RICHARD W. FESSENDEN
1808
Measurement of Short Radical Lifetimes by Electron Spin Resonance Methods'
by Richard W. Fessenden Radiation Research Laboratories, Nellon Institute, Pittsburgh, Pennsarlvania
(Received Januarg db, lQ6g)
Experiments have been performed demonstrating two methods by which it is possible to ineasure short radical lifetimes in electron spin resonance (em-.) experiments. These methods are an adaptation of the rotating sector or intermittent illumination technique, often used in photolysis studies, and a sampling technique. Both methods circumvent the limitations imposed by long output time constants and operate with only a moderate loss of signal-to-noise ratio over the steady-state experiment, even when direct observation of the radical formatioil and decay is not possible. These methods are used in a study of the reaction kinetics of ethyl radicals produced in liquid ethane by radiolysis with a 2.8-Mev. electron beam from a Van de Graaff accelerator. For these studies the accelerator was equipped with electronically controlled pulsing circuits, The two methods were compared and found to agree within 7% in experiments performed under the same conditions. At -177" and an electron beam current of 0.100 ~ a (6.9 . x 1017 e.v. ml.-I sec.-l) the ethyl radical lifetime in the sampling experiment was 7 . 3 rnsec, A plot of data showing the second-order radical decay in this experiment (reciprocal coiicentration vs. time) showed excellent linearity over the whole range studied (a factor of ten in concentration). The rate constant found for the second-order disappearance of ethyl radicals in liquid ethane is 1.7 f 0.4 x 108 M-I sec.-l. Over the temperature range -140 to -177" this may be expressed as 1.3 X 101oe-vao'RT M - I sec.-l.
Introduction A previous paperZ described e m - , studies of the alkyl radicals present during radiolysis of liquid hydrocarbons. In these experiments radicals were produced directly in the e.s r. cavity and studied at the steadystate concentration produced by a 2.8-Mev, electron beain f ~ o i na Van de Graaff accelerator. The signal-tonoise ratio (-100 or less with a 3-see. time constant) and radical lifetime (-5 insec.) were always such that it was not possible to observe directly the decay of the e.s.r. signal when the electron beam was i n t e r r ~ p t e d . ~ The inability to observe fast decays is a result of the long time constant necessary to give acceptable signalto-noise ratios when detecting small concentrations of radicals [S/K (RC)''z].4 Thus, the e.s.r. methods cannot be used in such experiments to study the kinetics of radical disappearance in a way completely coniparable to the use of optical techniques in flash photolysis and pulse radiolysis. (With the optical methods The J o trrnal of Physical Chemistry
it has not been necessary to use long time constants t o achieve satisfactory signal-to-noise ratios.) The present paper illustrates the use of two methods by which the limitations of a long time constant can be partly circumvented allowing the kinetics of disappearance of these reactive radicals to be studied iiz a direct fashion. The first method is an adaptation of the intermittent illumination or rotating aector technique used in photolysis and allows the lifetime of the radicals t o be determined. The second method utilizes sampling (1) Supported, in part, by the U. S.Atomic Energy Commission. (2) R. W. Fessenden and R. H. Schuler, J . Chern. Phys., 39, 2147 (1963). (3) With a time constant of 1 msec. it is calculated that ethyl radical should give a signal-to-noise ratio of - 3 , Little information could
be gotten by observing this decay. (4) With disappearance rate constants somewhat smaller than those pertaining here and radical lifetimes of 0.1 see. and above, L. H. Piette and W. C. Landgraff [ J .Chern. Phys., 3 2 , 1107 (1960) 1 were able to observe radical decays directly.
MEASUREMENT OF SHORT RADICAL LIFETIMESBY E.s.R. METHODS
techniques to determine the complete curve of the radical formation and decay. These methods require the use of a pulsed beam from the Van de Graaff accelerator. It should be emphasized that both methods involve an inherent reduction in signal-to-noise ratio over the d.c. (steadystate) experiment. For an initial experiment it is desirable that, for comparison purposef,, there be other data concerning the particular radic,al lifetime or disappearance rate constant. Also, proper use of the rotating sector technique requires that the dependence of radical concentration upon (in this case) electron beam current be known. For these reasons the liquid ethane system seems to be a good choice because the rate constant for ethyl radical disappearance has been determined from the radical production rate and concentration2 and because the ethyl radical signal has been shown to be proportional to the square root of the electron current below about 0.1 pa.2 In addition, it is known that essentially only ethyl radical is present giving a rate constant for this species alone and not an average for several species. Finally, the excellent signal-to-noise ratio for this system, even with the necessary reduction, permits taking good data.
Experimental The irradiation arrangement consisting of the Van de Graaff electron accelerator, axial-hole magnet, and e.s.r. cavity was as described previously.2 The TE103 cavity was used with the lower two electrical half wave lengths filled with ethane. The irradiation zone was in the center of this volume and all of the -1 cm. diameter of the electron beam was effective in irradiating the liquid. The e.s.r. spectrometer was a Varian unit with 100 kc./sec. field modulation. For measuring the average radical concentration in the intermittent radical production experiment, double-field modulation at 100 kc./sec. and 200 C.P.S.was used since with this method there is no base-line drift and only the height of the line a t its cen1;er need be measured. If drift of the cavity resonance frequency and magnetic field are sufficiently small it if, even possible to sit on the top OF the peak and record the height for diff erent pulse repeti tion rates. I n this way no scanning of the line js necessary and a very long time constant can be used. With the sampling method a relatively high frequency response is necessary in the circuits before the sampling unit so that only the lOO-kc./sec. modulation is used. The output of the lOO-kc./sec. unit ordinarily used for oscilloscope presentation was found to have sufficient response. Since a first-derivative presentation was used in this type of experiment, base-line drift did occur. It was necessary to operate under conditions, such as ab
1509
stable temperature and a relatively low beam current, which minimized this drift. Since only one line need be recorded, conditions are not as severe as if a whole spectrum were desired. To average out the base-line drift the line of interest was recorded with field scans in both directions and the average peak-to-peak height used. With the excellent signals available for ethyl radical at the lower temperatures, drift was not a major consideration. Pulsing of the Van de Graaff beam is accomplished electronically with direct external control of both the beginning and end of the electron pulse. The basic means of conveying the timing information to the high voltage terminal is by a light pulse from a neon bulb (;*\E-2H).6 The light pulse travels through a Lucite light pipe to the terniinal where it is detected by a photomultiplier. The signal from the photomultiplier is amplified in a video amplifier which is d.c.-connected through the use of Zener diodes. The output of this amplifier, after some clipping to square the pulse, controls the voltage on the electron-emitting filament. The d.c. coupling is necessary because of the very low frequencies, -0.1 c.P.s., needed for some of the experiments. The observed rise and fall times of the electron pulse were about 1 psec. Because of the requirements of the rotating sector technique that the ratio of beam pulse period to the repetition period be constant for various repetition rates, individual control of these periods is not desirable. Consequently, a control circuit was designed which used digital techniques. The separation between consecutive pulses in a pulse train derived from the output of a square wave generator determines the length of the beam pulse while the spacing between every second, fourth, or eighth pulse in the train determines the repetition period of the beam pulses. The output of this control unit increases the current through the neon bulb for the beam pulse duration. For beam pulse lengths longer than 100 psec. the duty cycle of the beam is within 0.5% of the expected value of l / 2 , l / q , and becomes even closer for longer pulses. or The beam measuring equipment was of the usual electronic type with the output read on a chart recorder. I t was verified that the correct average current was read with pulsed currents whenever the frequency was high enough that the recorder did not respond to the individual pulses. The ethane used was Phillips research grade. Samples were transferred to a vacuum line and de(5) This method had been used previously in similar equipment developed by Llr. s. Wagner of Brookhaven National Laboratory. The author is indebted to Dr. H. A. Schwarz for pointing out that such a pulse has sufficiently fast rise and fall times.
Volume 68, Number 6
June, 1964
RICHARD W. FESSENDEN
1510
gassed by pumping a t liquid nitrogen temperature. Two bulb-to-bulb distillations were carried out a t -78’ to remove any high boiling impurities. From studies of the ethane-ethylene system2b6it is known that small amounts of ethylene, either as an initial impurity or that produced during irradiation, have no detectable effect on the ethyl radical concentration. The sample temperature was measured and controlled during irradiation by a copper-constantan thermocouple with the sample junction soldered to the rear cavity wall. The temperature measured is more nearly that of the sample than of the rest of the cavity and heat sink because of the low conductivity of the cavity wall. The temperature at this position can be maintained to within *0.5’ although it is felt that during irradiation there may be temperature differences of several degrees within the sample. Differences of 1 or 2’ will not significantly affect the results.
The Method of Intermittent Radical Production Themy. The effects of intermittent illumination (or the rotating sector technique) have been used for years in photochemical studies to determine the lifetime of active species. The theory has been presented a number of times.’ In the original method, radicals are produced photochemically by a mechanically chopped light beam. The chopping system, for instance a slotted disk rotating at an adjustable rate, is constructed to maintain a constant ratio of light to dark periods while the length (and therefore repetition rate) of the light periods can be varied. The average light input is, therefore, independent of chopping rate. For there to be any effect of chopping rate, the species of interest must disappear by other than a first-order reaction. The discussion to follow will be limited to a simple systeni containing one kind of radical which disappears by second-order reaction with itself. A t slow chopping rates the radical concentration will closely follow the light pulse and will remain at the steady-state concentration [R], for essentially the whole time the light is on. The average concentration [R J over one cycle is then given by [R]/ [R], = P, where P is the fraction of the time the light is on. At very fast chopping rates the radical concentration changes very little over one cycle and the effect is the same as using a light intensity smaller by the factor P. Because of the dependence of the steady-state concentration upon the square root of the light intensity, the average concentration at this limit is [R]/[R], = P”’. As the chopping rate is varied from slow to fast, [R]/[R], increases by p-‘”. The detailed analysis’ gives for a particular P an explicit equation for the sigmoid curve of [R]/[R], as a function of m, the ratio of the light The Journal of Physical Chemistry
period to the steady-state radical lifetime. The steadystate radical lifetime, T,, is defined by T~ = [R],/PR, where PR is the rate of radical production, and is the time it takes for the concentration to drop by half after interruption of the light. I n the chemical experiments the average radical concentration is measured by the rate of a pilot reaction which is first order in the radical concentration and hence is unaffected by the intermittent radical production. Conditions must be adjusted so that the presence of the pilot reaction does not significantly affect the radical concentration to be measured. I n the variation of the method used here the electron beam for the radiolysis is pulsed electronically and the average radical concentration measured directly by the height of the recorded e.s.r. signal. The method is, therefore, simpler in concept in that nothing of the nature of a pilot reaction, with attendant complications, is needed. In Fig. 1 are shown typical results giving the height of a line in the e.s.r. spectrum of ethyl radical (in irradiated ethane) as a function of the length of the electron-beam pulse. The solid curves give the theoretical behavior expected for the two values of P ( l / k and 1/8). These curves have been translated along the time axis to give
to-
0.s-
E
-
.P
P- 0.60 0,
Ir
0.2
~
001
t
(0
(00
Pulse Duration (m s e d
Figure 1. Ethyl radical signal heights as a function of electmn and 0 , p = l/*. The smooth beam pulse duration: 0, p = curves are theoretical and have been translated along the time axis to give the best fit. The vertical bars show the position of rn = 1.
(6) R. W. Fessenden and R. H. Schuler, Discussions Faraday Soc., 36, 147 (1963). (7) See (a) H. W. Melville and G. M . Burnett in “Technique of Or-
ganic Chemistry,” Vol. VIII. S. L. Friess and A. Weissberger, Ed., 1953, p. 138 ff.; (b) R.. G. Interscience Publ., Inc., New York, N. Dickinson, in “Photochemistry of Gases, W. A. Noyes, Jr., and P. A. Leighton, Ed., Reinhold Publ. Co., New York, N. Y., 1941, p. 202 ff.
T’,
h4EASUREMEN'T O F S H O R T
RADICAL IiIFETIMES
BY
E.s.R. METHODS
the best fit. The lifetime a t the steady state is given directly by the value of the pulse length which corresponds to m = 1 on the theoretical curve. I n the e.s.r. experiment it proves easier to compare signal heights to the value a t high pulse repetition rates than to the height for a steady d.c. beam; for this reason all points are normalized to the height a t a fast pulse rate where no departure from the limiting value is expected. Up to this point it has been assumed that the recorded output of the double modulation-double phase detection system does, in fact, represent the average radical concentration even though the radical concentration is varying. By writing the radical concentration as a Fourier series in the beam-pulse repetition frequency and considering the integrals which represent the output after the lield modulation process, it is pos.. sible to show that the output (averaged by the output time constant) does represent the average concentra-. tion as long as no integral relation exists between the field modulation and pulse repetition frequencies, I n practice no "beats" in the output have been observedl which might be evidence for the approach to such ab condition. The excellent agreement of the observedl curves with the theoretical ones also suggests that no difficulty is encountered in practice. Results and Discussion. The relative ideality of the ethane system referired to above suggests that the measured curve of ethyl radical signal height against electron beam pulse duration should follow the theoretical one very closely. The points shown in Fig. 1 for /3 = 1/4 and '/a do agree well with the theo-. retical curves and the lifetimes determined for the two experiments, 7.7 and 7.9 msec., also agree. These stated values have been corrected for slight differences in the beam current a t which the experiments were performed and refer to a beam current of 0.100 pa. and a temperature of - 177'. Once the general (agreement of the data with the theoretical behavior is accepted, further study shows a slight departure of the /3 = '/8 data from the curve. This departure is quite reproducible; the experimental curve tends to be a bit flatter than the theoretical. The reasons for this behavior are not apparent although possible causes are suggested later. Because of this deviation, determination of the proper way to overlap the two curves is a bit more difficult than the scatter of the points would suggest. With data such as those of Fig. 1 the lifetime can be determined to better than f50j0. Relative measurements can be made even more accurately if a smooth curve is drawn through the p = '/* data of Fig. 1 and this curve used to overlap further data. As was stated above, normalization is always to the
1511
signal height a t a high pulse repetition rate. This method is used because the heating caused by a d.c. beam makes accurate comparison to these conditions difficult. Severtheless, it is important to demonstrate that the expected behavior occurs. A comparison of the signal height was made for fast pulse repetition rates a t the same pulse current with p = 1, and '/8. The good agreement between the expected and observed heights is shown in Table I. (A pulse duraTable I : Ethyl Radical Signal Heights a t the High Pulse Rate Limit for Various Values of B B
Theory
Measurement4
1.000 0.707 ,500 ,354
1,000 0,679 ,527 ,365
1 '/Z 1/4
'/a
and
Measured for a pulse duration of 0.1 msec. at 0.100 pa. - 174'.
tion of 0.1 nisec. was used and is short enough to ensure being a t the limiting signal height.) No attempt was made to compare directly signals a t the low repetition rate limit because the output time constant is no longer sufficient to average the signal over an on and off cycle. An indirect comparison can be made through the data of Table I and of Fig. 1. To check the dependence of lifetime on beam current, experiments were performed a t three different pulse currents, 0.0235, 0.111, and 0.432 pa. The three sets of points (at -174') with the theoretical curves fitted to them are given in Fig. 2. Taking the lowest current as the base value, the lifetimes can be calculated from the expected square root dependence. These values (A) are given with the measured values in Table 11. Clearly there is a departure from the expected dependence. To help in determining the Table I1 : Ethyl Radical Lifetimes for Several Electron Beam Currents
- - - ~ Lifetime------
Calculated-----
Current, m.
Measured
Am
Bb
0.0235 0.111 0,432
14.4 7.4 4.4
14.4 6.62 3.36
14.4 7.16 4.01
Calculated from the reciprocal of the current. The first value is defined. Calculated from the curve of signal height vs. beam current in ref. 2. The first value is defined.
Volume 68, Number 6
June, 1964
RICHARD W. FESSENDEN
1512
1.0-
0.8-
E
-
f - 0,6-
w
-
,? L Q
0.4-
LT
c
3.1.
0.2
00,L
l
I
8
9
I IO
1 0 ~ 1 ~ I
,
1 1 1 1 1 1 1
1
I
I
I , , , , ,
1
I
1
10 Pulse Duration (m s8c)
,,,,,,,
100
,
, , , , , , ,1
1000
Figure 2. &thy1 radical signal heights as a function of electron beam pulse duration: 0, 0.0235 pa.; 0 , 0.111 pa.; 0, 0.432 pa. Curves are theoretical.
reasons for this, lifetimes are also shown (B) which represent the previously measured2 curve of signal height as a function of beam current. The similar behavior of the two types of data suggests a common cause for the departures. The suggestion made earlier2 that ionization in the vapor above the sample reduces cavity Q could not explain both sets of data. Possibly convection currents caused by the heating of the irradiated portion sample could distribute the radicals over a larger volume and increase the radical lifetime although it is not clear why the effects should be the same for similar d.c. and pulse currents rather than fdr the same average current. I n any case the previous curve of signal height against beam current shows that the lifetime measured at 0.0235 pa. can safely be taken to represent the true second-order disappearance. A very important study that can be made with this technique is the measurement of radical lifetinie as a function of temperature in a determination of the activation energy for the disappearance reaction. This method is much superior to that of measuring signal height as a function of temperature because all other temperature-dependent factors affecting the signal height need not be known. It is only necessary that the spectrometer sensitivity not depend on the pulse repetition rate and that the absorbed dose rate as a function of temperature be known. These two requirements are easily met. The measured lifetimes for temperatures from - 140 to -177' are given in Fig. 3. Small corrections have been applied for changes in the absorbed dose rate beThe Journal of Physical Chemistry
Figure 3. Ethyl radical lifetime as a function of 1/T; 0,108-pa. beam current. The lifetimes have been adjusted by the square root of the density so they all refer to the absorbed dose rate a t -177". The straight line is determined by the least-squares procedure and corresponds t o an activation energy for the disappearance reaction of 830 cal./mole ( k a T - ~ ) ) .
cause of density changes and to correct all values to a common current. The straight line was determined by the least-squares method and corresponds to a value of E , in IC = Ae-Ea/Er of 830 cal./rriole with a probable error of 70 cal./mole. If the expression k = e-Ea''RT is used, E,' is 720 cal./mole with a similar probable error. The value corresponding to E,' determined previously2 was 780 cal./mole in excellent agreement despite several possible sources of error in the earlier work. X o change in the discussion given there is necessary.
The Sampling Method Theory. The intermittent production method avoids the problem of long time constants by detecting only the time average radical concentration. To use this method the dependence of radical concentration upon electron beam current should also be known. Direct measurement of the growth and decay of the radical concentration such as can be done optically is superior in that, in principle a t least, the decay curve shows the order of the reaction and any changes in it during the decay. It was shown in the introduction that the direct measurement cannot be carried out with the type of system under consideration. However, if a number of similar formation and decay curves can be added together the signal-to-noise ratio improves as the square root of the number added and measurements become practical. A number of methods of doing this exist. I n the method used here the e.s.r. signal is observed or sampled for a short interval at a given time after the start or finish of the irradiation pulse. This point
1vEASUREMEKT OF
--
SHORT RADICAL LIFETIMESBY E.s.R. P\/IETHODS
on the formation-decay curve is then represented by a pulse the amplitude of which is just the signal level plus any noise. The series of such pulses which occur when the irradiation pulse is repeated and the signal sampled with the same intervening time delay can be added electrically. After a number of pulses, the noise components tend to cancel and the d.c. level represents the averag;e signal during the sampling interval. This procedure is illustrated in Fig. 4. Two formation-decay curves which might occur are shown on the upper line wit,h the sampling interval indicated. The series of pulses resulting after the sampling process are shown on the lower line. The dashed line shows the the average level. To get the whole formation-decay curve, the amplitude of the selected e m . line must be determined at a number of delay times.
Figure 4. The basic sampling idea. The upper line shows two hypothetical formation-decay curves with the sampling interval marked. The lower line shows the pulses resulting from the sampling procedure with the average height of a number of such pulses indicated by the dashed line.
The sampling unit used is the "boxcar" integrator described by Blume." With this circuit a capacitor i, charged through a resistance during the sampling interval. A number of samples are necessary before the voltage across the ca,pacitor reaches the final level so that the noise component on any one sample changes the voltage very little. The detailed theoryg shows that the signal-to-noise ratio at the boxcar output is less than that of a steady-state experiment by the square root of the ratio of the sampling period to the repetition period. A more elegant technique has been used by PiettelO to determine simultaneously a number of points on the decay curve of a photolytically generated radical. In this method the voltage in successive time intervals is converted into a digital form and stored in a number of core memories. A number of decays are sampled and the results for the various time intervals added to the appropriate memories. The process continues until
1513
sufficient data are accumulated to provide a satisfactory signal-to-noise ratio. This method is more efficient than a point by point determination in that many less irradiations periods are needed for the same final result. Fortunately, in the radiolysis studies reported here the total irradiation time available is not a limiting factor. To use the sampling method for determining formation-decay curves where the radical lifetimes are short, it is necessary that the output of the lOO-kc./sec. phase detector used to feed the boxcar integrator be such that little distortion of the curve occurs by loss of high-frequency components. Measurements performed by gating an input to the lOO-kc./sec. unit show that the output rises to 90y0 of the steady-state value in about 0.4 msec. Most of this delay occurs in the filtering circuits after the phase detector. Analysis shows that for a ramp function, which is a better description of portions of the formation-decay curves than a step, little distortion other than a 0.1-nisec. delay is expected. With radical lifetimes of the order of 5-10 msec. these delays are of little consequence. Results and Discussion. The formation and decay curve of ethyl radical in liquid ethane at - 177" and 0.11 pa. beam current, as measured by the sampling technique, is shown in Fig. 5 . Line 8 of the spectrum, which is one of the two most intense lines, was used to provide
'.Ot
00L---L--
I
Y-l
2b
1
I
40
I
I
1
60
I 80
Time (m sec)
Figure 5. Growth and decay of the ethyl radical signal a t -177" and 0.110 Ma. as measured by the sampling technique. The beam-on and beam-off intervals are indicated. Solid curves are theoretical and are based on r8 = 6.93 msec. derived from the decay (see Fig. 6). (8) R. J. Blume, Rev. Sci. Instr., 32, 1016 (1961). (9) J. L. Lawson and G. E. Uhlenbeck, "Threshold Signals," McGraw-Hill Book Co., New York, N. Y., 1950, p. 273 ff. This theory applies to a slightly different type of boxcar but the form of the result should apply here. (lo) L. H. Piette, presented at the Sixth International Symposium on Free Radicals, Cambridge, England, July, 1863.
Volume 68, hiumber 6 June, 1964
RICHARD W. FESSENDEK
1514
the best signal-to-noise ratio. As is indicated in the figure, the beam was on for 20 msec. and off for 60 insec. The sampling period was 0.5 msec. for the first 30 nisec. and 1msec. for the rest. Points are correctly plotted a t the center of the sampling interval because there is little curvature over this interval. The solid curves are theoretical curves based on the steady-state lifetime, T,, of 6.93 msec. determined from the first 18 msec. of the decay (see below). These curves are for the formation, [R]/[R], = tanh ( t l ~ , ) and , for the decay, [R]/[RIs = (1 t / ~ ~ ) - l .The curve is normalized to the expected value of 0.995 for [R]/ [R], a t 19.75 msec. Agreement of the data with the theoretical curves is excellent. A more severe test of the decay curve is to plot it in the form appropriate to give a straight line for a secondorder decay. This plot ([R],/[R] against time) is given in Fig. 6. The straight line here is a leastsquares fit to the data of the first 18 msec. of the decay
+
,
'01
ative estimates of the expected errors based on the recorded noise level. The reciprocal form of the plot causes the error to increase rapidly near the end of the decay. The value of 6.93 nisec. corrects to 7.3 msec. a t 0,100 pa. This value is to be compared to those of 7.9 and 7.7 msec. determined by the sector technique for the same current and temperature. The differences are a bit outside the sum of the expected errors and may be associated with some slight iniperfections in one or the other method but nevertheless the agreement is quite gratifying. In view of the various small departures of the data up to this point from the theoretical behavior, particularly the departure of the lifetime from the square root dependence, it is somewhat surprising to find such an excellent fit of the data to the expected second-order decay. It is quite possible that, in fact, two coinpensating effects are at work. Although it is not at"al1 evident what these effects may be, the magnitude is very likely measured by the difference between the lifetime determined by the two methods. This difference, about 7% is hardly cause for great concern but does seem to place a limit on the accuracy possible for either method.
Absolute Rate Constant
/ A
I 50
Time After Pulse (m s e d
Figure 6. Plot of reciprocal concentration against time for the decay of the ethyl radical signal (data from Fig. 5 ) . The straight line was determined from the first 18 msec. of the decay by the least-squares procedure. Its slope corresponds to T~ = 6.93 maec.
and gives the value mentioned above for re. The straight line agrees well with thc data over the whole range which extends over a factor of 10 in concentration. The vertical bars at 12 and 56 nisec. are conservThe Journal of Physiea,l Chemistry
In the earlier workZthe rate constant for disappearance of ethyl radicals was determined from the relation, 2k = PR/[R],~,in which PR is the production rate of ethyl radicals and [RIBtheir concentration. Because of the difficulties associated with the measurement of [R], by e.8.r. techniques the value of 3 X lo8M - l set.-' found for k was believed to be good to only a factor of two. I n the present work the lifetime can be combined with PRaccording to the equation, 2k = ( P R 7 s 2 )-I, to give a much more accurate value for k . As in the earlier work, the absorbed dose rate was nieasured by the rate of disappearance of galvinoxyl froni a solution in cyclohexane. Two separate determinations were made at different total doses and were found to agree to within 370. Since the cavity contains about 6 mi. of solution the whole cross section of the beam is effective. A t 0.100 ha. beam current the nieasured absorbed dose rate in cyclohexane is 9.0 X lo1' e.v./sec. (This value is in perfect agreement with the figure of 1.8 MeV. (g./cm.2)-1 for the energyloss parameter for electrons at these energies.) This dose is absorbed in a volume of 1.15 nil. defined by the 1-cm. cavity dimension and the diameter of the beam. The beam diameter in the sample was determined to be 1.2 cni. by the bleaching of colored cellophane in a phantom arrangement. This active sample volume is probably the least accurately known factor in the equa-
MEASUREMENT OF SHORT RADICAL LIFETIMES BY E.s.R. METHODS
tion for k. With a value for the yield of ethyl radicals of 4.4 radicals/100 e.v.,2 the production rate becomes 5.0 X A!! sec. -l at 0.100 pa. in our geometry. After making a correction for the presence of a small concentration of methyl and vinyl radicals" the effective values become 4.7 radicals/100 e.v. and 5.4 X M sec.--l. Because of the slight departure from the square-root dependence at the higher currents the lifetime a t 0.0235 pa. (see Table I) should be the best figure to use in determining the rate constant. The value so determined is 1.68 X lo8M-l sec.-1 at - 177'. (This value has been corrected to -177' by the curve of Fig. 3.) Values of 1.80 X lo8and 1.56 X lo8are found from the sampling experiment and from the curve of Fig. 3, re-. spectively. A value of 1.7 It 0.4 X lo8 M-l sec.-l is chosen as the final value a t -177'. The stated error (20%) comes from a consideration of the errors expected in both PRand ra2. Combination of the final value for k with the activation energy determined from the temperature study gives k = 1.3 X 1010e-8SoIRT 1M-l set.-'. Implicit in the calculation of the rate constant and, in fact, in the use of both methods of measuring lifetime, is the assumption that the radical concentration is uniform over the irradiated volume. This assumption cannot be completely valid and the extent to which it affects the lifetime measurements and ultimately the calculated rate constant is not readily apparent. However, calculations show that for a system with two subgroups containing equal numbers of radicals which have respective lifetimes of 1 and 1.5, the form of the sector curve is essentially unchanged and gives the average lifetime. This method, then, is not sensitive to the concentration distribution. The decay curve would, however, be expected to show any large spread
1515
in the radical lifetimes. In fact, no curvature is observed in Fig. 6 which might be attributed to such a distribution. I n the absence of any evidence to the contrary it will be assumed that this effect does not change the stated error of the rate constant. The radical concentration calculated for this experiment is 1.7 X lo-' M a t -177' and 0.100 pa. This value cannot be compared directly to the previously measured value because of the differing geometries, but is of a similar magnitude. A more interesting comparison is possible if the previously measured radical concentration (through the calculated rate constant) and the value for the lifetime measured in this experiment are combined. I n this way the production rate of ethyl radicals can be determined by purely physical methods. The value obtained, 3.8 radicals/ 100 e.v. gives better than expected (30%) agreement with the chemically measured value of 4.4. Although this calculation is not of practical importance because of the 30% error associated with the measurement of the concentration, it reaffirms confidencein the chemical methods (scavenger studies and product analysis) used to measure radical yields. Acknowledgment. The author gratefully acknowledges the assistance of W. L. Siegmann in treating several of the mathematical problems encountered in this work.
(11) The presence of these radicals should not affect the form of the results in the two experiments. However, their contribution to the total effective radical production rate should be included. For this small correction the assumption can be made that these radicals react with the same rate as do ethyl radicals. The correction is then just an increase in the radical yield by the factor 1.07 to take into account the 7% concentration of vinyl and methyl radicals.
Volume 68, Number 6 June, 1964
