J. Phys. Chem. 1083, 87, 3179-3181
charged porphyrins (H2TPPS", ZnTPPS", H2TMPyP4+, H2TAPP4+,and ZnTAPP4+(TAPP = tetrakis(trimethy1aminophenyl)porphyrin)), complications due to groundstate aggregation preclude any detailed kinetic model using the present model.
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Acknowledgment. The authors are grateful to Dr. Pierre P. Infelta for many hours of assistance in the computer manipulation and simulation of data. This work has benefitted from grants from the Swiss National Funds for Scientific Research and from Ciba-Geigy.
Incubation Times in Strong-Colllsion Thermal Unimolecular Reactions H. 0. PrRchard Centre for Research ifl Experimental Space Science, York Universw, Downsview, Ontario, Canada M3J 1P3 (Received: April 21, 1983)
It is shown that, for a strong-collisionthermal unimolecular reaction, the incubation time measured in a shock-wave heating experiment becomes independent of pressure at the low-pressure limit.
Introduction Bunker defined a strong collision as one in which "so much energy is transferred... that the subsequent condition of the reactant molecule may be chosen at random (with appropriate weighting factors for energy) from all its possible states";' similar verbal statements were made by Tardy and Ftabinovitch,2 and by N ~ r d h o l m . This ~ definition was given mathematical precision by Nordholm, Freasier, and Joll3p who (after transcription into the present notation) gave the strong-collision rate constant qij for a transition from grain i to grain j as qij = PPj (1) where Pj is the equilibrium population of grain j and CL is the relaxation rate, which differs only from the collision rate w by the usual collision efficiency factor A, as discussed in detail e l ~ e w h e r e ;p~is? ~proportional to the pressure p . The rate constant for a strong-collision thermal unimolecular reaction is then given as the smallest eigenvalue yoof a reaction matrix [Q - D ] where the elements of Q are (2) [Qlij = ~u((1 - 8ij)Pi - & j ( l - P i ) ) and the elements di = [DIiiof the diagonal matrix D are the decay rate constants for the reactive grains. Q, and therefore [Q - 03,are clearly unsymmetric, but are easily symmetrized by a similarity transformation to give R = -=1/2[Q - D]E'/2 = [ P ( 1 -Po) + Dl (3) where [Elij = 8ijPj, and the elements of p o are [ p O l i j = /3;12/j{/2. Let q0be the eigenvector of R corresponding to yo, and Sothe eigenvector of p(1- po) corresponding to its zero eigenvalue; the elements of So are (So)i = and (So,So)= 1 where the notation (,) denotes the scalar product of two vectors. Incubation Times The incubation time for reaction is a function of the initial distribution n(O),and may be written in one of two forms6#' Tint = yo-' In [(S,,lC/o)(E-1/2n(o),J/,)1 (44 (1) D. L. Bunker, 'Theory of Elementary Gas Reaction Rates",Pergamon, Oxford, 1966,pp 48-9. (2) D. C. Tardy and B. S. Rabinovitch, Chem. Reu., 77, 369 (1977). (3) S. Nordholm, Chem. Phys., 29, 55 (1978). (4)S.Nordholm, B.C. Freasier, and D. L. Jolly, Chem. Phys., 26,433 (1977). (5)H.0.Pritchard, "Quantum Theory of Unimolecular Reactions", Cambridge University Press, London, 1984. (6) A. W. Yau and H. 0. Pritchard, Can. J. Chem., 57, 1723 (1979). 0022-385418312087-37 79$01.50/0
if the normalization of q0 is such that (#o,lC/o) Tint = YO-' In
= 1, or
[(E-1/2n(o),lC/o)/(~o,~o)l (4b)
if the normalization is such that = 1. (It is important to recognize that both normalizations are convenient for different purposes,6 and, as the quantity inside the logarithm can be extremely close to unity, the use of the incorrect formula may sometimes cause errors of several orders of magnitude in the calculated values of qnC) When an incubation time is measured in a shock-wave experiment, the initial distribution can usually be taken to be that of the cold gas, with all the molecules in the lowest energy grain of the reactant molecule, Le., n(0) = [1,0,O,. .IT. This being the case, it is possible to derive mathematically rigorous upper and lower bounds for the quantity qnC, which behave quite differently between the strong-collision and weak-collision cases.' In the weakcollision case, the incubation time is expected to exceed the internal relaxation time by an amount which correlates with the "weakness" of the collisions. On the other hand, in the strong-collision case, the limits are
where k , is the infinite-pressure rate constant; thus, in the strong-collision case, T~~~IT,,~, and pqncneed not necessarily be constant once the reaction is in the falloff regime. Most shock-wave measurements of incubation times have been made for the ionization of inert gas atoms, or for the dissociation of diatomic mole~ules,~ but there has been one study of a weak-collision unimolecular reaction, the thermal dissociation of nitrous oxide;*also, there is one set of data from which an approximate incubation time can be inferred for a strong-collision reaction, that of the thermal isomerization of cycl~propane.~In the nitrous oxide case, T~~~> T,,~ as expected; also notice that these measurements were made in the second-order region, and the results were quoted in the form of implying that prhCwas a constant. Moreover, this set of results has been the subject of three modeling exercises,6"0J1and while the question of the constancy of pqncwas not addressed specifically in the first two, it must almost certainly have (7) H. 0. Pritchard and S. R. Vatsya, Chem. Phys., 72,447 (1982). (8)J. E. Dove, W. S. Nip, and H. Teitelbaum, Symp. (Int.)Combust., [Proc.],15th, 903 (1974). (9)E. A. Dorko, R. W. Croseley, U. W. Grimm, G. W. Mueller, and K. Scheller, J.Phys. Chem., 77,143 (1973). (10) W. Forst and A. P. Penner, J. Chem. Phys., 72, 1435 (1980). (11) J. E. Dove and J. Troe, Chem. Phys., 35, 1 (1978).
0 1983 Amerlcan Chemical Society
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Letters
The Journal of Physical Chemistry, Vol. 87, No. 17, 1983
been noticed had there been a variation of pqncwith pressure; in the third case, crude formulae were given which imply that 7inc/7,elwould vary only slightly between the low-pressure and the high-pressure limits. Another recent examination of incubation times in model weakcollision systems has shown that the ratio 7inc/7,e1 is insensitive to pressure.12 In the cyclopropane isomerization, the reaction was measured in the region of 1100 K at total pressures of 2-5 atm, and from the spectral density vs. time plots presented, it appears that 7inc N 7rel; since, at these pressures, the reaction would be approaching its highpressure limit, this result is not inconsistent with eq 5.
Incubation Times in Strong-Collision Systems Equation 5 defines qncvery closely if the reaction is at or near its high-pressure limit, but the lower bound is a very wide one if it is in the falloff region. In the limit as p 0, the strong-collisionrate constant yo pC'&, where the prime denotes that the summation is taken only over the reactive g r a i n ~ ; thus, ~ J ~ eq 5 becomes
-
-
C'Pi
I
h
0 QJ
VI u
-2
1
1
-I7inc I-
CPidi c1 In fact, at the low-pressure limit, it is possible to write an exact solution for qncfrom the equations given in ref 7, simply by replacing the matrix C in that general analysis by the matrix D. Equation 4 of that paper becomes -8
-6
-4
shift
which tends to the quantity A as both p and y o tend to zero; the quantity A itself is given by ( [ p - 7 0 D]-'D'/'SO,[p - yo + D]-1D1/2So) A= (8) ( [ p - yo + D]-'D'/2So,D'/2So)
+
and as the pressure becomes so low that p, y o can be neglected in comparison with the elements of D , then (D-'/2So,D-'/2So) C'P;/ di =rinc A (9) (D-1/2So,D'/zSo) C'Pi Thus, we have the remarkable result that 7incis a constant at the low-pressure limit for a strong-collision reaction initiated in a shock wave! This conclusion is confirmed by the numerical calculations shown in Figure 1 for C3H8 at 1500 K, CH,NC at 1000 K, and (for illustration only) C02treated as if it were a strong-collisioncase, at 4000 K; the unimolecular rate calculations on these molecules were performed by the standard strong-collision method, as described e l ~ e w h e r e . ~ J ~ The exact rate constant yo can be found from the equation
-
+
by iteration; the corresponding eigenvectors are13 ($o)i = P i 1 " / { [ F - Y O+ d i l [ E ( ~ YO + dj)-2Pj]"2] (11) J
where the normalization (the second term in the denominator) is chosen such that ($o,$o) = 1,and thus is to be used with eq 4a. The top panel of Figure 1 shows the falloff curves for these three reactons, as solid lines, and the values of qnc/7,el, as dashed lines; the separations (12)H.0.Pritchard, Can. J. Chem., in press. (13)S. R. Vatsya and H. 0. Pritchard, Can. J. Chem., 59,772(1981). (14)A. W.Yau and H. 0. Pritchard, Can. J. Chem., 56,1389(1978).
-2
0 +
2
4
6
8
Log p ( T o r r )
Flgure 1. Falloff and Incubation tlme curves for strongcollislon models of (from left to rlght) C3H6,CH,NC, and C02. Top panel: solM lines, klk,; dashed Ilnes, 7hc/7R(. Bottom panel: solM Ilnes, 7hc;dotted Ilnes, T , ~ .Note that in the C3H6case only, all curves are shifted leftward by 3 log unlts to avoid overlap with other curves.
between these pairs of lines at their respective low-pressure limits reflects the fact that (Z'&/di)/(Z'fli) in eq 9 is very much greater than the (C'&)/(CPidi)of eq 6. The lower panel of Figure 1 shows the behavior of 7,1 (dotted line) and of qnc(solid line) for the same calculations: each qnc curve has the property that it becomes constant as soon as y o becomes second order, and the limiting value is exactly that predicted by eq 9.
Discussion Strong-collision behavior, as defined by Bunker,' carries with it the requirement that the incubation time for a shock-wave-initiated thermal unimolecular reaction shall become constant in the second-order region. Intuitively, it does not seem reasonable that this could happen, simply because it is always thought that internal relaxation is predominantly a cascading or sequential process, and therefore that reaction must follow along after relaxation. In contrast, strong-collision behavior, as defined by eq 1 and 2, means that any departure from equilibrium of some state, anywhere in the system, is counteracted by a simultaneous relaxation from all other states in the system, with rate p; in the present calculation, the reactive grains are fed directly from the ground state alone by the strong-collision mechanism. That qncI1/p, as is shown in the bottom panel of Figure 1, can easily be understood in qualitative terms. First, in the strong-collision regime, all states below reaction threshold are in equilibrium with the heat bath;5 the rate constant for the achievement of this equilibrium distribution, starting from the 0 K initial distribution n(O), is p. As p m, the steady distribution of the reactive states above threshold also tends to the equilibrium one; the rate constant for the filling of these states is also p ,
-
J. Phys. Chem. 1983,87,3181-3183
and so rhCN 1/11.. At lower pressures, when the rate is fallen off, the steady distribution above threshold is depleted, and can be attained in a shorter time than for the proper equilibrium distribution, hence rinc< 1/11.. What is not obvious is why pint should tend to a constant value as the second-order region is reached, nor why the limiting value should be the mean decay time constant for the manifold of reactive states. From a more practical viewpoint, it would seem that the incubation time, measured in a shock-wave heating process from a low starting temperature, could provide a rigorous method of classifying reactions into strong-collision and weak-collision types. Strong-collision behavior would be identified with a value of qncI ~,1, but, as true strong-
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collision behavior can never be observed since all relaxations must contain some sequential character, rincN rrel, as seems to be the case in the cyclopropane isomerization experiment, would appear to be the more likely occurrence. On the other hand, it is well established6p8J+12that in a ~ the 1 , indications are' weak-collision system, rinc> ~ ~ and that the magnitude of the disparity between these two quantities will increase with increasing departure from the strong-collision behavior of eq 2 or 3.
Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council of Canada; it is also a pleasure to thank Raj Vatsya for his helpful criticisms.
Multiple Adsorbates on Copper Surfaces in Formic Acid Vapor Observed by Polarization Modulation Infrared Spectrocopy Toshlmasa Wadayama, Klyoshl Monma, and Wataru Suetaka Laboratory of Interface Science of Metals, Faculty of Engineering, Tohoku University, Sendai 980, Japan (Received: April 28, 1983)
IR spectra of formic acid adsorbed on copper surfaces were observed in the presence of ita vapor at a relatively high pressure. The spectra show that formic acid dissociatively adsorbs on the metal at room temperatures to form bridged and monodentate formates. A new band, which appeared only in the presence of gas-phase formic acid, was observed at 1160 cm-' (1200 cm-I in deuterated species) in addition to the bands due to the formates. The weakly adsorbed species giving rise to the new band is discussed. The vibration spectra of adsorbed species on metal surfaces may provide information crucial for elucidating the mechanism of catalytic reactions on the metal. High-resolution electron energy loss spectroscopy is a powerful tool for obtaining vibration spectra of adsorbed species on solid surfaces in ultrahigh vacuum, but cannot be used for the observation of metal surfaces in gaseous medium. The polarization modulation (PM) technique, which has been successfully applied for improving the sensitivity of infrared absorption and emission spectroscopies,1*2is feasible for obtaining infrared absorption spectra of species on metal surfaces in a gaseous or liquid medium without hindrance of the absorption of the med i ~ m . ~ ~ ~ In catalytic reactions, weakly adsorbed species are generally present on the catalyst in addition to strongly adsorbed ones. Since not only the latter but also the former may play important roles in the reaction, in situ observation of both the species probably yields information valuable for shedding light on the reaction mechanism. However, infrared spectra of adsorbed species have generally been obtained from solid surfaces after evacuation of the gas-phase species so as to remove the absorption of infrared light by the gas. As a consequence, only the spectra of strongly adsorbed species have been obtained at room temperatures, because weakly adsorbed species (1) H. Pfnllr, D. Menzel, F. M. Hoffmann, A. Ortega, and A. M. Bredshew, Surf. Sci., 93,431-62(1980). (2) K. Wagatauma, K. Monma, and W. Suetaka, Appl. Surf. Sci., 7, 281-5 -- - - (1981). - - - - ,. (3) D. S. Dunn, M. W. Severson, W. G. Golden, and J. Overend, J. Catal., 65,271-80 (1980). (4)J. W. Russel, J. Overend, K. Scanlon, M. Severeon, and A. Bewick, J. Phys. Chem., 86, 3066-8 (1982).
.
0022-3654/83/2087-3181$01.5QIO
were desorbed in the course of the evacuation. We applied the above-mentioned modulation technique for the in situ observation of adsorption of formic acid on evaporated copper films in the presence of the acid vapor, and were successful in detecting a weakly adsorbed species as well as strongly adsorbed formates. In the present Letter, the results obtained will briefly be reported. Figure 1shows the schematic diagram of the PM spectrometer used in the present work. The plane-polarized infrared beam was incident onto the sample at a high incident angle (ca. 80'). The polarizer was rotating at a constant speed of 1650 rpm to give a modulation frequency of 55 Hz. A thin KBr plate (1 mm in thickness) was inserted into the light path at an angle near the Brewster angle so as to compensate for the difference in reflectivity of the sample and mirrors for the p and s components of the infrared light. The resolution of the monochromator* filter used is about 15 cm-' over the region of measurement and is rather low. However, the filter does not change the intensity of the infrared light upon rotation of the plane of polarization in contrast to grating monochromators. A stainless-steel vacuum cell, which was equipped with NaCl infrared windows, was pumped out with an oil diffusion pump. Pure copper (purity 99.999%) was deposited onto a smooth glass plate in the cell at a background pressure of 1 X lo4 torr (1.33 X Pa). The temperature of the substrate glass was monitored with an alumel-chromel thermocouple embedded in the plate. Formic acid (purity 95%) and dideuterioformicacid (DCOOD, purity 99.9% ), which were used without further purification, were introduced into the cell after a degassing procedure in a glass vessel. Figure 2 shows the spectra of formic and deuterioformic acids adsorbed on copper films in the range of 1250-1800 0 1983 American Chemical Society
