Anal. Chem. 1995,67,365-373
Time=ResolvedPhotothermal Grating Calorimetry for Investigating Kinetics of Free=RadicalChain Reactions R. T. Cambron and J. M. Hawis*
University of Utah, Department of Chemistty, Salt Lake City, Utah 841 12
Dfiaction from photothermal gratings is monitored on a nanosecond time scale to determine the rates of a propagating free-radical chain reaction. The kinetics of photochlorination of chloroform, initiated by 355 nm laser radiation, are investigated. Monitoring these reactions on a nanosecond time scale avoids chain termination and enables the time-resolved *action transients to be interpreted using a simple kinetic model for the two propagationsteps of the chain reaction. From the pseudofirst-order rate parameter of this simple kinetic model, individual rate constants for the steps that propagate the chain are determined from Stern-Volmer relationships. The rate constant for hydrogen abstraction from chloroform by a chlorine atom in carbon tetrachloride is determined, k l = (7.8 & 0.6) x lo6M-' s-l, and the chlorine atom transfer rate constant is found, k2 = (4.6 & 0.4) x lo8M-l s-l. The reported rate constants do not depend on the concentration of radicals generated in the laser pulse. This direct approach is an attractive alternative to stationary-statetechniques that generally o d y yield composite rate constants. Time-resolved monitoring of the kinetics of free-radical chain reactions is an important but challenging analytical problem. The photolytic initiation of free-radical reactions is important to many areas of technology, including the synthesis of organic compounds, the synthesis and modiiication of polymer materials, and the photobindmg of molecules to surfaces. In these applications, the performance of materials depends sensitively on their reuctivity, where the rates of reactions dictate the product yield, molecular weight, or degree of curing or cross-linking. Despite the importance of measuring the rates of free-radical chain processes, few analytical techniques are capable of direct time-resolved monitoring of the progress of these reactions due to the combined need for speed and sensitivity. Techniques such as photoinitiated electron spin resonance'-3 and laser flash photoly~is~-~ have had limited success elucidating the kinetics of free-radical chain reactions. ESR lacks the speed to follow the (1) Bresler, S. E.; Kazbekov, E. N.; F o d c h e v , V. N.; Shardrin, V. N. Mukromol. Chem. 1972,157,167-177. (2) Bresler, S . E.; Kazbekov, E. N.; Shardrin, V. N. Mukromol. Chem. 1974, 175,2875-2880. (3) Atherton, N. M.; Melville, H.; Whiffen, D. H. J. Polym. Sci. 1959,34,199207. (4) Scaiano, J. C. J. Am. Chem. SOC.1980,102, 7747-7753. (5) Nigel, J. B.; Ingold, IC U.; Landers, J. P.;Lusztyk, J.; Scaiano, J. C. J. Am. Chem. SOC.1985,107,5464-5472. (6) Kaiser, E. W.; Rimai, L.; Schwab, E.; Lim,E. C . J. Phys. Chem. 1992,96, 303-306. 0003-2700/95/0367-0365$9.00/0 0 1995 American Chemical Society
individual chain propagation steps on a submicrosecond time scale, while flash photolysis lacks the sensitivity to monitor submicromolar radical concentrations required to sustain long chains and high product yields. The concept of investigating free-radical chain reactions by monitoring the heat that they release into the sample was f i s t demonstrated by Magee and Daniel~.~ In their experiments, the heat deposited by the free-radical photobromination of toluene was detected using 50 thermocouple junctions mounted in a quartz photocalorimeter cell. Since this initial work, several other attempts to develop calorimetry for investigating photoinitiated chain reactions have been reported; some have involved pseudosteady-state methods for monitoring heat deposited from photcinitiated gas-phase free-radical chain reactions between halogens and hydrogen using an optoacoustic techniq~e.83~ These methods are based on generating a series of non-stationary-state periods by periodic interruption of photochemical initiation, similar to the rotating sector technique,'OJ1 yielding a ratio of composite rate constants of the form k,2/kt, where k, is the composite rate constant for chain propagation and kt is the composite rate constant for bimolecular chain termination. To estimate absolute rate constants for chain reactions using these techniques, the initial radical concentrations must be accurately known, and termination rates must be independently determined. Calorimetric monitoring of photoinitiated free-radical chain reactions in liquids using thermal lens spectroscopy was reported by Daree early in the development of the thermal lens technique.12 This method is based on the optical detection of heat deposited by a photoinitiated chemical reaction through its influence on the refractive index of the sample. Heat deposited from the freeradical photobromination of toluene was detected during intermittent photoinitiation by a helium-neon laser, similar to a rotating sector approach. From the kinetic signature of the thermal lens signals collected, composite reaction rate constants for chain propagation and chain termination were determined. Despite the high sensitivity of a thermal lens experiment for detecting heat from chain reactions in liquids, the temporal resolution of the steady-state measurement is not fast enough to determine the individual rate constants for the reactions that propagate the chain. To determine the rate constants of these reactions, the temporal resolution of the experiment must be (7) Magee, J. L.;Daniels, F. J. Am. Chem. SOC.1940,62, 2825-2833. (8) Diebold, G . J.; Hayden, J. S. Chem. Phys. 1980,49,429-437. (9) O'Connor M. T.;Diebold, G.J. Nuture 1983,301,321-322. (10) Briers, F.;Chapman, D. L;Walters, E. J. Chem. SOC.1926,562. (11) O'Driscoll, K F.; Mahabadi, H. IC]. Polym. Sci., Polym, Chem. Ed. 1976, 14,869-881. (12) Darke, IC Opt. Commun. 1971,4,238-242.
Analytical Chemistry, Vol. 67,No. 2,January 75,7995 365
capable of directly monitoring the kinetics of the chain propagation steps on the time scale of their approach to steady-state, zeroorder kinetics. Although thermal lens experiments that employ pulsed laser excitation have sufticient speed to monitor radical recombination reactions,13these experiments lack the nanosecond response time needed for monitoring the chain propagation steps free of the intluence from chain termination; as discussed below, avoiding termination kinetics in the signal greatly simplifies modeling the propagation kinetics. Both the speed and sensitivity requirements for monitoring chain reactions under these conditions are readily satisfied by measuring diffraction from photothermal gratings. Early applications of photothermal grating spectroscopyinclude the study of excited-state lifetimes, energy transport, and optically excited ultrasonic waves.14-18 More recently, the method has been developed for time-resolved monitoring of the kinetics and energetics of photochemical reactions in l i q ~ i d s . l ~Time-~~ resolved photothermal grating spectroscopy has been used to investigate the photodissociation and nongeminate recombination of iodine,21 bromine,Z2 and ch10rine.l~ The temporal resolution of this technique is well suited to investigate photoinitiated processes on a nanosecond to microsecond time scale, including the monitoring of propagating free-radical chain reactions. Recombination of submicromolar concentrations of radicals, which terminates the chain, occurs on a time scale much longer than microseconds; monitoring a chain reaction on a nanosecond time scale, therefore, enables modeling the kinetics of the chain reaction using only the chain propagating steps of the reaction. The resulting kinetic model contains a single, pseudo-first-order kinetic parameter. Using this simple kinetic model to describe time-resolved diffraction transients, we have investigated a simple model chain reaction, the photochlorination of chloroform in carbon tetrachle ride. From the kinetic signature of the time-resolved diffraction transients as a function of chloroform concentration at constant chlorine concentration, the rate constant for the hydrogen abstraction from chloroform by chlorine is determined. From the kinetic signature of diffraction versus chlorine concentration at constant chloroform concentration, the rate constant for the atom transfer reaction between the trichloromethyl radical and molecular chlorine is determined. The reported rate constants do not depend on the concentration of radicals generated in the laser pulse; this direct approach is an attractive alternative to determining rate constants by stationary-state techniques that yield composite rate constants. THEORY Since photothermal grating spectroscopy is based on detecting heat deposited into the sample from nonradiative processes, (13) Cambron, R T.; Harris,J. M. J. Phys. Chem., in press. (14) Phillon, D. P.; Kuizenga, D. J.; Siegman, A. E. Apfil. Phys. Lett. 1975,27, 85-87. (15) Jarasiunas, D.;Gemtsen, H.J. Appl. Phys. Lett. 1978,33,190-193. (16) Eichler, J. J.; Eichler, J.; Knof, J.; Noack, C. H.Phys. Status SolidiA 1979, 52, 481-486. (17) Salcedo, J. R.; Siegman, A. E.; Dlott, D. D.; Fayer, M. D. Phys. Rev. Lett. 1978,41, 131-134.
(18) Nelson, K A; Miller, D.; Lutz, D. R; Fayer, M. D. J. Appl. Phys. 1982,53, 1144-1149. (19) Zimmt, M.B. Chem. Phys. Lett. 1989,160,564-569. (20) Morais, J.; Ma, J.; Zimmt, M. B. J. Phys. Chem. 1991,95,6490. (21) Zhu,X. R; Harris, J. M. Chem. Phys. 1991,157,409-422. (22) Zhu, X. R; Harris,J. M. Chem. Phys. Lett. 1991,186,183-188.
366 Analytical Chemistry, Vol. 67, No. 2, January 15, 1995
interpreting the calorimetric data requires relating the timeresolved heat release to the molecular photophysics. Several assumptions simpQ modeling heat deposited in this system. The thermophysical constants of the medium and reaction rate constants are assumed to be invariant with respect to changes in temperature and composition. This assumption is reasonable since the maximum increase of the temperature due to photoexcitation and subsequent photochemical reactions is less than 0.1 K in the present experiments. The error introduced in the physical constants by the temperature change is smaller than other experimental uncertainty. The change in sample composition during the reaction is also small for the submicromolarradical concentrations generated by the laser pulse, so the physical properties of the solvent deviate little from their initial values. Heat Deposited by Photodissociation of Chlorine. Following absorption of ultraviolet radiation, the chlorine molecule dissociates into two atoms,
c:1,
2c1'
where f$d is the photodissociation yield at the photolysis wavelength. After dissociation, a fraction of the separated atoms combine geminately within the cavity of the surrounding solvent cage on a picosecond time scale.23324 Another fraction of the dissociated atoms may succeed in breaking out from the solvent cage such that they are separated by at least one solvent molecule. The probability of an encounter between these atoms on a subnanosecond time scale is h i t e due to a higher local concentration of at0ms.2~Based on a recent investigation and development of a theory for these geminate rebinding the probability of diffusive geminate recombination for chlorine should become negligible when the atoms have been separated for 2 1 ns. The remaining dissociated atoms move in an unperturbed solvent until they encounter either another radical species or a reactant from which the chlorine atom may abstract a hydrogen. Since dissociation and geminate recombination generate heat in the sample on a time scale faster than the nanosecond rise time of the diffraction response (limited by the propagation of sound across one fringe spacing or the laser pulse, whichever is largerz6), their associated heat amplitude, Qr, is not time resolved. Heat associated with photodissociation,geminate recombination, and further photoexcitation of chlorine atoms during the laser pulse appears as an initial rapid rise in the diffraction signal prior to the time-resolved component of the signal. The relationship between this heat and the photophysics of chlorine has been presented elsewherel3 and may be used to determine the photodissociation yield from its amplitude compared to the signal from nongeminate recombination. Heat Deposited by Nongeminate Recombination of Chlorine Atoms. Nongeminate recombination of chlorine atoms occurs on a time scale that is slower than the instrument response time. The kinetic signature of this heat release is governed by the second-order recombination of the chlorine atoms,
2
2 c1' c1, where k3 is the recombination rate constant defined as follows: (23) Harris,A L.; Berg, M.; Harris,C. B. J. Chem. Phys. 1986,84, 788-806. (24) Zhu, X. R; Harris,J. M. Chem. Phys. 1991,157,409-422. (25) Mier, J. B.; Postlewaite, J. C.; Zyung, T.; Chen, S.; Roemig, G. R; Wen, X.; Dlott, D. D.; Szabo, A J. Chem. Phys. 1990,93,8771-8776. (26) Genberg, L.; Bao, Q.; Gracewski, S.; Miller, R J. D. Chem. Phys. 1989, 131,81-97.
(3)
Integration of eq 3 yields the time dependence of the chlorine atom concentration describing nongeminate recombination following the excitation pulse, t 2 to. The rate of heat released by nongeminate recombination is predicted by multiplying the time dependence of the chlorine atom concentration by the heat deposited upon nongeminate recombination, Ma; dQ,ci,(t > to> dt
d[C1'] (t > to> dt
2
- mk3[c1'l,to2
[ l + 2k3[C1'l,tg2
(4)
tion reactions contribute to the overall calorimetric signal, manipulating the concentration of the reacting species and photolysis conditions enables temporal separation of the rates of chain propagation and termination by more than an order of magnitude. This allows monitoring this system under conditions where heat deposited by chain termination makes a negligible contribution to the observed signal. The total contribution to the time-resolved calorimetric signal from chain termination is described by a rate constant, kt, that is an enthalpy-weighted composite rate constant for eqs 8-10. In these experiments, the rate of chain propagation, k,, is more than an order of magnitude greater than the second-order initial rate of chain termination, 2kt[CPlI,t,. Under the conditions t to> = [cl'],to
[l - exp(-k,t)]
+
(7)
Termination:
C1'
+ R'?.
RCl
2R'5R2
(9)
+
where k, = kl[RH] kz[Clz] is the pseudo-first-order rate constant of chain propagation. The rate of heat released by chain propagation is then given by
(10)
Heat deposited into the sample by free-radical chain propagation and termination occurs on a time scale longer than the instrument response. While both chain propagation and terminaAnalytical Chernistty, Vol. 67, No. 2, January 15, 1995
367
where AH1 and A H 2 are the heat released by hydrogen abstraction and atom transfer respectively, Q,' = [Cl'lt,t,,(AH~k~2[RH12 AH&~[RHlkz[Cl~l)/k, is the amplitude of the heat from the pseudo-first-order component of the kinetic model, and Q," = [Cl'lt,to(AH~ AHz)kl[RH]kz[Clzl/k, is the amplitude of the heat from the zero-order kinetic component of the model. The accumulated heat produced by chain propagation is found by integrating eq 15:
+
Q,,(t > to> = Q,'[1
- exp(-k,t)l
+ Q,"t
(16)
DifIraction from Photothermal Gratings. Optical excitation of radical precursors in an interference pattern initially creates a periodic spatial distribution of radicals that form a population grating. As the radicals participate in a free-radical chain readion, a thermal pattern is generated that changes the density by the thermal expansion of the sample matrix, which changes the refractive index of the matrix. In photothermal grating experiments, the refractive index gradient caused by nonradiative relaxation and density change is selectively detected by using a nonresonant probe wavelength that is well removed from any optical absorption. Under these conditions, the modulation of the refractive index is dominated by the density change due to the thermal expansion, and the contribution from population gratings changes accompanying chlorine can be n e g l e ~ t e d . Volume ~~ dissociation and radical and/or atom recombination can contribute -15% to the amplitude of the refractive index m0du1ation.l~The chain propagation kinetics, however, involve a constant number of molecular and/or atomic species, so the change in volume accompanying these reactions is much smaller and can be neglected. The modulation of the refractive index is defined through its relationship to the interference pattern created at the intersection of the two equal intensity laser beams, I = l o [1+cos(qx)I, where l o is the total intensity of the two excitation beams and q is the grating wave vector which is related to the grating spatial period, A = 2x/q = ,le/(2ttosin(Oe)). The time-resolved temperature modulation, T@,t),induced by the free-radical chain reaction is modeled using the thermal diffusion equation,28which is solved by using a spatial-domain Fourier transform followed by a timeWe obtain a set of equations to domain Laplace describe the time evolution of the heat flow into a series of uncoupled spatial harmonic modes. The time-dependent heat source, dQ(t)/dt, may be partitioned into fast and slow components, where the fast heat source may be approximated as an i m p u l ~ e ;the ~ ~fast * ~heat ~ impulse can be related to the photodissociation of chlorine, which is presented e1~ewhere.l~ The slow component of diffraction provides information about the rates of reactions of chlorine atoms and the subsequent chain reaction. To predict the efficiency of diffraction at the Bragg angle, the first (m= 1)spatial Fourier component of the slowly deposited heat, dQSm=l(t> to)/dt, contains the kinetic signature of the photothermal transients: (27) McGraw, D. J.; Hams, J. M. Phys. Rev. A 1 9 8 6 , 10, 4829-4842. (28) Desai, R C.; Levenson, M. D.; Barker, J. A Phys. Reu. A 1 9 8 3 , 2 7 , 19681976. (29) McGraw, D. J.; Michaelson, J.; Harris,J. M. J. Chem. Phys. 1987,86,2536. (30) Zhu, X. R; Harris, J. M. /. Phys. Chem. 1 9 8 9 , 93, 75-83. (31) Tellinghuisen, J. J. Chem. Phys. 1 9 8 2 , 76, 4736-4744.
368 Analytical Chemistty, Vol. 67, No. 2, January 15, 7995
The time-dependent heat source produces a refractive index change, An, in the fundamental spatial harmonic given by the convolution of the heat source with impulse response of the thermal diffusion,An = [(dn/dT)/~oC~] exp(-D&?)*d@"l (t > to)/dt, where Dth = ,l/eoCp is the thermal diffusivityof the medium and * represents convolution. S i e g m a ~has ~ ~derived ~ an expression for the diffraction efficiency of a crossed Gaussian volume grating in the weak Bragg limit probe at Bragg incidence by a Gaussian beam as a function of the change in the refractive index, An. The time-resolved diffraction efficiency from the propagating steps of a free-radical chain reaction is proportional to the square of the refractive index modulation and given by
where Af = &r"='(dn/dT)/eoCp is the amplitude of the refractive index modulation of the first harmonic grating due to photodissociation and geminate recombination of chlorine. The convolution in eq 18 results in the following expression for the time dependent diffraction efficiency:
12
where&' = QSm=l'(dn/dT)?leoCpand A," = QSm=l"(dn/dT)/& are the amplitudes of refractive index modulation of the first harmonic grating due to pseudo-first-order and zeroorder chain propagation reactions and kth = Dfiq2 is the first-order rate constant of thermal decay. EXPERIMENTAL SECTION
Photothermal Grating Experiment. The photothermal grating experiment used in these investigations is shown in Figure 1. Two excitation beams, derived from a frequency tripled, Qswitched Nd:YAG laser, 1, = 355 nm, with a pulse duration of 5 ns fwhm, are crossed in the sample with an angle of 4.2" (in air) near their waists, each having a l/e2 spot size of me = 410 f 15 pm. The excitation laser pulse was triggered at a repetition rate of 10 Hz to prevent the heat building up from shot-to-shot and to allow diffusion and convection of photoproducts out of the excitation zone between pulses. The radiation from a 25 mW HeNe laser, ap = 632.8 nm, is used as the probe beam, which has a measured spot size of up = 211 f 8 pm. The probe beam is focused to intersect the grating at Bragg angle so that the three beams cross near their waists. The diffracted beam is angularly (32) Siegman, A E. J. Opt. SOC.Am. 1 9 7 7 , 67, 545.
n n a
I
" 1
'g
I
Filter
9
4.0
A
Finer
Figure 1. Photothermaldiffraction experiment; detailed description in text.
0
4
8
12
16
20
Time (p)
resolved, spectrally filtered through an interferencefilter, imaged through a pinhole to remove scattered light, and then reimaged onto a photomultiplier tube (PMT). Intersection of the thermal grating by the probe beam at its waist prevents the probe beam divergence from being affected by the formation of a thermal lens withii the excitation zone.33 At the highest excitation intensity, the temperature rise at the center of the thermal grating is 0.1 K. The current output of the PMT is directed to a LeCroy 9450 digitizing oscilloscope for signal averaging. The oscilloscope was triggered using an unbiased silicon photodiode to detect a reflection from the pump beam. One hundred transients were collected, averaged, and transferred from the scope to a PC-XT via an IEEE-488 interface. Data Analysis. Photothermal transients were fit to their kinetic model by a nonlinear least-squares routine employing a Marquardt algorithm compiled in Microsoft FORTRAN. Uncertainties in the fitted parameter are estimated at the 95%confidence level using Student's t statistics. Reagents and Solutions. Chlorine samples were prepared by bubbling Matheson UHP (299%) grade chlorine gas into carbon tetrachloride (Berdict & Jackson, glass distilled). The concentration of chlorine in each sample was determined by measuring each sample's absorbance at the absorption maximum, 330 nm, using a Hewlett Packard UV-vis diode array spectrometer. The corresponding concentration of each sample was calculated using a previously reported absorption cross section of chlorine.34 A fresh sample of chloroform (Omni-solve, glass distilled) was used for each experiment. Samples were prepared in a darkened room under a 30 W incandescent red light to prevent photoinitiation of the chain reaction by ambient room light prior to the measurement. Photothermal signals were collected immediately after preparing each sample. Heat deposited by a propagating free-radical chain reaction, eq 16, follows a mixed-order kinetic model. The observed rates of both the pseudo-first-order and zero-order components of this model may be separated in space (or concentration) and time. As a result, gradients in radical concentration, caused by exponential attenuation of the excitation light by absorption across the path length of the sample cell, do not iduence the observed rate of chain propagation. At the highest chlorine concentration,13.3 mM, the absorbance of the sample along the 1mm beam path was A , = 0.07, resulting in a 15%drop in the excitation intensity and the resulting concentration of chlorine atoms by the end of the sample cell. The sample is assumed to be weakly absorbing, (33) Hu, C.; Winnery,J. R Appl. Opt. 1 9 7 3 , 12, 72-79. (34) Czech, F. W.; Juchs, R J.; Antczak, H. F. And. Chem. 1961,33,705-707.
Figure 2. Photothermal diffraction signal collected from a mixture of chlorine and chloroform superimposed on a signal collected from 2-hydroxybenzophenone. Both signals were collected in carbon tetrachloride. (A) 0.1 mM 2-hydroxybenzophenone; (B) 0.5 mM Clg.
and the effect of this small concentration gradient on the observed kinetics cannot be detected. RESULTS AND DISCUSSION
Rate Constantfor Hydrogen Abstraction. Shown in Figure 2 is a photothermal diffraction signal collected from a solution containing a mixture of chlorine, [Clz] = 0.51 mM, and chloroform, [HCC131 = 0.12 M. Superimposed is a signal collected from 2-hydroxybenzophenone in carbon tetrachloride to illustrate the prompt photothermal grating response, limited by the temporal width of the excitation pulse. Since the excited states of 2-hydroxybenzophenone are not reactive and decay to the ground state in picosec0nds,3~heat delivered to the sample from excitation of 2-hydroxybenzophenone occurs on a time scale faster than the instrument response. Following this impulse of heat, the effect of thermal diffusion smearing out the refractive index modulation is observed as an exponential decrease in the diffraction signal with a time constant of zth = 7.2 f 0.2 ps. The measured thermal decay of the diffraction transient is in good agreement with the time constant predicted by thermal diffusion, 1/D&q2= 7.8 f 0.5 ps, where Dth = UQOCP.Since heat generated by a propagating free-radical chain reaction is detected on a time scale where the contribution to the signal from thermal diffusion cannot be neglected, the time constant of thermal decay is fixed to the value measured for 2-hydroxybenzophenone to reduce the uncertainty in fitting the chain reaction data to eq 19. Comparing the instrument response from Z-hydroxybenzophenone to the signal collected during the photochlorination of chloroform clearly shows excess heat derived from the chain reaction. The early rise in diffraction collected during the reaction (< 1ps) is derived from the coupled propagating steps of the chain reaction as they relax to a steady-state, zero-order kinetic condition. As'the free-radical chain reaction evolves in time, the contribution to the signal from termination becomes detectable. Using previously reported values for the photodissociation yield of chlorine at 355 nm, $d = 0.35,13 and the absorption cross section of chlorine, u1 = 1.9 x cm2,34together with the measured laser intensity, IO = 2.5 x W4cm-2 s-l, and pulse duration, to = 5x s, to estimate the chlorine atom concentration generated in the laser pulse, [Cl.]t=to= 2[Cl&,J,y~lto,~~ a second-order kinetic mode113,21~22 predicts that nearly 7% of the initial radicals are ~~
(35) Hung, R R; Grabowski, J. J. J. Am. Chem. Soc. 1 9 9 2 , 114, 351-353.
Analytical Chemistry, Vol. 67, No. 2, January 75, 1995
369
8.0
-
7.0
-
6.0
-
5.0
-
4.0
-
m
8 -2
iij
.g
23
E .
Ccl
.%
n
0
D -
c . B A -
100
200
400
300
500
600
700
800
Time (ns) Flgure 3. (a) Diffraction signals collected during the photochlorination of chloroform superimposed on signals collected from the nongeminate recombination of chlorine atoms and 2-hydroxybenzophenone. (b) Diffraction signals corrected for the index modulation from chain termination superimposed on a diffraction signal collected from 2-hydroxybenzophenone. (A) 0.1 mM 2-hydroxybenzophenone; (B)0.5 mM Clp; (C) 0.5 mM Clp 0.12 M HCC13; (D) 0.5 mM Clp 0.49 M HCC13; (E) 0.5 mM Clp 0.87 M HCC13.
+ +
+
consumed by chain termination within 10 ps, using a composite rate constant collected from the photodissociation of carbon tetrachloride at 325 nm for modeling chain termination, (k,) = 4.2 x lo9 M-' s-l.I3 Since a simple kinetic model is developed for interpreting chain reactions with the assumption that chain termination can be neglected, significant heat deposited from chain termination and changes in radical/atom concentrationsmake the kinetics of signals on the full 10 ,us time scale shown in Figure 2 difficult to interpret. Chain termination makes a significantly smaller contribution to the kinetics of chain reactions on a submicrosecond time scale. At submicromolar r a d i d a t o m concentrations, the probability of two radicals or atoms encountering each other within several hundred nanoseconds is insignificant. At the same time, the probability of an encounter between a radical or atom and HCCb or Cl2 can be increased by using high reactant concentrations; these conditions produce diffraction signals from chain reactions that are dominated by the coupled chain propagating steps in the first several hundred nanoseconds. Shown in Figure 3 are photothermal diffraction signals (C, D, and E) obtained from three solutions containing different concentrations of chloroform at constant chlorine concentration, [Clz] = 0.51 mM, in carbon tetrachloride superimposed on the best fit of the time-resolved diffraction transient to eq 19. The initial rapid rise of each of the diffraction signals in Figure 3 has been offset to the same initial amplitude from 2-hydroxybenzophenone (A) for comparison. To estimate the contribution of chain termination to the diffraction signals collected during the chain reaction on a submicrosecond time scale, a representative diffraction signal observed for the photolysis of chlorine, [Cl~l= 0.5 mM, is included in Figure 3. The additional time-resolved heat, compared to the 370 Analytical Chemistry, Vol. 67, No. 2, January 15, 1995
decay of the instrument response (from 2-hydroxybenzophenone), is from nongeminate recombination of chlorine atoms. Comparing the index modulation (proportional to the square root of the diffraction amplitudes) from nongeminate chlorine atom recombination (l3) and the instrument response from 2-hydroyxbenzophenone (A) yields an estimate for the heat deposited by chain termination to the total diffraction signal collected during the photochlorination of chloroform. Since chlorine atoms diffuse through solution faster than trichloromethyl radicals,13 the heat deposited from the chlorine recombination overestimates the contribution of chain termination to the chain reaction signals. The 60%slower rate of diffusion of CCl3 compared to CP l3 yields an estimated contribution of the coupled chain termination reactions, eqs 8-10, to the photochlorination diffraction signal of less than 9%at the lowest concentration of chloroform, [HCClJ = 0.12 M, and less than 3%at the highest concentration. The small contribution to the index modulation from chain termination does not vary with chloroform concentration and may be subtracted from the chain reaction diffraction signals. The resulting corrected diffraction signals, obtained by subtracting the index modulation due to chlorine recombination from the photochlorination reaction signals, are shown in Figure 3b, along with the best fits to eq 19;the effect of subtracting the small recombination signal from the chain reaction data does not change the fitted first-order kinetic parameters outside their range of uncertainty. While the effects of recombination on the kinetics of chain termination are small, the recombination signal can Muence the amplitude of the zero-order signals, A,". The heat accumulation from the zero-order (steady-state chain propagation) kinetics in eq 16 increases linearly with time. Since less than 2% of the radical population is depleted by chain termination in the 750 ns observation time window, the second-order chain termination kinetics closely resemble a zero-order process on this short time scale. Superimposed on the diffraction transient collected from the nongeminate recombination of chlorine atoms in Figure 3a is the best fit to both a second-order kinetic mode113~21~22 and a zeroorder kinetic model. Both models accurately describe the kinetic signature of chlorine recombination in this time window, producing equivalent fits to the data within the noise level of the diffraction measurement. As a result, the zero-order amplitude, A,", captures the kinetic signature of chain termination in the diffraction transients collected from chain reactions yielding a composite zero-order amplitude that is difficult to accurately interpret. The ability of the mixed zero- and first-order model to account for observed kinetics of chain reactions, including a small contribution from chain termination, is shown by varying the rate of termination at a fixed rate of chain propagation. The pseudofirst-order rates of chain propagation, obtained by the best fit of these diffraction transients to eq 19, is plotted as a function of chlorine atom concentration at constant chlorine and chloroform concentration, [ C l ~ l =3.8 mM and [HCC$I = 0.12 M, respectively, in Figure 4. The concentration of chlorine atoms was varied in these experiments by changing the excitation laser intensity. The fraction of chlorine atoms depleted by chain termination in these experiments varies from 0.8% to 2.8% from the lowest to the highest laser intensity. Over this range of chlorine atom concentration, the pseudo-first-order rate of chain propagation, k,, varies less than 5% about its mean value, as shown in Figure 4a. This result is consistent with the definition of the pseudo-first-order
8.0
-
h
;o
v
6.0
'90 CI
3
4.0
2.0
0.0
0.2
0.4
0.6
0.8
1 .o
IHCC131 (M)
Figure5. Stem-Volmer plot of the pseudo-first-orderrate constant of chain propagation, 4, as a function of chloroform concentration at constant chlorine concentration, [C12] = 0.5 mM.
[C1,1 (PM) Figure 4. Parameters from mixed-order kinetic model as a function of chlorine atom concentration at constant chlorine, [CI,] = 3.8 mM, and chloroform, [HCC13] = 0.12 M, concentration. (a) Pseudo-firstorder rate constant of chain propagation, 4, as a function of chlorine atom concentration.(b) Amplitude of the pseudo-first-ordercomponent of the mixed-order kinetic model, A i , as a function of chlorine atom concentration. (c) Amplitude of the zero-order component of the mixed-order kinetic model, A / , as a function of chlorine atom concentration.
+
rate for chain propagation, k, = kl[RHI kz[Clzl, indicating that this rate should be independent of chlorine atom concentration. The amplitude of the pseudo-first-order kinetic component of the mixed-order kinetic model, A,' = [Cl'lt,t,(AH1k1~[RHl~ - AH&IRHlk2[ClzI) (dnldT)lkgoC,, is plotted versus chlorine atom concentration in Figure 4b. A linear relationship is observed for AI as a function of chlorine atom concentration, as predicted. The zero-order amplitude of the mixed-order kinetic model, A[ = [Cl'lr,to(AH~+ AZ-Z2)k1[RHlkz[Clzl (dn/dT)/kg&,, is also expected to grow linearly with chlorine atom concentration. This amplitude captures the additional heat deposited by chain termination, as shown in Figure 4c,and deviates strongly from linearity at radical concentrations above 1pM. This result confirms that the zeroorder component of the mixed-order model indeed captures the heat amplitude from chain termination as described above. To determine the rate constant for the hydrogen abstraction reaction, the pseudo-first-order rate for chain propagation, kp = kl[RHI kz[Clzl, is examined as a function of chloroform concentration. The fust-order rate parameters at constant chlorine concentration,[Clz] = 0.51 mM, obtained from the best fit of each
+
photothermal transient corrected for the recombination heat as in Figure 3b, are plotted in Figure 5 versus chloroform concentration. A linear relationship between the observed rate of chain propagation and chloroform concentration is observed,where the slope is the rate constant for hydrogen abstraction from chloroform by a chlorine atom, kl = (7.8 f 0.6) x lo6 M-' s-l. This value is similar in magnitude to reported values for the rate constants of hydrogen abstraction from methane by chlorine atom in dinuorodichloromethane,kl = (5.0 f 1.7) x lo5 M-l s-1,36 and hydrogen abstraction from ethyl chloride extrapolated to 293 K, kl = 2.0 x 1 06 M-l s-1.36,37 The smaller rate constant for hydrogen abstraction from methane, compared to chloroform, by chlorine atom is consistent with a larger C-H bond energy for methane, 105 kcal/mo1,38 than for chloroform, 95 kcallm01.3~ Rate Constant for Atom Transfer. To determine the atom transfer rate, the reaction conditions are adjusted so that this step dominates the pseudo-ht-order rate of chain propagation, kp;this requires a higher concentration of chlorine and a much smaller concentration of chloroform than used to determine the hydrogen abstraction rate. A photothermal signal collected under these conditions, [Cl,] = 9.5 mM and [HCC131 = 24.8 mM, is shown in Figure 6 along with the instrument response from 2-hydroxybenzophenone. The time-resolved signal from chain propagation follows a slower initial rate of heating than the example in Figure 2 (dominated by hydrogen abstraction); this reaction eventually produces a larger amplitude signal in the 5 ,us region due to the Cfold higher enthalpy associated with the atom transfer step and the greater contribution from chlorine atom recombination. Using the same values cited above for @d, (71, and to, along with the measured laser intensity, l o = 1.8 x loz4cm-2 s-l, to predict the chlorine atom concentration generated in the laser pulse, a secondorder kinetic model predicts that nearly 50%of the chlorine atoms generated in the excitation pulse are consumed by chain termination within 10,us. The influence of chain termination on the signal collected during the photochlorination of chloroform is again minimized by monitoring the chain reaction in a shorter time window. (36) Sergeev, G. B.; Smirnov, V. V.;Porodenko, E.V.; Pukhovsky,A V. Inf. J. Chem. Kinef. 1985, 17,1321-1331. (37) Sergeev, G. B.; Smirnov, V. V.; Shlyapnikova,E.A; Nemtsoy, H. Vestn. Mosk. Uniu., Ser. Khim. 1978, 19,281. (38) Baghal-Vayjooee, M. H.; Colussi, A J.; Benson, S. W. J. Am. Chem. SOC. 1978, 100, 3214. (39) Mendenhall, G. D.; Golden, D. M.; Benson, S. W.J. Phys. Chem. 1973,77, 2707.
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6.0 30
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B A -
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40.0
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E -
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D ' C
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A
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100
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300
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500
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Time (ns) Figure 7 . A series of scaled photothermal diffraction signals collected from solutions of varying chlorine concentration at constant chloroform concentration superimposed on a diffraction signal collected from the nongeminate recombination of chlorine atoms and a diffraction signal collected from 2-hydroxybenzophenone.(A) 0.1 mM 2-hydroxybenzophenone;(B) 1.9 mM Clp; (C) 1.9 mM Cln 24.8 mM HCC13; (D) 5.7 mM Clz 24.8 mM HCC13; (E) 9.5 mM Cln 24.8 mM HCC13.
+
IO
5
15
[W (mM)
Figure 6. Photothermal diffraction signal collected from a mixture of chlorine and chloroform superimposed on a signal collected from 2-hydroxybenzophenone. Both signals were collected in carbon tetrachloride. (A) 0.1 mM 2-hydroxybenzophenone;(B) 9.5 mM Cln 24.8 mM HCC13
M
/ i
+
+
Shown in Figure 7 are photothermal diffraction signals (C, D, and E) obtained from three solutions containing different concentrations of chlorine at constant chloroform concentration, [Clz] = 24.8 mM, along with the best fit of the time-resolved diffraction transient to eq 19. The contribution of chain termination to the diffraction signals collected under these conditions is again estimated by comparing the time-resolved heat from nongeminate recombination of chlorine atoms to the heat deposited by the photochlorination of chloroform. Included in Figure 7 is a diffraction signal observed from the nongeminate recombination of chlorine (B), [Clz] = 1.9 mM, collected under the same photolysis conditions used to photoinitiate the chain reaction. By comparing the amplitudes of the refractive index modulations (from the square root of the diffraction signals), the contribution of chain termination to diffractionin the photochlorination reaction is estimated to vary from 14%to 39%from the lowest concentration of chlorine to the highest. Unfortunately, the recombination rate varies with the chlorine concentration, which prohibits a single recombination transient to be subtracted from the data. As discussed above, the amplitude of the zero-order component in the kinetic model, A:, captures most of the diffraction amplitude associated with chain termination at low atom concentratiocs. As the rate of chain termination approaches the time scale where 372 Analytical Chemistry, Vol. 67, No. 2, January 75, 1995
Figure 8. Stern-Volmer plot of the pseudo-first-orderrate constant of chain propagation, kp,as a function of chlorine concentration at constant chloroform concentration, [HCC13] = 24.8 mM.
chain propagation is monitored, however, the heat accumulation from chain termination is poorly modeled as a linear function of time, and the zero-order amplitude cannot reliably capture all of the amplitude associated with chain termination. Superimposed on the diffraction signal collected from the recombination of chlorine atoms in Figure 7 are the best fits to both a second-order kinetic model and a zero-order kinetic model. The best fit of the signal to a second-order kinetic model contains a detectable amount of curvature, making differences in the two kinetic models detectable. As a result, recombination kinetics are expected to influence the chain propagation data collected at the highest chlorine concentrations (see below). To determine the rate constant for the atom transfer reaction, the rate of chain propagation, k, = k l [ R H ] kz[Clzl, is measured as a function of chlorine concentration at constant chloroform concentration. The observed first-order rate parameter, obtained from the best fit of each photothermal transient to eq 19, versus chlorine concentration at [HCClJ = 24.8 mM, is plotted in Figure 8. A linear relationship between the pseudo-first-order rate of chain propagation and chlorine concentration is observed for the lowest five concentrations in Figure 8. The deviation in linearity observed at high concentrations of chlorine is expected due to incomplete capture of the chain termination amplitude by the zeroorder component in the model. At a b e d laser intensity, a higher chlorine concentration leads to a greater concentration of atoms and radicals that recombine on a faster time scale, which eventually affects the first-order kinetic parameter. Using the data at the lowest five concentrations of chlorine in Figure 8, where this effect is small, the rate constant for the atom transfer reaction is determined from the slope of this plot, kz = (4.6 k 0.4) x lo8 M-l SI. This value is in good agreement with the rate constant for the atom transfer reaction of the methane radical, 'CH3, with chlorine,4OKz = 2.48 x 108 M-1 s-1. The intercept from Figure 8 also provides an estimate of the hydrogen abstraction rate constant, k l = (3.8 f 0.3) x lo6 M-l SI, which is in reasonable agreement with the estimate of this rate derived from the slope of the data in Figure 5 . In general, determining rate constants from the intercept values is much less reliable since these results do not depend on the variation in a rate with reactant concentration but on the absolute rate at b e d conditions. The low chlorine concentration limit of the intercept of Figure 8 gives a fairly reliable absolute reaction rate since the
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(40) Timonen, R; Kalliorinne, K; Koskikallio. J. Acta Chem. Scand. A 1986, 40, 459-466.
chain termination rate is extrapolated to zero under these conditions. This stands in contrast to the intercept of Figure 5, where the chlorine concentrationremains finite, [Clzl = 0.5 mM. The intercept of this plot corresponds to an atom transfer rate constant, kz = (3.1 f 0.6) x lo9 M-l s-l, which is larger by a factor of 7 than the rate constant derived from the slope of Figure 8. The result from Figure 8 is reliable since it is determined from the variation of the first-order rate with chloroform concentration. The intercept of Figure 5, however, extrapolates to a fist-order rate parameter that is dominated by heat deposited from chlorine recombination and represents conditions where the approximations for the simple first-order model are not reliable. Summary. Photothermalgrating spectroscopy is found to be a useful method for investigating the kinetics of photoinitiated chain reactions in liquids. The technique is quite general and applicable to a variety of reactions since the method detects heat released into the sample during the course of the reaction and since most chain reactions are strongly exothermic. Using this technique, we have monitored the chain propagation steps of a simple photoinitiated free-radical chain reaction on a submicro-
second time scale, where contributions to the signal from chain termination reactions can be neglected. Neglecting chain termination enables the time-resolved photothermalgrating transients to be interpreted using a simple kinetic model derived with only the two propagating steps of the chain reaction. Individual rate constants for the H atom abstraction and the C1 atom transfer steps were determined without requiring information of the concentration of radicals produced in the sample. This direct approach to monitoring free-radical chain reactions is an attractive alternative to determining rate constants by stationary-state methods yielding composite rate constants. ACKNOWLEDGMENT This research was supported in part by the National Science Foundation through Grant CHE9G06667. Received for review September 12, 1994. November 11, 1994.@
Accepted
AC940907D @Abstractpublished in Advance ACS Abstracts, December 15, 1994.
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