- a -r = Kr C* (1 + -- - ea2t erfc a Vt + --- eb2t erfc b Vtj

m a-b b-a

For other rate laws only numerical solutions can be used. Recently, such anumerical solution for the FFG type of adsorption has been used for theinterpretation of adsorption kinetics of Triton-X-100 at the mercury-sodiumchloride interface (hanging mercury electrode with nearly spherical geo-metry)?".

In Figure 5 log-log plots of e (1- e) vs. Care compared for cases ofpure diffusion and intrinsic kinetic control of the Langmuirian type ofadsorption. Obviously, the shapes of there two plots are different at highercoverages and shorter time of adsorption. Therefore, it is possible to distinguishthese two cases from the experimental results50,73.

Time dependence of adsorption of sodium dodecyl sulfate at the mercury--sodium chloride interface has not been successfully interpreted with thesame model. Discrepancies have been observed between the theory andexperiment at longer times of adsorption'". Experimental results on the time

'1:70 1. RUZIĆ

CD,

//

I/

/:»: kdt =0.2/

///

//

//

//

10 //

/v

c*= lIK

10 KC

c

Figure 5. Plot of 61(1 - 6) vs. C (KC for theoretical predictions) in alog-log diagramfor the Langmuirian type of adsorption and two different values of Gm = TmK/Y Dt(full lines Gm = O and 10) under equilibrium conditions as well as one plot forkdt = 0.2 and Gm = O (dashed line). Obviously, the kinetic and diffusion controlled

adsorption can be distiriguished from the shape of these plots.

10

dependence of adsorption for some fatty acids at the mercury-sodium chlorideinterface suggested that adsorption of these substances proceeds in severalsteps including strong lateral interactions between the adsorbed molecules-'"!",The extremely high FFG type of inter action coefficients suggests that thereis a strong probability of new phase formation at higher times of adsorption.The final adsorption equilibrium in the case of valeric acid has not beenattained up to three hours of adsorptionwšs. Similar behaviour has beenreported also for some polymeric adsorbates'<'", A theory of these processeshas not yet been developed at a satisfactory level. A simple model of thefirst order secondary surface process has been developed recentlyš-, Assumingthe Langmuirian type of adsorption, such a model can be described by thefollowing eqns:

KCo = 6/(1- 61- 62) and d6/dt = k161 - k262

where el is the surface coverage for the first attachment of the adsorbate tothe interface (very fast process), and e2 the surface coverage for adsorbedmolecules in the secondary transformed state (formed by a relatively slowprocess). The final adsorption equilibrium can then be described by thefollowing relations:

61 = k16/k2 and KCo (1+ k/k2) = 61(1- 6)

n


where (9 = (91 + (92 and Co is the volume concentration of adsorbate near theinterface which can be obtained from the integral eqn. for mass tr ansport'',In this way the total surface coverage can be obtained:

FJ = 6100 + (1- 61"X» kl S 61 e-E(t-"t) d-c:where

and

In cases where kz» kJ, the final adsorption equilibrium involves only thesecondary transition state of adsorbed molecules (9= (92with the adsorptionequilibrium constant Kkl/lcc. Then measurable surface coverages are attainedin conditions where (9100 =KCo and consequently:

6= kl"( S Coe-E(t-"t) dr

and

The overall rate constant of adsorption is in this case equal to klK. Theprim ary attachment to the interface represcnts an undetectable intermediatestate of adsorption process and the over all process is equivalent to the caseof single step adsorption kinetlcs. In the case of the FFG type of adsorptionone can write the following eqns.š-:

KCo =6lcbllGl >b12G,/(l- 61- 62)

and

In cases where k2 » k, and consequently (9= (92,these eqns. reduce to simplerforms:

andd6/dt= kjKCo (1- 6) e-(a12+ b,,)G - ka6ea22G

which is also equivalent to the case of asingle step FFG type of adsorptionkinetics. The mixed type of adsorption kinetics could be observed in this caseonly if the apparent strength of adsorption for the final state due to theintrinsic kinetics is comparable to the strength of the first attachment of theadsorbate to the interface because (92=(9. The influence of mass transporton adsorption kinetics for these complex models can be obtained only bynumerical methods.

CONCLUSIONS

The time dependence of adsorption can be controlled by intrinsic adsorpt-ion kinetics and mass-transport of the adsorbate from the bulk of homoge-neous phase to the interface. Theoretical predictions suggest that the limitfor the influence of mass transport by natural diffusion on the time depen-dence of adsorption can be expressed in the following way Gm = r III Ki Y Dt ==0.1. The strength of adsorption clearly has a profound influence on thislimiting condition, because rm and D do not vary significantly for different

472 1. RUZrĆ

adsorbates. However, in the case of hydrodynamic contributions to the masstransport in the form of eddies of different time scales, diffusion coefficientD should be repI aced by the so called dispersion coefficient which can bechanged depending on the environment for many orders of magnitude. There-fore, the time dependence of adsorption in natural environment canriot bestraightforwardly predicted from the results obtained in the laboratory.Although the rate of the first attachment of the adsorbate to the interfaceis assumed to be very high in many cases, the time dependence is observedalso under the conditions when mass transport cannot control the adsorptionprocess. For interpretation of the results obtained under such conditions awhole variety of rate laws have been developed corresponding to differenttypes of adsorption equilibrium and surface equations of state. The rateconstants obtained by fitting the experimerital results with theoretical pre-dictions obtained using a certain rate law are probably not intrinsic rateconstants for the first attachment of the adsorbate from the hornogeneousphase to the interface, but rather some overall paramet.ers which are theresults of a whole series of secondary surface processes. The model of adsorpt-ion which corresponds to the FFG type adsorption isotherm can be used fortheoretical predictions for adsorption of most organic as well' as same poly-meric adsorbates at the solid-liquid interfaces. The same model could beused for the interpretation of metal ions at hydrous oxide surfaces. Severalmethods for the interpretation of experimental results using the FFG modelof adsorption have been developed so far in literature. The type of adsorptionisotherm can be selected very effectively on the basis of a plot of log (efC)vs. log (1 - el. Theoretical predictions are not very sensitive to the valueof the kinetic coefficient I"~in the rate law corresponding to the FFG typeof adsorption, which can therefore be set to the value of 0.5 without pro-ducing significant errors. Theoretical predictions about the limit of the influ-ence of intrinsic adsorption kinetics on the time dependence of adsorptioncan be expressed as kd t = 10 in the absence of mass transport on adsorption.This limit is shifted to higher kd t valu es if the mass transport plays amoresignificant role (kd t = 30 for Cm = 1). Adsorption parameters can be deter-mined from the comparison of theoretical predictions with experimentalresults in a plot of log eeb*Gl/(l - e) vs. log C (log KC for theoretical pre-dictions) where the b* coefficient is selected arbitrarily in order to increase theaccuracy of determination of a true FFG interaction coefficient »b«. The adsorp-tion equilibrium constant K should be determined with great caution. The ab-sence of the time dependence of adsorption should be supported by the shape ofthe adsorption isotherm in order to avoid possible errors which could appearin cases of a very slow establishment of the adsorption equilibrium. Anydeviation of the shape of the adsorption isotherm from the one expectedfor the state of equilibrium indicates a need for additional experiments in awider range of adsorption times. Sometimes, in a wide range of adsorptiontimes different steps of the adsorption process can be observed which cancomplicate the interpretation of experimental results. Simple first order kineticmodels of multistep adsorption have not been successful in predicting accu-rately enough this type of adsorption processes. Therefore, more complexmodels of surface processes should be developed in the future.


Aeknowledgement. - The author is grateful to prof. Werner Stumm and Dr. B.Cosović for helpful discussion, as well as to Dr. Nikola Batina and Hans Jacob Ulrichwhose experimental work stimulated a part of this paper. This work has beensupported by the National Council for Scientific Research of Croatia. Part of thework was done during the time when the author visi ted the Swiss Federal Institutefor Water Resource and Water Pollution Control in Dubendorf-e-Zur ich, Svitzerland.

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SAŽETAK

Vremenska ovisnost adsorpcije na granici faza kruto-tekuće

I. Ruži,ć

Raspravlja se o vremenskoj zavisnosti adsorpcije na granici faza čvrsto-tekućeza slučajeve kada je proces adsorpcije kontroliran transportom mase ili kinetikomadsorpcije. Pokazano je da vremenska ovisnost adsorpcije u slučaju adsorpcije kon-trolirane transportom mase uglavnom ovisi o jakosti adsorpcije (prirodna difuzija),a ponekad i o koeficijentu transporta mase (koeficijentu disperzije za slučaj kadahidrodinamika kretanja medija dolazi do izražaja, kao što je to slučaj u prirodnommediju). Za pravu kinetičku kontrolu procesa adsorpcije prikazan je čitav niz kine-tičkih zakona. Prikazane su metode za interpretaciju eksperimentalnih rezultata.Spomenuta je također i mogućnost razlikovanja slučajeva kada je adsorpcija kon-trolirana transportom mase ili kinetikom adsorpcije. Opisani su jednostavni modelidvostepene adsorpcije gdje je prvi stupanj prethodno vrlo brzo prianjanje adsor-biranih molekula uz granicu faza. Zaključeno je da za ispravnu interpretaciju ekspe-rimentalnih rezultata utisak da je adsorpcija u uvjetima ravnoteže treba obveznopotvrditi detaljnim proučavanjem oblika adsorpcijske izoterme. U neravnotežnomslučaju potrebno je proširiti mjerenja adsorpcije na širu vremensku skalu.

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Time Dependence of Adsorption at Solid Liquid Interfaces*

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