the heterogeneous decomposition of nitric oxide ... - ACS Publications

packed with zirconia pebbles. Slaughter5 ... through a time chamber 685 ml. in volume, made of Pyrex tubing ..... 12 times more surface area than does...
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Feb., 1958

HETEROGENEOUS DECOMPOSITION OF NITRICOXIDEWITH OXIDECATALYSTS

215

THE HETEROGENEOUS DECOMPOSITION OF NITRIC OXIDE WITH OXIDE CATALYSTS BY JAMES M. FRASER AND FARRINGTON DANIELS Contribution from the Department of Chemistry, University of Wisconsin, Madison 6 , Wisconsin Received September 0 , 106'7

The catalytic decomposition of nitric oxide was measured from 740 to 1040' by slowly passing the gas, diluted with nine volumes of helium, over pellets of CaO, CrzO3,Ga208,ZrO2, FezOs, Ti02 and ZnO. The extent of the decomposition, up to 50%, was determined from the concentration of NO2 formed subsequent to decomposition as measured with a photocell. All reactions were zero-order. The Arrhenius activation energies ranged from 16 to 39 kcal. per mole. The results are interpreted in terms of activation energy, and surface area, as determined for each catalyst by the BET theory using low temperature nitrogen adsorption.

The thermal decomposition of nitric oxide 2 N 0 Nz Oz has been studied in this Laboratory from 600 to 1900°.2 Above 1400" the reaction was found t o be a homogeneous, second-order gas phase reaction, between 1000 and 1400" it was partly homogeneous and partly surface catalyzed, and below 1000" it was entirely heterogeneous and zero order under the conditions used in this research. The purpose of the present contribution was to study, quantitatively, the catalytic effect of various metal oxides in pellet form. Preliminary experiments in this Laboratory by R. J. Williams and E. L. Yuan showed that various oxides have different catalytic activities for the decomposition of nitric oxide. Diluent gases with different adsorption properties were found t o affect the rate of the surface-catalyzed reaction.2 A qualitative test for the heterogeneous decomposition of nitric oxide at room temperature has been going on over a 39 year period,-so far with negative result^.^ Koerner4 obtained an activation energy of 24.5 kcal. per mole using an alumina tube packed with zirconia pebbles. Slaughter5obtained similar results. Wise and Frech6 reported a second-order activation energy on quartz which varied from 21.4 kcal. a t 627" to 56.6 kcal. at 1000" and proposed a complex surface reaction. Kaufman and Kelso,' while disagreeing with the mechanism of Wise and Frech, conclude that the heterogeneous decomposition is second order with an activation energy of 21.4 kcal. when carried out on quartz. The results described here are confined t o catalysts of metal oxides. Nitric oxide is very stable at room temperature and its decomposition is of interest in the general study of catalysis and in the possibility of reducing the concentration of nitrogen oxides in the exhaust fumes of automobiles and flames. Such studies are of interest in connection with the reduction of smog in certain cities.

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Experimental A flow system was used and the extent of the decomposition was foliowed spectrophotometrically. Nitric oxide, (1) More complete details of this work may be found in the Ph.D. thesis of James M. Fraser filed in the Library of the University of Wisaonsin in 1956. (2) E. L. Yuan, J. 1. Slaughter, W. Koerner and F. Daniels, forthcoming publication. (3) C. 8. Howard and F. Daniels, J . Phys. Chem., 6 2 , March (1958). (4) W. Koerner, Ph.D. Thesis, University of Wisconsin, 1949. ( 5 ) J. I. Slaughter, Ph.D. Thesis, University of Wisconsin, 1953. (6) Wise and J. Freah, J . Chem. P h y s . , 20, 22 (1952). (7) F. Kaufman and J. Kelao, ibid., 18, 1702 (1966).

diluted with helium, was contained in a 20-liter stainless steel tank under pressure and metered through a calibrated, floating-ball flow meter to a Vycor reaction chamber. The vessel, about 60 cm. long and 1.73 cm. in diameter, was heated with a "globar" furnace manufactured by the Harry W. Dietert Company. The middle portion of the tube, 16.5 cm. in length, comprised the reactor zone and was packed with 300 metal-oxide catalyst pellets, each of which was 5.20 mm. in diameter and 3.25 mm. in length. The pressure drop through the system was measured with an open-end manometer since i t was necessary to apply pressure corrections to the flow meter calibration. Needle valves and a pressure reduction valve (tank pressure was 200 lb.) were used to control the flow of the 10% NO in He. Before making a determination, the apparatus was evacuated to remove air and gases adsorbed on the catalyst pellets. The reaction product gases and diluent gas passed through a cooling condenser and then through a time-chamber of Pyrex to a spectrophotometer. Oxygen produced as a result of the decomposition 2 N 0 +Nz On (1) reacts with nitric oxide in the time chamber at the lower temperature to form nitrogen dioxide by the reaction

+

+

2N0 On +2N02 (2) The brown color of NOz affords an excellent means of measuring the extent of decomposition of nitric oxide. Reaction 1 was never allowed to proceed to more than 50% of completion, and thus the NOn concentration was a measure of the 02 produced and a measure of the extent of completion of reaction 1, after consideration of the rapid equilibria 2N02

and NO

2Nn04

+ NOz Jr NzOs

(3)

(4)

Reaction 2 is relatively slow, especially when the reactants are in low concentration, and it is not complete a t high temperatures. Accordingly, the product gases from the decomposition (reaction 1 ) were passed over cooling coils, which were maintained at room temperature, and then through a time chamber 685 ml. in volume, made of Pyrex tubing, 2 cm. in diameter and 210 cm. long, before entering the absorption cell of the spectrophotometer. Thus almost sufficient time was provided for completion of reaction 2. When necessary, corrections for incompleteness of the reaction were made on the basis of the data of Treacy and Daniels.* The s ectrophotometer, const,ructed by J. I. Slaughter, R. J. d l i a m s and E. L. Yuan, was of the split beam type in which one half of the incident light beam was reflected to a photocell and the other half passed through the absorpticn cell to a second photocell. Monochromatic light of 4359 A . was produced by a mercury AH4 lamp and appropriate filters. The spectrophotometer was calibrated by plotting chart readings against the NO2 concentrations, as obtained by titration of the nitric acid formed by adding 10 ml. of 10% HZOZand an excess of oxygen to known volumes of NO9 at various pressures and a t constant temperature. The acid was titrated with standardized NaOH using bromcresol green. It is very important to allow ample time for (0) J. C. Treaoy and F. Daniels, J . Am. Chsm. Xoo., 71, 20aa (1966).

JAMES M. FRASER AND FARRINGTON DANIELS

216

the conversion of nitric oxide to nitric acid with water, oxygen and hydrogen peroxide, because in the reaction 3N02 H20 + 2"Os NO the reoxidation of the nitric oxide to NO2 at low concentrations is very slow. It was found that a 3-hour waiting period was necessary before titration to insure complete conversion of the NO to nitric acid. A t each flow rate of NO-He through the reactor a steady state was produced at which the concentration of NO2 in the spectrophotometer remained constant. The attainment of the steady state was indicated by the plateau drawn by the pen on the recorder chart. Thus for each flow rate at each temperature a concentration of NO2 was obtained to which the appropriate corrections were applied. The residence times of the nitric oxide in the reaction chamber were calculated from the flow rates and the volume of voids between pellets in the reaction zone. The temperatures of the reaction zone were measured with a platinum-platinum and 10% rhodium thermocouple and potentiometer. They agreed within about + l % with the indicated furnace temperatures and the reaction zone was uniform in temperature. The metal oxide catalysts were prepared in pellet form by making a paste of the powdered oxide with ethyl alcohol containing a little stearic acid and then pressing them out in a pellet mold. The stearic acid prevented wumbling and was decomposed and eliminated by evacuation and heating before the catalyst was used. The oxides were reagent quality oxides produced by Merck, Baker and Mallinckrodt companies with the exception of ZrOz which was a spectrographically-standardized material supplied by Johnson, Matthey and Co., Limited. The oxides used were A1203, CaO, Cr203, G&Os, ZrOz, Fe203, TiO,, ZnO, NiO and MgO. No results were obtained with N i 0 because of instability of the oxide and none with MgO because of the crumbling of the pellets. These difficulties were not encountered with the other oxides. Surf?ce areas were determined on used catalysts only, because sintering probably occurred to varying extents a t the high temperatures used, which decreased the surface area of the fresh catalyst. The kinetic experiments were always carried out at the highest temperature first since additional sintering would not be expected to occur upon lowering the temperature. A conventional high vacuum system employing a mercury diffusion pump, gas buret and a precision mercury manometer was used to determine the volumes of nitrogen adsorbed by the catalysts at liquid air temperature and a t various pressures. From these data the surface areas of the catalysts were calculated using the BET theory.9 The catalyst pellets were outgassed a t 400' to pressures of less than 1 p before adsorption data were obtained. The "dead volume" or the volume not occupied by the catalyst was determined by bleeding a known volume of helium a t a known pressure into the system and observing the new pressure after thermal equilibrium was attained. The calculations involve simple P V T relationships.

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Vol. 62

posed to an extent greater than 50% of its original concentration, and no data were taken under these conditions. Adding reactions 1and 2 4N0

--+

N1

+ 2NOz

Each mole of NO2 found in the spectrophotometer cell indicates that two moles of NO have disappeared-one by chemical decomposition in the high-temperature reaction chamber, and one by subsequent reaction with the oxygen produced by decomposition. The residence times were calculated from the flow rates, reactor temperature and void volumes as (5)

where t = residence time; V , = volume of empty reactor zone; V, = volunie of catalyst; V v = void volume; F = 380w rate through flow rater corrected for pressure drop in system; TI = absolute temperature of gas a t flow meter; and T2 = absolute temperature of gas in reactor zone. These simple calculations are possible because there is no change in the number of molecules in the reaction zone. Reaction 1 involves no change in the number of molecules and reaction 2 does not occur at the temperatures of the reaction zone. It occurs only after the gases have left the reaction chamber and have been cooled. Corrections.-To the NOz concentrations obtained from the recorder charts three corrections were considered. The first involves loss of NOz by conversion to colorless Nz04through reaction 3. However, in calibrating the spectrophotometer chart readings, allowance was made for this reaction. The nitric acid titrations were converted into total amounts of NO2,assuming no association to N204, and these NO2concentrations were plotted directly against the optical densities, obtained as chart readings, to give a calibration curve. The second correction involves N203 which forms to a small extent by eq. 4 and is also n p absorbing a t the wave length used here (4359 A.). The dissociation constant for N z 0 3(for the reverse of equation 4) at 25" is

Results K - CNoCNos - 0.0965 mole/literlp10 (6) The experimental data included the flow rate of CNaOa 10% NO in He (4.628 X mole NO/liter) From reactions 1, 2 and 3, it follows that the conthrough the Vycor reactor tube, and the NO, con- centration (CNO) of NO a t any time is equal t o the centration, as indicated by the spectrophotometer initial concentration ( ~ 0 ~ 0of) NO minus twice the chart readings. For each catalyst used decom- concentration (CNOJ of NO,. position experiments were carried out at four acCONO was determined by analysis of the tank gas curately known temperatures (from about 740 t o and CNO, was determined from chart readings and 1040"). The maximum flow rate used for each the known dissociation constant for Nz04 (reaccatalyst at each temperature was governed by the tion 2).l. lo Then minimum detectability of the spectrophotometer CNOCNOz (CON0 - ~ C N O I ) ( C N O Z ) = = consistent with accuracy while the minimum flow CNpOg = 7 K rate was set by the fact that for analytical calculaC'NOCNOI - 2 c 2 N 0 2 (7) tions it was necessary t o decompose something less K than 50% of the initial NO. This minimum flow The maximum concentration of Nz03 will occur rate was ascertained from the chart readings. If when the concentration of NO2 began t o decrease upon C'NO - ~ C N O=~ decreasing the flow rate, the NO was being decomK -

(9)

3

8. Brunauer, P. R.Emmett and E. Teller, J . Am. Chem. SOC.,

60,309 (1938).

(10) I?. € Verhoek I. and F. Daniels, dbid., 68, 1250 (1931).

"I

HETEROGENEOUS DECOMPOSITION OF NITRICOXIDEWITH OXIDECATALYSTS

Feb., 1958

217

Therefore C N ~ O is , at a maximum when C N O ~= ~ / ~ c O N O

=

oq

= 0.001157 mole

1.-1

(8)

Substituting this result and the value for CONO back into equation 7 indicates that the maximum concentration of N20a possible a t 25" with about 10% nitric oxide is 2.77 X 10-6 mole/liter which is only 2.4y0 of the concentration of NO$. Although neglecting the presence of N203 leads to low results as reported by Kaufman and Kelso' the error for the concentrations involved here is usually too small t o consider. However, calculations for correcting for N20sformation are straightforward and were Bpplied wherever significant. The third correction involves the incompleteness of reaction 2 in spite of the time chamber. The rate expression for reaction 2 is where k = 7.5 X lo3liters2 moles-2 set.-*.* At time zero when the gases from the reaction chamber enter the time chamber CNO

=

CoNO

- 2c0,

= 4.628

x

10-3

- 2coz = A ; and Cor = B

At time t cNO% =

x, CNO

=

A

- X;

and

C O ~=

B

0 TIME

Fig. 1.-Graph

TABLE I TYPICAL DATAFOR THE HETEROQENEOUS DECOMPOSITION OF NO ON METALOXIDECATALYSTS Metal oxide catalyst

Temperature in O K . 1313 1233 Residenoe NO9 oonon., Residenoe NO8 oonon., CNOP X 108, time, t C N O ~X 101, time, t molefl. mole/l. (sec.) (sea.)

Abos

0.38 .49 .60 . .75 .88

AlzOa

0.74 0.93 1.18 1.61 2.24 2.60 3.91

ZrOz

0.65 1.15 1.88 2.98 5.03

ZrOz

0.78 1.37 3.44 13.3 16.8

and kt =

for estimating the correction for the inoomplete oxidation of NO by 0 2 .

Typical corrected data for the decomposition of NO on A120aand ZrO2 appear in Table I.

- X3

Then

-22

(CONO

- ~ B ) ( c O N O - 2B - Z)(C"NO - 2B) + 2B(C0N0 - 2B - 2) 4.606 (@NO - 4B)' log (@NO - 2B)(2B - X )

(SECI.

1.16 1.36 1.66 2.11 2.38

0.41 .47 .53 .75 .88 1.07 1048

1128

(9)

Since NO is always in excess, completion of the reaction indicates that all of the oxygen has reacted and x = 2B which would require an infinite length of time. The fraction of completion f, a t any time in the time chamber is given byf = .x/2B. Values of t in the integrated expression 9 were calculated for various fractions (f) for different values of x (or C N O J as determined spectrophotometrically with corrections for N204 and N2Oa. The results of these calculations are plotted in Fig. 1 and from these plots the small corrections for incompleteness of reaction 2 were made when necessary. Experimental Data.-In measuring the rates of nitric oxide decomposition the concentrations of NO2 are calculated for standard temperature and pressure, but the rates of flow are calculated for the experimental conditions at the high temperatures. In all experiments it was found that the decomposition of nitric oxide on the various oxide catalysts followed zero-order kinetics. When CNO% (which is equivalent to the concentration of NO which has reacted) is plotted against the time of residence in the reaction chamber straight lines were obtained.' Other functions of CNO did not give straight lines. The rate equation then is

0.82 0.95 1.09 1.34 1.57 1.98

0.37 .48 .59 .78 1.00 1.25 1.69

0.83 1.21 1.96 3.57 5.40 10.1

0.098 .I41 .24 .41 .61 1.14 1213

1313

0.135 ,279 ,414 .648 1.085

0.71 1.39 3.16 5.90 8.64

1013

1113

0.042 .061 .147 .519 .669

0.097 .216 .462 .848 1.247

0.83 1.51 3.78 10.8 17.0

0.015 .018 .044 .I28 .188

I n Figs. 2 and 3 the concentration of NO2 formed (which is equivalent to the moles per liter of NO which has decomposed) is plotted against residence time. The straight lines for &O3 and ZrOz show that the rate of reaction is a constant, independent of the concentration of nitric oxide and hence the reaction is zero-order. Figure 4 in which log k is plotted against 1/T gives the Arrhenius activation energy for the catalytic decomposition on A&OS and ZrOz.

JAMESM. FRASER AND FARRINGTON DANIELS

218

Vol. 62

each oxide is given in terms of the activation energies, obtained from graphs similar t o those of Fig. 4, and the pre-exponential terms or frequency factors.

-I .

-J

a W

TABLEI1 HETEROGENEOUS DECOMPOSITION O F NITRIC OXIDE

2

~ U M M A R I Z E D KINETIC DATAFOR

dH

k =

AlzOs Temp.,

m

O K .

0

x

28.2 x 18.5 x 4.57 x 1.35 x

1313 1233 1128 1048

' N

P

0

k =

CaO Temp., OK.

0

2

4

6

TIME (SEC.1. Fig. 2.-Decomposition of nitric oxide on AlzOs. Zero order constants klo4ao= 28.2 X ksso" = 18.5 X kst,bo = 4.57 X 10-4; k?no = 1.35 X mole 1.-l see.-'.

-

w

-I

0

ZrOp Temp., OK.

I m

0

k

N0.4

0

z

0

k

ZnO Temp., OK.

0 0

8 12 16 20 Time (sec.) Fig. 3.-Decomposition of nitric oxide on ZrOz. Zero order constants klo*oo = 2.17 X h 4 0 " = 1.47 X leproo = 3.95 = 10-6; k740" = 1.14 x 10-6 mole 1.-1 sec.-l.

10-4 10-4 10-4 10-4

(mole 1.-1 sec.-1) Zero-order rate constant k (mole I.-' sec.-1)

3.48e-19*mO/RT

10-3 10-3 10-4 10-4

(mole 1.-1 seo.-l) Zero-order rate constant k (mole 1. -1 sec. -1)

= ll.Ze-2'!800/RT

(mole I.-' see.-') Zero-order rate constant k (mole 1. - 1 see. - 1 )

= 4.9e-319O/RT

2.30 x 10-6 1.14 X

1313 1213

4

,

2.17 x 10-4 1.47 x 10-4 3.95 x 10-6 1.14 X

1313 1213 1113 1013

X

10-4

10-4 10-4

(mole1.-1 sec.-1) Zero-order rate constant k (mole 1.-1 s m - 1 )

2.45 x 1.33 x 7.30 x 2.44 x

1313 1213 1113 1013

I2

Id

k =

Gal03 Temp., OK.

10-4

43.6e-9V'00jRr

56.2 x 25.2 x 15.7 x 5.2 x

1313 1218 1123 1033

IO

8

5 8 9 e - 3 1 E 0 0 / R T (mole 1.-1 seo.-l) Zero-order rate constant k (mole 1.-1 sec.-I)

Measurements were made also on Crz03, Ti02 and Fe203,but the decomposition of nitric oxide was so slight that the accuracy was poor and the data are not recorded here. At 1313"K., C1-203 (mole 1.-' set.-'); gave a value of k of 3.2 X TiOz about 4 X lo+; and Fe@3 3.7 X The Fe2O3gave a very lorn activation energy of 16,000 calories per mole, but at these temperatures a chemical action apparently occurred. The catalyst changed color and shrank in size. Surface Areas.-Catalyst surface areas were determined by nit,rogen adsorpt)ion and the Brunauer, Emmet,t, and Teller (BET) theoryg as shown in Table 111. The surface mens are small because the oxides

2 1

TABLE TI1 CATALYST DAT.4 7

8

9

IO

lo4/ T, Fig. 4.-Activation energy for catalytic decomposition of NO on AlzOs and ZrOr. k.41~0, = 580e-31W'ffTmole 1.-' sec.-l; kZpo = ll.2e-27,800'RT mole I.-' sec.-l.

Table I1 gives the values of the zero-order rate constants k a t various temperatures for other oxide catalysts which were studied. The value of k for

Catalyst

AlzOz CnO Cr20s ZrOz GszOs ZnO Fez03

Total wt. of catalyst used (300 pellets) (P.)

30.105 16.028 59.475 39.701 12.390 84.425 41.040

Area per g. of catalyst (m.?

10.2 7.9 1,55 2.81

4.00 0.87 1.54

Total area

of catalyst

(m.2)

307 127 92.4 111.6 50.6 30.0 63.3

Feb., 1958

HETEROGENEOUS DECOMPOSITION OF NITRIC OXIDE WITH OXIDECATALYSTS

have been partially sintered by heating to the highest temperature used in the experiment before determining the areas. Discussion The rate of decomposition of NO on Ti02 was low even at the highest temperature used in this work (1313°K.). At low temperatures the concentration of NO, formed was too small to measure accurately and lower flow rates could not be achieved in the apparatus used. The same difficulty occurred in some of the other experiments, especially with Fez03. Whenever the concentration of NO2 was less than 0.05 X loF3mole/liter, the accuracy of the spectrophotometric method was inadequate. Experiments with these poor catalysts, however, show that the decomposition on the walls of the Vycor reaction chamber is negligible. The fact that the catalytic decomposition of NO on these oxides is zero order can be explained in terms of the Langmuir adsorption isotherm. If the rate of reaction is proportional to the fraction of the surface covered by gas molecules, then

- derio = at

k’lCNo

I

+

k’’CNO

where k is a constant, and IC” is a constant related to the rate of adsorption and desorption. If the concentration of the reacting gas is high enough, 1c”cNo becomes large compared to 1 and the rate of reaction reduces to which is the expression for zero-order kinetics. It is concluded, then, that the concentration of NO was high enough and that the active sites on the surface of the catalyst were always completely occupied. At lower concentrations of NO the reaction rate would undoubtedly “fall off” and give a higher order. The mechanism for the decomposition Of No On these surfaces probably involves-two adjacent surface sites with NO molecules adsorbed on them and the reaction is thus a bimolecular process. A unimolecular process involving only one molecule of nitric oxide adsorbed on one surface site would have t o desorb nitrogen atoms and this process would require a much greater activation energy. Interaction between two adjacently adsorbed NO molecules seems more reasonable since nitrogen would then be desorbed as diatomic molecules. Since interaction between two like molecules adsorbed on adjacent surface sites is proposed, the velocity of the reaction would be proportional to the square of the fraction of the surface covered by reactant molecules but a t sufficiently high concentrations, the equation reduces to a zero-order equation. It is concluded that bimolecular decomposition of NO a t adjacent adsorption sites is a more likely mechanism than unimolecular decomposition and that this is not in conflict with the zero-order results obtained experimentally. I n comparing the kinetic data with the surface areas the activation energies, the rates of reaction per gram of catalyst, and the areas per gram of catalyst are to be considered. Since the reaction tube was always packed with 300 pellets of uniform size,

219

the weights of the catalysts were different and the rates per gram are probably more significant. Table I V lists the rates of reaction per gram of catalyst, the areas per gram of catalyst, and the activation energies. TABLE IV RATESOF DECOMPOSITION OF NO PER GRAMOF CATALYST AT 1313’K., SURFACE AREAS A N D ACTIVATIONENERGIES

Catalyst

Ga203 A1203

CaO

ZrOz Ct-203

Fez03

ZnO

Rate of dec., moles 1. - I ?,eo.-’ per g. of catalyst, x 108

Surface area, m.2 per g. of catalyst

Activation energy kcal. mole -1

108 94.8 32.3 5.46 5.38 0.85 0.668

4.09 10.2 7.9 2.81 1.55 1.54 0.87

19.2 31.6 23.0 27.8 40 t o 60 16 31.5

The relative activities of catalysts depend on the number of active sites on the surface per gram of catalyst and on the relative activities of the sites. The activities of catalysts would not be proportional to the surface areas given above because the activity of a catalyst depends also on the fraction of the surface area which is active. A catalyst may have a large surface area but only relatively few active sites. However, catalysts which are similar or differ only in the methods of preparation or activation can a t times be compared on the basis of surface area. If one assumes that the fraction of the total surface area which is active is.approximately the same for two catalysts then a measure of their surface areas can be used to determine which catalyst is better for a given reaction. The activation energy of a surface catalyzed reaction as compared t o the activation energy of the uncatalyzed reaction is also a measure of the value of a catalyst. In Table IV the metal oxides are listed in order of decreasing rate of reaction per gram of catalyst at 1313°K. It will be noticed that in general the surface areas decrease from Ga203to ZnO with the exception that Gaz03has the third largest surface area. The activation energy associated with Ga203 however is appreciably lower than the activation energies associated with the other oxides with the exception of Fe2O3which is unique as shall be discussed presently. The activation energies associated with A1203 and ZnO are the same while the rate of reaction on is about 140 times as fast as the rate on ZnO. Since the activation energies are the same, the difference in rates of reaction is attributed in part t o the fact that A1203 has about 12 times more surface area than does ZnO: The pre-exponential factors in the Arrhenius expression for the rate constant given in Table I1 can be associated with the number of adsorbed molecules according to the theory of absolute rates of surface reactions. It can be shown that the rate law for zero-order kinetics is =

C, kT h

e-EIRT

(12)

where C, is the concentration of adsorbed molecules and the other terms have their usual meanings.