Vibrational Excitation from Heterogeneous Catalysis
composition (S/D), (ii) decomposition-decomposition (Dl/D2), and (iii) stabilization-stabilization (Sl/S2). Both S / D and D1/D2 are appropriate for single and multichannel decomposition systems and are dependent on the properties of the transition state and reactant (vibrational frequencies and thermochemistry). S/D studies are most sensitive for large values of the excess energy and in the lower pressure region while the D1/D2 systems provide information at moderate and high pressure for low values of the excess energy, The S1/S2 technique requires a reversible isomerization and at low pressure is sensitive (exoergicity and vibrational to reactant properties (Moo frequencies); the sensitivity increases as Eo for isomerization decreases and as laEooIincreases. In fact, the S1/S2 competition provides direct values of ( a independent ) of model type for low levels of excitation that have not been availablle by the other techniques. At higher pressures the properties of the isomerization transition state are important. Table I1 summarizes the features of the three types of' competitive systems that are amenable to experimental study in external activation systems. The application of these models to real systems is generally straightforward. However, differences in detailed behavior of the different model types are reduced since a mono-energetic input is not realized in practice. This is due to the "extra" averaging which is present in systems with a distribution of energies. The S / D and D1/D2 systems have been demonstrated previous1y.l For the S1/S2 example (the isomerization of pentyl radicals) the low pressure intercelpt ratio should be attainable at =lo-* torr. We are presently deconvoluting such experiments. The added complication is that the radicals can also decompose so that the amount of S1 and S2 at low pressures is small, thus analytical problems arise. For this system both lIl/D2 and S1/S2 information will provide unique data. Nonetheless, S1/S2 systems can provide information on energy transfer for low levels of excitation that cannot kie realized by S / D or D1/D2 studies.
Acknowledgment. Monies from the University of Iowa Graduate Colllege for computer work performed at the Weeg Computing Center were greatly appreciated, along
The Journal of Physical Chemistry, Vol. 83, No. 8,
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with a University of Iowa Development Award for the summer of 1977. Appendix The following vibrational frequencies (cm-l) and degeneracies (enclosed in parentheses) were used in calculating rate constants and internal energy eigenstate densities as a function of energy. These frequencies were calculated by an algorithm developed in our laboratories.12 Pentyl-1 Radical. 2942(6), 2864(5), 1461(6), 1375(3), 1336(1),1287(3), 1240(1),1164(2), 1052(2),953(3), 870(1), 742(2), 402(2), 215(1), 188(1), 103(2), 88(1). Pentyl-2 Radical. 2950(6), 2864(5), 1462(6), 1375(3), 1336(1), 1287(3), 1164(2), 1043(3),953(3), 870(1), 742(2), 402(2), 215(1), 188(1),150(1), 88(2). Pentyl-Dl Decomposition Complex. 2953(7), 2866(4), 1457(6),1359(3),1287(3),1178(2), 1000(5), 742(2), 644(2), 389(2), 209(2), 100(1), 94(1), 50(1). Pentyl-D2 Decomposition Complex. 2966(8), 2872(3), 1462(6),1362(4),1301(2),1164(2),1000(4),900(1), 742(2), 649(2), 389(2), 209(2), 150(1), 94(1). Pentyl-1 /Pentyl-2 Isomerization Complex. 2950(10), 1902(l), 1606(l), 1520(1),1465(l),1461(l),1456(1),1407(l), 1366(1),1333(1),1287(1), 1258(1),1221(1),1162(1), 1144(1), 1088(1), 1037(1), 1029(1), 992(1), 932(1), 892(1), 883(1), 867(1), 841(1), 777(1), 583(1), 485(1), 409(1), 268(1), 209(1), 111(1),89(1). References and Notes (1) D. C. Tardy and B. S. Rabinovitch, Chem. Rev., 77, 369 (1977). (2) R. E. Harrington, B. S. Rabinovitch, and M. R. Hoare, J. Chem. Phys., 33, 744 (1960). (3) G. Khlmaier and B. S. Rabinovitch, J. Chem. Phys., 38,1692, 1709 (1963). (4) M. Hoare, J . Chem. Phys., 38, 1630 (1963). (5) D. C. Tardy, C. W. Larson, and R. S. Rabinovitch, Can. J . Chem., 46, 341 (1968). (6) C. W. Larson and B. S. Rabinovttch,J. Chem. fhys., 51, 2293 (1969). (7) W. P. L.Carter and D.C. Tardy, J . Phys. Chem., 78, 1579 (1974). (8) R. A. Marcus, J. Chem. Phys., 20, 359 (1952). (9) W. Forst, "Theory of Unimolecular Reactions", Academic Press, New York, 1973. (10) W. P. Carter and D. C. Tardy, J. Phys. Chem., 78, 612 (1974). (1 1) D. C. Tardy and B. S. Rabinovitch, J. Chem. Phys., 48, 5194 (1968). (12) W. P. L. Carter, Ph.D. Dissertation, University of Iowa, Iowa City, Iowa 52242, 1973.
Vibrational Excitation from Heterogeneous Catalysis George D. Purvis, 111, Michael J. Redmon, and George Woken, Jr." Chemical Physics Group, Battelle Columbus Laboratories, Columbus, Ohio 4320 1 (Received October 6, 1978) Publication costs assisted by the Air Force Office of Scientific Research
Classical trajectories have been used by numerous researchers to investigate the dynamics of exothermic chemical reactions (atom + diatom) with a view toward understanding what leads to vibrational excitation of the product molecule. Unlike these studies, we consider the case where the reaction is catalyzed by a solid surface. The trajectory studies indicate that there should be conditions under which considerable vibrational energy appears in the product molecules without being lost to the solid during the course of the reaction.
I. Introduction There have been experimental indications that, for atomic recombination on solid surfaces, energy is disposed of in highly specific ways.l?*For example, in recombination of oxygen on a variety of metal surfaces, the data indicated that most of the reaction energy was often deposited into the gas phase diatolmic molecule.2 This was in contrast 0022-3654/79/2083-1027$01 .OO/O
to conventional ideas that there was essentially equilibrium partitioning of energy between gas and solid for atomic recombination. Recent data by Halpern and Rosnerl clearly confirm the specific nature of energy disposal in heterogeneous reactions for recombination of atomic nitrogen on a variety of metals. The data are sufficiently precise and reproducible that some hypothesis about the 0 1979 American
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The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
chemical dynamics of recombination reactions on surfaces could be proposed. It had been known for some time that when molecules react in the gas phase, various modes of energy disposal (i.e., as vibration, rotation, or translation in the products) are not all e q ~ i v a l e n t .Whether ~ or not a large fraction of the reaction energy appears as product translation or vibration is known to be determined by the detailed energy contour along the reaction path, relative masses of the atoms, and other subtle features of the p ~ t e n t i a l . ~These experiments showed that adsorbed atoms under conditions of high reaction exothermicity can adsorb, react, and desorb and lose only modest energy (510%)into the substrate. A reaction occurring on a solid surface shares many of the same features. A reaction path can be constructed, the energetics along this path can be described, and the dynamics of the resulting reaction can, in principal, be studied. The sole complicating feature is the presence of a solid surface upon which the reaction occurs. Therefore, it is not surprising that heterogeneous reactions would exhibit preferred modes of energy disposal. Precisely how the solid surface complicates the dynamics of heterogeneous reactions is far from clear, but the fact that interesting dynamical effects exist in heterogeneous reactions seems beyond dispute. One of the most useful consequences of specific energy disposal in gas phase reactions has been the formation of vibrationally excited product molecules and resulting infrared chemical lasersa5 The question naturally arises whether reactions promoted by a heterogeneous catalyst can lead to equally useful vibrational population inversions. Various experimental groups1s2have been led to speculate that a large fraction of the available reaction energy does appear as product vibration, but the experimental support is extremely fragile. In analogy with similar studies for gas-phase dynamics, we have undertaken a series of computational studies attempting to model vibrational excitation in heterogeneous reactions. Our method for approaching this problem has followed closely the theoretical methods (Le., classical trajectories on a series of model potentials) whereby the behavior of gas phase chemical lasers was elucidated. Since these methods were productive for gas phase chemical lasers, there is reason to believe they will also lead to important insights into heterogeneous chemical dynamics and vibrational excitation. Because of this close analogy, we think it is productive to review briefly the history of gas phase chemical lasers to point out the reasons for our current studies and, by learning from history, we hope to indicate likely future directions for heterogeneous chemical dynamics. Polanyi and co-workers observed vibrational population inversion via infrared chemiluminescence in HC1 formed Clz. The possibility of confrom the reaction of H structing an infrared laser was explicitly discussed in a paper submitted for publication in 1960.6 An extensive series of classical trajectory studies on a model potential surface (LEPS) was undertaken in order to understand the conditions under which vibrational population inversion could be expected in exothermic reactions. The first results were submitted for publication in 1962,7'and a detailed discussion of energy release in exothermic reactions appeared in 1966.4 Various features in the potential energy surface in addition to exothermicity were shown to be necessary for efficient population inversions to occur. Acting on Polanyi's 1960 prediction, Kasper and Pimente16 produced the first chemical laser in 1965 using the H2/Clz reaction. The function of a heterogeneous catalyst is to alter the energy profile along the reaction
+
G. D. Purvis, M. J. Redmon, and G. Wolken
path to accelerate the rate of formation of products. For gas phase reactants and products the overall exothermicity cannot be affected by the catalyst. However such catalysts alter reaction rates drastically by altering the energetics of intermediate reaction states. Therefore, from atomdiatom collisions in the gas phase we know the importance of details in the energy path for determining final state populations and can ask, how will a catalyst affect these final state population distributions? Clearly, there must be an effect since the energy profile along the reaction contour changes, but what can be expected in detail? Following what was done for gas phase chemical lasers, we chose to use classical trajectories to attempt to elucidate these features in the gas-solid interaction, most likely to lead to population inversions. Don Bunker was one of the pioneers in classical trajectory methods and would not be surprised to find the method applicable, essentially unchanged, to the emerging area of dynamics of heterogeneous catalysis. In this paper, we present some recent results concerning chemical dynamics at the gas-solid interface via classical trajectories. This represents the culmination of a series of papers whose goal was to assess the possibility of obtaining vibrational population inversion by means of a heterogeneous chemical reaction. Therefore, in addition to reporting our recent results concerning the dynamics of adsorption, we will provide an overview and rationale for the recent work in this area. In order to model realistic chemical dynamics, a reasonable model potential is needed. We used a modified four-atom London-Eyring-Polanyi-Sat0 (LEPS) model potential to account for the interaction of a diatomic molecule with a solid s ~ r f a c e .In ~ our first studies,'O the surface was assumed to be rigid and provide simply a substrate upon which the reaction could occur. This is consistent with the simplest picture of a catalyst as providing merely a lower-energy pathway from reactants to products but (as a rigid surface) not adding or removing energy from the reaction. A very similar model potential was developed independently by Gelb and Cardilloll and used to investigate isotope effects for hydrogen, and cluster formations on a copper surface.12 Consistent with the rigid-surface model, we have assumed that the reaction energy to be partitioned in the product molecules must come entirely from the reactants rather than from the substrate. For efficient population inversion to occur, considerable energy must be available. Since catalysts are typically heavy metals, one does not expect a very efficient heat transfer between the surface and rapidly reacting molecules on the surface as an intermediate state in a fast reaction. Therefore, the source of the reaction exothermicity is assumed to be the reaction itself and the surface plays a minor role dynamically (but a major role in altering the energetics of the reaction path). This assumption has been tested numerically and found to be accurate to As discussed above, experimental results have appeared in which the fractional energy deposited in the solid has been measured. While the experiments are not precisely comparable with the present calculations, the data clearly show that strong chemical bonds are frequently formed without significant energy transfer to the solid.'J This is perhaps surprising since it is expected that energy transfer between the adsorbing species and the surface would be relatively rapid. However, the calculations reported here seem to indicate that two effects can come into play tending to cancel this effect. The primary effect is that the energy of chemisorption (kinetic and potential) is only a part of the overall
Vibrational Excitation from Heterogeneous Catalysis reaction energy. If most of the available energy comes in the recombination step, then energy losses to the solid may not be severe. Our calculations indicate that much of the reaction energy musrt, in fact, come from the atomic recombination for subrrtantial population inversion to occur, consistent with expectations. Also, many reactions occur under conditions of high coverage in which the reaction takes place with a direct collision with an adsorbed atom (Rideal mechianism) or after a few collisions with the surface. Under such conditions not much energy is expected to be transferred. Section I1 provides a description of our model for chemical dynamics a t the gas-solid interface, emphasizing the possibility of olbtaining vibrational population inversions from such a system. Section I11 discusses our most recent results\ for the dynamics of adsorption as related to the possibility of obtaining vibrational population inversions and connects these with previous results for the dynamics of lieterogeneous recombination. Conclusions are drawn and the possibility of heterogeneously catalyzed reactions leading to useful population inversions are discussed in Section IV. 11. Description of the Model We envision studying a reaction of the general form
+ 132 A2 + B2
A,
gas phase
catalyst
2AB(V,J)
BAB(V',J?
In practice, direct reaction l a frequently has a substantial activation barrier and correspondingly slow rate. The catalyst is chosen specifically to remedy this situation but, by altering the energy contours between reactants and products, the (V,J) distributions will also change. Conceptually, it is convenient to discuss catalytic reaction l b in two steps. In the first step, a diatomic molecule (A2or B,) collides with the solid surface and dissociatively chemisorbs. In the second step, the adsorbed atoms A and B encounter one another on the surface and desorb as AB. We do not consider the case in which a gas phase species adsorbs as an undissociated diatomic molecule and this species, rather than chemisorbed atoms, participate in the reaction. Nor do we consider direct reactions between gas phase A, and adsorbed B atoms (or the inverse). Both of these cases (particularly the second) could be important for practical catalytic systems operating at high pressure. However, the additional complications introduced into the reaction path by having to follow more atoms materially increase the computational effort. Since our understanding of the dynamics is rather crude, we believe it is justified to study the simplest realistic system. Hence, we consider reaction l b tlo proceed via dissociative chemisorption of both A and B, a subsequent encounter of the adatoms on the surface, and, finally, the desorption of AB. Such a model would be appropriate at low pressures and low coverages of adsorbed species and (obviously) for a system in which Az and B2 do dissociatively chemisorb. In summary, our two-step process is A2,B2(gas) A,B(chemisorbed atoms)
; z
AB(V,J)(gas) (2) The requirement that dissociative chemisorption takes place means step I should be exothermic for both A, and B2 As argued above, we require AB to receive its internal energy from the overall reaction exothermicity, rather than from the catalyst. Therefore, we have the further restriction that the overall reaction steps I + I1 be exothermic. The detailed procedure for modeling the overall
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
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reaction steps I and I1 is as follows. Step I. A single diatomic molecule (either A2 or Bz, for simplicity we will assume A,) collides with the surface of the solid. Energy may be exchanged with the solid, the Az bond may break, or the molecule may rebound into the gas phase. This calculation involves just one diatomic molecule colliding with a moving surface and is well within the computer technology developed previously. This part of the calculation provides estimates of the efficiency with which dissociative chemisorption occurs, the distribution of kinetic energies for the atoms adsorbed on the surface, and the relative fraction of energy transferred into the solid (and thus lost to the reaction). All are calculated for a variety of potential surfaces constructed consistent with our assumptions (Le., exothermic for step I). Step II. Two adsorbed atoms A,B encounter each other on the surface, recombine, and from a gas phase diatomic molecule, Again, we must consider only two atoms interacting with a moving surface to compute the state distribution of the products and the energy lost to the solid. The only direct connection between step I and step I1 is through the distribution of kinetic energies of adsorbed atoms on the surface. However, judging from gas phase dynamics, the formation of vibrationally excited product molecules seems to be more sensitive to the energy profile of the reaction contour in the formation step rather than to the detailed initial conditions of the reactants. That is, for an exothermic reaction A + (B,C) the product state distribution is largely determined by the energy profile along the reaction path, and effects due to the initial state of the (B,C) or due to the kinetic energy of collision are not as i m ~ 0 r t a n t . l ~ To this approximation, steps I and I1 can be decoupled and computed separately. Detailed studies of recombination dynamics (step 11) have already appeared138and will be briefly summarized below. In this paper we report computational results for adsorption step I. It is important to emphasize that the overall reaction must be highly exothermic from gas phase reactants to gas phase products. Computations performed for neutral reactions (Le., H2/D2 exchange) or exothermic reactions in which the desorption step is endothermic (i.e,, recombination of H2/Dz on surface) do not indicate the effects we are seeking either computationally or experimentally. 111. Results and Discussion. Dynamics As indicated, there were reasons to feel that step 11, the recombination step, would be the most crucial in determining the population distribution of the final molecules. Therefore, this step was studied first and in some detail.l3* We briefly summarize the results here. Using a model potential based on the London-Eyring-Polanyi-Sat0 (LEPS) method, nine potential surfaces were generated (Figures 1 and 2). The original rigid-surface model was generalized to allow for motion of the surface atoms, and, hence, gas-solid energy exchange. Three of the nine surfaces were endothermic for recombination (no. I, 11, and 111), while the remaining six were exothermic for recombination. Various barriers and modes of energy release (early, late) were included. The results for the nine surfaces are summarized in Tables I and 11. The three endothermic surfaces did not lead to efficient recombination nor to significant reaction energy appearing in vibration even though kinetic energy considerably in excess of the minimum was supplied. Hence, one requirement for efficient energy deposition in AB seems to be although A2,B2dissociatively chemisorb on the material, AB should not dissociatively chemisorb (i.e., AB recombination in the surface should be exothermic). This should be a useful
TABLE I: Mean Distribution of Energy in Product Molecules: (a) Rigid Surface; @ ) Two Surface Atoms Free to Movea
TABLE 111: Energy Lost to t h e Solid for Potentials I, 11, I11 Defined in Figures 1 and 2
%
potential
potential surface
molecule forma% tion E*
IV
(a) (b) V (a) (b) VI (a) (b) VIII(a) (b)
41 72 36 78 42 65 30 69
H, t W(OO1) Erot
Et,
( a ) Equal Masses 41 15 49 11 60 20 52 26 51 10 57 8 68 18 62 18
44 34 20 15 39 30 14 13
5.24
4.79 3.86 5.89 4.26
TABLE 11: Product Vibrational State Distribution (%) Summed Over Rotational SubleveW
IV V VI VI11 IX
18 11 8 1 14
IV V VI VI11 IX
1 ( a ) Equal 39 17 36 20 29
( b ) Light-Heavy 11 20 15 21 10 10 18 23 39 22
2 Masses 30 36 39 48 32
3 12 34 15 26 24
I1 I1 I1 I11 I11 I11 I11 I11
6.69
a Reprinted with permission from ref 13. Copyright 1977 by the American Institute of Physics.
u=0
I
6.32
7.04
4
5
1 3 2 5 1
Mass Combination 20 30 15 26 22 7 23 23 25 13 13 27 3 14 20
0 0 0 0 0 4 9 10
8 2
a Reprinted with permission from ref 13a. Copyright 1977 by the American Institute of Physics.
guide in choosing candidate systems for experimental studies. We see that some of the systems studied (i.e., VII) produce vibrational population inversions, quite comparable to the HF laser. Also, system VI1 has a satisfactory amount of energy in V L 2 , which is another important characteristic. Unfortunately, the best results were obtained for the case of equal mass which is likely to be a homonuclear molecule not suitable for IR emissions (although CO could approximately qualify). In summary, the results for the recombination seem to indicate a good population inversion may be obtained if (1) the recombination is exothermic, ( 2 ) atoms have roughly equal masses, and (3) from potential VII, the energy is released early in the recombination with no barrier to recombination. That vibrational excitation is produced more readily by an attractive potential (i.e., VII) is consistent with the work of Polanyi14 on gas phase A + BC collisions. It remains to examine the adsorption step to see if significant energy loss to the surface occurs. We have shown in previous studies that adsorption will proceed efficiently even without energy transferred to the solid if it is exothermic and does not encounter an activation
I 1
%energy transfer
%
%
( b ) Light-Heavy Mass Combination 13 40 IV ( a ) 21 47 51 10 34 (b) 73 40 49 30 21 V (a) 40 19 79 37 (b) 13 36 VI (a) 40 51 17 32 69 47 (b) 26 37 40 23 VIII(a) 46 27 23 (b) 70
potential
G. D. Purvis, M. J. Redmon, and G. Wolken
No. 8, 1979
The Journal of Physical Chemistry, Vol. 83,
1030
“ B r 2 ” t W(OO1)
I I I I1 I1 I1 I11 I11 I11
E*& eV 0.3 0.3 0.3 0.3 0.3 0.3 0.9 0.9 0.9 0.3 0.5 0.3 0.3 0.3 0.3 0.3 0.3 0.9 0.9 0.9
AE,I
V
J
0 3 0 0 3 0 0 3 0 3 3
0 0 3 0 0 3 0 0 3 0 0
3.2 1.6 2.9 4.5 2.3 4.3 3.0 2.3 2.9 1.9 2.4
0 31 0 0 31
0 0 30 0 0 30 0 0 30
6.2 5.0 7.0 10.2 13.1 10.6
0 0 31 0
Etot, %
6.5 10.3 8.4
barrier.loc Also, we do not expect the detailed motion of the atoms A and B across the surface to be as important as the potential in determining the final state distribution of AB product molecules. Therefore, our chief concern is with the energy transferred to the solid during adsorption and whether or not such energy transfer could be a serious loss of energy otherwise available for AB excitation. We have previously studied the energy transfer for H2,and HD colliding with a tungsten surface using a potential that had a barrier to adsorption.1° Here, we report results for a few potentials lacking such activation barriers, as well as for some heavier particles with early and late energy release, and for heavier gas molecules. Potentials 1-3 in Figures 1 and 2 are exothermic to adsorption (endothermic to recombination), provide both early and late energy release, and these were used for our study. Both light and heavy mass combinations were used (to simulate Hz and a system with the mass of Br, colliding with tungsten) and the results are reported to Table 111. Both isolated diatomic molecules were described by a Morse curve with potential parameters given previo~sly.~ This potential was chosen to reproduce the properties of the isolated H2molecule and only the mass was increased to 80 amu to mimic a heavy molecule colliding with the surface. Hence, the system labeled ‘‘Br2” has only the mass of “Br” and the other properties of dissociation energy, force constant, and equilibrium separation appropriate for H,. The Morse parameters for the isolated diatomic molecule affect the LEPS potential in a complicated, nonlinear ~ a y . ~ - l l Hence, attempting to change the Morse parameters to model more closely a physical Br2 molecule would change the total potential and the energetics of the reaction. We believe it is more important to produce a comparison on the same potential with simply a heavier molecule. Both vibrational and rotation were included. Approximately 1.75 eV of vibrational energy is present in both systems (H,(V = 3), “Br2”(V = 31)). Similarly, the rotational quantum numbers were selected such that equivalent rotational energy (0.09 eV) was present for both molecules. For all cases studied, nine surface atoms were permitted to oscillate about their lattice sites, according to the Einstein model discussed previ0us1y.l~” The parameters for this oscillation about the lattice are the same as those used in the recombination study of ref 13a.
Vibrational Excitation from Heterogeneous Catalysis
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
T
1031
-
-? 0
v
I I
I E
4
3 2 I
01
i J b 5-k
'd 8 9
7
7l 6
'
-4.5
I
Flgure 1. Equipotential contours (in eV) for the approach of H, to W(OO1) for the different potential surfaces studied. The contour levels are equally spaced for any given potential surface. X indicates the position of the reactants, two atoms adsorbed in adjacent 1CN sites (5CN for IX). For details of the construction of these surfaces, see ref 13a. (Reprinted with permission from ref 13a. Copyright 1977 by the American Institute of Physics.)
Etotin Tablle I11 i s the total energy available to be deposited into the solid: adsorption energies plus kinetic and internal energies of the gas molecule. The surface temperature was initially 300 K. The energy gained by the solid because of this adsorption is AEs. It is calculated by summing the kine tic and potential energies of the solid atoms as they move about their lattice sites. This is not a precisely defined number because the couplings between the surface atoms and adsorbed atoms will cause energy to flow back and forth until the adsorbed atoms recombine and leave the surface, or the adsorbed atoms leave the region of moving surface atoms. In this case, the trajectories were followed until the final separation of the adsorbed atoms exceeded the region of moving surface atoms. From Table I11 we see that the larger mass will increase the energy transfer by roughly a factor of 2-3 although the mass increase by a factor of 80. For the worst cases, the energy transfer is approximately 10% of the total energy.
However, there are recent indications that the Einstein model used here tends to underestimate energy transferred to the solid when compared with generalized Langevin (GL) computations.15 The GL computations considered only one moving surface atom, and it is not clear how much this affected the results. That is, several Einstein oscillators may provide a closer approximation to a more accurate GL model (there being more degrees of freedom in the solid in which to store energy) than a single oscillator would provide to a single oscillator GL model. Until these calculations are performed, the results must be too small by a factor of 2. For H2 W(OO1) the net energy loss to the solid is still within the original estimate of 10%. However, for the mass 80 system, 20-25% energy loss could occur.
+
V. Conclusion In conclusion, the current series of computer simulations indicates that vibrational excitation should be possible for
1032
The Journal of Physical Chemistry, VoL 83, No. 8, 1979
G. D. Purvis, M. J.
I
I ' I -
G. Wolken
??
-3.9-4.2
Redmon, and
1-45 -4E
l
P
L
l
m
5 1
2
3
I
!
I
!
m l
I-
t 3
l
I ' i
4
5
6
1
,
l
7
8
9
DISTANCE (a u ) Figure 2. Energy profiles along the reaction paths for the potential surfaces given in Figure 1. The dashed line marks the region of transition from reagents to products. (Reprinted with permission from ref 13a. Copyright 1977 by the American Institute of Physics.)
heterogeneous reactions. The conditions seem to be that both steps in the reaction, dissociative chemisorption and recombination, should be exothermic. Efficient population inversion occurs for early energy release in the recombination step and for nearly equal mass atoms. A question that has not been fully explored is the energy loss to the solid for all conditions of interest. The present computations indicate that this could be a serious loss (i.e., -20%) for heavy systems, but should be quite tolerable for light gases. There are indications that heating the substrate would tend to reduce the net energy loss into the s01id.l~ One is tempted to speculate about the possibility of constructing a "catalytic chemical laser" given the estimates of population inversions given above. However, this endeavor is not likely to be fruitful at this stage because there is simply too little data, either experimental or theoretical, with which to do a realistic estimate. The present research indicates that a vibrational population inversion should be possible in reactions catalyzed by a solid surface. There are also indications16 that collisions of vibrationally excited molecules with surfaces do not invariably lead to deactivation. Deactivation probabilities
as low as are not uncommon. This would be the chief loss mechanism not already present in chemical lasers, and it may be possible to overcome it by an appropriate design. The overriding need now in heterogeneous chemical dynamics appears to be for good measurements of final state population distributions for a variety of prototype systems. If classical trajectory studies such as these in any way stimulate interest in such measurements, or provide guidance as to how they should be performed, the work and career of Don Bunker in developing the technology must receive due credit. Acknowledgment. This research was sponsored by the Air Force Office of Scientific Research (AFSC), United States Air Force, under Contract No. F49620-77-(2-0004.
References and Notes (1) 6. Halpern and D. E. Rosner, J. Chem. SOC., Faraday Trans. 1 , 74, 1883 (1978). (2) G. A. Melin and R. J. Madix, Trans. Faraday SOC.,67, 198 (1971). (3) R. D. Levine and R. B. Bernstein, "Molecular Reaction Dynamics", Oxford University Press, New York, 1974. (4) P. J. Kuntz, E. M. Nemeth, J. C. Pohnyi, S. D.Rosner, and C. E. Young, J. Chem. Phys., 44, 1168 (1966). (5) For a brief review see, K. L. Kompa, "Chemical Lasers", Springer-Verlag, New York, 1973.
Theoretical Study of Hot-Atom Chemistry (6) (7) (8) (9) (10)
The Journal of Physical Chemistry, Vol. 83, No. 8, 1979
J. C. Polanyi, J . Chem. fhys., 34, 347 (1961). J. C. Polarryi and S.D. Rosner, J . Chem. Phys., 38, 1028 (1963). J. V. Kasper and 01.C. Pimentel, Phys. Rev. Lett., 14, 352 (1965). J. H. McCreery and G. Wolken, Jr., J. Chem. fhys., 63, 2340 (1975). (a) J. H. McCreery and G. Wolken, Jr., J . Chem. Phys., 63, 4072 (1975); (b) ibid., 64, 2845 (1976); (c) ibid., 65, 1310 (1976); (d) Chem. Phys. Lett., 39, 478 (1976); (e) Chem. Phys., 17, 347 (1976); (f) J . Chem. Phys., 66, 2316 (1977).
1033
(11) A. Gelb and M. Cardillo, Surf. Sci., 59, 128 (1976). (12) A. R. Gregory, A. Gelb, and R. Silby, Surf. Sci., 74, 497 (1978). (13) (a) J. H. McCreery and G. Wolken, Jr., J. Chem. Phys., 67, 2551 (1977); (b) G. Wolken, Jr., Chem. Phys. Lett., 54, 35 (1978). (14) J. C. Polanyi, Acc. Chem. Res., 5, 161 (1972). (15) A. C. Diebold and G. Wolken, Jr., Surf. Sci., in press. (16) G. Black, H. Wise, S. Schechter, and R. L. Sharpless, J. Chem. Phys., 60, 3526 (1974).
Theoretical Study of Hot-Atom Chemistry. The T iHD Exchange Reactions James S. Wright,” Stephen K. Gray,+ Department of Chemistry, Carleton University, Ottawa, Canada K I S 586
and Richard N. Porter” Department of Chemistty, State University of New York, Stony Brook, New York 11790 (Received September 5, 1978)
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The hot-atom exchange reactions T + HD TH + D and T + HD TD + H have been studied by trajectory calculations on the Porter-Karplus potential surface up to a laboratory collision energy of 60 eV. The integral reaction probability (IRP) equations for product yields are extended to include the effects of inelastic collisions. The IRP formulas are then reduced to their Monte Carlo form for use with the trajectory calculations. A realistic moderating function P(E,E’) is obtained and resolved into elastic and inelastic contributions. Reactive, nonreactive, and dissociative cross sections are given as a function of energy up to 60 eV. The reaction probabilities are given and reaction mechanisms are discussed. The IRP equations are solved numerically using the realistic and hard-sphere moderating functions to obtain absolute TH and TD product yields and the TH/TD product ratio, and compared to the kinetic theory and experimental results. All three theoretical models agree with the experimental result of 0.70 at 2.8 eV. The realistic IRP model gives an inverted isotope ratio of 1.78 at high energy, whereas the simpler models predict an isotope ratio of 0.80, in much closer agreement with experiment. The differences are related to the detailed shape of P(E,E’) and it is shown that the hard-sphere and kinetic theory models predict the right result by a combination errors. The need for consideration of secondary collisions in a rigorous treatment is emphasized.
I. Introduction Hot-atom reactions, in which translationally excited atoms react with thermal atoms or molecules, have been the subject of extensive experimental and theoretical study. Experimental studies of hot-atom reactions in the gas phase have been recently reviewed by Urchl and by Spicer and Rabinovitch.2 The theoretical treatment of hot-atom reactions began with the “kinetic theory” of Estrup and Wolfgang3 and was further developed by W ~ l f g a n g .This ~ theory required only knowledge of the energy dependence of the reaction prolbabilities to obtain product yields, and has been widely used. The Estrup-Wolfgang kinetic theory is expected to be most accurate when the initial hot-atom energy is large and when inelasticity can be ignored. An early Monte Carlo calculation by Rowland and C o ~ l t e rwhich ,~ used an isotropic scattering model, gave results in good agreement with the kinetic theory prediction. More sophisticated theoretical treatments considered how hot atoms lose energy in inert6-10 and reactive11-14media, and treated approximately the effect of inelastic collisi~ns.’~A general phenomenonological theory for calculating integral reaction probabilities (the IRP theory) has been devised to treat cases in which the conditions for the Wolfgang model are not strictly met, such as systems with low initial hot-atom energies with appreciable inelastic scattering. The IRP equations were Department of Chemistry, University of California, Berkeley, Calif. 94720.
0022-3654/79/2083-1033$01 .OO/O
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applied by Porter and Kunt” to the T H2 and T + D2 reactions, using simplified models for nonreactive scattering. The effect of quantum-mechanical hard sphere nonreactive collisions was considered by Adams and Porter18 in a study of T + H2, T + D2, and T + HD reactions. Even though the formalism now exists to allow its treatment, the effect of inelastic collisions on product yields has received very little attention in the literature. Using the IRP equations, Malerich and Spicerlg obtained a formula which contained the effect of inelasticity on the “moderating function” P(E,E?,and applied the theory in approximate form to the system T + D2. Here the treatment of inelasticity was restricted to dissociative collisions. The analysis by Porter and K u n P of different models for P(E,EE’? led to the conclusion that product yields were relatively insensitive to the shape of P(E,E’). However, a trajectory study by Malcolme-LawesZ0of the T HF system showed a complicated dependence of the average logarithmic energy loss vs. collision energy, indicating that inelastic effects may be significant. All accurate theoretical treatments of hot-atom reactions require knowledge of the excitation functions for the various possible hot products. The usual way to obtain these functions is by use of quasi-classical trajectory (QCT) calculations (for a widely read review by Bunker, see ref 21). Excitation functions have been computed in this way by Karplus, Porter, and Sharma22for T + H2 and T + D2, and reported by Adams and P o r t e P for T + HD. Cross sections as a function of collision energy were given for the
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0 1979 American
Chemical Society
