J . Phys. Chem. 1987, 91, 5445-5451
5445
= 0.46 eV then no reactions can occur for Y b near 90” while for a widely distributed Yb reactions can take place via near collinear conformations. 3. Figure 7 shows that the steric effect vanishes at E,, 1.2 eV. This means that the corresponding energy E2 is already sufficiently large to admit reactions via both the near bent and the near collinear conformations (cf. Figure 8). Then the different ya preparations lead to a reaction with the same probability. Using these arguments one would predict for a reaction with a bent transition state that u , ~< ul. No experimental data are available so far but the result of a recent QCT study for Li HFI4 based on a PES featuring a bent transition state13 is in accord with this prediction.
3). For a system with a realistic anisotropy the final Yb distribution is of course altered. But using the “rotational sliding mass model”15 it can be easily demonstrated that due to the moderate anisotropy of the PESs of the M HX-systems the range of Yb still remains roughly the same as for Y ~ . Following these arguments one is led to the conclusions: 1. Steric effects for the present system arise from collisions with small b. As a consequence the ratio ( u , ~- uI)/a must be small in accord with the experimental results. 2. The finding ull> u1 means that near central collisions are less likely reactiveJor ,a Y~preparation which predominantly leads to,Yb near 90” (ElV) than for one providing no preferred Y b (Ell V). This result can be easily rationalized if one assumes that the system exhibits a collinear transition state with a Yb dependence of the barrier height like the one shown in Figure 8. Is E l the energy available for passing the barrier at a collision energy E,,
-
+
+
Acknowledgment. Support of this work by the Deutsche Forschungsgemeinschaft (SFB 2 16, P5) is gratefully acknowledged. Registry No. K, 7440-09-7; HF, 7664-39-3.
(15) Loesch, H. J. Chem. Phys. 1986, 104, 213.
+
Isotopic Variants of the H H, Reaction. 1. Total Reaction Cross Sections of the H D, and H HD Reactions as a Function of Relative Energy
+
+
G. W. Johnston, B. Katz,+K. Tsukiyama, and R. Bersohn* Department of Chemistry, Columbia University, New York, New York 10027 (Received: January 26, 1987)
+
The total cross sections for the reactions H D2= HD + D and H + HD = H2 + D have been measured as a function of relative initial translational energy in the ranges 0.87 to 2.70 and 0.82 to 1.86 eV, respectively. The technique for measuring the rate constant exploits the fact that the atomic product and reactant have very similar absorption wavelengths and that nearly monoenergetic hydrogen atoms can be generated by photodissociation of particular small molecules. Detection of the hydrogen atoms was carried out by laser-induced fluorescence at such short times that single collision conditions prevailed. The measured cross sections are in excellent agreement with the classical trajectory calculations of Blais and Truhlar and Schechter and Levine. As a byproduct of these experiments it was shown by Doppler spectroscopy that at the wavelengths 248 and 193 nm the transition dipole moment for H2S is perpendicular to the molecular plane and that almost all of the available energy in this photodissociation is released into translation.
+
The reaction H H2 = H2+ H (and its isotopic variants) is the central elementary chemical reaction. The elementary chemical reactions A BC = AB C are at the heart of theoretical reaction dynamics because of their simplicity. The subset of exchange reactions A BA = AB A have a special simplicity in that the potential surfaces are symmetrical with respect to reactants and products. H + H2is the only such system for which an accurate potential surface has been calculated.’S2 Although quantum mechanical calculations have so far been carried out only at energies near the threshold30 extensive calculations have been carried out of the classical trajectories on the best potential ~urface.~-’*~~ Ever since the early work of Londons an immense theoretical literature has been devoted to this reaction (see ref 9 for a review). Classical kinetic experiments have been carried out on isotopic versions of H + H2 in which the temperature-dependent rate constant k( r ) was measured.lo The corresponding energies are near the threshold for the reaction, that is, in the region 0.4 to 0.6 eV. More recently a new class of experiments have been carried out involving some degree of state selection. Differential reactive scattering cross sections have been measured for D + H2 at 0.48, 1.0-, and 1.5-eV relative The H D state distribution has been measured for the H + D2 reaction at 0.98, 1.1-, and 1.3-eV relative energy by coherent anti-Stokes Raman scattering and by multiphoton ionization (MPI).13,14
+ +
From these experiments we have learned that near threshold the reactive scattering for D H2 is backward in the centerof-mass system but at higher energies the scattering is mainly sideward. At least up to 1.3 eV the H D product of the H + D2 reaction is mainly in the u = 0 state. The rotational distribution broadens and shifts to higher J values with increasing relative translational energy. A minor failing of these superb experiments is that the total reaction cross section as a function of initial relative translational energy ( E )cannot be easily measured. In general, there are major difficulties in measuring u(E) or k ( E ) for a given chemical re-
+
+ +
(1) Liu, B.; Siegbahn, P. J . Chem. Phys. 1978, 68, 2457. (2) Truhlar, D. G.; Horowitz, C. J. J . Chem. Phys. 1978, 68, 2466. (3) Blais, N. C.; Truhlar, D. G. Chem. Phys. Left. 1983, 102, 120. (4) Mayne, H. R.; Toennies, J. P. J . Chem. Phys. 1981, 75, 1794. (5) Schechter, I.; Kosloff, R.; Levine, R. D. Chem. Phys. Lett. 1985, 121, 297, (6) Schechter, I.; Levine, R. D. Int. J . Chem. Kinet. 1986, 18, 1023. (7) Schechter, I.; Kosloff, R.; Levine, R. D. J . Phys. Chem. 1986, 90, 1006. (8) London, F. Z . Electrochem. 1929, 35, 352. (9) Truhlar, D. G.; Wyatt, R. E. Ado. Chem. Phys. 1977, 27, 50. (10) Ridley, B. A,; Schulz, W. R.; LeRoy, D. J. J . Chem. Phys. 1966, 64, 3344. (11) Geddes, J.; Krause, H. F.; Fite, W. C. J . Chem. Phys. 1974, 56, 3298. (12) Gotting, R.; Mayne, H. R.; Toennies, J. P. J . Chem. Phys. 1960, 80, 2230. 1986, 85, 6396. (13) Gerrity, D. F.; Valentini, J. J . Chem. Phys. 1983, 79, 5202. 1984, 81, 1298. 1985, 62, 1323. 1985, 83, 2207. (14) Marinero, E. E.; Rettner, C. T.; Zare, R. N. J . Chem. Phys. 1984, 80. 4142.
On leave from the Department of Chemistry, Ben Gurion University, Israel.
0022-3654/87/2091-5445$01.50/0
0 1987 American Chemical Societv
8-
5446
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987
action. In molecular beam studies ratios of reactive and elastic scattering intensities are measured. The elastic scattering cross sectior, can be calculated theoretically and thus the reactive scattering cross section is obtained. The method does not give highly accurate values. Energy selection in a bulb can be accomplished by photodissociation of a diatomic molecule, thus generating atoms of known, usually high speeds. If the measurement is based on a product analysis, the results are blurred t y the fact that nonreactive collisions as well as reactive collisions can occur. Thus reactions take place at a variety of energies. The experiments described here use the photochemical technique of K ~ p p e r m a n n for ' ~ preparing the reactive atom with a known translational energy. They also exploit recently developed laser techniques for probing hydrogen atoms at short time~.'~J'.'~The pressures and times are chosen so that most hydrogen atoms have not made two collisions. Those that have collided nonreactively will have a low probability of colliding reactively during the short time before detection. Even these conditions are not, in general, sufficient to measure a rate constant. In the general reaction A + BC = AB + C, a swift spectroscopic measurement of a quantity proportional to the concentration of C is not sufficient to determine the rate constant. The kinetic equation d(C)/dt = K(A)(BC) is integrated at short times t to give
k = (C)/((A)(BC)t)
(2)
The reactant BC is present in excess and therefore its concentration is essentially constant and measurable before the generation of A. In general, one does not know the absolute values of the concentrations (C) and (A). In a hydrogen atom exchange reaction we have the great advantage that by laser-induced fluorescence (LIF) or MPI one can determine the (D)/(H) ratio. The isotope shift of the 1s to 2p transition of the H and D atoms is 22.38 cm-'. Inasmuch as the third harmonic of light from a dye laser is used to excite the transition, the fundamental wavelength needs to be changed by only 7.46 cm-' to go from the excitation of H to the excitation of D atoms. Experimental Section The experimental procedure and apparatus have been described in outline in previous publication^.^^^'^ A mixture of H atom precursor (H2S or HC1) of about 3 mTorr and Dz or H D (40-150 mTorr) is irradiated with a pulse of light from a Lumonics HE-440 excimer laser of 10-14-11s duration (fwhm) at 248, 193, or 157 nm at a repetition rate of 8 Hz. The energies/pulse were 550, 400, and 10 mJ, respectively. To achieve the latter, preliminary passivation with a He-F2 mixture was required. After a delay of 50-150 ns a pulse of vacuum-UV light at 121.6 nm, the Lyman a wavelength, was used to excite fluorescence of the H atoms. The light was produced by Wallenstein's technique.I6 364.8-nm light was generated by using a Lambda Physik 201 MSC excimer laser at 308 nm to pump the dye DMQ in an FL2002E dye laser. Typically the pulse energy was about 25 mJ with a bandwidth of 0.06 cm-'. This light was tripled by focussing with a 70-mm diffraction-limited lens in a 20-cm-long cell containing 100 Torr of krypton. The fluorescence of the H and D atoms was monitored by a solar blind photomultiplier with a CsI photocathode (EMR 5420-08-1 7) whose axis was perpendicular to the two laser beams. After preamplification the photomultiplier pulse was fed to the A channel of a boxcar integrator (Princeton Applied Research 162, 164). A short nitrogen-flushed brass tube mounted between the LiF window of the cell and the photomultiplier contained a LiF lens mounted in an iron ring. A pair of external magnets adjusted the position of the lens to maximize the signal. The (15) Kuppermann, A. Isr. J . Chem. 1969, 7, 303. (16) Wallenstein, R. Opt. Commun. 1980, 33, 119. (17) Schmiedl, R.; Dugan, H.; Meier, W.; Welge, K. T. 2.Phys. A 1982,
,4304, 138. (18) Tsukiyama, K.; Katz, B.; Bersohn, R. J . Chem. Phys. 1985,83,2889. (19) Tsukiyama, K.; Katz, B.; Bersohn, R. J . Chem. Phys. 1986,84, 1934.
Johnston et al.
C
I
V
I
.
no
I
I
::
I ,
A
- i
Figure 1. Experimental and theoretical (Schechter and Levine) cross sections for the H + D2reaction. Experimental results are represented by rectangles and classical trajectory results by open circles. The line joining the calculated results is drawn to aid the eye.
Lyman a light whose duration is about 8 ns passed through the cell and impinged on a small vacuum-UV monochromator (Minuteman) set at 121.6 nm and was then detected at the exit slit by another solar blind photomultiplier (EM1 RFI/8215FV) whose amplified output was fed to the B channel of the boxcar integrator. The fraction of this light absorbed in an experiment was negligible. The ratio of the A to B signals was sent simultaneously to a chart recorder and a LeCroy 3500 microcomputer. While this ratio should compensate for variations in the intensity of the exciting light, nevertheless the dye laser was adjusted so that intensities at the H and D lines did not differ by more than 15%. A crucial part of the experiment (see eq 2) is an accurate measurement of the pressure which was accomplished with a 0-1-Torr capacitance manometer (MKS) accurate to 0.1 mTorr which was more accurate than the one used previo~sly.'~ This is the main reason for the discrepancy between one of the data here and in a preliminary report.19 The difference in speed of the reactant H and the product D atoms causes a problem. The rate constant measurement depends on an accurate measurement of the H / D ratio but the H atoms may escape much faster thus altering the ratio. This problem is solved by making the probing beam much larger or smaller than the dissociating beam by using baffles. In practice, in the absence of D2 or HD, an H signal was measured at a delay time of 80 ns. The delay time was then lengthened to 180 ns and the signal remeasured. The position of the unfocussed Lyman a beam was adjusted until the decrease of H atom signal was less than 15%. Then the H / D ratio for a reaction was measured by using a delay time of 80 ns and it was then assumed that no time-of-flight problem exists. The delay time between the firing of the two lasers had to be accurately known. It could not be too short because of the jitter in the firing times of the two lasers and the need to accumulate sufficient product for an adequate signal. It could not be too long because of the need to eliminate multiple collisions and loss of atoms from the volume under observation. In general, for fast H atoms shorter delay times and higher pressures were used; for slower H atoms longer delay times and lower pressures were used. Digital pulse generators were used to set the delay times but these were always measured by displaying the two optically detected laser pulses on the screen of a fast oscilloscope. Results Reaction Cross Sections. One sees at once from Figures 1 and 2 that there is a remarkable agreement between the experimental cross sections and those calculated from classical trajectories on the Liu-Siegbahn-Truhlar-Horowitz potential surface. No parameters enter the theoretical calculations. The theoretical points shown in Figures 1 and 2 through which a line has been drawn are those calculated by Schechter and Levine.6 For H + D2, Blais and Truhlar obtained almost identical results from their classical
Isotopic Variants of the H
+ Hz Reaction
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5447
TABLE I: Rate Constants and Cross Sections
1.08 f 0.11 1.95 f 0.05 2.46 f 0.12 3.36 f 0.03
0.87 f 0.13 1.57 f 0.13 1.97 f 0.11 2.70 f 0.17
k , 1O-Io cm'/(molecule s) H D, = H D 1.25 f 0. is 2.47 f 0.17 2.77 f 0.30 3.17 f 0.31
1.08 f 0.11 1.95 f 0.05 2.46 f 0.12
0.82 f 0.13 1.47 f 0.14 1.86 f 0.18
H H D = H2 0.70 f 0.07 1.18 f 0.07 1.48 f 0.19
EH,' eV
eV
+
+
+D
no. of runs
ET,
u,C km/s
14.6 19.3 21.8 25.3
0.88 f 0.12 1.28 f 0.09 1.27 f 0.14 1.25 f 0.12
20 11 43 14
14.6 19.3 21.8
0.48 f 0.05 0.60 f 0.04 0.68 f 0.09
14 4 22
+D
'EH is the translational energy of the H atom in the laboratory frame. b E is~the relative kinetic energy for the given reaction whose rate constant is k and whose cross section is u. c u is the relative speed of the reactants.
I . I
HD 110 miorr H2S 2 mtorr At = 95 nsec
J /
/"
/"
0
Classicol Trajectories Experiment
I
I
I
I
O
I
I
l
l
0 I 2 v - v 0 ( cm- I )
- 7 - 6 - 5 - 4 -3 - 2 - 1
l
l
3 4
l
l
5 6
l
7
Figure 3. The fluorescence excitation line shape for H atoms generated from H2S a t 193 nm in the presence of 100 mTorr of H D at a delay of 95 ns.
L---.-.--.-L
I IO 20 R E L A T I V E TRANSLATIONAL ENERGY ( e V I
Figure 2. Experimental and theoretical (Schechter and Levine) cross sections for the H + HD reaction. Notation as in Figure 1.
trajectory calculations. Mayne and Toennies have carried out similar trajectory studies but for relative energies, ET, lower than used in our experiment^.^ The essential results are the seven rate constants, k(ET), as a function of initial relative translational energy, ET, which are listed in Table I. The E T values are average values with relatively large uncertainties arising mainly from the thermal motion of the diatomic reactants, Dz and HD. Intuitively, when one compares the laboratory kinetic energy which is 1-3 eV with the average kinetic energy of the diatomic at 300 K, 0.039 eV, one tends to assume that the thermal motions can be safely ignored. However, the relative kinetic energy, ET,depends on the difference of velocities of the H atoms and the diatomics. It is the square root of the ratio of the thermal energy to the laboratory H atom energy which must be much less than one in order that the thermal velocity can be ignored. This question was discussed some time ago by BernsteinZoand Chantry.zl In order to represent the experimental results fairly, they are presented in Figures 1 and 2 as points at the center of a rectangular box. The height of such a box is two standard deviations of the cross section and the width is the full-width at half-maximum of the E T distribution. The details of the calculation of the width are given in the Appendix. Doppler Spectroscopy of H2S and HCI. Figures 3 and 4 show the Doppler-broadened fluorescence excitation spectra of the H atoms dissociated from HzS at 193 and 248 nm. The dynamics of the photodissociation of HzS has been studied by Hawkins and (20) Bernstein, R. B. Comments At. Mol. Phys. 1970, 4, 43 (21) Chantry, J. J . Chem. Phys. 1971, 55, 2746.
Houstonz4and by van Veen et aLZ5 The former workers showed that the S H radicals are generated in a rotationally cold state; in other words if the parent HzSmolecule has an average of 3kT/2 rotational energy, the diatomic fragment will have on average only a little more than kT of rotational energy. The latter workers showed that at 193 and 248 nm about 10% and 5% of the SH radicals were in states with u > 0. They also showed that the electronic transition moment at these wavelengths was perpendicular to the molecular plane. Much but not all of the above information can be obtained by Doppler spectroscopy, that is, by measuring the shape of the Doppler broadened excitation line. The observations as shown in Figure 4 are that the absorption curves look like inverted parabolas superimposed on rectangles whose sides are located at approximately 4 and 6 cm-' above and below the center of the line when the HzS is dissociated at 248 and 193 nm, respectively. On adding krypton the parabolic part of the curve disappears and the absorption becomes quite flat. Figure 5 shows the results of a typical experiment in this case carried out by photodissociating HCl at 157 nm. The signal/noise ratio is poorer than for HIS at 193 nm because the excimer laser is weaker at 157 nm. One does see that the line width is significantly greater than in Figure 4a and that under these conditions after 125 ns only a small fraction of the H atoms have reacted to produce a D atom.
Discussion Reaction Cross Sections. Figures 1 and 2 show that there is virtually complete agreement between the experimental and classical theoretical cross sections. The cross sections rise with increasing energy above the threshold and thewlevel off. We will (22) Xu,Z.; Koplitz, B.; Buelow, S.; Baugh, D.; Wittig, C. Chenz. Phys. Lett. 1986, 127, 534. (23) Tsukiyama, K.; Bersohn, R. J . Chem. Phys. 1987, 86, 745. (24) Hawkins, W. L.; Houston, P. L. J . Chem. Phys. 1980, 73, 287. (25) van Veen, G. N. A,; Mohamed, K.; Baller, T.; deVries, A. Chem. Phys. 1983, 74, 261.
5448
Johnston et al.
The Journal of Physical Chemistry, Vol. 91 No. 21, 1987 ~
\
(
248nm Kr 0 mtorr
m A t = 120ns
+6 t8 -6 -4 -2 0 +2 + 4 +6 V U V LASER TUNING (cm-') Figure 4. The fluorescence excitation line shape for H atoms dissociated from HzS (a) at 193 nm, (b) at 193 nm with addition of krypton, (c) at 248 nm, and (d) at 248 nm with addition of krypton. Note that Figure 4a is indistinguishable from Figure 3. -6
-4
-2
0
+2
+4
H
I
I
I
I
I
82260 82280 VUV L A S E R WAVENUMBER ( c m - I ) Figure 5. This is the result of a typical experiment. The curves shown are the fluorescence excitation spectra for the H and D atoms in the reaction H + D2 = HD + D. The measured quantities are the delay time ( t = 125 ns) between the dissociating and probing laser pulses and the ratio of the areas under the D and H excitation spectra. HCI at a pressure of 13 mTorr was dissociated at 157 nm in the presence of D2 (129 mTorr).
summarize the reasons for these two effects based on the trajectory calculation^.^-^ The potential barrier for the reaction H + D2 = H D + D is 8 kcal/mol when the reaction path is collinear. When there is a sidewise approach the barrier to reaction is usually said to be in the range 60-80 kcal/mol. However, this barrier is calculated by fixing the D-D distance at the equilibrium value of 0.7 A for the isolated molecule. As pointed o u t by Schechter
and Levine a more realistic and lower barrier is obtained by allowing the deuterium atoms to reach a new equilibrium value (=1 A) in the presence of the sidewise approaching H atom.6 These workers showed that reactive trajectories could be categorized according to whether the D2was compressed or expanded at the transition state. The former corresponds to abstraction and the latter to insertion. Just above the threshold energy only nearly collinear reactions are possible and the reactive cross section is correspondingly small. As the relative energy increases larger angles of attack can lead to reaction and the crms section increases. Insertions become possible. The kinetic energy over the low barrier is so great that the system often crosses the barrier, bounces off the potential wall, and is reflected across the barrier. In other words, the system cannot turn the corner on the potential surface. In the 2-3-eV range insertions dominate because the system has low kinetic energy as it passes over the higher barrier. Above 3 eV the cross section is predicted to decrease again because of reflection. As Malcolm-Lawes has at higher energies there is a double collision. The attacking H atom is slowed by collision with one of the atoms of the diatomic and it can then form a bond with the second atom. At sufficiently high energies the incident H atom cannot slow sufficiently to react. Also, above 4.5 eV the channel for collision-induced dissociation will open. The cross sections for the reaction H HD = H2 + D are very close to half the values for the reaction H + D2 = H D + D. Again (see Figure 2) there is excellent agreement with the values calculated from classical trajectories. Another reaction can occur
+
H
+ HD = H D + H
(8)
which is predicted to have a higher cross section6 but which we are unable to measure. However, if a change in the nuclear spin direction of the H atom proton could be measured, this would be a proof that a reaction had occurred. The hyperfine interaction is too weak and the duration of the collision too short t o cause a spin flip of a nonreacting H atom. ( 2 6 ) Malcolm-Lawes, D. J. J . Chem. SOC.,Faraday Trans. 2 1975, 71, 1183.
Isotopic Variants of the H
+ H2 Reaction
Single-Collision Conditions. The claim is made for these experiments that they are carried out under single-collision conditions. This can be proven in several ways. Let us assume that the cross section for the scattering event in which the H atom loses considerable energy is at most the geometrical cross section because the forces are of short range. Taking 4 A’ for the cross section, a typical number density of 3 X 1015molecules/cm3, and a typical relative speed of 2 X lo6 cm/s, one finds that the number of collisions per second is FWU = 2.4 X lo6. Assuming the same speed and a reactive cross section of 1 A2,one finds that only 5% of those which have collided nonreactively with a D2 molecule are able to collide reactively subsequently with a Dz molecule within a typical time of 100 ns. An even stronger proof is given by a comparison of Figures 3 and 4a which show H atom shapes in the absence of added gas and in the presence of 110 mTorr of HD. There is no appreciable change which shows that, under the quoted conditions, the velocity distribution does not have time to relax. Choice of the Hydrogen Atom Precursor. The ideal molecule which would generate hydrogen atoms on photodissociation would be a molecule H X such that on photodissociation the fragment X would be left in a single quantum state. In this case the H atom kinetic energy would be known as well as possible. Recent work on the photodissociation of larger molecules such as NH3,” PH3,2’ C2H2,” and toluene, C6H5CH3,23has shown that there is a broad distribution of kinetic energies of the H atoms produced by photodissociation. This is natural in view of the many vibrational modes of the X fragment. It has been found that for HzO at 157 nm and H2S at 193 nm there is little vibrational or rotational excitation of the O H or SH fragments. Likely candidates are therefore diatomic and triatomic hydrides, that is, the series HzO, H2S, H2Se, H2Te, and the hydrogen halides, HI, HBr, HCl, and HF. The hydrogen halides all suffer from the problem that they are dissociated into halogen atoms which are in both the and the ’P3/’ states. For HC1 the difference in energy between these two states is only 0.1 1 eV and a theoretical calculation predicts that at 193 and 157 nm 81% and 71% respectively of the C1 atoms will be in the ground 2P3/2state.’* There is a systematic rather than a random error in the neglect of the minority of excited C1 atoms but it is smaller than the random errors discussed in the Appendix. We are helped further by the fact that the cross section is a relatively flat function of the energy. HBr and H I have two drawbacks. Their spin-orbit splittings are large, 0.46 and 0.94 eV, respectively, and the population ratio is not known at the wave lengths 193 and 157 nm. In addition they exhibit a strong vacuum-UV fluorescence when irradiated a t 121.6 nm. For H I a Lyman cy filter was found to remove some but not all of the fluorescence. In an MPI experimenti4 this emission would not be a problem. Doppler Spectroscopy of H2S and HCl. The translational distribution of the H atoms dissociated by 193-, 222-, and 248-nm light from HzSmolecules in a molecular beam has been measured by using a time-of-flight method by van Veen et al. The H atoms were shown to have a sharp translational energy distribution. The Doppler-broadened profiles (see Figure 4) of the fluorescence excitation spectrum leads to the same conclusion. An atom moving with velocity component v, = v cos 0 toward the observer will have an absorption at v = uo( 1 v cos 8u,k/C) where 80,k is the angle between the observation direction (beam of the probing laser) and the velocity. For an isotropic distribution of velocities, cos Ou,k takes on all values uniformly between -1 and +1 and therefore the absorption should have a retangular shape. The spectra of Figure 4, a and c, clearly do not have a rectangular shape. The reasons are that the incident light comes from a unique direction and not from all directions a t once and that the H2S molecule absorbs light anisotropically. The transitions excited at both 248 and 193 nm is believed to be a promotion of
+
(27) Wcdtke, A. M.; Lee, Y. T. J . Phys. Chem. 1985,88,4744. (28) Givertz, S. C.; Baht-Kurti, G. G. J . Chem. SOC.,Faraday Trans. 2 1986, 82, 1231.
The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 5449 F(B):I-1/4 Pz ( C O S 8 )
0 */2 7r - 8 Figure 6. The solid line is a plot of the function 1 - ‘/4P2(cose). The dotted line is a rectangle with area equal to that under the function.
a 3 p r electron on the sulfur atom to an antibonding u* orbital. Thus the transition dipole moment is normal to the molecular plane. The probability of absorption is proportional to ( 3 / 4 ~ )COS’
OE,n
+
= ( 1 / 4 ~ ) { 1 ~ P ~ ( c OE,n)) os
(3)
where BE,n is the angle between the E vector of the excimer light and the normal to the plane of a given molecule. Of more interest is the angle the angle between the velocity of the H atom and the E vector. Using the addition theorem for Legendre polynomials one finds (PZ(COS O E , ~ ) )=
COS b , u )
PZ(COS ou,n)
(4)
The average is over all azimuthal angles of n which causes the remaining terms (not shown) on the right-hand side of eq 4 to vanish. Because the H atom velocity is in the molecular plane, P ~ ( C OOU,J S = -l/z. What is finally needed for the distribution over frequency of absorption is the distribution over angles Ou,k. To obtain this distribution we use the Legendre polynomial addition theorem once again with the result (pZ(cOs O E , u ) ) = (pZ(cOs o E , k ) ) pZ(cos
ou,k)
= ‘/,p2(cos ou,k)
(5) The value is used for (Pz(cos 8E,k)) because in our experiment the probing Lyman a laser beam is perpendicular to the unpolarized excimer laser beam. Combining eq 3, 4,and 5 we find that the desired distribution function is (6)
y4r(1 - ‘/,p2(c0s o u , k ) )
This equation can be transformed into an absorption line shape by substituting cos 8U-k = c(v - VO) /YOU One then has F(u) = (1/4,,0
-
Y4(
3cyv 2v2v02 - vo)’
-
;))
(7)
This function, plotted in Figure 6, is the superposition of an inverted parabola on a rectangle which is the general shape of Figure 4, a and c. A further confirmation of this theory is furnished by an experiment in which sufficient krypton gas is added so that within the measuring time of 100 ns about one collision occurs. The potential energy of a H atom and a rare gas atom is weakly attractive at large distances and strongly repulsive at short distance^.'^ The fast H atoms considered here are hardly affected by the weak attraction and are scattered by a potential very close to that of a hard sphere. This scattering is isotropic and therefore one collision is sufficient to make the velocity distribution isotropic. At the same time because the krypton atom is about 80 times heavier than the H atom, the collision is like that of a ping pong ball with a bowling ball; little kinetic energy (29) Toennies, J. P.; Welz, W.; Wulf, G. J . Chem. Phys. 1979, 71, 614. (30) Truhlar, D. G.; Wyatt, R. E. Annu. Reu. Phys. Chem. 1976, 27, 1. (31) Schechter, I. Ph.D. Thesis, Hebrew University, 1987.
Johnston et al.
5450 The Journal of Physical Chemistry, Vol. 91, No. 21, 1987 TABLE 11: Energies as a Function of Dissociation Conditions
parent molecule H 2s HCI HIS HCI
A,
nm
248.4 193.3 193.3 157.6
EREL,'eV 1.11 f 0.11 2.01 f 0.03 2.54 f 0.1 1 3.46 f 0.03
E,, eV 1.08 0.11 1.95 f 0.05 2.46 f 0.12 3.36 f 0.06
*
ERELis the relative kinetic energy of the two photodissociated fragments.
is lost. The shape of the curve is significantly flattened but the overall width is negligibly altered. The flattening is a confirmation that the original distribution was peaked in the center and that therefore, following the line of reasoning expressed by eq 3-7, the transition dipole at 193 and 248 nm is indeed perpendicular to the molecular plane. The same result was obtained by van Veen et aLZ5from the variation of H atom fragment intensity with the angle of polarization of light relative to the direction of a molecular beam. The kinetic energies of the H atoms dissociated from H2S at 248 and 193 nm (see Table I) are 1.08 f 0.1 1 and 2.46 f 0.12 eV with corresponding speeds u of 14.4 f 0.7 and 21.7 f 0.8 km/s and maximum Doppler shifts (u/c times 82259 cm-I) of 3.95 f 0.21 and 5.95 f 0.22 cm-I. The reasons for the lack of perfect rectangular shape of the spectra in Figure 4, b and d, are ( 1 ) there is a variation in H atom speeds due to thermal effects, (2) the dye laser fundamental width of 0.06 cm-I leads to a vacuum-UV width of 0.2 cm-I, the same magnitude as the thermal width, and (3) 5-10% of the H atoms have slower speeds and therefore smaller shifts than those calculated above. Nevertheless, the good agreement between the calculated and observed maximum shifts (see Figure 4) shows that indeed almost all of the available energy in this photodissociation is released as translational energy. Measurements on HC1 were not as extensive as those on HzS; in particular, the curves shown in Figure 5 were obtained without an etalon. Nevertheless, there is agreement with a width of 6.95 cm-I predicted for H atoms dissociated from HC1 at 157 nm.
Conclusions Under favorable circumstances it is possible to study the dependence of reaction cross section on translational energy using a gas contained in a bulb rather than a pair of crossed molecular beams. The techniques used here can be applied to other H,D exchange reactions as well. A future paper will present information which is extracted from the line shape of the atomic product and its variation with polarization direction of the dissociating light. Acknowledgment. This work was supported by the U S . Department of Energy.
Appendix. Breadth of the Distribution of Relative Translational Energies, ET The relative incident translational energies listed in Table I are average values. Their uncertainty arises from the fact that the distribution of the energy, EH, of the H atom in the laboratory frame is broadened by several effects and that for any given EH there is a distribution of relative translational energies because of the thermal velocity distribution of the hydrogen molecules, D, and HD. First we consider the dissociation process. The conservation of energy law is expressed by the equation hu + Eint(HX) = DO+ EREL+ E,nt(X)
(All
That is, the photon energy plus the internal energy of the H X parent are equal to the dissociation energy from the lowest quantum state plus the relative kinetic energy of the H and X fragments and the internal energy of the X fragment. The desired E R E L is a sum of four quantities, each of which has some uncertainty, random or systematic. The photons with energy hu are derived from excimer lasers nominally at 157.6, 193.3, and 248.4 nm which are not perfectly monochromatic. Typical full-widths
at half-maximum are 0.3 nm so that the corresponding uncertainties in energy are f0.003,fO.005, and f0.007 eV, respectively. The internal energy of the H X molecule is rotational energy because at r c " temperature the vibrations are negligibly excited. The average rotational energies of HCl and H2S molecules at room temperature T are k T and 3/2kTof rotational energy; one could assume that the departing SH radicals carry off about k T of rotational energy. There will be a net contribution of k T and kT/2 to E R E L for HCl and H2S, respectively. The rms fluctuations in these quantities are also k T and kT/2, Le. 0.026 and 0.013 eV, respectively. The dissociation energies of HCI and H2S are 4.433 f 0.002 and 3.84 f 0.1 1 eV, respectively. The internal energy of the CI atom is uncertain to the extent that a fraction of C1 atoms are produced in the 2Pl,2state 881 cm-' or 0.109 eV above the ground 2P3jzstate. There are no experimental values for the partitioning of the C1 atom states when HCl is dissociated at 193 and 157 nm. However, a theoretical calculation by Givertz and Balint-Kurti2' predicts that at these two photon energies 81 and 71% of the C1 atoms are produced in the ground state. Photodissociation studies of jet-cooled HzS have shown that 90 to 95% of the H atoms are produced with SH companions which are rotationally and vibrationally cold. As a consequence of the above information we have assumed that all of the X fragments have zero internal energy for the purposes of this experiment. Because of the slight dependence of cross section on energy, this turns out to be a harmless assumption. At this stage we find that the square root of the sum of the squares of the uncertainties in the relative energy E R E L discussed above is +0.026 eV for the HC1 dissociation and +O. 117 eV for the H2Sdissociation. The major uncertainty for the HCI is just the thermal fluctuation in the rotational energy whereas the major uncertainty for the H2S dissociation is due to the fact that the bond energy of H2S is not known to high accuracy. Having determined the uncertainty in the relative translational energy produced by the dissociation, we must now calculate the energy of the H atom in the laboratory frame, E H . The velocity of the hydrogen atom in the laboratory system is just the vector sum of its velocity with respect to the center of mass of the HX molecule and the velocity of the center of mass. The two velocities are uncorrelated so that there is an additional uncertainty induced in E H . The equation for conservation of momentum is
P
= p'H
+ p'x
(A21
and the relative momentum p' is defined by the equation m n G = -mGH
+ mGX
(A3)
The kinetic energy of the H atom in the laboratory system is given by
where p is the reduced mass mHmXJmHX. The second term on the right-hand side of eq A4 is negligible and will be discarded but the third is substantial. An average value of the magnitude of the momentum P of the H X molecule is (8kTmHx/r)'lZand the magnitude of the relative momentum is (2pi?REL)1/2. Substituting these values in eq A4 one finds
cos Bp,p, the cosine of the angle between the relative momentum of the fragments and the momentum of the parent molecule takes on all values uniformly between -1 and + l . To calculate the The values of E H and uncertainty, cos Bps is replaced by E R E L are listed in Table 11. Having determined the energy of the hydrogen atom in the laboratory frame, we must now calculate the relative translational energy, ET, of the hydrogen atom and the diatomic molecule with which it reacts. If we let PH and p D , be the momenta of the H atom and D2 molecule, respectively, then
J. Phys. Chem. 1987, 91, 5451-5455
= mD1/mHDgH2/2m~ +~
H / ~ H D # D ~ ’ /~ 8rpfD2/mD2 ~ D ,
= mD1/mHDlEH + m H / m H D ~ -
@ H I ~ I ~ I / ~ cos H D ‘%,D2 ~
(A6)
The thermal kinetic energy of the D2 molecule appears in the second term on the right-hand side of the equation. Its average is 3/2kTwith a mean square fluctuation of the same amount. cos 6 H D l is the cosine of the angle between the momenta of H and D,. Because the scattering cross section is a function of the relative energy this cosine does not take on all values uniformly between -1 and + l . However, the energy dependence is weak and we will assume this merely to estimate the uncertainty in ET. For lpD21
5451
we take an average (8kTmD2/r)‘/2. The energy ET in eq A6 is the relative kinetic energy of an H atom and a D2 molecule. To obtain the relative kinetic energy for an H and H D collision, mDl is replaced everywhere by mHD in eq A6. The values of EREL, EH, and ET calculated from equations in this appendix are given in Tables I1 and I. The outstanding conclusion is that the relative incident kinetic energy in these experiments is uncertain to the extent of +10 to 15%. There would be some improvement if the dissociation energy of H2S were more precisely known but the major problem is that the thermal velocity makes an appreciable and varying contribution to the relative kinetic energy. Registry No. H, 12385-13-6; D,, 7782-39-0; HD, ‘ 1983-20-5; HIS, 7783-06-4.
Dynamical Stereochemistry in Nonreactive Collisions A. J. McCaffery School of Chemistry and Molecular Sciences, University of Sussex, Brighton BNI 9QJ, U.K. (Received: March 6, 1987)
Information on stereochemical forces in molecular collision dynamics may be obtained by using high-resolution polarized fluorescence in molecules in thermal cells. Particular attention is paid to diatomics in Z states where laboratory-frame experiments in conjunction with theoretical models, rigorous or simple, allow a collision-frame determination of a preferred collision geometry. In the case of II-state diatomics and triatomics, the presence of optically accessible molecule-frame vectors permits direct determination of stereochemical preferences from simple thermal- or flow-cell experiments.
Experiments that are sensitive to the angle dependence of the intermolecular potential in reactive and nonreactive collisions are generally both difficult and expensive, and a perusal of contributions from Fifth Fritz Haber Symposium rapidly will convince the reader of this. Molecular-beam techniques in conjunction with inhomogeneous electric and magnetic orienting fields are often essential in any attempt to obtain stereochemical data at the single-molecule level. However, this level of sophistication is not always necessary, and this contribution demonstrates that useful stereochemical information may be obtained from precise polarization measurements in conjunction with very high spectroscopic resolution on molecules in simple thermal cells. The use of narrow-line, tunable lasers enables the experimenter to select a wide range of input channels, and high-resolution detection using either a second laser or emission spectroscopy allows many exit channels to be sampled. Velocity dependence in the collision frame may be obtained through laser scanning of line profiles. Furthermore, although it is intuitively clear that direct interpretation of laboratory-frame spectroscopic experiments in the molecular frame is most readily achieved when the electric vector of the light wave can be “tied” to or correlated with a vector that is fixed in the molecule frame, here we demonstrate techniques for interpreting laboratory-frame observations in atom-diatom collisions in terms of molecule-frame events.
and NaK,3 and Se, and Te; demonstrated that the phenomenon is widespread. Double-resonance work on Ba05 and a combined laser-molecular-beam investigation of Na? have confirmed that the m quantum number is remarkably slow-changing following collisional interactions. Theoretical interest in this problem revolved around the evolution of criteria for evaluating approximations to the full quantal treatment of the atom-diatom scattering problem. For example, exact close-coupled calculations on a variety of atom-molecule systems7-” indicated a strong propensity to conserve m in space and collision frames for elastic collisions, but no propensity was observed in calculations on inelastic processes until Khare et chose either the kinematic or the geometric apse as the quantization axis. An interesting feature of these calculations was that sets of u(jm -j’mq for other quantization axes could be obtained by rotating the dominant u(jp -J’p’)Am = 0 cross section from the geometric or the kinematic apse into the appropriate frame. This conservation of angular momentum projection for the case of strongly repulsive interactions may be demonstrated by using classical a r g ~ m e n t s that ’ ~ show that there can be no change of rotor orientation along the momentum transfer vector in the “sudden” limit. However, when long-range anisotropic interactions become important, the situation
1. Polarization in Elastic and Inelastic Collisions With the availability of stable single-mode dye lasers for studies of energy transfer and the consequent ability to prepare single quantum levels in the input channel, fully resolved polarized fluorescence investigations revealed a surprising result. This was that the magnetic quantum number m in the laboratory frame is changed only very slowly as a result of collisions. This result NaLi was shown first for the case of IZ.l Later studies on
(3) McCormack, J.; McCaffery, A. J. Chem. Phys. 1980, 51, 405. (4) Ibbs, K. G.; McCaffery, A. J. J . Chem. SOC.,Faraday Trans. 2 1981, 77, 637. ( 5 ) Silvers, S.; Gottscho, R.; Field, R. J . Chem. Phys. 1981, 74, 6000. (6) Mattheus, A.; Fisher, A.; Ziegler, G.;Gottwald, E.; Bergmann, K. Phys. Rev.Lett. 1986, 56, 712. (7) Kouri, D. J.; Shimoni, Y . ;Kumar, A. Chem. Phys. Lett. 1977, 52, 299. (8) Monchick, L. J. Chem. Phys. 1977, 67, 4626. (9) Alexander, M. H. J. Chem. Phys. 1977, 66, 59. (10) DePristo, A. E.; Alexander, M. H. J . Chem. Phys. 1977, 66, 1334. (11) Schaefer, J.; Meyer, W. J . Chem. Phys. 1979, 70, 344. (12) (a) Khare, V.; Kouri, D. J.; Hoffman, D. K. J. Chem. Phys. 1981, 74,2275. (b) Khare, V.; Kouri, D. J.; Hoffman, D. X. J . Chem. Phys. 1982,
(1) Kato, H.; Jeyes, S . R.; Rowe, M. D.; McCaffery, A. J. Chem. Phys. Left. 1976. 39. 573. (2) (a) Rowe, M. D.; McCaffery, A. J. Chem. Phys. 1978, 34, 81. (b) Rowe, M. D.; McCaffery, A. J.; Chem. Phys. 1979, 43, 3 5 .
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76. 4493.
(13)McCaffery, A. J.; Proctor, M. J.; Whitakel, B. J. Annu. Reo. Phys. Chem. 1986, 37, 223.
0022-365418712091-5451$01.SO10 0 1987 American Chemical Society
