562
J U X E F. ZIJIMERJIAN
(17) SHIKATA,XI., AND \ ~ A T A N A B E ,11.:J . Agr. Chem Soc. Japan 4, 02-1 (1928). (18) SMITH,L. I . , I i O L T H O F F , 1. XI., R A l \ ZOSEK, s.,. W D I I U O F I . , P. AI.: J . .in]. Chem. 63, 1018 (1941). (19) TEISISGER, J . : c a s . lek. cesk 1937, 325. (20) TEISIKGER, J . . Mkrochemie 26, 151 (1935). (21) WERTHESSES,S . T., ASI) BAKER,C. F.: Endocrinology 36, 351 (1945). (22) WIZIXEL,h.,ASD PROSKE,G : Ber. 71, l i e 5 (1938).
SOC.
DIFFUSIOX AXD ACTIVATION COSTROL Iir; HETEROGEKEOUS KEACTIOSS JUXE F. ZIIIIIEIIIIAiS Dalton Hall, Rryn X a u r C o l l e g e , Rrijn Mawr, Pennsylvania l?eceiced Jiilii 17’. 1948
The study of heterogeneous reactions is concerned chiefly lvith the determination of the nature of the processes controlling the observed velocity of the reaction. It was Sernst (40) who first postulated that reactions of this type are controlled by diffusion, either the diffusion of the “corrosive” medium to the phase boundary or the diffusion of the reaction products away from the phase boundary. This postulation involves the tacit assumption that the reaction a t or with the boundary occurs extremely rapidly, so rapidly, in fact, that its rate has little or no effect on the observed velocity of the reaction. Bruner (5) and Koyes and Whitney (41) liken ise made similar suggestions. Upon publication of these viens there arose t n o schools of opinion, the one (49) maintaining that the principle had only limited validity, and the other (see, e g . , 22) maintaining that the ~eriist-Koyes-Bruner theory or a modification of it is applicable to most heterogeneous reactions. There have been data contributed for countless reactions in which diffusion is rate-determining ; however, there are also data for other reactions IT hich indicate that the observed velocity cannot be identified with the velocity of diffusion alone, but rather that the speed of’ the chemical process, per se, may exert a considerable influence. These facts suggest that the t n o viewpoints (i.e., diffusion control 1 s. activation control) represent limiting extremes of behavior. The follov ing generalities may be set up: 1. Diffusion to or from n phase boundary may be important in determining the rate of a heterogeneous reaction, and it i b in many of them. 2 . The rate of the chemical reaction proper, that is, the rate of activation of the reactants, may be important in determining the rate of a heterogeneous reaction, and it is in a fairly large numher of linovn reactions. 3 . The rate of diffusion t o or from a phase boundary may be comparable in magnitude t o the rate of reaction a t or \\ith the phase boundary. hs uill he seen later, some fen. reactions seem to fall into this category. 4. -1particular reaction may be predominantly diffusion-controlled under a
D I F F U + I O S . I S D .ICTIY.iTIOS
COSTROL I S RE.LCTIO1-S
563
particular set of conditions and predominantly reaction-controlled under another set of conditions (variation of temperature, concentration, condition of the phase boundary, etc. j . In the follon-ing the characteristics of the first two types of reactions will be reviewed, cariteria for the recognition of the third type will he developed, and a mathenxitical expression for the xiriation in observed 1-elocity as the velocity of diffusion and the velocity of reaction also vary ivill lie introduced. D I F F U S I O S - C O S T R O L L E D REACTIOSS
Criteria for the characterization of these reactions Tvere first set up by Scinst (40 and have been discussed in detail by Moeln-pi-Hughes (39) very rcwntly. For this reason the characteristics. of which there are foui., ~ 3 be1 mentioned only briefly. The first, incapable of c h e r t experimental pi,oof except hy inference n.hen the other three criteria are obeyed, is that the chemical reaction, coil of the transfer of material across the phase bouiidai,y, occurs \\.it11 a very great velocity. From general considerations this must niean that the free energy of activation is extremely loiv, or that the reacting molcciilnr species are already present in an activated state. Secondly, when the velocity of difiusion determines the over-all rate of the process, the speed of agitation of the reactants influences the speed of the observed reaction rate. I n general, the observed velocity becomes proportional to the stirring velocity, or the stirring velocity raised to a constant poi\-er:
I\ here L'
reprebentz the oljserved reaction velocity, P' the proportionality conbrant, S the rate of stirring, and a another constant. Thirdly, if diffusion is the most important rate-determining factor, it is to be expected that the velocity observed obeys the equation for first-order kinetics, for diflusion is a process of the first order (24, 26); as a ,special case, Boguski's rule (21 is also obeyed, for it i b in a w w e a restatement of the Ficlc Inn- in hich 2,
=
Djd
(2)
that is to say, the observed velocity of a heterogeneous reaction is equated t o he ratio of the diffusion coefficient, D, to the thickness of the diffusion layer, d. rhis has been T-erified for many reactions (3, 5 , 6, 7 , 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 28, 29, 30, 31, and 41, for example). -kctually, it is this fact hat poses the greatest difficulty in the theory of dflusion control, for calculaions show that the thickness of the diffusion layer should be appreciable. Spangenberg (43), Stanton (451, Fage and Ton-nend (23), and Shern-ood (42). among Ithers (27), have offered solutions to the problem. I n particular, Tovbin (47) ind Zdanovskii (50) recently offered partial explanations, although these latter KO m-orkers are not reconciled with respect to the initial assumptions concerning he concentration of the diffusing material in the diffusion layer. The final criterion is that of the temperature coefficient. If diffusion control 3 predominant, the temperature coefficient should be about equal to the tem-
perature coefficient of diffusion, that is, roughly of the order of 1.3 for each 10" rise in temperature. .ICTIVATIOS-COXTROLLED REACTIOKS
The caharacteristics of this type of reaction are to a certain degree the cont-erse of those of diffusion-controlled reactions. *Is Brgnsted and Kane (4) and Iiilpatrick and Rushton (34) have pointed out on the basis of the extended acidbase theory, it may be expected that the order of such a reaction will be 1 or 2 or n-ill assume an intermediate d u e . Langmuir (38) has shown that the order of heterogeneous reaction Ti-here surface adsorption occurs map be 0, I , 2 , or intermediate between these values. The temperature coefficient of an artivation-controlled reaction should be similar to that of the ordinary homogeneous chemical reaction, that is, the velocity should be about doubled for every 10" rise in temperature. This is, of course, in sharp contrast t o the temperatiire Coefficient of diffusion-controlled i*eactions (9, 10, 44). The third criterion is that the observed rate of reaction of this class is uninfluenced by stirring or agitation of the system, for stirring and agitation can upset diffusion relationships only and not pure chemical processes. The fourth feature of this type of reaction can be detected, again, only by inference 11-hen the other three are obeyed; &.. ivhen the observed velocity is controlled by the velocity of the chemical reaction, this latter must be much smaller in magnitude than the velocity of diffusion. R E A C T I O S s C O S T R O L L E D Ill- D I F F U S I O S YSD RT ACTIV.ITIOS
In addition to the t x o clearly defined types of heterogeneous reaction> mentioned above, there have been some reported which do not fall clearly into either category. Ytoelu-yn-Hughes 139) and others (28 and 29, for instance) have pointed out that these very probably represent reactions in which the velocity of the chemical reaction and the velocity of diffusion are comparable in magnitude. However, the case has iecaeived little treatment from a theoretical *tadpoint. Frank-Kamenetskit ( 2 5 ) treated the problem by determining what the order expected from an heterogeneous reaction of this kind should beTu4ng n method not unlike Bodenstein's for quasi-stationary reactions in purely homogeneous system>. He concludes that if the surface reaction is of the first order, the reciprocal of the characteristic time of the reaction is equal to the hum of the reciprocals of the characteristic times of the diff'usion and activation reactions. K h e n the surface reaction is of the second order, the reaction velocity observed can no longer be predicted by a relationship of such simplicity. With respect to the influence of velocity of agitation on the observed velocity, it xould be expected that variation in speed of agitation should have little or no eff'ect on the rate of reaction, providing that other factors did not enter in. Figure 1 shows the effect of varying linear velocity on the speed of dissolution of sheet cadmium in T.3457 S hydrochloric acid. It has been postulated that thiq reaction is both diffusion- and activation-controlled (51). From the curve it
caii be seen that in this case increased linear velocity of the metal qiecimen raised the observed solution velocity. Lane and JIcDonald (37) report data on the effect of agitation on the velocity of dissolution of copper sheet in nquew? ammonium hydroxide. This reaction appears to consist of t n o sets of reactions (as the authors have pointed out), the first set of vhich is controlled tjy 1)oth diffusion and activation. In this caF;e also, the velocity seems to be affected Iiy the velocity of agitation of the liquid medium. Hon-ever, it n-ill be shonn luter that initially the criterion of the agitation effect is of little m l u e for this type of reaction. The first case cited ib an extremely short-lived reaction. In the kecond c.a.e the portion in question endured for less than 40 min in moht c a + ~ .
Spec.ftc Reoclion 9 o l e C o n s t a n t
FIG.1. Vwiation in specific reactioii r:Lte constant (zero order) n-it11velocity of agitation for the dissolution of cadmiuni shect in 7.3457 S hydrochloric acid. Volume of acid, 250 nil. Area of c:idniiuni sheet,, 17.90 c ~ i i . Temperature, ~ X0C.
The kinetics of the biphasic reaction between benzoyl-o-toluidine (in benzene bolution) and potassium permanganate (in water solution) were studied hy Hell i 1) and confirmed, essentially, 11y Kassel and Schaffer (33). These u . o ~ , l m ~ found that the velocity of the oxidation reaction was not accelerated even though the velocity of the agitation n a s increased fourfold, after an initial tiansient period had elapsed. In this connection it is interesting to note that I
I n the most favorable case, the total ieaction nxiy be uritten ai: the transfer of niolecules of 11 acio- the phase boundary into the "dx-atcd" products of the reaction, that is, the total re:tction may be T\ ritten as the slim ot
- ttt1I.hi
[~>I'I
/ t i 1 1 i p ~ i a a e1)
t/iAI
(phase 2 )
- 0-1+ [/till b-1')
m3I (pha-c 2
!I 1)
(~Ivationi
(lla1
K n o u ing that 2J'jk = R1' In K!,
(121 subtitiition of the T-alue, for the equilibrium cvn-tant. for reactions 11 and 1 l a into 10 and 12 yield- :in eXprcGm for the ob+m-ecl T-eloritv a- a function, 6 . oi (131 F~oinrelationship 13 it 1- t c i tie expected that in the txginriing oi a heterogeneous reaction of the type \\here hoth diffubion and activation velocitie< are ot comparable magnitude, the rate of btirring should have a great effect on the reaction T-elocity Other things being equal, n hen the cmcentration of the corroding medium i.j very great the effect of diffusion should be overshadoned, that is, the effect of stirring might be negligible and the reacation should appear to tie tfiffuiioncwntrolled, as mentioned previously. THI: GEYER I L E Q T \ T I O \
F O R O B 3 E R V E D V1:LOCITI OF 1 HETEHOGF.\ E.OC> Iil:
\CrLo3
-1mathematical de iptiori of the obseri ed velocity of a reaction in a heterogeneous system must aatkfy three condition.: ( 1 ) -4. the velocity of diffusion hecomes very great (or? as the xelocity oi the reaction itself become3 very small), the observed velocity must approach the i-elocity of the chemical reaction: that is to say \\hen
i'd,f,
>>
then
zjrx
I ' ~ I , ~ , I=
z rx
( 2 ) -1s the velocity of the reaction becomes very great, the observed velocity milst approach that ot cliffu4on, that is to say,
when
itrr
>>
l'dlir
then
iol,sd
= 2
'
~
~
~
(3) If the velocity of the reaction, per se, should equal the velocity of diffusion, then the three velocities must all be equal to each other, or tn: =
Ld,ff
=
t'ob9d
A function satisfying these three conditions may be nritten in the form:
lifferentiation show* hon- the observed velocity may be espected to vary with -ariation in diffusion and reaction velocity:
568
JI’SE F. ZIMMERMAS
For the special cabe of metals dissolving in acids this equation reduces to the relationship derived by Kimball (35, 36). The data reported by Tu, Davis, and Hottel (48), while incomplete for an analysis according to equation 15, indicate that the predicted trend may be follon-ed. The same statement may he made concerning the n-ork of Damliohler (19). SGMMART
The characteristics of reaction- and diffusion-controlled heterogeneous reactions are revjewd. Criteria for the recognition of a heterogeneous reaction under joint actimtion and diffusion control are set, up. -1general equation is introduced to show the variation in observed velocity with variation in the velocity of diffusion and in the 1,elocity of activation. REFEREKCES BELL,11. P.: J. Phys. Chciii. 32, 882 (1928). BOGGSKI, J . J.: Ber. 9, 1646 (18i6). BOGUSXI, J. J., . 4 S D 1I.: Itec. t r n r . chim. 42, 579, 1065 (1923). (15) CESTSERSZWVER, 31.:Iioczriiki Chcm. 6, 383 (1926). (16) CESTKERSZTT-ER, 31.;%. physik. Chem. 131, 214 (1928). AI. : %. physik. Chem. A137, 352 (1928). (17) CESTKERSZWER, (18) CEXTSERSZV-ER, 11.:%. physik. Chem. A141, 297 (1929). (19) DAMKOHLER, G. : I n Euckrn-,J:~kob’sDer Clie,nie-lngenicu,., Vol. 111, pp. 413, 4 3 - 6 8 .kkademische ~ e r l n g s ~ e s e l l s r h : i f1.cipzig t, (1937). (20) DRCCBER, C.: %. physik. Chcm. 36, 693 (1901). (21) DRVCBER. C.: Z. x i o ~ g .Chcm. 29, 459 (1902). (22)ERRER, W.:Z. : ~ n o i g .Chein. 250, 145 (1942). (23) FACE,X.,.%SI) TOWSESI). H. C. €1.: h o c . Roy. 80c. (London) A135, 656 (1932,. (24) FICB,A . : Pogg. Ann. 94, 59 (1SS5). (25) F R . ~ s K - I < . ~ U E K E TD. J ~.I,: I ~ , .$?tit I’liysicochim. c.lt.s.a.12, 9 (1940). (26) GLASSTOSE, S.,I I . ~ ~ ~ ~ ~ ~EYRISG,II.: , I < . , . iThc ~ ~l’heoi!joJRnle ) Processes. AlcGranHill Book Company, Ine., Sen- York (1911). (27) HURT,D. 3 l . : Ind. Eng. Chem. 35, 522 (1943). (28) .J.~BLYCZYXSKI,I
