3400
J. Phys. Chem. 1983,87,3400-3404
defining pl(?)as $he equilibrium polarization for the reactant state and P2(3for the product. Substituting this expression into equation A21 and rearranging, we find that Hll* = H22* yields The quantityd23/dc12will be positive because we have chosen c1 to minimize 9. To avoid confusion, we will write it as an absolute value. The interesting eigenvalue, which we will call r, is where A3- is the free energy difference between the fully solvated reactant and the fully solvated product. Following Marcus and Hush, we define the free energy of repolarization as 1eo-1
1
=(J!?,'(O - ~ ! ? ~ ~ (d3r r ) ) ~ (A25) 24TEO
Notice that in the abnormal free energy region c1 will be outside the range 0 < c12 < 1. The nature of this critical point will depend on the sign of the eigenvalue that we mentioned previously. The quantities that we need to evaluate are
which will essentially be negative when the free energy of repolarization exceeds the off-diagonal matrix element. This is a statement that the barrier must exceed the splitting. At the other possible critical points cI2= 0 or c12= 1, the expression shows that this eigenvalue will be positive, so those two critical points correspond to local minima.
Energy Dependence of the Reaction of Laser-Excited Sulfur Hexafluoride with Potassium Vapor M. Eyal, F. R. Grabher,*+ Department of Chemistry. Tel-Avlv Unlversw, Ramet-Avlv 69978, Israel
U. Agam, and L. A. Gamss B p r t m n t of Chmktry. Ber-Ilan UniversnY. Ramet-Gen, Israel (ReceIvM: June 22, 1982; I n Flnel Form: March 29, 1983)
The rate of reaction between potassium and IR-laser-excited sulfur hexafluoride has been studied as a function of vibrational energy absorbed from a pulsed COz laser and total thermal energy in the range 340-390 K. In this range the rate increases with vibrational energy but is not dependent on added translational energy. This result is in agreement with an earlier temperature-dependent molecular beam study and is also consistent with our previous study of the reaction between sodium and sulfur hexafluoride.
Introduction The relative importance of vibrational and translational energy in promoting chemical reactions has been the subject of several experimental and theoretical studies.' We have recently reported on the infrared laser enhancement of the reaction of sodium atoms with sulfur hexafluoride. By studying the vibrational energy dependence of the react rate at various temperatures, we have compared the importance of vibrational and translational energy in promoting the reaction.2 In the present work we report on the reaction K + SF, studied by essentially the same diffusion cloud technique employed in the previous work. The temperature dependence of the reaction suggests that vibrational energy is significant in promoting the reaction, while additional translational energy has little, if any, effect. This reaction has previously been studied +Present address: Allied Corp., P.O.Box 1021R,Morristown, NJ
07960. 0022-3654/83/2087-3400$01.50/0
in a molecular beam,4is and the temperature dependence of the cross section also suggests that only vibrational energy is important in promoting the r e a ~ t i o n . ~ The diffusion cloud technique used for this study is a modification of the technique developed by Hartel and Polanyi? which is suitable for the time-resolved study of fast reactions. In this method an inert gas carries alkali-metal vapor from an oven through a nozzle into a background of a reactant gas. The diffusion and flow conditions are adjusted to ensure that all the metal vapor reacts within a short distance from the nozzle, so that a (1) M. Kneba and J. Wolfnun, Annu. Rev. Phys. Chem., 31,47(1980). (2)M.Eyal, F. Grabmer, U. Agam, and L. Gamss, J. Chem. Phys., 75, 4396 (1981). (3)M. V. Hartal and M. Polanyi, 2.Phys. Chem., Abt. B, 11,97 (1930). (4)T.M. Sloane, S. Y. Tang, and J. Ross, J. Chem. Phys., 57, 2745 (1972). ( 5 ) S. J. Riley and D. R. Hershbach, J. Chem. Phys., 58, 27 (1973). (6)R. D.Levine and J. Manz, J. Chem. Phys., 63,4280 (1975).
@ 1983 American Chemical Society
Reaction of Laser-Exclted SF, wlth Potassium Vapor VACUUM PUMP
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983 3401
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Flgurr 1. Experimental apparatus.
102 t [WCI
Flgure 2. Sample of signal averager output. The rising part of the curve represents a loss of potassium. The rates obtained from the curve are as follows: steady-state rate, k , = 42 ms-'; energydependent reaction rate, kE = 2.3 ms-'; and the deactivation rate, k , = 5.5 ms-'. The dotted curve is synthetised via eq 1 with the same rates.
spherical cloud of unreacted metal vapor is centered at the nozzle. In our apparatus this method is used to prepare a steady-state reacting system of a polyatomic molecule with an alkali-metal atom. The system is perturbed by a rapid input of vibrational energy into the polyatomic reactant due to absorption of IR pulses from a Q-switched COz laser. The effect of added energy on the reaction rate is determined by monitoring the decrease in alkali-metal concentration after the pulse of laser. The effect of addition of total energy is determined by measuring the change in reaction rate as a function of reaction temperature.
of potassium with SF6,in which a positive signal represents a loss of potassium in the cloud. Within 2 ps of the laser pulse the potassium concentration starts to decrease at a rate determined by the reaction conditions and absorbed energy, and then returns to its steady-state value on a longer time scale determined by resupply of the potassium into the reaction region. The differential equation, evalSF6, is valid also for the uated for the reaction Na present reaction: d[K]/dt = -[ks k~e-~"t][K] + ks[K]s (1)
Experimental Section The experimental apparatus shown in Figure 1has been described previously.2 In the present work a potassium hollow cathode resonance lamp and an interference filter are used. Good experimental conditions are achieved by varying the pressure of SF6admitted into the cell until the potassium vapor forms a spherical cloud about 1 cm in diameter above the nozzle. Potassium has no strong fluorescence in the visible spectrum, and, thus, the diffusion cloud cannot be seen by illumination, as was in the case of sodium. Therefore, the size and shape of the cloud can be determined only by measuring the absorption of the potassium resonance lamp light. Since its size is determined by a dynamic balance between the reaction rate and the diffusion rate of potassium atoms from the nozzle, and since the observed rate of reaction K + SF6is much larger than that of Na + SF,, reasonable steady-state cloud sizes are obtained with a much lower pressure of SFG.The melting point of potassium is considerably lower than that of sodium (328 vs. 378 K). Consequently, the lowest vapor pressure suitable for the experimental conditions is reached at a lower temperature, 335 K, as compared to 380 K for sodium. Typical working steady-state pressures are 2-15 mtorr of SF6 and 1.2-1.5 torr of argon, while the potassium pressure is about 10 ptorr. Because the reaction system is flowing, the pressure measured during the experiment is used only as an indication of the conditions. The pressure used in the calculations is determined by the absorption of the laser radiation by the SF, compared to the measured absorption of static SF6/Ar mixtures. The dependence of the reaction rate on vibrational energy at a particular temperature is determined by measuring the rate of change of potassium concentration as a function of absorbed laser power. The dependence of the rate on total energy is determined by varying the temperature of the reaction. Analysis of Experimental Signals. The curve in Figure 2 represents typical signal averager output for the reaction
This equation assumes that, in the absence of the laser excitation, a steady-state reaction with rate ks is occurring, so the rates of reaction and resupply of potassium are both equal to ks[K]s, where [K] represents the potassium concentration, and the subscript, s,refers to the steady-state conditions. For the present reaction conditions, the vibrational equilibration (V-V) time is about 3 ps, the vibrational relaxation (V-T) time is about 300 ps, and the time for the reaction is on the order of 20 ps. In the time scale of the laser-enhanced reaction, the energy absorbed from the laser, E, is therefore considered to be distributed among the vibrations in a Boltzmann distribution corresponding to an elevated vibrational temperature. This increases the rate constant by an amount kE, which then decays exponentially back to the steady-state value with rate constant kd. This rate, although primarily ascribed to the vibrational relaxation of the SF,, may be also influenced by diffusion and flow in the reaction cell. It has been experimentally verified that the signals decay as single exponentials. Since the change in potassium concentration due to the laser-enhanced reaction is small, it is further assumed that the rate of resupply of potassium to the reaction center is constant. The validity of these assumptions is borne out by the excellent fit of computer-generated simulations to the experimental signals. The experimental curve shown in Figure 2 was analyzed to have ks = 42 ms-', kE = 2.3 ms-',and k d = 5.5 ms-'. The computed simulation obtained by substituting these values into eq 1 is shown as dots overlaying the experimental curve in this figure. The method of evaluation of these rates was different from the method used for evaluation of Na + SF, reaction rates. It is based on measuring the maximum strength of the signal instead of measuring the rate of rise of the signal. Introduction of this method follows from a need for a technique to determine the rates that is more sensitive to small variations of kE than the "direct" method of measuring slopes of the experimental signals. In the case of the reaction Na + SFe, the direct method proved to be sufficient because the variation of kE with absorbed energy was relatively large. However,
+
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The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
Eyal et al.
where Isis the steady-state absorption. Substituting into eq 3, we get
In IO
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to
t”
c t
Figure 3. Pictorial description of definitions from eq 5, 6, and 9.
in the case of K + SF,, this dependence appears to be roughly 5 times smaller than for Na + SF,. The direct method bears an uncertainty of about f 2 % of the measured rise time. Therefore, the kE,which is no more than 5% of the steady-state value of ks for the K SF, reaction, cannot be determined from the direct method, especially for the low absorbed energies, when it is less than 1% of ks. The “height” method, as we shall show, has two important advantages over the earlier method. The first is the increased sensitivity for small variations of KE. Since the signal strength is almost directly proportional to the increase of the rate, kE,as a function of vibrational energy, while the rate of rise of the signal is proportional to ks + kE,ks being more than 90% of this sum, the measurements based on signal strength are 2-3 orders of magnitude more sensitive than the direct measurements. The second advantage is that the linear approximation of the BeerLambert law, the basic approximation of the direct method, becomes unnecessary in the height method. The starting point for the evaluation is eq 1. The derivative of potassium concentration with respect to time equals zero at two times: at t = 0, when the reaction rate is at its steady-state value, and at some time t = t,, when the rate of disappearance of the potassium is balanced by the rate of supply of “fresh” potassium atoms from the oven. At this point of time the potassium concentration, [K],, is furthest from its steady-state value, and its time derivative is zero. Hence
+
-[k, + kEe-kdtm][K]m+ ks[K]s = 0
The quantities kd, t,, Is, and Io are easily measurable. k d is measured directly from the “tail” of the signal, and t, is easily determined as the time of the maximum of the signal; Is and Io are steady-state quantities. ks is determined as in the direct method. It is the value of the intercept of the graph of the rate of rise of the signal plotted against energy absorbed from the laser. In the reaction with potassium, the slope of this graph is small enough that the rate of rise of any of the low-energy experiments gives ks. The remaining quantity I, has two components: Is, which is independent of time, and A, which is a measure of the height of the signal: I , = Is A (7)
+
Since the maximum height of the signal is measured in digital units from the display screen of the averager, and the averager has a linear response to the analog signal acquired from the photomultiplier, the A must be related to the maximum height, Hm,by a simple factor: A = CH,
(8)
The factor C is determined by comparing the known output of a wave-form generator with its “picture”acquired by the signal averager. Hence, once the C is calculated, the “working”formula of the height method is
(2)
or (3)
This equation, however, can be useful only when the concentrations [KIs and [K],, or a t least their ratios, are known. The absolute concentrations are hard to determine, but the corresponding ratio is calculated relatively easily, as follows (see Figure 3). The Beer-Lambert relationship for [K], can be expressed by the formula In Um/Zo) = -atK]m (4) where Io is the intensity of the potassium lamp light transmitted through the cell with potassium vapor absent, I, is the attenuated intensity at t = t,, and a: is the absorption coefficient for the cell filled with potassium. For small changes of potassium concentration a: can be considered constant. Thus, a similar relationship can be written for the steady-state concentration of potassium, and
The validity of this method was checked by simultaneously measuring the rise times and the maximum heights of signals obtained for the laser-enhanced reaction of Na + SF,. The result is displayed in Figure 4. The quantity obtained from the height method is kE/ks, while the quantity obtained from the direct method is k(E) = kE + ks. The two results are mutually normalized by subtracting In ks from the In (kE + ks) obtained from the direct method and by adding 1to the k E / k s obtained from the height method. That is In (kE + ks) - In ks = In (1 + k E / k s ) (10) Comparisons of the slopes show that the two methods yield results falling within each other’s experimental errors. Results The dependence of the rate of reaction K + SF6 on vibrational energy absorbed by SF6is presented as a plot of In (k, + k E ) vs. AEv (Figure 5 ) . This particular experiment was done with 12 mtorr of SF6 and 1.2 torr of argon. The temperatures of the reaction zone were varied
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
Reaction of Laser-Excited SF, with Potassium Vapor 0.6I
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FlQve 4. Comparison of the two methods for determining the reaction rate of sodium with sulfur hexafluoride. The X's are calculated by measuring the rate of rise of the experimental signal and yleM the line labeled S,. The points are calculated by measuring the height of the signal and yield the line labeled.:S I
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Figure 6. Comparison of vibrational with total energy dependence of the logs of the reaction rates. Details in text.
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AE [ cm-'/molec] Figure 5. Dependence of the rate of reaction of potassium with sulfur hexafluoride on temperature and vibrational energy absorbed per molecule. k = k , KE.
+
from 344 to 378 K. Potassium pressure was estimated to be 10 rtorr, and the average steady-state absorption, Is, of the potassium hollow cathode lamp light was 1-15% of the total intensity. Each set of points represents an experiment that was done at constant temperature. The quantity kE/kShas been evaluated by the height method for each absorbed energy, while ks has been evaluated directly. The experimental points form straight lines that are parallel within experimental error with an average slope, A In k/hE,, equal to (3 f 0.1) X 104/cm-'. The average dependence on vibrational energy for four independent sets of experiments is plotted as k(E,) in Figure 6. The average slope is (3.0 f 0.15) X lo4 cm-'. The intercepts of the lines in Figure 5 are the logarithms of the steady-state rates, In ks, for each temperature. In order to compare the effects of vibrational and total energy in promoting the reaction, the procedure developed for the reaction Na + SF6 is followed.2 The dependence of the reaction rate on total energy is given by the change in the logarithm of the steady-state reaction rate as a function of total thermal energy added by increasing the temperature from that of the lowest temperature experiment. The conversion of temperature to added energy is done by using the calculated heat capacity of the gas for the given temperature range. The average dependence on total
0 00285
0.00290
[OK-']
Flgure 7. "Arrhenius" plot of the logs of thermal rates obtained in the experiment vs. the reciprocal of the temperature.
energy is (2 f 0.2) X 104/cm-' and is plotted as k(E,) in Figure 6. The thermal reaction constants, obtained by dividing the ks's by the pressure of SF6, are on the order of 3200-3500 ms-l torr-'. This is 1 order of magnitude larger than the reaction constants of 350 ms-l torr-' obtained for Na + SF6at somewhat higher temperatures. The reaction cross section calculated from this rate is 25-30 A2 in the tem erature range of 340-380 K. This is smaller than the 55 calculated from a molecular beam reaction! The activation energy for the reaction is determined by measuring the slope of an Arrhenius plot (Figure 7) based on the same set of experimental results. The average of three determinations equals 0.58 kcal mol-'. There is no literature value available for comparison. It is, however, much lower than the activation energy for the reaction Na SF6, which is 3.3 kcal mol-'.
fi
+
Discussion The slope of the logarithm of the steady-state reaction rate vs. total energy is about 67% of the slope of its dependence on vibrational energy. This ratio is approximately equal to the ratio of the vibrational to the total heat capacity in the temperature range of the experiments. This result implies that within experimental error the reaction K + SF6 has little, if any, translational-rotational energy dependence, as shown below. The total derivative of log k in respect to total energy is d In 12 d In k aE, +- 8 I n k a In E t r / r o t (11) =tot aEv aEtr/rot aEtot where Ebt, E,, and Etrlrotare the total, vibrational, and translational-rotational energies, respectively. The translational and rotational energies are taken together because in the experiments their effects on the reaction rate are not resolved. The total derivative, d In k/dE,,,
3404
The Journal of Physical Chemistty, Vol, 87, No. 18, 1983
and the partial derivative, a In k/dE,, have been measured in the experiments. The derivatives a In E,/dE,, and dEtrlrot/dEtotare simply the ratio of the respective heat capacities. The computed heat capacities (direct count of vibrational levels) at the mean temperature of the experiments are as follows: total heat capacity, CPbt = 9.185 an-'molecule-'; vibrational heat capacity, C = 6.414 cm-' molecule-'; and translational-rotational fieat capacity, C?lrot = 2.771 cm-' molecule-'. After algebraic manipulations one obtains
or
As can be seen, the logarithmic dependence of reaction rate on translational-rotational energy is several orders of magnitude smaller than the dependence on vibrational energy. The accuracy of the experiment does not enable us to determine that number exactly. What can be concluded, however, is that it is much smaller than the dependence on vibrational energy. The line denoted K(Eb&" in Figure 6 has a slope equal to that of the vibrational energy dependence, K(E,), multiplied by the ratio of vibrational to total heat capacity (70%). This is the line expected experimentally for the total energy dependence if there is no dependence on additional translational-rotational energy. It falls close to the line K(E,J and is equal to it within experimental error. Since the reaction rate does not depend on translational energy in the range studied, the increase in reaction rate with temperature is due to the increase of the average vibrational energy of the molecules with increasing temperature. The experimental dependence of the reaction rate on vibrational energy has been compared to the theoretical model formulated by Zamir and Levine' for the treatment of polyatomic-atomic exothermic reactions. For a very exothermic reaction, the total reaction rate, KO,is expressed as
where Ei is the energy of harmonic level i, -AEo is the reaction exothermicity, k is Boltzmann's constant, gi is the degeneracy of level i, and Q, and T, are the vibrational partitional function and temperature after vibrational equilibrium of the energy absorbed from the laser, calculated by using accepted energy levels of SF6. For a polyatomic molecule N is equal to S + 4, where S is the number of vibrational degrees of freedom in the polyatomic product; for a diatomic reactant6 N equals 5/2. The exothermicity of the reaction has been calculated to be 9000 cm-', based on published values of bond strength^.^,^ Theoretical total reaction rates based on this (7) E. Zamir and R. D. Levine, Chem. Phys. Lett., 67, 237 (1979). (8)C. D. Cantrell, S. M. F r e u d , J. L. Lyman in 'Laser Handbook", Vol. 3, M. L. Stitch, Ed., North Holland Publishing Co., Amsterdam, 1979.
Eyal et al.
model have been computed, and the derivatives of the logarithms of the rates with respect to mean vibrational energies added to the system fit the experimental results closely when N is chosen to be 5/2, as it would be for a diatomic reactant. If all vibrational modes were participating in the reaction, N would be equal to 16. The behavior as a diatomic reactant implies that only one stretching vibrational mode is important in the reaction and the molecule behaves like a diatomic SF5-F. The theoretical In ks, as calculated here, has also been plotted against reciprocals of the temperature. The activation energy calculated from the slope is equal to the experimental activation energy. The results reported here are qualitatively similar to those previously reported for Na SF6. In both cases it has been determined that, for the temperature ranges used in the experiments, the reaction rates depend on vibrational energy but not on translational energy. In both cases the data fit the same theoretical model, but with reduced numbers of active vibrational modes ( N equals 10 for the reaction with Na). In the present case of the reaction with potassium, the best fit is derived with an assumption that only one stretching mode is active in the reaction. The reaction with potassium, which is more exothermic, is faster and has a smaller dependence on additional vibrational energy. This also means that the activation energy, which is a measure of the dependence of the reaction rate on added energy, is smaller for the reaction with potassium. The present results are also in qualitative agreement with a previous molecular beam study of the temperature dependence of the same reaction? In that case it was also determined that the reaction rate depends on vibrational energy, but not on translational energy. The best fit to the data occurred by assuming that all stretching modes were important in promoting the reaction. The cross section for the reaction predicted by the present study is within a factor of 2 of the previous value.
+
Conclusions The reaction rate of the reaction K + SF6 depends weakly on translational energy but is significantly enhanced by the vibrational energy added to the SF6. This does not imply that the reaction does not depend on translational energy in general. Our conclusions have been drawn from experiments that have been carried out in a relatively narrow region of temperatures and are valid only for this region. The calculated number of degrees of freedom participating in the reaction is much lower than the corresponding number for the reaction Na + SF6 ( N = 5//z as opposed to 10). This is the result expected if only one SF6stretching vibrational mode is important in the reaction, and it behaves like a diatomic molecule SF5-F. These results are in qualitative agreement with a molecular beam study of the same reaction and with our previous study of the reaction of Na + SF6. Acknowledgment. This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. Registry No. K,7440-09-7; SF6,2551-62-4. (9) R. S. Berry in "Alkali Halide Vapors", D. Davidowita and D. L. McFadden, Eds., Academic Press, New York, 1979, p 86.
