Determination of Multiquantum Vibrational Transition Moments in Hydrogen Iodide and their Significance in its Vibrational Photochemistry Citation Reiner, Robert Henry (1976) Determination of Multiquantum Vibrational Transition Moments in Hydrogen Iodide and their Significance in its Vibrational Photochemistry. Master's thesis, California Institute of Technology. doi:10.7907/ay4m-0474. https://resolver.caltech.edu/CaltechTHESIS:04072010-075012259 Abstract Estimates of the band intensities of transitions between the ground and upper vibrational levels (1-9) were made for HI. Calculations were performed with both Morse and RKR oscillator wavefunctions, using various functional approximations for the dipole operator fitted to known experimental results. Absorption measurements using a multiple pass cell placed upper bounds on transition intensities, which were not inconsistent with the calculated values. (Band strengths for HI ΔV = 8,9 are less than 5 x 10(-4) cm^(-2) atm(^1)). The feasibility of using these multiquantum vibrational transitions to pump the reaction HI + HI*(v - 8,9) → H_2 + 2I is discussed. Item Type: Thesis (Master's thesis) Subject Keywords: (Chemistry) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): Kuppermann, Aron Thesis Committee: Unknown, Unknown Defense Date: 18 December 1975 Record Number: CaltechTHESIS:04072010-075012259 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04072010-075012259 DOI: 10.7907/ay4m-0474 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5696 Collection: CaltechTHESIS Deposited By: Tony Diaz Deposited On: 19 Apr 2010 18:10 Last Modified: 16 Aug 2024 22:01 Thesis Files Preview PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

Copyright © by Robert Henry Reiner 1976 Determination of Multiquantum Vibrational Transition Moments in Hydrogen Iodide and Their Significance in its Vibrational Photochemistry by Robert H. Reiner In Partial Fulfillment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California 91125 1976 (Submitted December 18, 1975) Ti ACKNOWLEDGMENTS This thesis is dedicated to my parents, Mary and Bob, for their many years of sacrifice on my behalf. Professor Aron Kuppermann has been the principal investigator in this research and I thank him for his support. I would also like to thank George Schatz, Jackie Berg, and Tom Orlowski for their friendship and professional interaction during the course of this research. ABSTRACT Estimates of the band intensities of transitions between the ground and upper vibrational levels (1-9) were made for HI. Calculations were performed with both Morse and RKR oscillator wavefunctions, using various functional approximations for the dipole operator fitted to known experi- mental results. Absorption measurements using a multiple pass cell placed upper bounds on transition intensities, which were not inconsis- tent with the calculated values. (Band strengths for HI Av = 8,9 are less than 5 x 10 * cm 2 atm !.) The feasibility of using these multiquantum vibrational transitions to pump the reaction HI + HI*(v = 8,9) +H, + 21 is discussed. Il, Il. IV. VI. VII. VIII. IX. iv TABLE OF CONTENTS Introduction Theory Calculation of Band Strengths A. Potential Energy Curves B. Wavefunctions C. Dipole Moment Operator Experimental Determination of Band Intensities Results Discussion Appendix I Appendix II References Page 11 11 28 33 34 40 57 59 I. Introduction The availability of high intensity, tunable dye lasers has created a renewed interest in visible and near UV photochemistry (300-800 nm). In addition to studying the photochemistry of molecules with low lying electronic states, the availability of an intense, very monochromatic light source in this region could conceivably use "forbidden" multi- quantum rotational, vibrational transitions to populate upper vibrational levels of the ground electronic state, allowing one to study the effect of vibrational energy on chemical reactions. Indeed, photochemistry of this type has been done, using a chemical laser to excite an “allowed” transition in Her. The best choice for such a molecule would be a diatomic. This would simplify problems of intramolecular energy transfer after an excited ro- tation vibration state is produced, making the kinetics tractable. Since the intensity of a vibrational transition decreases markedly as Av (the. change in vibrational quantum number) increases, the vibrational spacing should be large so that av can be kept small when attempting to pump a reaction involving this molecule with visible light. Finally, the activa- tion energy for.a bimolecular reaction involving it should be sufficiently small for the added vibrational energy to be a significant fraction of the activation energy. Hydrogen iodide was chosen with the above considerations in mind. The fundamental band absorption is observed at ~2300 cm !. Observation of the first, second, and third vibrational overtones indicates that absorption of visible light would correspond to transitions in which Ay = 758 or 9.° mole of vibrational energy. The activation energy for the reaction HI + HI + Hy + 21 (A) is 44 kcal/mole. °?* If it is possible to produce a significant steady state population of HI*(v = 7, 8 or 9), then one could study the vibrationally excited analog of reaction A, HI* + HI + Hy + 21 (B) The feasibility of observing reaction B depends on the magnitude of the transition moment, <8|M(R)|0>, where M(R) is the dipole operator as a function of internuclear distance, R. Calculations and experiments performed to measure this moment are presented. The feasibility of studying reaction B in bulk gas phase is The 0-8 absorption corresponds to approximately 45 kcal/ then discussed, and results of preliminary attempts for measuring its rate are presented. I. THEORY The strength, sv . > , of a vibrational transition between states v and v’, produced by interaction of an electromagnetic field with a molecule, can be shown (by time-dependent perturbation theory) to be proportional to the square of the transition moment. 5,6 That is, sv Po M « |v, J,M|ylv, J’,M’) |? 2 = |f¥5 3 mB DER) mn Bar|, (1) where quantum numbers J and M specify the rotational angular momen- tum and its projection on the laboratory z axis. Since eigenfunctions wy, J, Mik; r) and ¥y J,’ (R,r) are functions of nuclear and electronic coordinates, the integration specified in Eq. (1) must be performed over both sets of coordinates (d7 = dRdr). 6 In the Born-Oppenheimer approximation, the electronic motion is assumed to be independent of nuclear motion; this means yy, 3,M‘B; r) may be separated into a product of a pure electronic wavefunction Up(r) and a nuclear wavefunction Xy, g,M{B); HR r) = VRMXy, 3,M‘8)- Xy, g, M8) is written as Xv 3,MR) = O/B, RY yy (8, 9), (2) a product of a spherical harmonic Yy Mies o), whose arguments refer 3 to the orientation of R with the laboratory axes, and VY, yf) a solution > to the radial Schrédinger equation dy, 3(R) a. + 2M fe - vin) -BIG+D py yr = 0. (3) dR fi 2mR ? In Eq.(3), V(R) is the mechanical potential, J is the rotational quantum 7 number, and m is the reduced mass. Similar statements hold for Wyn J’ , M’ (R, r) . : In a diatomic molecule, the dipole moment, p, must be directed along the internuclear axis. From Eqs. (1), (2), and the Born-Oppen- heimer approximation we get Wi TM|nIV IM) © XY, 5 RMR) Xy, 3, yp (BAR (4) x {Wy (RMR), y(R)GR [YF (6 9) x RY yr yy (8, d)d2 (5) « (v, 3|M(R) |v’, 3")¢3,M|R| J’, M’) 2 (6) where MR) = fUR@uR,NURdr = MIR. (7) When Eq. (6) is squared, summed over all M’, and averaged over all M, v’,.J’ Sy) | total band strength, sy , of a vibrational transition between states v and the intensity of a particular rotational line, is obtained. The v’ is defined as the sum of the intensities of all rotational vibrational transitions in the band and is denoted by S(v~v’) or simply S. There- fore, knowledge of M(R) and the nuclear radial wavefunction Wy, (8) suffice to evaluate S. If V(R) is known, Eq.(3) can be integrated to give Vy jfR). Band strengths may then be computed using Eq.(7) and an assumed form of M(R). Rather than compute wavefunctions and transition moments for each rotational state in the vibrational band, we assume that in the rotational term in Eq.(3), R can be replaced by its equilibrium value for J=0. The corresponding Wy. jf) is then essentially independent of J and the total nuclear energy becomes the sum of vibrational and rotational energies with no vibrational rotational coupling. This simplification is a very good approximation. Herman and Wallis” h have shown that this approximation introduces into the gt rotational transition moment a relative error of order J-B °/ where @? B e is the rotational constant at equilibrium nuclear separation, and w e is the vibrational constant at equilibrium position. For HI, B /# e® 0.003. Calculations performed on the first three vibrational bands showed that this simplification produced less than 10% error in HI band strengths. Since this is much less than the error produced by extrapo- lation of M(R) to transitions of higher Av, this simplification is entirely justified. - Computation of transition moments becomes a straightforward problem of determining eigenfunctions of the potential, V(R), which are then used in Eq. (7) to‘evaluate (v|M(RY|v’) for a particular choice of M(R). III. Calculation of Band Strengths A. Potential Energy Curves One of the spectroscopic problems with the hydrogen halides is the absence of a low lying bound electronic state which emits to the ground state, 10 As a result, emission spectra cannot be used to determine the energies of all the vibrational eigenstates. Hence, the RKR method can- not be used to generate a complete potential energy curve (as, for example, has been done for Na>).t! Instead, using >» 0X, OS e> WeZo determined from absorption ee 2,12 experiments, the positions of the first eleven eigenstates were calculated. They are listed in Table I. The RKR procedure of Demitroder, i was then used to determine the classical turning McClintock, and Zare points for each calculated eigenvalue. The smoothed potential curve drawn through these points using a cubic spline method is shown in Figure 1. For comparison, the HI Morse potential was used. The parameters listed in Table II were used to construct the Morse curve also shown in 13 While both curves agree near the bottom of the well, above Figure 1. v=3 they begin to diverge. This is not unexpected, since it is well known that the Morse potential fails to describe the molecule accurately 14 The RKR potential may not be exact near the dissociation limit. either, even if the eigenvalues used to generate it are exact, since a potential which reproduces the energy levels of a molecule is not unique. !° However, as shown Jater, use of either potential below v=10 Table I. Eigenvalues of HI(cm'!). Calculated from Integration of Integration of spectroscopic data* Morse Potential RKR Potential 0 1144.59 1140.90 1144.04 1 3374.17 3344.70 3372.86 2 5523.86 5480.64 5519.62 3 7592.63 7439.43 7584.13 4 9579.31 9331.24 9565.15 5 11482. 20 11119.42 11463.02 6 13299. 22 12804.35 13275.39 7 15027. 93 14386. 84 15001.45 8 16665. 48 15866.10 16641.21. 9 18208. 63 17242.92 18193.06 * See reference 2. Table II. Morse Parameters for HI. V(R) = D,(1 - @7P{R-Re) y2 e A where g =1.2177 x 107 eV a De For HI Do = 25843.41 cm } Ro = 3.04069 bohr 8 = 0.92520 bohr ! Figure 1. HI potential energy, V(R), is plotted as a function of internuclear separation, R. The Morse parameters are listed in Table II. RKR turning points shown in solid circles,@. Eigenvalues are shown for each potential as well as the first five Morse eigenfunctions. V (R) (cm! x 10%) 2.0 10. ll. is justified in this work, because the error in approximating the dipole moment operator exceeds the difference between calculations using the different potentials. B. Wavefunctions Equation 3 is integrated for both potentials by the finite differ- ence method. The resulting matrix equations are solved using the 16 Eigenvalues calculated in this manner are Givens-Householder method. shown for both potentials in Figure 1 along with some Morse eigenfunc- tions. The integration procedure was checked by comparing the eigen- values used to generate the RKR potential with those calculated from equation 3 using that potential. As shown in Table I, they agree to within 0.2 percent. C. Dipole Moment Operator Determination of M(R) as a function of internuclear distance R is the greatest source of error in the band strength calculation. We have fitted several trial functions, M(R), to measured band intensities. Other band strengths are then calculated using this M(R) and compared. First we attempted to approximate M(R) by M(R) = M(R) = A-eRe7B°R (8) Equation 8 satisfies the necessary boundary conditions 12. M(O) = 0 (9) and M(o) = 0 (10) 17 However, when A and B were adjusted to yield the measured fundamental 18 and first overtone intensities, a third condition; <0|M(R) |O> = Mg® . (11) where Mj° is the permanent dipole moment, was not met. We next tried an operator of the form M(R) = M(R) = ARMe~BR | (12) where A,B and n were fitting parameters. When A,B and n were adjusted to give curve A in Figure 2 (chosen to match the leading coefficients in the Taylor series expansion of M(R) about R,--see below), a disap- pointing fit to experimental results was observed (see Figure 3, curve A). Better results are obtained by expanding the dipole operator in a Taylor series about R,, the equilibrium internuclear separation, M(R) = Mo + M)(R-R,) + Mz (R-R,)? +o... (13) In this treatment, equation 13 is truncated at the appropriate order, My is approximated by My? and the remaining coefficients are expressed in terms of the fundamental and higher order band strengths. This yields a set of linear equations. -” Since the measured intensities are related Table IIIa. Dipole Moment Operator Approximations, M(R). 13. ; (Fig. 2-6) Potential A(R) for Mo? My* Equation A Morse Exponential ---- wre ee 12 B Morse Polynomial woe eee 13 C Morse Polynomial 2 woe nee 13 D Morse Polynomial 3 + =-~ 13 E Morse Polynomial 3 - ose 13 F Morse Polynomial 4 + + 13 G Morse Polynomial 4 + - 13 H - Morse Polynomial 4 - + 13 I ‘Morse Polynomial “4 - - 13 J Morse Polynomial 4 (Average F-I) 13 K Morse/RKR Pade wran + wae 16 L RKR Polynomial 2 eee een 13 M RKR Polynomial 3 + wae 13 N RKR Polynomial 3 - eee 13 14. ae 19/50 °0 €96600 *0 gv200°0 pG9T 0 eT N an GZ9T0°0- 6v£20°0 £8200 °0 vS9T 0 eT W lalate iia 7S€20°0 £61900 °0 vS9T "0 eT 7 (€289°0- = 4 ‘I¥GE'O = G ‘9PH6°0- = 9 ‘HETT*O- = 9 *8920°0 = Y) vS91°0 9T yt (I - 4d “$0 abeuany) vS9T ‘0 eI c 8LbT0 0 Z8TS0°0 £68900 0 G79200°0 pS9T “0 El I z6tt*0 9T6£00 *0 fO0E00 *0- £58S00°0: pS9T “0 eT H £L160°0- £1920°0 L8Z€0°0 T06£00 °0 vS9T "0 &L 3) €09900 *0 ET1Z0°0- L6420°0 OTTZ00°0 pS9T 0 eT J iat 09850 °0 GZ€800 "0 0€2200°0 vS9T*0 el | aie L08T0°0- 29820°0 €v6900 *0 vS9T*O eT q ae nooo GST0°0 TZOT0°0- vS9T 0 el ) woo nian GZ10°0 £9010 °0- vS9T ‘0 el q (8S62°0 = U *88TE°O = d “088T"O = Y) ral ¥ hwy EW can TW OW uot zenby peLaL (*n-e) squezsuo) (Y)W “suo. euwLXxouddy AozeuadQ JuawmoW aLodiq ‘“qII] algqeL Figure 2. Various approximations of the dipole operator, M(R), are plotted as functions of internuclear distance, R. Curve labels, 0, refer to Table III. 15. 0.9% 0.8K 16. 17. Figure 3. Total band strength, S, is plotted versus change in vibrational quantum number, Av, for Morse oscillator transitions from the ground vibrational state to state v. Curves A, B, and C are calculations described in text and Table III. Solid circles, @ , are experimental results and approximations explained in text. Bars shown for Av = 8 and 9 are experimentally established upper bounds for S. 18 1o7~e = lomd > 3.4 567 8 9 O Av 19. to the square of the matrix elements, there exists an ambiguity in the sign of the higher order moments, which can be resolved by measurement of the individual rotation lines in each band.°° Benesch has used this method to analyze data on the fundamental and first overtone. 18 Curve B in Figure 3 was obtained using Benesch's values for M, and M, (higher terms in equation 13 being omitted) and the Morse potential wavefunctions. It slightly overestimates the first overtone intensity. This is not unexpected since his operator was optimized for a different potential, V(R) = F(R-Rg)? - G(R-Ra)$ (14) The fortuitous agreement of curve B with experimental results is, in fact, only because Benesch's potential, equation 14, and the Morse potential are identical about R,. Choosing coefficients M; and Mj which give proper fundamental and first overtone band intensities for the Morse potential, one gets curve C in Figure 3. M, and M, so obtained are well within the experimental uncertainty proposed by Benesch. As seen in Figure 3, the curves are in good agreement, even when extrapolated to av = 9. The danger of using this type of approximation for the dipole operator in computing transitions beyond the data base is discussed by Truschka and Salwen.“! The truncation of a Taylor series expansion of any function about Ro becomes a poorer approximation to the function as (R-R, ) becomes larger. This is clearly seen in Figure 2 where several 20. polynomial approximations are plotted as functions of internuclear separation. When an nth order polynomial is found which fits the first n transition moments, the approximations agree quite well for small Ar, Ar = [R - Ral (15) As Ar increases, however, the various approximations diverge. Unfor- tunately, successively higher transitions "sample" the dipole operator at larger Ar. From experimental data on the 0+3 transition,-< two cubic polynomial approximations are generated (curves D and E in Figure 4) depending on the sign of My? = <O|M(R)|3>- The quadratic polynomial underestimates the 0+3 band intensity by an order of magnitude. Yet, the quadratic and cubic polynomials do not differ by more than that when extrapolated to AV = 9, 12 to Using the experimental conditions given by Haeusler and Meyer make an order of magnitude estimate of |My*|, four quartic approximations of M(R) are computed (curves F, G, H, I in Figure 5). Since the approx- imation of |Mo*| is crude,and since the signs of Mg? and My* are not known, perhaps it is better to compare the quadratic and cubic approxima- tions of M(R) with the average of all the quartic results (curve J). Not surprisingly, as the order of the polynomial approximation increases, the calculated curves begin to converge. Herbelin and Emanuel <2 have suggested using a Pade approximation for the dipole moment operator, . A(R-R,)* + B(R-R,) + C*Mo M(R) = (16) D(R-R,)? + E(R-Ra) + C Figure 4. Total band strength, S, is plotted versus change in vibrational quantum number, Av, for Morse oscillator transitions from the ground vibrational state to state v. Curve labels, 0, are described in text and Table III. 21. 22 i a © 1Q79 = rom 2 3 4 5 6 7 8 9 Av Figure 5. Total band strength, S, is plotted versus change in vibrational quantum number, Av, for Morse oscillator transitions from the ground vibrational state to state v. Curve labels, 0, are described in text and Table III. 23. 24 ‘anand | — | | 1O7e L 1073 = (2 1o~* tuo) WD) S IO7I KF lIO-6 1o7? 0 2345 6 7 8 9 Av 25. where the coefficients A, B, C, D, and E are known functions of the polynomial coefficients used in equation 13. Herbelin and Emanuel have shown that for HF, the Pade approximation gives better agreement with experiment than a cubic polynomial for transitions outside the data base of the polynomial. Presumably, the Pade approximation behaves more like M(R) than does the cubic approximation for large Ar. However, one should note that the Pade approximation puts an unphysical singularity in the dipole curve at 2 bohr (Figure 2). Evidently, one must still use caution when extrapolating the Pade results to higher Av. Curve K of Figure 4 shows the band intensities predicted by the Pade method. While only data from the fundamental, first and second overtones determine curve K, note that the Pade method predicts the third overtone strength previously guessed. !? Up to this point, only results for the Morse potential solutions of equation 3 have been discussed. Figure 6 compares band intensities cal- culated from Morse and RKR eigenfunctions. It is apparent that the form of the HI potential is not as critical as the form of the dipole operator.“ Indeed, the Pade results are identical for both potentials, and even for the quadratic polynomial, the intensities agree when Av < 6. While it is true that the form of the potential may change the calculated band strengths for large Av, it seems unproductive to search for better representations of V(R) until more information about M(R) becomes available. Figure 6. Total band strength calculations using the Morse potential are compared to calculations using the RKR potential. See Table III. 26. lon! 1o7-& Tomé 27 28. IV. Experimental Determination of Band Intensities The experimental measurement of weak optical transitions using a multiple pass or White cell is a well documented, thirty-year-old pro- cedure. °3 Herzberg, for example, was able to obtain an equivalent path length of 55 km atmospheres while looking at the quadrupole allowed rotation-vibration spectrum of hydrogen. 24 However, at the two to three atmosphere pressures necessary in this experiment, HI proved to be too corrosive for the mirrors to be mounted inside the multipass cell. This necessitated our modification of the method developed by White and Herzberg to locate weak transitions in non-corrosive gases.” The simple act of moving the mirrors outside the multiple pass cell complicates the experimental procedure enormously. Since the light must pass through two windows on each traversal of the cell, the reflective losses at the window surfaces are large. For the 1.5 cm thick plate glass windows used on our cell, reflective losses were 0.17 per traversal. -In a 40 pass experiment, the incident intensity, Ip , is reduced by, i," (1 - 0.17)#9 = 6.3 x 107" , (17) more than three orders of magnitude compared to a multiple pass cel] with mirrors inside. Attempts to find a protective mirror coating that would allow mirrors to be mounted inside the cell were not successful. The apparatus used to measure line strengths consisted of three components: (1) a source of tunable monochromatic radiation; (2) the - multiple pass cell; (3) a detection system. A schematic is shown in 29. Figure 7. A tunable dye laser (Coherent Radiation 490), pumped by a CW argon jon laser (Spectra Physics 170), was used as a monochromatic light source. The argon laser supplied a maximum of eighteen watts input power to the dye laser. The dye laser produced a well-collimated, two to four watt beam with a 0.05 nm bandwidth (FWHM). The dye laser wave- length drive was calibrated using a 1.83 meter scanning spectrometer (Jarrel-Ash 78-400). The beam was chopped at 40 hz and focused by a 50 cm F.L. lens upon mirror A of the multiple pass cell. The multipass cell consisted of two spherical mirrors, placed out- side a 144 cm long, 15 cm diameter absorption cell. The mirrors (A and B in Figure 7) were 150 cm radius of curvature, 15 cm diameter, spherical, front surface, silvered mirrors arranged as suggested by Bernstein and Herzberg. 2° Mirror A was fixed, while split mirror B could be rotated to change the number of traversals. Without the absorption cell in place, a maximum of 164 traversals between A and B were obtained, corresponding to an optical path of 246 meters. With the absorption cell in place, scattered and reflected light limited the number of cell traversals to 32 before the signal at photodiode P was too weak to be detected. The chopped signal at silicon PIN photodiode, P, (United Detector Technology 6DP) was normalized to that of reference diode R and processed by a lock-in amplifier (P.A.R., HR-8). The output of the amplifier was recorded as the dye laser was scanned from 660 nm to 560 nm. Hydrogen iodide (Matheson Gas Products, 98% purity) was vacuum dis- tilled three times from an acetone slush bath (180 K), the center one Figure 7. Schematic diagram of the experimental apparatus used to measure weak transitions. 30. 31. Y3AqQHOO3Y 4JQOIdOLOHd VWW3S9 SSVd AIdILINW WNOIS d—Gy V YONI — es ee ome _—— = — — eee ee — <— -_ —— = ——_—e _ — ——_—— = — —— ee — —o ee — _— one — ee — —aomemenre ere. —_—_ae ew ee oe ee —— = ! HALLS 30010 Waa ~---Q-¥ “SONS YalsITdWY ey NI 4907 w/a |’ 4 J @ YOMNIW ! : nsw ~~ aft Nese qd Z---\ yasvi Noouv 7% YdddOHD SAIYQ ONINNVOS 32. third being retained after each distillation. Between two and three atmospheres of HI were added to the cell. A cold trap on the absorption cell was maintained at 263 K to remove any I, formed from HI photolysis or reactions with stopcock grease. 33. V. Results No absorption (100% transmission +5% noise) was observed for an effective path length of 90 meter atmospheres. This placed an upper limit of 5.0 x 10% atm ! cm 2 on the total band intensities of the Av = 8 and 9 transitions shown in Figures 3-6. An upper limit on the line strength of the strongest rotational vibrational transition in either band is 5 x 10° atm! cm 2. Curves J and K in Figure 4 form upper and lower bounds for the band strengths for Av < 9. The median of J and K may be the best estimate of the band strengths. This agrees with all experimental band strength for Av < 4 and is not inconsistent with the experimental upper bounds set for Av = 8,9. Therefore, the value “4 “1 om72 Sosg < 5x 10 * atm * cm is used in our discussion of HI vibrational photochemistry. 34. VI. Discussion The feasibility of using a multiquantum vibrational transition HI(v = 0 + 8) to pump reaction B in the gas phase is now considered. Suppose a vessel of HI is irradiated with an intense laser beam, pro- ducing absorption from the ground state to the eighth vibrational level in an attempt to observe reaction B. It is important to note that observation of increased reaction rate in the photolyzed sample (above that in a thermal one) is a necessary but not a sufficient indication of rate enhancement due to vibrational excitation@© (see Appendix I). Indeed, a detailed analysis of the reaction kinetics is necessary to determine the effect of vibrational excitation. Assume the following mechanism describes the above experiment: HI + hv + HI*(v = 8) (C) HI* + HI + HI(v < 8) + HI (D) HI¥ + HI > Hy + 21 (B) HI + HI > H, + 21 (A) Making the steady state approximation on {HI*}, one obtains Tpreyy°% kp + kp {HI*}.. = (18) where £2 is the absorption path length, eny is the extinction coefficient for the pumped transition (0,J > 8,J') kp and ky are the rate constants 35. for reaction B and reaction D. Using reaction A, reaction B, and equation 18, the rate of H» formation is Tey 2kp {HI} Rate, = + k,{H1}2 . (19) H> ky + kp A Two important facts are displayed in equation 18 and equation 19. First, {HI*}.. is pressure independent. This means that reaction A and reaction B obey different rate laws, reaction A is second order while reaction B is first order. The contributions of each reaction may be separated (see Appendix I). Second, the contribution of reaction B in equation 19, Toe 2kp {HI} Rate, = ——————— (20) B ky + ky not only depends on laboratory parameters (I,,2%,{HI}) and kp and Eyp> as anticipated, but also on kp (the relaxation process). Thus, information concerning kp is vital in the analysis of equation 20. In general, the measured first order rate constant in equation 19 is proportional to kp/kp + Kp: That is, k -f¢. B 2 Ratey = Ce {HI} + k,{HI} (21) where C = Ipe,72- If, for example, k, >> kp» equation 20 reduces to Rate = Tpe, {HI} (22) B(k,>>k B p) 36. Under conditions of equation 22, the measured rate is independent of kp» making the proposed experiment meaningless. Equation 20 cannot be used to solve Rate, for kp since ky is not known. There are, in fact, several processes by which the vibrationally excited HI molecule can relax to the ground vibrational state. Thus ky is a weighted average of the rate constants for each of those processes. The processes include infrared fluorescence, collisional deactivation via vibrational to translational energy transfer (V>T) and collisional deactivation via vibrational to vibrational energy transfer (V-V). If one assumes that the most important deactivation step in reaction D is HI*(v=8) + HI(v=0) + HI(v=7) + HI(v=1) ’ (E) the following approximation may be made: ~ k (23) E Justification for equation 23 is based on two assumptions: a) At pressures used to perform the photochemistry ({HI}>5 torr), infrared fluorescence can be neglected. Using the Einstein coefficient of HI(v=1) . 27 = 1 A, 7 0.038 sec ; (24) with the assumption that the radiative lifetime of HI(v=8), tg»is the same order of magnitude as 28 37. one finds that te 7 oT = 26 sec (26) 0<1 At pressures greater than five torr, HI*(v=8) will undergo more than 3 x. 108 collisions during tae Since only 10° collisions are needed to deactivate HI(v=1) (and HI*(v=8) should deactivate faster) fluores- cence can be ignored. 22 b) Collisional deactivation is therefore the important process for HI* relaxation. Deactivation cross sections have been measured for HI(v=1) and HI(v=2) 39734 They are = 2 = 2 The relaxation of HI(v=1) can proceed only by a V+T,R process while HI(v=2) can also relax by a V>V transition. This dramatic increase in relaxation cross section going from equation 27 to equation 28 shows that the V+V process is much faster than the V>T,R in HI. This is true in general. >! The V+V transition is therefore assumed to be the dominant relaxation process in HI*(v=8). Unfortunately, equation 23 still does not help one solve equation 20 since k- has not been measured either. However, if HI were a harmonic E oscillator, then 8k F E 2 (harmonic) 38. where ke refers to the measured reaction _ _ . 33 HI(v=2) + HI(v=0) > 2HI(v=1) . (F) HI, of course, is not a harmonic oscillator, thus equation 29 is not strictly valid. Indeed, the V+V transfer rate may be significantly slower than that predicted by equation 29 since the mechanical anharmon- icity of HI prevents reaction E from being a resonant energy transfer process (see Table I). Since a better approximation is not available, we use equation 29, obtaining ~ Ak. = 7.7 x 105 torr’! sec ! (30) ky F This value of Ky is used in Appendix II to evaluate equation 20 for kp. Under the experimental conditions cited there, it is required that k, > 7 x 102 torr ! sec"! B in order to observe the vibrationally enhanced reaction. An upper bound may be placed on kp using data on ky: If vibra- tional enhancement is important in reaction A as Anderson's experiment suggests, >" the following mechanism may be proposed for the thermal reaction: HI + HI 2 HI*(v=8) + HI (G) HI*(v=8) + HI +H, + 21 (B) Since reaction B proceeds slowly compared to the fast equilibrium 39. reaction G, a Boltzman distribution can be assumed. Using rate data for k an upper limit on Kp» A? k, > 2.7 x 107 torr’! sec} B may be set. This is an upper limit on kp» since it was assumed that reactions G and B were the only source of thermally produced H>. It is seen that the experimental conditions proposed in Appendix II would allow one to measure this rate ( provided that the transition fre- quen@y is known within the accuracy needed to tune the laser). 40. VII. Appendix I Vibrational Photolysis of HI A molecular beam study done by J. B. Anderson“ showing that transla- tion#t energy is unimportant in reaction A suggests that vibrational energy plays an important role in his reaction. Continuum radiation (suitably filtered to prevent electronic excita- tion of HI) from a 2.5 kW xenon arc Tamp (Hanovia 975 C-98) was focused into a photolysis vessel. The vessel, a 10 cm hollow quartz sphere, was gold plated on the exterior (except for the entrance window) to maximize the optical path. This vessel was mounted in a resistance heated oven whose temperature was regulated with a platinum resistance thermometer bridge (Hallikainen Thermotrol 1472 A). Temperature was measured with. six iron constantan thermocouples placed around the sphere. A mercury- free vacuum line attached to the vessel was used to purify HI (Matheson, 98%) and transfer to the vessel. After photolysis, HI + I, from the reaction mixture were frozen over liquid nitrogen. The remaining H, was measured in a precalibrated mass spectrometer (C.E.C. 21-103 C). When the rate of H, formation was compared to that of a similarly thermostated but not irradiated vessel, it was discovered that the reaction rate in the photolyzed vessel was higher. It is extremely important to note that the effect of vibrational energy on the reaction rate is not necessarily the difference between the rates of the irradiated and non-irradiated samples. Since the experiment was conducted in the gas phase rather than by crossed molecular beams, it al, is quite possible that vibrationally excited HI, produced by multiquantum absorpfion, may be collisionally deactivated before reaction with another molecule, pumping rotational and translational energy into the system as well as vibrational energy. Thus, the experiment does not distinguish between a rate increase due to vibrational excitation and a rate increase . due to heating effects which may be produced by the intense light source used in the photolysis. Because of this ambiguity, results of many recent experiments concerned with the rate of vibrational energy on reaction rate are questionable. “© Using the mechanism developed in the Discussion, equation 18 and equation 19 are obtained. If I) and & are fixed, {HI*}.. is constant. Equation 19 may be simplified to Rates kp tHI*}. {HI} + k {HI} k' {HI} + k,tHI} . (32) According to equation 32, a plot of Rate, /{HI} should be a straight line of slope Ky and intercept k'. Results for the photolyzed samples are shown in Figures 8-13. _ Shown in Figures 14 and 15 are results for thermal samples. Notice that both sets of data (photolyzed and thermal) have non- | zero intercepts. If the intercepts are plotted versus observed temperature, measured at the vessel wall, there appears to be a difference between the photolyzed and unphotolyzed results (see Figure 16). Since the slope of the lines in Figures 8-15 is ky» the well known thermal rate constant, an internal reaction temperature may be calculated from the Arrhenius expression for reaction a4 As shown in Table IV, temperatures calculated in this manner are between 8K and 20K higher than 42. Table IVa. Summary of HI Photochemistry. Figure > = Photolyzed) os "eale (k) (T = Thermal) 8 P 670.9 711.9 40.9 9 Po 650.7 687.9 37.2 10 P 693.1 728.3 35.2 ll P 639.9 677.5 37.6 12 P 653.7 667.1 13.4 13 P 664.0 695.3 31.2 14 T 668.6 676.7 8.1 15 T 697.8 713.0 15.2 43. Tabye IVb. Summary of HI Photochemistry. . Kp Op k' Op Figure Conditions (liter mol! sec" )n 10°*) (sec } x 10 7) 8 P 9.34 + 0.64 6.93 + 1.2 9 P 3.15 + 0.14 2.49 + 0.29 10 P 18.9 + 0.90 16.1 + 1.0 ll P 1.91 + 0.23 1.13 + 0.38 12 P 1.15 + 0.10 2.91 + 0.17 13 P 4.43 + 0.45 5.68 + 0.74 14 T 1.85 + 0.15 4.08 ¢ 0.32 15 T 9.82 + 0.45 8.73 + 0.85 44, 8 aunbi4 (o_O! X) (494 /1OW) NOILVYLNAONOD Ov a ve 9'| 80 O TT TT 1 | a | > m = ce fa) O = Te 2 Oo — — Vw 466¢9 AYNLVYSdWSAL GSAYSSEO @ o MG229 SYNLVEAdWSL GALVINIWO Jo 1 AIdWVS G3ZA1OLOHd = a | a | | 9 45, Ov 6 ainbi4 (o-O1 x) (48411 1OW) NOILWYLNSONOD oe vee 9" 8°O O TT TI a es eee O ~—- —O- —O—— = 4 | - ae = mn = 40 AH2¢SG9 JAYNIVYAdNW SAL AAAYSASEO a M1299 SYNIVEAdWSAL GALVINOIWI _|e¢ AIdWVS GAZAIOLOHd | | | | on | 9 (;-99S) ONOO/SLVY 46. O| aunBig (oO X) (J8HI/1OW) NOLLVYLNZONOD O'p Ze po 91 8°O O | T T | | T T To? zo —— | Pe) — 7! $5 oO a [T] = jen, C) oO = 4¢ 2 ©) — 42'0S9 JYNLVYAdWSL GSANSSS8O 4b @& 46289 JYNLVYAdWAL GSLVINDIWO o Z AIdNVS GAZAIOLOHd 4g Lb l l l I | l | L l 9 I] eanbi4 (o_O x) (42H//10W) NOLLVYLNSONOD 47, Ov o& De 9° | 8°O O | j | | | | | 5 nad a _f§ om vee O — 4OvE9 AYNLVYEAdWSL GSAYSSEO 4 €°SG69 JYNIVYHSdW4Al GSLVINOIWO AFIdWVS GSZAIOLOHd | I | | L | | l | (; 98S) ONOO/SLVY 48. Ov 2] a4n6) 4 (¢-Ol X) (4241/10W) NOILVYLNSONOD o'€ '2 9"| 8'0 I | | | ii | ! | | 46029 AYNLVYSAdNSL AGSAYSSEO AO 112 SYNLVYSdW3L GSLVINOIVSO JIdWVS GSZAIOLOHd | l l l l l l | ().28S) ONOO/ 3LVY | eunb14 (o_O1X) (49411 710W) NOILVYLNAONOD 49. O'v ce ve 9° | 8°O O | T l Tr T T I O o A|1°S69 SYNLVYSdWSL GSANSSEO 4 AD AHF 8clL SYNLVYSdDWSL AGSLVINOIVI a JIdWVS GSZAIOLOHd (M] —O © — 4A7¢ ZF < = ADD CD ‘?) - 4g 1 50. O'V | einb! 4 (¢_O] X) (494! /10W) NOILVYLNSONOD oe v2 9"| 8°0 49°899 JYNIVYAdDNAL GSAYSSEO A2929 JAYNLIVYSKUWNSL GALVINIIVO JIdWVS IVWWYSHL l | | | | | J | O aN +t ™ ().98S) ONOO/SLVY Ww) ce) 51. O'V G] eunb1 4 (o-Ol X) (494|/710W) NOILVYLNAONOD oe b'2 9" 8°0 | 1 qT | | | | | 48°269 SYNLVYSdWSL GSAY3SSEO HOEIL JYNLVEYSdWSAL ASLVINIIVS JIdWVS IWNYSHL | | | t | (;_98S) ONOO/SLVY Figure 16. The intercepts, k', from Figures 8-15 are plotted versus measured cell temperature. 52. kK (sec) L 1o7? Porerrrrl io 8 B® THERMAL O PHOTOLYZED io | jj i td ! 1.43 1.45 1.50 I/T (K)"! (x 1074) |.55 34. those measured at the cell wall. If the intercepts in Figures 8-15 are shown in an Arrhenius plot using T calculated rather than T observed, it is immediately apparent that the observed rate increase in the photolyzed samples is not due to vibration excitation but is a heating effect (Figure 17). The temperature dependence of k' seen in Figure 17 may be described by k' = 2416 x exp(-30.93/RT). The origin of this first order thermal reaction component is unclear. Later experiments showed that H, envolved slowly from the walls of the quartz vessel. However, when quartz wool was added to the vessel, increasing the surface to volume ratio by 30, k' was unchanged. It is interesting to note that a similar non-zero intercept is seen in the thermal data of Crist and Taylor’ when plotted as in Figure 8. The procedure outlined in Appendix II can be used to evaluate equation 20 under the conditions of this experiment (I, = 10!3 photons sec ! cm, 2 = 100 cm). If ky 7 x 105 torr”! sec’!, the maximum rate of Bo This is well below the minimum detectible rate (3 x 101! molecules/sec, see equation 34). Hy production would be 6 x 10° molecules/sec even if k Figure 17. Figure 16 is replotted using the calculated reaction temperature (as explained in text). 55. kK’ (sec!) 720 700 680 ! @ THERMAL QO PHOTOLYZED |.40 l.45 1.50 [OOO K/T 57. VIII. Appendix II Observation of Vibrational Excitation Case i kp < ky ~ 7x 10° torr! sec } Choosing representative values of I, = 10!9 photons sec ! cm (dye laser 3 watts, bandwidth 0.05 nm), 2 = 50 cm, and e, = 5 x 10° atm ! cm 2, equation 19 yields a value for {HI*} 06> {HI*} = 4.2 x 107 molecules . (33) The minimum number of H, molecules detectable by the mass spec- trometer (C.E.C. 21-103C) is 3 x 10!© molecules. Using a reasonable photolysis time of 10° seconds (1.1 days), the minimum detectable rate of H. production is Rate > 10!! molecules/sec . (34) Equations 33 and 34 are used to solve equation 20 for kp. Using {HI} ~ 10 torr, one finds that the minimum detectable value of kp under the above experimental conditions is k, > 7 x 102 torr’! sec } B Case ii. kp > ky ~ 7x 105 torr! sec ! Equation 22 may be used immediately to give Rate, ~ 3 x 10!4 molecules/sec . (35) This is three orders of magnitude above the minimum detectable rate, equation 34. 58. 59, IX. References 1. Truman J. Odiorne and Philip R. Brooks, J. Chem. Phys., 51, 4676 (1969). 2. Larry A. Pugh and K. Marahari Rao, J. Mol. Spectrosc., 37. 373 (1971). 3. John H. Sullivan, J. Chem. Phys., 46, 73 (1967). 4. Alfred H. Taylor, Jr. and R. H. Crist, J. Am. Chem. Soc., 65, 1377 (1941). 5. Leonard I. Schiff, Quantum Mechanics (McGraw-Hill Book Company, New York, New York, 1968) third ed., p. 403. 6. Gerhard Herzberg, Spectra of Diatomic Molecules (D. Van Nostrand, Inc., Princeton, New Jersey, 1950) second ed., pp. 19-21. 7. J. R. Oppenheimer, Proc. Camb. Phil. Soc., 23, 327 (1926). 8. R. E. Meredith and F. G. Smith, J. Quant. Spectrosc. Radiat. Transfer, 13, 89 (1973). 9. R. Herman, J. Mol. Spectros., 2, 369 (1958). 10. G. Herzberg, op. cit., p.540. 11. M. McClintock, W. Demtroder, and R. N. Zare, J. Chem. Phys., 51, 5509 (1969). 12. Claude Haeusler, Claude Meyer, and Pierre Barchewitz, J. Physique, 25, 961 (1964). 13. B. Rosen, Spectroscopic Data Relative to Diatomic Molecules (Pergamon Press, Oxford, England, 1970), p. 187. 14. G. Herzberg, op. cit., p. 99. 60. 15. R. Jost and W. Kohn, Phys. Rev., 88, 382 (1952). 16. Bruce Carnahan, H. A. Luther, and James 0. Wilkes, Applied Numerical Methods (John Wiley and Sons, New York, New York, 1969), p.267. 17. George Ameer and William Benesch, J. Chem. Phys., 37, 2699 (1962). 18. William Benesch, J. Chem. Phys., 39, 1048 (1963). 19. William S. Benedict, Robert Herman, Gordon E. Moore, and Shirleigh Silverman, J. Chem. Phys., 26, 1671 (1957). 20. J. M. Herbelin and G. Emanuel, Aerospace Report No. TR-0074 (4530)-5, Aerospace Corp., Los Angeles, California, 1974, p. 6. 21. J. Trischka and H. Salwen, J. Chem. Phys., 31, 218 (1959). 22. This has also been shown true for HF for v< 10. See R. J. Fallon, J. T. Vanderslice, and E. A. Mason, J. Chem. Phys., 32, 698 (1960). 23. John U. White, J. Opt. Soc. Am., 32, 285 (1942). 24. G. Herzberg, Nature, 163, 170 (1949). 25. H. J. Bernstein and G. Herzberg, J. Chem. Phys., 16, 30 (1948). 26. This point is evidently overlooked by many researchers in the field. See, for example, H. B. Kaplob and 0. M. Ctel' max, J.E.T.P. Letters, 11, 135 (1970); M. Brunet and F. Boimer, C. R. Acad. Sci. (Paris), 271C, 1116 (1970); A. Yogev and J.R.M.J. Loewentein, J. Amer. Chem. Soc., 94, 1091 (1972). 27. This is calculated using Sp, = 0.37 atm! cm 2 and reference 28. This is true, for HF, HCl, and probably true for HI. See reference 20, Table V. 6l. 29. A. Guthrie, Vacuum Technology (Wiley, New York, 1965), pp. 12, 505. 30. H. L. Chen, J. Chem. Phys., 55, 5551 (1971). 31. Measurements of V>T,R and V-+V processes in HC] indicate that V+V reactions occur at least ten times faster than V>T,R. These results are typical of other molecules studied. See, for example, H. L. Chen, J. Chem. Phys., 54, 4072 (1971). 32. J. H. Ahl and T. A. Cool, J. Chem. Phys., 55, 1980 (1971). 33. N. Cohen, Aerospace Report No. TR-0073(3430)-9, Aerospace Corporation, Los Angeles, California, 1974, p. 48. 34. S. B. Jaffe and J. B. Anderson, J. Chem. Phys., 51, 1057 (1969). 35. B. Lunelli, J. A. Betts, and Aron Kuppermann, Annual Report: Chemistry and Chemical Engineering, Caltech, Pasadena, California, 1972, p. 58.