(PEERS, A.M, 1965) Elovich Adsorption Kinetics and the Heterogeneous Surface

7/27/2019 (PEERS, A.M, 1965) Elovich Adsorption Kinetics and the Heterogeneous Surface

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JOURNAL OF CATAL YSIS 4, 49%503 (1965)

Elovich Adsorption Kinetics and the Heterogeneous SurfaceA. M. PEERS

From the Laboratoire Curie, Institut du Radium, Paris, Frarance

Received November 4, 1964

Earlier treatments of the rate of adsorption on a heterogeneous surface have givenresults which do not agree with the Elovich equation at very short times. It is shown thatthe familiar, simp le model of a nonuniform surface does, in fact, lead to a rate equationin good agreement with the Elovich equation at very short as well as at very long times.The model gives a physical significa nce to the constant to and, when to s appreciable,leads to Elovich plots having two linear se ctions. Similar experimental plots can befound. The common assum ption that the activation energy for chemisorption increaseswith the energy of adsorption is incomp atible with the model when the energy of chemi-sorption decreases with coverage and is thus difficult to justify: The apparent justifica tionto be found in earlier treatments of the simple model is shown to be unsound.

INTRODUCTIONThe rate of adsorption of gases on solidsis often satisfactorily represented by theexpression

dO/dt = LYexp (-be), (1)which is generally known as the Elovichequation (see Ref. 1). Here, 0 is proportionalto the quantity adsorbed and the parameters(Y and /3 usually depend on temperature andpressure. Equation (1) may be integrated togive

0 = (l/P> In (t + td - C, (2)where to = l/alp and C is a constant.* An-other form which will be found useful isobtained by differentiation of (2) whichgives

de/& = l/p(t + to) (3)* It frequently happens t.hat the process obeying

Elovich kinetics appears to be superimposed on arelatively instantaneous, limited, adsorption, sothat the linear plot of 0 against log (t + to) does not,extrapolate to 0 = 0 at t = 0. In such circum-stances , however, one may sti ll consider the Elovichequation to be satisfied , although not, for the over-allprocess down to zero time. The pre-exponentialfactor 01 = l/b& does not then give the initial rateof adsorption, but otherwise retains its signific ancewith regard to the process obeying Elovich kinetics.

Various authors (2, 3, 4) have derivedexpressions similar to Eq. (2) using a simplemodel of a nonuniform surface. In each case,the equation arrived at was of the forme = kl In t + i&, (4)where k, and k, are constants. Equation (4)differs from (2) by the absence of to from thetime-dependent term, so that a finite init ialrate, l//S, is not obtained. Both Stone (3)and Kubokawa (4) conclude that the simplemodel is at fault. The present communica-tion shows that the simple model leads, onthe contrary, to an expression which behaveslike the Elovich equation at very small aswell as at very large times. In addition, themodel gives a precise significance to to.The choice of a function to represent thedependence of the activation energy of ad-sorption, E,, on the energy of adsorption, H,will also be discussed. It is emphasized thatthe relation between E, and H need not beknown in order to derive (4). The choice ofsuch a relation is imposed, however, if theeffect of nonnegligible desorption rates is tobe considered. When applied to the simplemodel (2, 3, 4), the choice made by Halsey(6) is diff icult to reconcile with well-estab-lished observations. Another choice can bemade which is in more general agreementwith experiment.

499


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500 A. M. PEERSTHE SIMPLE MODEL AND EQ. (4)

Following Halsey (5), Porter and Tomp-kins (d), and Stone (3), the essential steps nthe derivation of Eq. (4) are briefly givenbelow, except that no dependence of E, onH is specified. The term simple modelrefers to a surface on which a fixed number ofadsorption sites are associated with a con-tinuous distribution of activation energiesof adsorption. The distribution function, N,is assumed to be constant (dN/dE, = 0)and desorption rates are taken to be negligi-ble. A set of sites is defined as all those sitesassociated with the samevalue of the activa-tion energy, E (where the subscript is drop-ped, E, is to be assumed). The rate ofadsorption on any set of sites is then,

f$f = /dhPexp(~)dB (5)

where= A(1 - 0~) (6)

A = XP exp (-E/RT) (7)and x and P, the pressure, are constant. In-tegration of (6) gives,

!9E= 1 - exp (-At) (8)The coverage ~9,over the entire surface, isgiven by the sum over all sets,

where E1 and Ez are the limits of the dis-tribution of activation energies. Integrationof (9) is simplified by assuming that thesets of sites are successively filled (0, = 1)in order of increasing values of E, no setbeginning to fill before the preceding set issaturated (in contrast to the correct pro-cedure where all sets are allowed to fillsimultaneously at different rates). With theabove assumption, Eq. (9) becomes simply*

* Note that Eq. (10) can be rearranged to E =El + (l/N)0, which ia often assumed a priori. Thesite-filling procedure assumed above, combined withthe constant distribution function, provides a par-ticular mechanism whereby a linear increase of Ewith coverage may be obtained. Brunauer, Love, andKeenan (6) substituted such an expression directlyinto (5). The result reduces to Eq. (1) for fI % dE

By differentiation of (8), one obtains(deE/dt) = A exp (-At)

and from (7),(dE/dA) = -RT/A

Substitution in (12) then gives


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ELOVICH KINET ICS AND HETEROGEN EOUS SURFACE 501

d8= -NETdt s A- exp (-At)dA (13)When the distribution of activation energyhas the finite limits El and Ez (where El > to, on the other hand, the approxima-tions dO/dt = l/Oft and l/Pt are valid. Atintermediate times, (15) gives slightly largerrates than (3) when the constants of bothequations have the same values. Slightlydifferent values for to and to, however, willmake the two nearly coincide when veryshort times are neglected, as in practicethey often are. In such cases, kinetic datawhich obey (15) will give an acceptablestraight line when 0 is plotted against log(t + to), in accordance with the integratedElovich equation, (2). Specifically, when t.is not more than three or four times greaterthan the time at which the first measure-ment is made, data which obey (15) will givea satisfactory Elovich plot using to = tfo/2.This is demonstrated by Curve A of Fig. 1,for which the data were obtained by graph-ical integration of Eq. (15), using arbitraryvalues of the constants. When to is con-siderably greater than the time of the firstmeasurement, more acceptable plots areobtained by taking to = to. The resultingcurve (B, Fig. 1) has two linear sections, that

0 1 2 3> LOG (ttt,)

FIG . 1. Elovich plots (coverage, 8, against t + t,) of data calculated by means of Eq. (15). Curve A:calculate d with to = 2; 0, plotted with to = 0; 0, plotted with to = to/2 = 1. Curve B : csalcu latedwith tfo = 20; 0, plotted with to = 0; 0, plotted with to = to = 20.


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502 A. M. PEERSof smaller slope appearing at higher times.The ratio of the two slopes (about 1.2) isindependent of the values of the constants.The second linear section almost disappearswhen Ez - El is less than about 5RT, anegative curvature then being observed.The curves of Fig. 1 show that to is relatedto tfO, the inverse of the init ial rate on theset of sites associated with the smallestactivation energy.Elovich plots showing several linear re-gions have frequently been reported (1). Ofparticular interest here are the results ofCimino et al. (7) for hydrogen adsorption onZnO, since their Elovich plots bear a re-markable resemblance to the calculatedCurve B of Fig. 1. The present author,incidentally, has obtained curves of type Bfor the adsorption of another reducing gas(CO) on GO, although these were obtainedwith a constant-volume apparatus and apressure which was allowed to fall to about10% of its ini tial value. Unfortunately, nosolution of the rate equation for constant-volume systems is available, so that onecannot conclude with certainty that thesimple model is unable to account for suchbehavior to within the accuracy of themeasurements. It is, of course, possible thatthe rate is very insensitive to the ambientpressure, as has been shown for a number ofsystems (1). The simple model, however,makes no provision for such a possibility.It is sometimes of interest to have aqualitative understanding of the behaviorof the simple model when desorption ratesare not entirely negligible . For this purpose,noting that the activation energy of desorp-tion is given by Ed = E, + H, it is usefulto know the relat ion between E, and theenergy of adsorption, H, so that Ed may beexpressed as a function of E,. When dealingwith the kinetics of adsorption on a heter-ogeneous surface, it appears to be customary(9, 8, 4, following Halsey (5), to choose therelation E, = rH (where T < 1 is constant).Since E,, however, must also increase as 0increases, the simple model thus leads toadsorption energies which increase withcoverage, the adsorbate being immobile. Asis well known (8), such a variation of H with8 is in contradiction with nearly al l calori-

metric measurements. Indeed, it is usuallyassumed (8, 9) that the decrease of H withcoverage brings about the increase of E,, sothat a relat ion such as E, = Ez - rH, forexample, might be proposed (13).It must be noted that the above objec-tions to the use of E, = rH are not ap-plicable to the analysis of Halsey (5), whichdoes not neglect desorption and does notcontradict calorimetric data provided thatthe total heat evolved at a given pressureis plotted against coverage for a series ofincreasing pressures. Such behavior is as-sured by allowing the lower l imit of theoccupied region of the H distribution todecrease with increasing pressure.* In addi-tion, Halsey justifies the assumption E, =rH to the extent that it leads to the predic-tion of rapid desorption followed by slowerreadsorption when the temperature of thenonequilibr ium system is suddenly in-creased, in agreement with the observationsof Taylor and Liang (11). Other workers[see ref. $)I, however, have studied systemswhere adsorption occurs without preliminarydesorption, so that the assumption of Halseycannot be universally true. The absence ofrapid desorption on thermal cycling is con-sistent, rather, with the assumption E, =Ez - rH, whereupon Ed for al l occupiedsites is greater than E, for al l sites remainingto be filled.Apparent justification for the applicationof E, = rH to the simple model is to befound in the analyses of Stone (3) andKubokawa (4), but can be shown to beillusory. If E, = rH, then the activationenergy for desorption is relat ively smallwhen E, is small, so that the error intro-duced in (11) by the neglect of desorptionwill be relatively more important at smaller

*At any given pressure, however, the modelimplie s that sites of larger H are last to be occupied ,so that (dH/dB)p should increa se with time, i.e.,de/& should fall more rapidly than dH/dt. Nocalorimetric evidence for such an effect appears tobe available, although dO/dt has been shown to fallmore rapidly than the rate of change of conductivitywhen O2 is adsorbed on NiO (IO). Such an effectcould be related to a variation of Z, with H, but ismore readily taken to indicate the occurrence of twotypes of bondin g (8, IO).


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times. Stone was thus led to conclude that at the lower end of the H distribution. Tothe appearance of to in the logarithmic term account for the above feature it is necessaryof (2) is a correction for this effect, providing to assume, for example, that the height of thean indirect justification of the assumption barrier to direct entry (E,), like that of theE, = rH, as well as giving physical signifi- diffusion barrier (Em), increases rapidly withcance to to. As has been shown, however, the H. Since E,, in addition, must always besimple model accounts for to independently considerably larger than Em, except at theof desorption or any assumed relat ion be- point of entry, the former is required totween E, and H. Kubokawa, on the other increase with H even more rapidly than doeshand, having assumed that E, increases with Em which must itself increase rather mark-H, argues that appreciable desorption rates edly in order to account for the site-fillingwill thereby cause ,6 to decrease at higher procedure. The diffusion model thus appearsvalues of P, in agreement with many ob- relat ively unattractive in view of the natureservations. The argument assumes, however, of the assumptions involved. This is particu-that higher P is synonymous with higher 0, larly so when one considers that a variationwhereas the experimental example given by of E, with H somewhat less extreme thanKubokawa (Hz on ZnO) shows that /? varies that mentioned above suffices by itselfwith P even over the same range of coverage. (5, 5) to permit the derivation of (4). InOne may therefore conclude that the ob- spite of the above criticism, however, theserved variat ion of p with P does not justify diffusion model would evidently merit at-the assumed variation of E, with H. tention in the event of Elovich kineticsAn act ivation energy which increases with being observed with a system where in-H is also a feature of the Porter and Tomp- dependent evidence for diffusion of thekins model (.2), where the rate-determining chemisorbed species was also available.process is not the primary act of chemisorp-tion but that of migration from one chemi- REFERENCESsorption site to another. This mechanism, 1. Low, M. J. D., C&m . Rev. SO, 267 (1960).although later discarded by its authors on 2. PORTER, A. S., AND TOMPKINS, F. C., Proc. Roy.experimental grounds (I??), is still cited (9) Sot. (London) A217, 529 (1953).and it would perhaps not be out of place here 3. STONE, F. S., in Chemistry of the Solid Stateto draw attention to its theoretical weak- (W. E. Garner, cd.), p. 367. Butterworth s,nesses.The Porter and Tompkins derivation London, 1955.of Eq. (4) replaces E, by Em, the activation 4. KUBOKAWA, Y., Bull. Chem. Sot. Japan 33, 734energy for migration between chemisorption (1969).sites. In order that Em may increase with 6. HALSEY, G. D., J. Phys. Coil. Chem. 55, 21coverage, a linear distribution of adsorption (1951).energies is assumed and a site-filling pro- 6. BRUNAUER, S., LOVE, K. S., AND KEENAN, R. G.,cedure is adopted such that sets are suc- J. Am. Chem. Sot. 64, 751 (1942).cessively occupied in the order of increasing

7. CIMINO, A., MOLINARI, E., AND CIPOLLIN I, E.,Actes Congr. Intern. Catalyse, be, Park,H by migration from one set to another. 1960, 2, 263 (1961).Such a procedure implies that the adsorbate 8. DE BOER, J. H., A&an. Catalysis 8, 18 (1956).may enter the chemisorbed (phase only 9. ASHMORE, P. G. Catalysis and Inhibition offrom the lower end of the H distribution. It Chem ical Reactions, p. 165. Butterworths,is necessary, n addition, to suppose hat the London, 1963.energy of activation for diffusion from the 10. KUCHYNKA, K., AND KLIER, K., Collectionoccupied site A to the vacant site B increases Czeclz. Chem. Commun. 28, 148 (1963).rather markedly as Hg increases. The justi- 11. TAYLOR, H. S., AND LIANG, S. C., J. Am. Chem.fication for such an assumption, however, is Sot. 69, 1306 (1947).far from self-evident. Further difficulties 12. GUNDRY, P. M., AND TOMPKINS, F. C., Trans.arise from the related requirement that Faraday Sot. 52, 1609 (1956).13. SCHOLTEN, J. J. F., ZWEITERING, P., KON-direct entry from the gas phase (or a very v.iLIxK.4, J. A., AND DE BOER, J. H., Trans.mobile adsorbed state) be prohibited except Faraday Sot. 55, 2166 (1959).

ELOVICH KINET ICS AND HETEROGEN EOUS SURFACE 503

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(PEERS, A.M, 1965) Elovich Adsorption Kinetics and the Heterogeneous Surface