J. Phys. Chem. 1996, 100, 9745-9750
9745
Collisional Deactivation of CDCl3 Excited with a TEA CO2 Laser C. A. Rinaldi and J. C. Ferrero* INFIQC, Departamento de Fı´sico Quı´mica, Facultad de Ciencias Quı´micas, UniVersidad Nacional de Co´ rdoba, Argentina
M. A. Va´ zquez, M. L. Azca´ rate, and E. J. Quel CEILAP(CITEFA-CONICET), Zufriategui 4380, 1603 Villa Martelli, Buenos Aires, Argentina ReceiVed: NoVember 13, 1995; In Final Form: March 25, 1996X
The decay of infrared fluorescence from IR multiphoton excited CDCl3 was studied as a function of incident fluence and pressure of added Ar. The decays were exponential and allowed for the calculation of the average energy transferred per collision, 〈〈∆E〉〉, with Ar and with CDCl3, as a function of the average internal energy of excited CDCl3 using a direct method. Identical information was obtained following the inversion procedure developed by Barker et al. (Int. ReV. Phys. Chem. 1993, 12, 2, 305). Both sets of values agree well for a single value of the internal energy. The experimental results were also modeled using a master equation formulation to obtain 〈∆E〉d, confirming the predictive power of the model calculations. The increase of temperature of the gas mixture and its effect on the IR fluorescence signal were also analyzed.
Introduction The CO2 laser decomposition of CDCl3 has been the subject of many studies,1,2 mainly due to the possibility of selective excitation of this molecule in natural isotopic abundance mixtures with CHCl3 and also because CDCl3 is one of the best candidates for deuterium enrichment. One of the undesirable effects of multiphoton absorption excitation (MPA) arises from the intermolecular redistribution of the absorbed energy, leading to heating of the reaction mixture and, consequently, to thermal reaction and loss of specificity. Collisions play a complex role in MPA processes. On one side, they may contribute to a more efficient pumping to high vibrational states of the absorbing molecule through collisional removal of rotational and anharmonic bottlenecks, as is usually the case for small molecules such as CDCl3. On the other side, collisions may be detrimental to the decomposition process due to deactivation of those species with internal energy above the reaction threshold. In addition, collisions will equilibrate the excitation energy within the various degrees of freedom, with thermalization of the system. Energy transfer processes remain one of the major problems in chemical kinetics. Numerous theoretical as well as experimental works have been reported up to the present providing information on the amount of energy transferred by collision, 〈〈∆E〉〉, its dependence on internal energy, and the nature of bath gas and parent molecules.3-9 IRMPA could provide a powerful procedure for preparation of vibrationally excited molecules in the ground electronic state suitable to study energy transfer processes. The advantage of this technique is that a variety of molecules can be easily excited to different initial energies by simply changing the incident laser fluence. On the other hand, a serious objection to the use of CO2 laser excitation arises from the dependence of 〈〈∆E〉〉 on average internal energy, which has been thought to impose the preparation of ensembles with very well-known energy distributions, while those obtained through MPA are poorly defined and, in the case of small molecules, could even be bimodal.9 However, it has recently been demonstrated that exponential energy decays are independent of the actual shape of initial unimodal distributions as well X
Abstract published in AdVance ACS Abstracts, May 15, 1996.
S0022-3654(95)03355-7 CCC: $12.00
as of bimodal functions.10 This means that MPA could be an adequate method of excitation for energy transfer studies, and one of the purposes of this work is to show its applicability to obtain the dependence of 〈〈∆E〉〉 on internal energy. The loss of energy from vibrationally excited molecules can be monitored with different techniques. One of them is timeresolved infrared fluorescence,11a,b,12a,b which has been used in the present paper. From the decay curves only the bulk average energy transferred, 〈〈∆E〉〉, could be obtained. However, the independence of the rate of energy decay on the distribution function allows us to equate the macroscopic and microscopic averages, that is, 〈〈∆E〉〉 ) 〈∆E〉, for exponential energy decay.10 One of the objectives of this work is to obtain a comprehensive model for the behavior of CDCl3 under IR laser irradiation in a variety of conditions. Toward this purpose, the experimental results, as a function of the laser fluence and the Ar pressure, were also modeled with a system of coupled rate equations. The mean average energy transferred per deactivating collisions, 〈∆E〉d, obtained from these calculations is compared with the values that resulted from the analysis of the IRF signals. In addition, the effect of temperature variation during the decay of the fluorescence signal is analyzed. The results obtained for CDCl3 as a collider provide further evidence for energy-pooling processes.2,13 Experimental Section The experimental setup has been previously reported.14 The experiments were performed with a pulsed TEA CO2 laser as IR radiation source, tuned at 10.91 µm, resonant with the ν4 mode of CDCl3 and with a pulse of 90 ns FWHM followed by a 200 ns tail. In order to increase the fluence, the diameter of the laser beam was reduced, keeping a parallel geometry, using a set of ZnSe lenses. The experiments were carried out under two different conditions of incident fluence and pressure of CDCl3. In one set of experiments the laser fluence, φ, was 2.5 J/cm2 and the pressure of CDCl3 was 3.0 Torr, whereas in the other the fluence was 4.6 J/cm2 and the pressure of CDCl3 was 1.0 Torr. In both cases the effect of Ar was studied with the addition of up to 70 Torr. © 1996 American Chemical Society
9746 J. Phys. Chem., Vol. 100, No. 23, 1996
Figure 1. IR fluorescence intensity vs collision number for experiments made with an incident laser fluence of 4.5 J/cm2, CDCl3 pressure of 1 Torr, and different pressures of added Ar: (b) 10 Torr of Ar and (0) 30 Torr of Ar. The solid lines represent master equation calculations considering the temperature change of the gas mixture: (s, bold) 10 Torr of Ar and (s) 30 Torr of Ar. The broken lines correspond to calculations made assuming a constant temperature of 300K: (- - -) 10 Torr of Ar and (‚‚‚) 30 Torr of Ar.
The collision numbers used to convert pressure to collision frequency were 1.186 × 107 and 1.620 × 107 s-1‚Torr-1 for collisions of CDCl3 with Ar and with CDCl3, respectively. The IR fluorescence was viewed through a NaCl side window with a HgCdTe Judson detector, sensitive from 5 to 12 µm, cooled in liquid nitrogen. A gas filter was used in order to isolate the 2ν5 overtone of CDCl3. The delay time was around 1 µs, and the maximum duration of the fluorescence signal was around 20 µs as shown in Figure 1. The laser pulse repetition frequency was 1 Hz. Depending on the signal strength, between 200 and 400 laser pulses were accumulated for each decay curve in order to improve the signal/noise ratio. The number of photons absorbed, 〈n〉, as a function of the Ar pressure, was also measured under the same conditions to determine the mean value of bulk excitation energy, 〈〈E〉〉. These measurements were made with a dual arrangement of pyroelectric detectors, and the ratio of the transmitted to incident intensity was obtained with a Scientech power ratio meter, averaging at least 10 signals in every case. Results and Discussion (a) Experimental Results. Typical fluorescence decay curves are shown in Figures 1 and 2. The curves follow a single exponential decay during the first 5 µs, which corresponds to approximately 600 and 1800 collisions for 10 and 30 Torr of Ar, respectively. At longer times the signal is more difficult to analyze, as the system approaches thermal equilibrium and consequently the emission is weaker. The average excitation energy of CDCl3 at both fluences and with varying pressures of Ar, represented by the average number of photons absorbed per molecule, 〈n〉, is displayed in Figure 2. As expected for a small molecule such as CDCl3, 〈n〉 increases with the pressure of added inert gas up to a fluence dependent limiting value at pressures above ∼40 Torr, due to collisional relaxation of the rotational and anharmonic bottlenecks to absorption.1,15 The processes undergone by a CDCl3 molecule irradiated by a CO2 laser in the presence of Ar, as deactivating gas, can be represented by the following scheme:
Rinaldi et al.
Figure 2. Average number of photons absorbed vs Ar pressure: (b) 2.5 J/cm2 and 3 Torr of CDCl3; (0) 4.5 J/cm2 and 1 Torr of CDCl3. The lines represent the master equation calculations at both fluences: (- - -) 2.5 J/cm2 and (s) 4.5 J/cm2.
CDCl3 + nhν f CDCl3* k1
CDCl3* + Ar 98 CDCl3 + Ar k2
CDCl3 + CDCl3* 98 CDCl3* + CDCl3 k3
CDCl3* 98 CCl2 + DCl
(1) (2) (3) (4)
The deactivation process (3) and the elimination reaction (4) have been previously studied.1,14 Under our present experimental conditions, the decomposition of vibrationally excited CDCl3 (reaction 4) can be neglected.1 The experimental results were used to obtain information on the energy transfer process due to collisional relaxation of vibrationally excited CDCl3 with Ar or in self-collisions. Different approaches were used. In one of them, called here direct method, the fluorescence intensity decay was converted to energy using the measured average number of photons absorbed per molecule, and the bulk average energy transferred, 〈〈∆E〉〉, was calculated from the slope of the fluorescence intensity decay. These results were then compared with those obtained with the inversion method developed by Barker.11 Finally, the fluorescence decay rate and the average number of photons absorbed, as functions of fluence and total pressure, were modeled with the use of a master equation that describes the various rate processes that take place under laser irradiation.1,16 (b) Direct Method. From the evolution of the average excitation energy, the average energy transferred, 〈〈∆E〉〉, can be calculated as a function of 〈〈E〉〉, using the following expression:
d〈〈E〉〉 ) ω〈〈∆E〉〉 dt
(5)
were ω is the collision frequency. For a linear dependence of 〈〈∆E〉〉 on 〈〈E〉〉, that is
〈〈∆E〉〉 ) a + b〈〈E〉〉
(6)
integration of eq 5 yields
〈〈E〉〉 ) -a/b + c exp(-bωt)
(7)
where c is an integration constant, showing that the energy decay should be exponential. If 〈〈E0〉〉 is the initial average excitation energy and 〈〈E∞〉〉 is the internal energy of the parent molecule
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J. Phys. Chem., Vol. 100, No. 23, 1996 9747
at a time long enough so that the thermal equilibrium has been reached, eq 7 becomes
〈〈E〉〉 ) 〈〈E∞〉〉 + (〈〈E〉〉0 - 〈〈E〉〉∞) exp(-bωt)
(8)
〈∆E〉 ) 0.41 ( 0.12 + (7.8 ( 1.6) × 10-4〈〈E〉〉 for φ ) 2.5 J cm-2 (14) and
so that 〈〈E〉〉∞ ) -a/b. From eq 7 it immediately follows that the value of b can be obtained directly from a measurement of the energy decay irrespective of a knowledge of the absolute value of 〈〈E〉〉. The value of a can then be calculated from 〈〈E∞〉〉, which could be obtained, in principle, from the asymptotic behavior of the energy decay. This requires the experimental measurement of a very weak signal, which is also affected by optoacoustic waves and mass transport out of the observation region. A more convenient and simple alternative is to calculate 〈〈E〉〉∞ from the equilibrium Boltzmann distribution. This reasoning leads to the conclusion that for an exponential energy decay both coefficients for the linear dependence of 〈〈∆E〉〉 on the average internal excitation energy can be calculated without any knowledge of the absolute value of 〈〈E〉〉, provided that the IRF intensity is linearly dependent on it. To apply the above arguments to a mixture of two gases, ω should include the contributions from both components. Assuming independent behavior, eq 5 can be written as follows:
〈∆E〉 ) 0.75 ( 0.11 + (1.4 ( 0.2) × 10-3〈〈E〉〉 for φ ) 4.5 J cm-2 (15)
d〈〈E〉〉 ) ω1〈〈∆E〉〉1 + ω2〈〈∆E〉〉2 dt
In this case, the difference between the results at both fluences is beyond the experimental error and is indicative of a more complex behavior than with Ar as a bath gas. One possible explanation could be an increase in the temperature of the gas mixture in the experiments with neat CDCl3. Master equation calculations, presented below, show that temperature increment is important only at low dilution and long times. Then, this undesirable effect can be easily avoided, considering only the decay of the IRF signal at early times, so that the value of b should not be affected by thermal heating. Another cause of this discrepancy could be an energy pooling mechanism, where excited CDCl3 collides not only with cold molecules of CDCl3 but also with excited ones,2,13 and whose contribution should be more important at higher fluences, when the fraction of excited molecules is higher, in agreement with the results obtained. It should be noted that the difference in the values of b directly affects the values of a, since it is calculated from b and 〈〈E〉〉∞. The values of 〈∆E〉 were then converted to 〈∆E〉d with the use of eqs 11-15 of ref 1b. The results are shown in Table 1. (c) Inversion Method. The intensity of the emission of infrared fluorescence is given by the following equation,17 which relates the observed IRF signal to the vibrational energy content of the excited molecules:
(9)
where ω1 and ω2 refer to CDCl3 and Ar as deactivators, respectively. If the dependence of 〈〈∆E〉〉 on energy is linear, i.e.
〈〈∆E〉〉1 ) a1 + b1〈〈E〉〉
and 〈〈∆E〉〉2 ) a2 + b2〈〈E〉〉 (10)
the rate of change of 〈〈E〉〉 results
d〈〈E〉〉/dt ) (a1ω1 + a2ω2) + (b1ω1 + b2ω2)〈〈E〉〉 (11) yielding, similarly to eq 7,
〈〈E〉〉 ) -(a1ω1 + a2ω2)/(b1ω1 + b2ω2) + c exp[-(b1ω1 + b2ω2)t] (12) and
〈〈E〉〉∞ ) -(a1ω1 + a2ω2)/(b1ω1 + b2ω2)
(13)
Calculations using the formulation of Durana and McDonald (see below, eq 19)17 showed that the IRF intensity from CDCl3 is linearly dependent on the average excitation energy (as well as on excitation energy), at least in the energy range of interest in this study. With this approximation and using 〈〈E〉〉∞ ) 525 cm-1, as calculated from the Boltzmann distribution at 300 K, the coefficients for CDCl3, a1 and b1, were determined with the above equations from the IRF signal decay from experiments with near CDCl3. Then, the corresponding parameters for Ar, a2 and b2, were calculated from the measurements with different pressures of added Ar, using eqs 12 and 13. The results obtained for Ar pressures in the range 10-70 Torr were in very good agreement. Considering that, for exponential decays, the bulk average of the amount of energy transferred equals the microscopic value,10 the following expressions for the dependence of 〈∆E〉 on 〈〈E〉〉 for Ar as a collider are obtained as averages of the results at different Ar pressures:
The results obtained at both fluences agree reasonably well and yield the following relationship for the depence of 〈∆E〉 on 〈〈E〉〉 as an average of eqs 14 and 15:
〈∆E〉 ) 0.58 ( 0.12 + (1.1 ( 0.2) × 10-3〈〈E〉〉 (16) For CDCl3 as bath gas, the following relationships were obtained:
〈∆E〉 ) 1.21 ( 0.12 + (2.3 ( 0.3) × 10-3〈〈E〉〉 for φ ) 2.5 J cm-2 (17) and
〈∆E〉 ) 7.35 ( 0.02 + (1.4 ( 0.2) × 10-2〈〈E〉〉 for φ ) 4.5 J cm-2 (18)
nmax
modes
I(E,t) ) [N(E,t)ex/Fs(E)]
∑ i)1
hνiAi1,0
∑ niFs-1(E - nihνi)
ni)1
(19) where N(E,t)ex is the number of vibrationally excited molecules is the Einstein coefficient for at energy E and time t, A1,0 i spontaneous emission for the 1-0 transition of the ith mode, hνi is the energy of the emitted photon and F(E)s and F(Enihνi)s-1 are, respectively, the density of vibrational states for all s oscillators at energy E and that for the s - 1 modes, omitting the emitting mode and energy contained in it. Following the method of Barker,11 this equation can be inverted to yield the average vibrational energy as a function of the IRF intensity. The procedure requires a knowledge of the initial excitation energy, so that the initial intensity corresponds to that
9748 J. Phys. Chem., Vol. 100, No. 23, 1996
Rinaldi et al.
TABLE 1: Values of 〈∆E〉d Obtained with Different Methods 〈E〉 (cm-1) 9600.0 16000.0
〈∆E〉d (cm-1) CDCl3 126.0 127.0 111.0 580.0 515.0 457.0
〈∆E〉all (cm-1) CDCl3 36.0 23.0 308.6 230.0
〈∆E〉d (cm-1) Ar 55.0 62.0 49.0 73.0 82.0 85.0
〈∆E〉all (cm-1) Ar 8.5 8.0 20.4 23.0
For CDCl3 as a collider we obtained
〈∆E〉 ) 2.71 ( 0.01 + (3.47 ( 0.03) × 10-3〈〈E〉〉 cm-1 at φ ) 2.5 J/cm2 (23)
method master equation inversion direct master equation inversion direct
calculated from eq 19. As shown in Figure 2, the average absorbed energy, calculated from the total absorbed energy and the number of molecules absorbing in the irradiated volume, is independent of Ar pressure at values higher than 40 Torr. This is indicative of a unimodal distribution, and it can be assumed that a Poisson distribution has been formed.18 The average initial energy, 〈〈E〉〉, was calculated from the average number of photons absorbed per molecule, 〈n〉, under the same experimental conditions as the IRF measurements. For an Ar pressure of 60 Torr, the collision frequency is large enough so that intermolecular energy redistribution could take place during the absorption process, producing the transfer of part of the energy absorbed by the parent molecule to the bath gas during the laser pulse. In this situation, 〈n〉 would not provide a direct measurement of the initial average excitation energy of CDCl3. However, master equation calculations, presented below, show that the absorption of the laser energy occurs in the first 200 ns and that it is not affected by energy transfer to Ar during the laser pulse. Considering the calculated temperature increment during the laser pulse, the number of photons transferred from CDCl3 to Ar could be estimated to be less than 10% of the total number absorbed. As a consequence, the total absorbed energy, given by 〈n〉, represents, to a good approximation, the initial internal energy of CDCl3. Then, the average energy of the ensemble of excited molecules is ∼9600 cm-1 at a fluence of 2.5 J cm-2 and ∼16 000 cm-1 at 4.5 J cm-2. These values were used as a calibration in conjunction with eq 19 to convert the IRF curves in energy decays. From the slope of these curves we calculate the values of 〈〈∆E〉〉 as a function of 〈〈E〉〉, and then a plot of eq 6 allowed us to obtain the coefficients a and b. As before, since 〈〈∆E〉〉 ) 〈∆E〉, the following relationship is obtained for the dependence of 〈∆E〉 with 〈〈E〉〉, for Ar as bath gas, at a fluence of 2.5 J/cm2.
〈∆E〉 ) 0.53 ( 0.02 + (8.30 ( 0.04) × 10-4〈〈E〉〉 cm-1 (20) The same treatment, at φ ) 4.5 J/cm2, yields
〈∆E〉 ) 0.40 ( 0.01 + (1.25 ( 0.01) × 10-3〈〈E〉〉 cm-1 (21) The parameters of these equations show good agreement and give the following average behavior:
〈∆E〉 ) 0.46 ( 0.02 + (1.04 ( 0.05) × 10-3〈〈E〉〉 (22) These results also show good coincidence with those obtained with the direct method, which is a consequence of the direct proportionality between the internal energy content and the intensity of the fluorescence, at least at the energies of the experiments.
and
〈∆E〉 ) 11.0 ( 0.2 + (1.86 ( 0.01) × 10-2〈〈E〉〉 cm-1 at φ ) 4.5 J/cm2 (24) In this case there is also a good agreement with the values of b of eqs 17 and 18. However, the values of a differ by a factor of 2, and the difference is much larger than the error limits for each method. An explanation of this discrepancy is not apparent, but it should be noted that in the direct method a is calculated from 〈〈E〉〉∞ and b, so that any error in these values, as well as temperature variation during the pulse in the experiments with neat CDCl3, directly affects the value of a. Similarly to the results of the direct method, the parameters for CDCl3 as deactivator are also different at each fluence. The values of 〈∆E〉 were also converted to 〈∆E〉d. The results are shown in Table 1. (b) Master Equation Calculations. The decay of IRF intensity is a measurement of the evolution of the vibrational level populations as the excited molecules lose energy by collisions. In the present modeling, these populations are obtained from the solution of the following coupled rate equations:1
dNi/dt ) I(t)[σi,i-1Ni-1 + σi,mi+1Ni+1 - (σi+1,i + σi-1,i)Ni] +
dE(ω1∑PaijNj + ω2∑PbijNj) - ωiNi - kiNi (25)
where I(t) stands for the intensity of the laser pulse, σi+1,j is the microscopic absorption cross section from level i to level i + 1, Ni is the population of level i, ω1 and ω2 stand for the collision frequencies of CDCl3 with Ar and with CDCl3, respectively, Paj and Pbij are the collisional transition probabilities from level j to level i for Ar and CDCl3 as deactivating gases, and ki are the microscopic rate constants for decomposition of the excited molecule. In this case the rate constants were not considered because reaction 4 can be neglected under the present experimental conditions. All modeling calculations were made using a graining of 10 cm-1 to assure convergence of the integration. The model calculations have to reproduce two different pieces of experimental information, which are the average number of photons absorbed per molecule and the rate of decay of infrared fluorescence, both measured as functions of the incident fluence and pressure of parent and added bath gas. All this information can be acquired from a knowledge of the level populations, which are obtained from numerical integration of eq 25. In order to do this the microscopic absorption cross sections used for the calculations were the same as before,1,16 so that only the values of the collisional transition probabilities had to be determined. The transition probabilities Paij and Pbij for collisional deactivation were calculated using an exponential deactivation model, and the probabilities for transitions to higher levels were calculated by detailed balance. In these calculations 〈∆E〉d is an adjustable parameter selected to reproduce the experimental results and depends linearly on average internal energy, i.e., 〈∆E〉d ) R + β〈E〉. In order to compare the experimental fluorescence intensities with those obtained from model calculations, eq 19 was numerically integrated over the levels population that resulted from eq 25. Temperature variation always occurs in multiphoton absorption processes as a consequences of intermolecular energy
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J. Phys. Chem., Vol. 100, No. 23, 1996 9749
Figure 3. Final temperature vs Ar pressure: (a) calculated from the average number of photons absorbed at (b) 2.5 J/cm2 and 3 Torr of CDCl3 and (0) 4.5 J/cm2 and 1 Torr of CDCl3; (b) Calculated from the master equation: (- - -) 2.5 J/cm2 and (s) 4.5 J/cm2.
transfer. These temperature changes have been observed experimentally using the mercury tracer method19 and theoretically modeled using a system of time-resolved coupled rate equations.20 The present model calculations were made considering the temperature rise of the gas mixture and also, for comparison, at constant 300 K. The increase in temperature affects the collisional relaxation process by changing the collision frequency, ω, and the transition probabilities elements, Pij, through the condition of detailed balance, i.e., Pij/Pji ) [F(Ej)/ F(Ei)] exp[-(Ej - Ei)/kT] The instantaneous temperature, during and after the laser pulse, was calculated from
∆T ∆t
)
∑ ∑ NiωPij(Ej - Ei) i)1 j)1 ∑CνkNk
(26)
k)1
where Cνk is the heat capacity at constant volume of the kth component in a mixture with density Nk. The results obtained are shown in Figure 3, where the temperatures obtained from the resolution of the master equation, at times long enough so that no further change is produced, are compared with the temperatures calculated from the energy absorbed and the heat capacity of the system, i.e., ∆T ) 〈n〉/ ∑Cν,k. It should be noted that for our present experimental conditions, the total energy absorbed by the sample, and hence the temperature of the system, is larger at the lowest fluence because at that energy the concentration of absorbing molecules is larger than at 4.5 J/cm2. A very good agreement between both results is observed, at pressures greater than 40 Torr. One consequence of this temperature effect is that, when properly considered in the master equation, model calculations allow for the extraction of 〈∆E〉d values without need of high dilutions of the absorbing molecule, which results in low intensity of the fluorescence signal and the large errors in the determination of the decaying rates constants. In the model calculations the collisional deactivation of vibrationally excited CDCl3 by collisions with both Ar and cold CDCl3 molecules was taken into account. Collisions between CDCl3 molecules were incorporated using 〈∆E〉d ) 50 + 9 × 10-3〈E〉 cm-1 at 2.5 J/cm2 and 〈∆E〉d ) 100 + 3 × 10-2〈E〉 cm-1 at 4.5 J/cm2, which are the values that produced the best agreement with the experimental results, for measurements without the addition of Ar. For the collisional deactivation by Ar, the expression for 〈∆E〉d that best reproduces the experimental results is 〈∆E〉d ) 25 ( 3 × 10-3〈E〉 cm-1 for the cases here studied. These values of 〈∆E〉d, for both Ar and CDCl3,
are in good agreement with those obtained by the direct and the inversion methods, but they are smaller than those extracted from model calculations involving molecules excited above the reaction threshold.1 All the results obtained are compared in Table 1. The model calculations can reproduce very well the experimental dependence of 〈n〉 with fluence and with pressure of added Ar above 40 Torr. At lower pressures only a fraction of CDCl3 molecules are in resonance with the laser photons, due to rotational and anharmonic restrictions, which results in lower absorption. These processes can be incorporated in the master equation to obtain an adequate description of the absorption process. However, the purpose of this work is to model the collisional vibrational relaxation which is a post-pulse phenomenon that depends only on the average energy absorbed and not on its distribution as determined by the absorption process. In addition, at low dilution of CDCl3, energy pooling generated by collisions between vibrationally excited CDCl3 can significantly contribute to the absorption process and should be properly taken into account.2,13 Typical model calculations of the IRF, with variable and constant temperature, are shown in Figures 1 and 2 together with the experimental curve, at φ ) 4.5 J/cm2 with 10 and 30 Torr of added Ar. It can be observed that when the temperature effect is not taken into account, the calculated intensity reproduces the experimental data well at very short times, but a departure is observed at longer times. This is simply the consequence that at very short times the transfer of absorbed energy from the parent to the bath gas does not produce any significant change in the temperature of the gas mixture. It can also be observed that, up to approximately 100 collisions, model calculations show that the decay of fluorescence follows a single exponential in coincidence with the experimental results. At both fluences, the results of the calculations with high pressure of Ar tend to be the same as those obtained with constant temperature, because the heat capacity of the gas mixture is high enough to avoid a large temperature rise (Figure 1). Hence, in experiments at high dilution, the temperature remains constant and the decay of IRF is exponential. It should be noted that the energy decay is independent of the transition probability model10 and, hence, the results obtained by resolution of the master equation should not be affected by the particular model of Pij used. In conclusion, the present study shows very good agreement between the values of the average energy transfer per collision obtained from the resolution of the master equation and those calculated from the decay of infrared fluorescence, directly or according to the inversion method developed by Barker. The direct method constitutes an important simplification to the treatment of the experimental data. It assumes that the IRF intensity is proportional to 〈〈E〉〉 and it is only applicable to exponential energy decays, which corresponds to linear dependence of 〈∆E〉 on average excitation energy. The inversion of the IRF signal to yield the energy decay of the excited molecules also allows the extraction of the dependence of 〈〈∆E〉〉 and 〈∆E〉d on internal energy, yielding results in good agreement with the direct method. At high dilution, heating of the gas mixture due to collisional relaxation of the absorbed energy is moderate, and a constant temperature can be assumed for data obtained at reasonably long times (∼2000 collisions). At low dilution, thermal heating is more significant and seriously affects the shape of the decay curve at times longer than those corresponding to ∼100 collisions. The results with Ar are in good agreement with previous reports.1 For neat CDCl3 the value of 〈∆E〉d is smaller than
9750 J. Phys. Chem., Vol. 100, No. 23, 1996 those obtained in ref 1, where the molecules were excited above the reaction threshold. However, in that work the collision partner was CHCl3 which is transparent to the laser frequency used, and hence remains cold. Therefore, it can be expected that 〈∆E〉d should be lower in the present case, because excited CDCl3 collides not only with cold molecules of CDCl3 but also with excited ones, leading to an energy-pooling process.2,13 These results confirm the power of the master equation in order to provide a consistent quantitative model of the various processes involved in IR laser multiphoton absorption. Acknowledgment. The authors wish to thank the CONICET and CONICOR for financial assistance. References and Notes (1) Azca´rate, M. L.; Quel, E. J.; Toselli, B. M.; Ferrero, J. C.; Staricco, E. H. J. Phys. Chem. 1986, 92, 403. (2) McRae, G. A.; Yamashita, A. B.; Goodale, J. W. J. Chem. Phys. 1992, 92, 10,5997. (3) Oref, I.; Tardy, D. C. Chem. ReV. 1990, 90, 1407. (4) Quack, M.; Troe, J. Gas Kinetics and Energy Transfer; Chemical Chemical Socity: London, 1977; Vol. 2. (5) (a) Flynn, G. W. Acc. Chem. Res. 1981, 13, 334. (b) Mullin, A. S.; Michaels, C. A.; Flynn, G. W. J. Chem. Phys. 1995, 102, 6032. (6) Tardy, D. C.; Rabinovich, B. S. Chem. ReV. 1977, 77, 369.
Rinaldi et al. (7) Krajnovich, D. J.; Parmenter, C. S.; Cattlet, D. L., Jr. Chem. ReV. 1986, 87, 237. (8) (a) Bollati, R. A.; Ferrero, J. C. Chem. Phys. Lett. 1994, 218, 159. (b) Bollati, R. A.; Ferrero, J. C. J. Phys. Chem. 1994, 98, 3933. (9) Laser Spectroscopy of Highly Excited Molecules, Letokhov, U. S., Ed.; Adam Helger: New York, 1989. (10) Corondao, E. A.; Rinaldi, C. A.; Velardez, G. F.; Ferrero, J. C. Chem. Phys. Lett. 1994, 227, 164. (11) (a) Yerram, M. L.; Brenner, J. D.; King, K. D.; Barker, J. R. J. Phys. Chem. 1990, 94, 6341. (b) Barker, J. R.; Toselli, B. M. Int. ReV. Phys. Chem. 1993, 12, 305. (12) (a) Zellweger, J. M.; Brown, T. C.; Barker, J. R. J. Phys. Chem. 1986, 90, 461. (b) Zellweger, J. M.; Brown, T. C.; Barker, J. R. J. Chem. Phys. 1986, 83, 6251. (13) Oref, I. J. Chem. Phys. 1981, 75, 1,131. (14) Va´zquez, M. A.; Azca´rate, M. L.; Quel, E. J.; Doyennette, L.; Rinaldi, C. A.; Ferrero, J. C. Las. Chem. 1994, 14, 191. (15) Rinaldi, C. A.; Lane, S. I.; Ferrero, J. C. Int. J. Chem. Kinett. 1994, 26, 705. (16) Toselli, B.; Ferrero, J. C.; Staricco, E. H. J. Phys. Chem. 1986, 90, 4562. (17) Durana, J. F.; McDonald, J. D. J. Chem. Phys. 1977, 64, 2518. (18) Nakashma, N.; Shimo, N.; Ikeda, N.; Yashira, K. J. Chem. Phys. 1985, 82, 5285. (19) Wallington, T. J.; Sheer, M. D.; Bruan, W. Chem. Phys. Lett. 1987, 138, 538. (20) Eberhardt, J. E.; Knot, R. B.; Pryor, A. W.; Gilbert, R. G. Chem. Phys. 1982, 69, 45.
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