From quantum-state-specific dynamics to reaction rates: the dominant role of translational energy in promoting the dissociation of D2on Cu(111) under equilibrium conditions

摘要

Faraday Discuss., 1993,96, 17-31 From Quantum-state-specific Dynamics to Reaction Rates : The Dominant Role of Translational Energy in Promoting the Dissociation of D2 on Cu(ll1) under Equilibrium Conditions Charles T. Rettner", Hope A. Michelsent and Daniel J. Auerbach IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, CA 95120-6099, USA We have calculated the rate of adsorption of isotropic D, gas on a Cu( 11 1) surface, using recently determined differential adsorption probabilities, as a function of translational energy, angle of incidence, and surface temperature for molecules in each vibrational-rotational state. If the D, gas is at the same temperature, T, as the surface, the mean probability of dissociation per collision, (So), is calculated to increase rapidly with temperature.Arrhenius plots of (So) us. 1/T are in good qualitative agreement with measurements for hydrogen dissociation on Cu, but display a distinct curvature over the range 300-1000 K. A detailed analysis of this temperature dependence 'reveals that the increase in (So) with T is due almost entirely to the increase in translational energy of the incident molecules. Increases in the populations of vibrationally or rotationally excited molecules are relatively unimportant, as are the changes in the adsorption with surface temperature. 'The dissociative chemisorption of hydrogen on copper surfaces has been the subject of considerable interest and study for at least 150 years.' Since the work of Taylor in 1931,2 it has been generally agreed that this process is highly activated. Measurements of the rate of dissociation of hydrogen and deuterium on copper films,7-'' f0ilS,9*12--14 and single crystal surfaces' '-' have typically yielded activation energies in the range 0.6 amp; 0.4 eV.Selected values of the activation energies are given in Table 1 and Fig. 1, beginning with the result of Melville and Rideal who made the first measure- ments yielding an activation energy for adsorption in this ~ystem.~ More recently, the hydrogen/copper system has been the subject of many dynamical studies that have sought to provide an understanding of the dissociation mechanism in terms of specific molecular motion^.''*'^ Molecular beam studies have shown that dissociation can be promoted with both translational and vibrational consistent with calcu- la ti on^.^^-^^ Recent desorption experiments have determined the effects of surface and rotational rn~tion~~'~~ on this process.Since one goal of dynamical measurements is to provide a detailed understanding of reaction rates, it is appropriate to reconsider the kinetic measurements on the hydrogen/ Cu system in terms of these new quantum-state-specific results. The question we address is this: How does thermal excitation lead to an increase in reaction rate? We find that by far the most important factor is the increase in the population of the high-energy tail of the Maxwell-Boltzmann distribution of molecular velocities. This same basic question t Present address : Harvard University, Department of Chemistry, 12 Oxford Street, Cambridge MA 132138, USA.17 18 year 1936 1948 1948 1949 1954 1956 1956 1968 1971 1972 1974 1987 1990 1992 Translation Energy Eflects on Dissociation of D, Table 1 Activation energies measured for hydrogen adsorption on copper authors and method Melville and Ridea13 pressure change Rienacker and Sarry7 p-H, conversion Kwan and Izu8 pressure change Kwan49' pressure change Mikovsky et a!.' isotope exchange Rienacker and Vorm~m'~ p-H, conversion Eley and Rossingtong p-H, conversion Volter et al." p-H, conversion Cadenhead and Wagner' surface oxide reduction and isotope exchange Alexander and Pritchard' ' surface potential change Kiyomiya et aL6 isotope exchange Gabis et uE.'~ permeation rates Campbell et surface oxide reduction Rasmussen et al.' temperature-programmed desorption ~~ ~ E,/ev" 0.85 0.54amp; 0.02' 0.86 0.86 1.oo 0.48 f0.03 0.42 f0.05 0.45 amp; 0.01 0.73 +_ 0.09 0.39 0.58 0.37 f0.06 0.53 f0.06 0.61 f0.004 0.62 k0.06 0.50 f0.013 surface T/K powder 345-443 film 623-893 film 573-673 powder 573-673 foil 583-623 foil 623-893 film 373-573 foil 353-453 (111)face 763-853 (100) face film 521-543 film 242-337 powder 3 13-363 foil 790-1020 (110) face 473-723 (100) face 21 8-258 a 1 eV = 23.1 kcal mol-'.'Where a range of values was given, error bars indicate range.25 1.c 20 0.f c I 15 5, 0.E -E---. IULu" 10 30.4 0.2 5 0.c 0 1930 1950 1970 1990 year Fig. 1 Summary of selected activation energies reported for the dissociation of hydrogen on Cu. See Table 1 for references. C. T. Rettner, H. A. Michelsen and D. J. Auerbach 19 has recently been addressed by Campbell et a1.'6,34who reached a qualitatively similar conclusion. These workers measured the rate of the dissociation of H, and D2 on Cu(ll0) in a 'bulb' experiment using a buffer gas to control which degrees of freedom became equilibrated to the hot surface temperature prior to collision. From the manner in which the dissociation rate increased with buffer-gas pressure, they concluded that translational energy is the most effective in overcoming the activation barrier.2. Methodology 2.1 Arrhenius Expressions We are concerned here with the dissociative chemisorption reaction D,(g) + Cu( 11 1) +2D(ads)/Cu( 11 1) (1) in the limit of low surface coverage. The rate of adsorption is given by where kad,(T) is the rate constant for adsorption, and the square brackets indicate number density. It is generally assumed in kinetic studies that the temperature depen- dence of this parameter can be described by the Arrhenius relationship kads(T) = A exp(-Ea/kE3 (2) where E, is the activation energy for reaction, A is the pre-expcnential factor, and kB is the Boltzmann constant. Dynamical studies, in contrast, are concerned with the determination of adsorption probabilities as a function of the collision conditions.We have recently shown3, how the dissociation of D, on Cu( 111) depends on kinetic energy, Ei, incidence angle, Oi, rota-tional state, J, vibrational state, u, and surface temperature, T,. Using this information, it is possible to calculate the mean adsorption probability per collision with the surface, (So), averaged over all of the distributions relevant to the incidence conditions, Here So refers to the dissociation probability in the low coverage limit. The rate constant for adsorption is then related to (So) by kads( = kcod T)(SO( (3) The collision rate of the gas with the surface at temperature T,kc,,,, is given by k,,,(T)= ill = (kB 7'/2nm)1/2 (4) where rn is the molecular mass. Recent studies of the temperature dependence of hydrogen dissociation on Cu surface^'^*'^ employed a variation on eqn.(2) based on (So) rather than kads. These workers assumed (So( T))= A' exp( -E'amp;, 7') (5) The activation energy for adsorption based on this equation is slightly different from that based on eqn. (2). In one case the activation energy is obtained from the slope of a plot of ln(kads) us. 1/T; in the other it is obtained from the slope of a plot of ln((S,)) us. 1/T. Clearly both plots cannot be linear. The expressions for kads based on eqn. (2) and (5)cannot both hold since they differ by a factor of The activation energy deduced from the slope of a plot of ln((So)) us.1/T will differ specifically from that from a plot of ln((kads))us. 1/T according to E, = EL + k, T/2 20 Translation Energy Eflects on Dissociation of D, In this paper, we chose to work with eqn. (5), for which EL and A' can have a simpler physical interpretation. In the following, we will drop the primes in referring to these quantities. Note, however, that for experiments with T in the range 300-1000 K, the factor of k, T/2 is relatively insignificant, amounting to only ca. 5 of E, for this system. 2.2 Evaluation of So The mean adsorption probability per collision with the surface can be evaluated from detailed knowledge of the variation of the adsorption probability with the collision con- ditions. Knowing the dependence of So on a given variable, one can calculate the average adsorption probability for molecules with the distribution of that variable appropriate to the gas/surface system.The overall mean adsorption probability is obtained by suitable averaging over all relevant variables. Only recently has sufficiently detailed information become available to perform such an analysis for the D,/Cu( 1 11) system, which is the only system for which such information is available at present. 2.2.1 Form ofS,(E,, Oi, u, J, T,)forDamp;u( 111) We have recently shown32 that the dissociation probability of D, on Cu(ll1) depends on Ei, Oi, J, u, and T,, in a manner that is well described by the function 0.75 0.60 0.45 -?. h0 0.30 0.15 bsol; bsol; bsol; bsol; bsol; 'bsol; bsol; 0.00 0 2 4 6 810 1 2 14 J Fig.2 Values of the translational threshold, E,, for the dissociation of D, on Cu(ll1) plotted against rotational quantum number, J, for vibrational states o = (a)0, (b) 1 and (c) 2. The lines represent quadratic fits to the points performed for each vibrational state. The dashed portions of the lines are extrapolated beyond the range of the data. The form of these fitted lines is given in eqn. (8). The data points are from ref. 32. C. T. Rettner, H. A. Michelsen and D.J. Auerbach where the effective translational energy E, = Ei COS" Oi (74 E, is the kinetic energy required for the adsorption probability to reach half its maximum value, and W is a width parameter that controls the steepness of the function.We have found that molecular beam adsorption results are best described by n = 1.8,35 but are also consistent with perfect 'normal energy' scaling, i.e. E, = En= Ei cos2 Oi. It has been found that E, varies considerably with both u and J but that W is essentially independent of J, with values of about 0.16 eV for u = 0 and u = 1, and 0.14 eV for u = 2 for T, = 925 K. Fig. 2 displays the Eq parameters determined in a recent The lines are quadratic fits to the points, with the form E,(u = 0) = 0.607 + 0.0235 J -0.00264 J2 eV (84 E,(u = 1) = 0.396 + 0.0167 J -0.00183 J2 eV (8b) E,(v = 2) = 0.218 + 0.0132 J -0.00171 J2 eV (W Fig. 3 displays the dependence of the dissociation probability on kinetic energy for molecules in different vibrational and rotational states for normal incidence.These curves are based on eqn. (7),plotted for = 925 K. The upper panel is for molecules in A 1.2 .g 1.0-.-a -$ 0.8 ti -0.6.-+--E 0.4 -0.2 0.0 -1.4 -I l'{'l'l't B translational energy/eV Fig. 3 A, Translational-energy dependence of the dissociation of D,(J = 2) on Cu(11 1) for mol-ecules with v = (a) 2, (b) l and (c) 0. B, Translational-energy dependence of the dissociation of D,(v = 0) on Cu(ll1) for molecules with J = (a)14, (b)12, (c) 0 and (d)4. Translation Energy EJgrects on Dissociation of D2 J = 2 with u = 0, 1, and 2. The lower panel is for molecules in u = 0 with J = 0, 4, 12, and 14. The curves in Fig. 3 have been normalized to unity for clarity of presentation.In fact the level at which the functions saturate at high kinetic energy is given by the parameter AO(v,J) in eqn. (7). For a given u, these parameters are found to be essentially indepen- dent of J, but vary with vibrational state. The values for u = 0, 1, and 2 are found to be in the ratio 0.54: 1.00 :0.77.32 Considering results of molecular-beam adsorption mea- surements, where absolute adsorption probabilities are obtained, we estimate that these numbers are about a factor of two too large. For the purposes of this paper, therefore we assume Ao(O, J) = 0.27 (94 Ao(1, J) = 0.50 (9b) A0(2, J) = 0.38 (94 We have recently shown that changing surface temperature is associated primarily with changes in the width parameters.We believe, therefore, that these values of E, can be considered independent of K. It has been shown3, that the dependence of the width parameters on T, can be described by W(U,J, K) = W(U,J, T, = 925K) + C,(T,-To) (10) where C, is equal to 5.6 x eV K-l, and To = 925 K. 2.2.2 Averaging over Incidence Conditions From the above expressions, we are able to obtain the dependence of the dissociation probability of D, on Cu(ll1) for all quantum states and kinetic energies relevant to a system at temperatures between about 100 and 1000 K. Given this information, it is a simple matter to evaluate adsorption probabilities for D, gas on a Cu(ll1) sample for conditions appropriate to a typical lsquo;bulbrsquo; experiment.In such experiments, the crystal is exposed to gas that impinges randomly with a Maxwell-Boltzmann kinetic energy dis- tribution and a Boltzmann distribution of internal states. The effective translational, vibrational, and rotational temperatures will generally be the same as the surface tem- perature because of energy transfer from the surface to the gas phase. If the pressure above the surface is too low, however, this may not be the case. In the low-pressure limit, where the mean free path of the molecules is greater than the chamber dimensions, the effective temperature of the incident molecules may be that of the chamber walls or other objects in the chamber such as filaments. With increasing pressure of D, or of additional inert lsquo;bufferrsquo; gases, translational energy will be the first degree of freedom to come to equilibrium with the surface temperature, followed by rotational and then vibrational energy.rsquo; 6334 This order reflects the fact that translational-energy transfer is more efficient than rotational-energy transfer, which in turn is more efficient than vibrational-energy transfer.We assume that the probability distributions for the trans- lational, vibrational, and rotational degrees of freedom can be characterized by tem- peratures lsquo;T;, Ti,,,and Tot,respectively. Specifically, the adsorption rate per collision is given for each u-J state by the ratio of the probabilities for adsorption and for collision with a Cu( 11 1) surface placed in isotropic D, gas. Thus r2LaSo(Ei,4, V, J, 7JE exp(-E/k, ?;) dE cos 6i sin Oi doi GO(U, J, T,)= (11)$rsquo;E exp(-E/kB7J dE cos 6i sin Oi dei C.T.Rettner, H. A. Michelsen and D. J. Auerbach The overall mean adsorption probability per collision (So(TJ) is then (So(v, J, T,)) averaged over u and J, so that The statistical weight of the v-J state, N(u,J), is given by N(u,4 = ~XP(EJkB T,id2J + l)~,~XP(--E,/kB lsquo;Tot) (13) where E, and E, are the vibrational and rotational energies, respectively, and the term gnis the relative nuclear spin degeneracy, which for D, is equal to 2 for even J and 1 for odd J. For the special case of true lsquo;normal energyrsquo; scaling (n = 2), which is a good approx- imation for the D,/Cu(lll) system,35 it is possible to obtain an analytical result for {So(Eo, W, T)),the mean adsorption probability per collision for a species with an adsorption function of the form of eqn.(7). Multiplying this adsorption function by the flux-weighted probability distribution function for En, P(E,, T)dE, cc exp(-E,/kB T) dE, (14) integrating over all positive En,and normalizing to the collision probability, we obtain -2 2k~TA,{1 + erf($) + exp(s)e~p(~)~1 -erf(amp; 51 (1) For Eo 24W, this expression approximates to 2 2k~T 2k,T W This equation gives identical results to the full numerical integrations over Oi and Ei using n = 2 for the work presented here. Numerical integration has the advantage of being applicable to an arbitrary dependence on Oi, but requires considerably more com- putation time.In order to use eqn. (15) to evaluate {So), it is still necessary to average over quantum states. 3. Results and Discussion Adsorption probabilities have been calculated for D, on Cu(ll1) by substituting the expressions for the quantum-state-specific dissociation probabilities given in eqn (7)-( 10) into eqn. (11)-(13). Fig. 4 displays the predicted variation of the mean adsorption prob- ability per collision with temperature. Results are plotted as ln((S,)) us. 1/T. They were obtained assuming full equilibration of the gas at a temperature equal to the surface temperature. The approximate linear form of Fig. 4 indicates that the adsorption rate approximately follows Arrhenius behaviour, eqn. (5),so that a linear fit to these results yields an activation energy, E,, of 0.44 eV and a pre-exponential, A, of 6 x lop2.This fit is shown as a dashed line on the figure. Since the plot is not strictly linear, these param- eters depend on the temperature range. Between T = 500 and 1000 K we obtain E, FZ 0.51 eV and A FZ 0.2, whereas between 300 and 400 K we obtain E, = 0.36 eV and A M 4 x lop3.Clearly the values of E, are consistent with the range of values reported previously for the H,/Cu system, as summarized in Fig. 1 and Table 1. (A discussion of the isotope effect is given in Section 3.5.) Calculations performed using the measured E, Translation Energy Efects on Dissociation of D, TIK 1000 600 400 300 I I I I -8 -1 0 -1 2 h A0 !? -14 --1 6 -1 8 -20 bsol; I I I I I I 1.0 1.5 2.0 2.5 3.0 3.5 103 KIT Fig.4 Plot of ln((So)) us. 1/T for Damp;u( 111).The (So) values were calculated as described in the text. The dashed line is a linear fit, which gives an effective activation barrier of 0.44 eV and a pre-exponential of ca. 0.06. and W parameters, rather than using eqn. (8) and assuming fixed W for each 0, gave very similar results. 3.1 Activation Energies We now address the question of what determines the slope of such a figure. How should we interpret the phenomenological activation energy? Upon raising the system tem- perature, there are several different important effects that serve to change (So). (1) Raising the system temperature increases the number of molecules with high translational energies.(2) The width parameters, W(u),increase with increasing T, according to eqn. (lo), thereby increasing the probability for dissociation for molecules with En Eo . (3) The rotational-state distributions also change with increasing temperature, again following eqn. (13). Fig. 5 displays rotational distributions for D,(u = 0) for tem-peratures of 300, 600, and 1000 K. (4) The populations of vibrationally excited species increases exponentially with tem- perature eqn. (13). Fig. 6 displays the populations of D,(v = 1) and (v = 2) as a func- tion of temperature (solid lines) together with the (So) values (dashed line). 3.1.1 Role of Translational Energy While all these factors contribute to the overall temperature dependence to some degree, we have determined that the effect of temperature on the distribution of translational energies is by far the most important.One way to ascertain the relative importance on the (So) of the different factors is to calculate the overall temperature dependence of C. T. Rettner, H. A. Michelsen and D. J. Auerbach 0.4 t 0.4 1 . 0 2 4 6 8 10 J Fig. 5 Equilibrium rotational distribution of D, molecules at temperatures of (a) 1000,(b)600 and (c)300 K. These distributions are normalized to a sum of unity in each case. this probability with each of the separate variations with temperature suppressed in turn. We find that the calculated values of (So) are very sensitive to the translational temperature. For example, the lower dashed line in Fig.7 shows the result of holding ?; at 300 K. Comparing this with the result of the full calculation in which is equal to T at all temperatures (solid line), it is seen that holding ?; at 300 K gives (So) values that are up to a hundred times lower. We find that fixing ?; at yet lower temperatures further reduces (So) compared with the full calculation. 3.1.2 Eflect of the T, Dependence of the Width Parameter In contrast to the dramatic effect of holding constant, we find that suppressing the effect of increasing temperature on the other temperature-dependent terms is relatively weak. Fig. 7 indicates the effect of holding the width parameter, W,at the value that we estimate for T, = 0 K. Using eqn.(lo), we obtain W(T,= 0 K) = 0,109 eV, which leads to a reduction in (So) over the whole range of temperatures compared with the case where W increases with T. Using this width causes the slope for the range T = 500 and 1000 K to increase to 0.55 eV, compared to 0.51 eV for the full calculation. We conclude that changing surface temperature has a significant effect on the form of the ln((So)) us. 1/T curves. The origin of this effect is that the spread in velocities of the surface atoms, which increases with increasing T,, causes a lsquo;softeningrsquo; of the sharp adsorption us. energy function. Surface atoms moving away from the surface increase the effective translational energy of the gas-surface collision, thereby allowing adsorption to occur for translational energies below E, .The low probability of such surface motions is more than offset by the increase in So with translational en erg^.^'.^^ Translation Energy Eflects on Dissociation of D2 TIK 2000 1000 600 400 300 3.1.3 Role of Rotational Energy We have performed calculations with Totset to zero. In this case, all molecules are confined to the J = 0 state. Comparison of the resulting calculated variation of (So) with temperature with the dependence obtained with T,,, = T allows the effect of increasing rotational energy on (So) to be assessed. The result of this investigation is shown in Fig. 7. The solid line is the result of the full calculation presented in Fig. 4, while the upper dashed curve gives the result for Tot= 0 K.It is seen that the effect of lsquo;freezingrsquo; rotation at 0 K is to increase the (So) values. Further study of this behaviour reveals, however, that the primary effect of increasing Totoccurs for rotational tem- peratures up to 300 K. Further increasing T,,, has little effect on (So). The initial decrease with increasing Totfrom 0 K results from the fact that the E, values (Fig. 3) increase with increasing J at low J, reaching a maximum at about J = 5. The mean value of E, for Tot= 300 K is considerably higher than the value for J = 0. Considering Fig. 3 and 6 together, it may be apparent that raising Totfrom 300 to 1000 K has little effect on the mean E, value. This observation is consistent with the fact that (So) values calculated for Tot= 300 K are almost indistinguishable from the full calculation.Raising the rotational temperature of D, from 300 to 1000 K has essentially no effect on the mean adsorption probability per collision. 3.1.4 Role of Vibrational Energy This same type of analysis has been applied to assess the relative importance of vibra-tional excitation. Specifically, we have calculated (So) values with all molecules con- C. T. Rettner, H. A. Michelsen and D. J. Auerbach TI 1000 600 400 I I I -6 -a 2-10 02 --1 2 -1 4 -1 6 bsol; bsol; bsol; I I I Ibsol; I 0.0 0.5 1.0 1.5 2.0 2.5 103 KIT Fig. 7 Comparison of a number of different calculations of the temperature dependence of (So) for the DJCu(111) system. (b) is the result of the full calculation given in Fig.4.This is to be compared to (a) which was calculated while confining all molecules to J = 0; (c) which was calcu- lated while confining all molecuIes to u = 0; (d)which was calculated using a W parameter appro- priate to 0 K; and (e) which was calculated while fixing the translational energy distribution to that for 300 K. fined to u = 0. The resulting temperature dependence of (So) is given by the dotted line on Fig. 7. For temperatures below ca. 500 K, the result are indistinguishable from the full calculation. Even at the highest temperature, the results are within ca. 30 of the full calculation. In order to gain a better understanding of this result, we have calculated the contribution to (So) of molecules in each vibrational state.The results are given in the upper panel of Fig. 8. It is seen that the contribution from u = 0 molecules domi- nates for all temperatures. This result can be understood as follows. The Eo values for u = 0 molecules are higher than those values for u = 1 and u = 2, but by an amount that is only ca. 60 of their respective vibrational energies. Thus the increase in the mean dissociation probabilities per collision for vibrationally excited species does not make up for their low populations. The lower panel in Fig. 8 shows (So(u)) values calculated for each u. The values for u = 2 are much higher than for u = 1, which are in turn larger than for v = 0. At 500 K, for example, (S0(2)) and (So(l)) are ca. 7 x lo3 and 3 x lo2 times greater than (So(0)), respectively, but the populations of u = 2 and u = 1 mol-ecules are ca.2 x lo7and 5 x lo3 times lower than for u = 0. The conclusion that adsorption occurs predominantly through vibrational-ground- state species under equilibrium conditions is an important one. There has been consider- able debate recently about the relative importance of vibrational and translational energy in promoting dissociation in molecular beam experiments. 9-29 In this case it is now generally accepted that adsorption at low beam energies is dominated by vibra- tionally excited molecules. Indeed, measurements for En up to ca. 0.5 eV can be inter- preted entirely in terms of the adsorption of vibrationally excited species.' The key difference in the beam case is that the spread of translation energies is reduced in the Translation Energy Eflects on Dissociation of D, TIK 1000 600 400 300 I I I I -4- A- 2-6 -8 v 0 -10C- C -16 -18 -20 -2 -4 AL v O-8 = -10 -1 2 -1 4 0 1 2 3 103 KIT Fig.8 A, Contributions to the overall mean adsorption probability per collision, (So), due to D, molecules in different vibrational states, (a)all vibrational states, (b)u = 0, (c)v = 1 and (d) u = 2. B, Values of the mean adsorption probability per collision for molecules in each vibrational state, (a)v = 2, (b)u = 1 and (c)u = 0. expansion process, but the vibrational distribution remains close to a Boltzmann dis- tribution at the nozzle temperature, Tnoz.Even though the molecules attain translational energies of over (5/2)kB KO=,the very high energy tails are not populated.3.2 Relationship between E,, E, and W The dependence of the form of the ln((S,)) us. 1/T curves on T, discussed above pro- vides an important clue to understanding the magnitude of the activation energy deter- mined for D,/Cu(lll) or a similar system. We have established that the primary contribution to the increase in (So) with increasing temperature is from the population of the high-energy tail of the translational-energy distribution. For a step function form for So(amp;,),we would then expect to obtain activation energies close to the E, values for the low rotational states of D,(u = 0). Setting the width parameter to 0 eV, we find that this is indeed the case.For this condition, we obtain an activation barrier of 0.65 eV, which is essentially identical to the mean value of E, averaged over population at 700 K. This result is consistent with eqn. (15b), which reduces to eqn. (5) when W = 0 eV, with A = A, and E, = E,. (In obtaining the mean value of E,, including vibrationally C. T. Rettner, H. A. Michelsen and D. J. Auerbach 29 excited states in the averaging lowers the mean by less than 1 compared to only averaging over the J states of u = 0. Moreover, the average E, value is relatively insensi- tive to the rotational temperature, falling by ca. 5 meV as this temperature is lowered from 1000to 300 K.) We conclude that the observed activation barrier is lower than the mean E, values because S,(E,) rises considerably before E, = E, .Considering the temperature range 500-1000 K, the activation barrier is reduced from 0.65 to 0.51 eV by using a width parameter of ca. 0.15 eV rather than 0 eV. Increasing this width to 0.2 eV causes the calculated activation energy to fall to 0.31 eV. The measured activation energy is thus quite sensitive to the magnitude of the width parameter. The specific dependence of the activation energy of the width parameter derives from the particular form of eqn. (7), which expresses the dependence of So in translational energy. Fig. 9 expands on this point. This figure shows plots of ln((S,)) us. 1/T for different values of W.Here we set T,,, = Ti,,= 0 K so that the temperature dependence reflects only changes in the translational-energy distribution of the incident molecules.The width parameter was set to be a fraction of the single E, parameter, that for D,(v = 0, J = 0). The curves are for E,/W = 2, 3, 4, 5, 10, and 100, labelled (a)-(f), respectively. The true results for the D,/Cu(lll) system are very close to curve (d), for W = E,/5, with a mean slope of 0.72 E,. Only when W is small compared with E, does the slope closely approach E,, such as for curve (f),which is for W = E,/100.For this case the slope is 99.8 of Eo . 3.3 Pre-exponential Factors Since plots of ln((S,)) us. 1/T are only approximately linear, the calculated pre- exponential factors vary with the temperature range under consideration, even more so TIK 1000 600 400 300 -6 -8 2 -10 0 v c bsol;bsol;bsol; c--1 2 -1 4 -1 6 1.0 1.5 2.0 2.5 3.0 3.5 103KIT Fig.9 Effect of changing the width parameter relative to the E, parameter on calculated plots of ln((S,)) us. 1/T.The curves show results for E,/W = (a)2, (b)3, (c)4, (d) 5, (e) 10 and (f) 100. In all cases E, = 0.607 eV. Only when E, W are the plots linear. 30 Translation Energy Eflects on Dissociation of D, than the activation energies. Examination of Fig. 7 and 8, however, indicates that the various plots converge as 1/T tends to zero on an intercept of ln((S,)) z -1 1. Again, we can understand this behaviour by examining the results of selected calcu- lations. In particular, we find that setting rot= Kib= 0 K and W = 0 eV, we obtain the expected result of a linear plot with slope of 0.607 eV =E,(u = 0, J = O) and intercept 0.27 =A,(v = 0, J = O).The lnA,(u = O) value therefore approximately gives the ln((S,)) intercept, which for a linear plot is equal to the pre-exponential. The curvature in the plots of ln((S,)) us. 1/T derives largely from the fact that W is not equal to zero (see Fig. 9). 3.4 Effect of Surface Roughness It has been stated that the dissociation of D, on Cu( 11 1) scales quite accurately with En, i.e. with Ei cos2 Oi. This scaling law is a good approximation only for flat surfaces, however. For a microscopically rough surface, results would be expected to scale simply with Ei. In addition to changes in the adsorption rate due to the fact that such samples would most likely be polycrystalline, we believe that the adsorption rates would be increased by the change in scaling law. Changing eqn.(7) from a dependence on En to one on Ei causes an increase in rate of about an order of magnitude for the parameters relevant to the D,/Cu(lll) system. Moreover, the activation energy is reduced by ca. 10 by this change. This effect should be considered in comparing results on single crystals with those on powder samples, for example. 3.5 Differences between D,/Cu(lll) and H,/Cu(lll) From what we have learned about the dependence of (So) on the collision conditions for D,/Cu( 11 l), we can estimate how we expect the H,/Cu(l 11) system to behave.Con- sidering the four temperature-dependent effects listed in Section 3.1, we can dismiss the rotational and vibrational effects at once. Rotational effects have been seen to be weak, and should be very similar for H, and D,. The vibrational excitation energy of H, is even higher than for D,, so there will be very little contribution to (So) from vibra- tionally excited species. Molecular-beam adsorption measurements indicate that the E, values are slightly lower for the H, case,' which would lead to correspondingly lower activation energies. The W parameters may also be smaller, however, particularly for low T,. Lower widths would result in an increase in activation energy. In conclusion, we do not expect the activation energy of the H2/Cu( 11 1) system to be substantially differ- ent from that for D2/Cu( 11 1).4. Summary and Conclusions We have calculated the temperature dependence of the mean probability of adsorption of isotropic D, gas on a Cu( 1 11) surface using quantum-state-specific adsorption prob- abilities. We have concluded that: The primary reason for the rapid increase in the adsorption rate with temperature is that increasing temperature causes an increase in the number of molecules with high translational energy. Plots of ln((S,)) us. 1/T are not necessarily linear, but may be nearly so over a limited range of temperatures. This curvature is expected, based on the smooth form of the dependence of So on En in the region of the threshold energy, E, . Linear plots with E, z E, are only obtained in the limit of E, $-W.The slopes of such plots give activation energies of ca. 0.5 eV, in agreement with measurements on the H,/Cu system. C. T. Rettner, H. A. Michelsen and D. J. Auerbach The effects of vibrational and rotational energy are relatively unimportant. We believe that similar results will hold for the H2/Cu( 111) system. We thank C. T. Campbell and R. N. Zare for useful discussions. We would also like to thank the ONR for partial support of H. A. M. under grant #NOOO14-91-J-1023. 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