J. Phys. Chem. 1984,88, 5083-5086 C(t) = K.c(t)
(A.1)
The probability of survival P,(t) is easily shown to be P,(t) = 1 -
I*
k.c(t? dt'
('4.2)
KD = DK*D' ('4.3) and the solution of the master equation (A.l) as given by30 c(t) = exp(Kt)-c(O) C(t) = D'*eXp(KDt)*DC(O)
('4.5)
Integration of eq A.2 and substitution of eq A.5 yields the following result: p,(t) = 1 - k*D'.KD-'.[eXp(KDt)
- 1]-I).c(o)
Expansion of K.X = c(0) into its individual component equations leads to the following set of relations: -(kl kl2 kl3 ...)XI k l 2 ~ 2 k l 3 ~ 3 ... = ~1
+
+
N
P k ) = CCi(t)
+
+
+ kl2 + kz3 + ...)~2 + kz3~3+ ... = ~2 (A.8) k l 3 ~ 1+ k23~2- (k3 + k13 + k23 + ...)x3 + ... = ~3
k l 2 ~ 1- (k2 etc.
Adding these relations, and noting that CKlci(0)= 1, yields the (30) See, for example, N. S. Snider, J. Chem. Phys., 42, 548 (1965).
Study of the Recornbinatlon Reaction NOz
(B.1)
i= I
Define the row vector of dimension N , each element of which is equal to one, as
1 = ( l , l , l ,...) (B.2) This yields a new expression for the survival probability P,(t) = l.exp(Kt).c(O)
(A.6)
This is simplified further by evaluating the quantity k-K-'-c(O). Let X = K-'-c(O). The scalar quantity sought is therefore given by n k.K-'.c(O) = k.X where K.X = c(0) (A.7)
(A.9)
Appendix B An alternate formulation of the quantity Ps(t) to that given in Appendix A involves the observation that
(A.4)
is transformed to the diagonal representation to yield
+
result n = -1. Substituting this back into eq A.6 yields the final result for the survival probability as a linear combination of exponentials with rate constants being given by the eigenvalues of the K matrix: Ps(t) = -k.D'.KD-'.eXp(KDt).DC(O)
Let the matrix D represent the unitary transformation matrix which diagonalizes the real symmetric matrix K . Then
+
5083
[
(B.3)
The quantity 1.K can be evaluated simply as ( l , l , l ,...)
;kl - k12 -
".
k12
k,,
- k , - k , , - ...
k , , ... k,, ...
-1
=
( - k , , - k 2 , - k , , . . .) (B.4).
or simply 1.K = -k. Solving for the row vector 1 and substituting into eq B.3 yields the final result Ps(t) = -k.K-'.exp(Kt).c(O) 03.5) as obtained by direct integration of the master equation. Diagonalization of the matrix K then produces the same result as in eq A.9. Registry No. HNC, 6914-07-4; HCN, 74-90-8.
+ NO3 + M
N20,
--+
+ M at High Pressures
A. E. Croce de Gobos; H. Hippler, and J. Troe* Institut fur Physikalische Chemie der Universitat Gottingen, 0-3400 Gottingen, West Germany (Received: March 21, 1984)
The recombination reaction NOz + NO3 + M Nz05 + M was studied in the pressure range 2-200 atm of the bath gas Nz. NO3 radicals were formed by laser flash photolysis of N 2 0 and subsequent addition of 0 atoms to NO2; NO3 formation and disappearance were monitored by absorption spectroscopy at 662 nm. The recombination reaction was found to be close to its high-pressure limit with a limiting high-pressure rate constant of (2.2 i 0.5) X cm3molecule-' s-l. For the reaction N O NO3 2 N 0 2 a rate constant of (2 1) X lo-'' cm3 molecule-' s-' was derived. -+
+
-
*
Introduction The thermal dissociation reaction Nz05 M NOz + NO3
+
-
and the reverse recombination NO2 NO3 M
+
+
-
+M
(1)
+
N205 M (2) have recently found a revival of interest because of their importance for atmospheric NO, profiles. N 2 0 s decomposition (eq 1) has been studied for decades as an example of a thermal unimolecular dissociation which is accessible at very moderate tem'Permanent address: Instituto de Investigaciones Fisicoquimicas Teijricas y Aplicadas, Universidad Nacional de la Plata, Argentina.
0022-3654/84/2088-5083$01.50/0
peratures. The most recent experimental investigations by Connell and Johnston,' and Viggiano, Davidson, Fehsenfeld, and Ferguson,2could well be reconciled in terms of unimolecular rate theory as shown by Malko and T r ~ e .When ~ these data were converted into recombination rate coefficients, there persisted some doubt about the reliability of either the equilibrium constant from Graham and Johnston4 used for this conversion or the dissociation data. However, recent direct studies of reaction 2 by Kircher, (1) Connell, P.; Johnston, H. S. Geophys. Res. Lett. 1979, 6, 553. (2) Viggiano, A. A.; Davidson, J. A.; Fehsenfeld, F. C.; Ferguson, E. E. J. Chem. Phys. 1981, 74, 6113. (3) Malko, M. W.; Troe, J. Int. J . Chem. Kinet. 1982, 14, 399. (4)Graham, R.A.;Johnston, H. S . J. Phys. Chem. 1978, 82, 254.
0 1984 American Chemical Society
5084
The Journal of Physical Chemistry, Vol. 88, No. 21, 19'84
Margitan, and Sander5have removed these doubts and confirmed equilibrium constants and falloff rate data of reactions 1 and 2 within small error limits. The only other direct study of reaction 2 by Fowles, Mitchell, Morgan, and Wayne6 has led to a factor of 3 disagreement with the very consistent set of data from ref 1-5 such that sources for this discrepancy have to be looked for. The experiments from ref 1, 2, 5 , and 6 have been limited to pressures below 1 atm. The falloff curves constructed in ref 3 indicate that, at temperatures below 330 K and in the bath gas N,, reactions 1 and 2 at 1 atm are close to the high-pressure limit (within about a factor of 2). It should, however, be emphasized that extrapolations of falloff curves of this type always remain uncertain because of unknown molecular parameters entering the theoretical formalism. Therefore, direct measurements under extreme pressure conditions are obligatory to provide a complete picture of the falloff curves. As a matter of fact, in the construction of falloff curves in ref 3 and 5 , earlier high-pressure experiments from ref 7 were neglected which suggested a much broader falloff curve and a continuous increase of the rate coefficient at pressures beyond 10 atm. In order to clarify this discrepancy between the old data from ref 7 and the recent constructions of falloff curves from ref 3 and 5, we have undertaken a study of the recombination rate coefficient of reaction 2 in the pressure range 2-200 atm. This study continues the series of high-pressure kinetic investigations from our laboratory which, in the extreme, have gone up to pressures of 7 kbar.* Laser flash photolysis provides a convenient technique to induce radical reactions under these conditions. The present work forms part of a program to approach high-pressure limiting rate coefficients of recombination reactions of interest in atmospheric kinetics and in combustion. Studies of other systems like 03,9 H02,10SOz," CH302,12and NZO4l3have been completed already and will be published soon.
Experimental Technique Whereas NO, radicals were generated in ref 6 by reaction of NO, with O, and in ref 5 by reaction of C1 from C12 flash photolysis with ClONO,, in the present work we produced NO, by addition of oxygen atoms to NO2 via the reaction 0
+ NO2 + M
+
NO,
+M
(3)
The source for oxygen atoms was the laser flash photolysis of N20 at 193 nm with O(lD) atoms deactivated quickly in collisions with the bath gas N,. NO, formed in this way recombined with NO2 via reaction 2 to form N205. Since reaction 3 competes with
0
+ NO2 + N O +
0 2
(4)
depletion of NO, concentrations by reaction with N O NO
+ NO,
-
2NO2
(5)
has to be considered also. Our experimental arrangement consisted of an ArF excimer laser (Lambda Physik EMG 200, 15-11s pulses of 50 mJ at 193 nm), a high-pressure photolysis cell made from stainless steel with two sets of perpendicular quartz windows for photolysis and analysis light, and a Xe-Hg high-pressure arc lamp, monochromator, photomultiplier, and oscilloscope for absorption measurements. Laser energies before and behind the photolysis cell ( 5 ) Kircher, C. C.; Margitan, J. J.; Sander, S.P. J . Phys. Chem., in press. ( 6 ) Fowles, M.; Mitchell, D. N.; Morgan, J. W. L.; Wayne, R. P. J. Chem. SOC.,Faraday Trans. 2 1982, 78, 1239. (7) Mills, R. L.; Johnston, H. S. J . Am. Chem. SOC.1951, 73, 938. (8) Hippler, H.; Schubert, V.; Troe, J. J . Chem. Phys., in press. (9) Croce de Cobos, A. E.; Troe, J. Int. J . Chem. Kinet., in press. (10) Cobos, C.; Hippler, H.; Troe, J. Phys. Chem., in press. (1 1) Cobos, C.; Hippler, H.; Troe, J., to be published. (12) Hippler, H.; Luther, K.; Ravishankara, A. R.; Troe, J., to be published. (13) Borrell, P.; Luther, K.; Troe, J., to be published.
Croce de Cobos et al.
U
20 ps Figure 1. Oscillogram of NO, formation and decay (absorption signal at 662 nm, 200 atm of N2, 56 torr of N20, 1 torr of NOz).
were measured with a joulemeter. Typical laser energies in the cell were in the range 15-40 mJ. More details of our experimental arrangement have been described in ref 9. After the flash photolysis of N 2 0 , the formation of NO, via reaction 3 and its subsequent removal via reaction 2 were monitored by NO3 absorption spectroscopy near 662 nm. NO, absorption coefficients in this range now are very well-known (ref 14 and earlier work cited) such that our yields of NO, could be determined as a function of the initial oxygen atom concentrations. These were calculated from the measured laser energies. In our experiments N 2 0 pressures between 20 and 60 torr and N O 2 pressures between 0.1 and 6 torr were used. The pressure of the bath gas N2 varied between 2 and 200 atm. Since the pseudo-first-order rate coefficients of NO, formation and decay depended on the NO2 concentrations, the pressure of NO2 had to be measured carefully. Pressure measurements were made by capacitance manometry which was controlled by colorimetric measurements. The high dilution of the reactants by N2assured isothermal conditions of the reaction unperturbed by the laser flash. In a series of additional experiments, we added NO to the reaction mixtures in the range of N O / N 0 2 ratios between 0 and 0.4. All gases were from Messer Griesheim. The purity of the bath gas N, (99.996%) was of particular importance. We made sure than no N O impurities were present in NOz. The temperature of our experiments was 293 K.
Results After the laser pulse, the NO, absorption signals increased at first with rise times typically of the order of 500 ns. Later, on a much slower time scale of the order of 10 p s , the NO, signals decayed toward zero. A typical oscillogram in Figure 1 demonstrates this behavior. Because of the very different time scales, the two parts of the signal could be separated easily, the rise being represented by and the decay by The two apparent rate constants kf and kd are related to reactions 3 and 2, respectively. For this reason, they are represented (14) Ravishankara, A. R.; Wine, P. H. Chem. Phys. Lett., in press.
NO2
+ NO3 + M
-
+ M at High Pressure
N205
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 5085
TABLE I: Measured Rate Coefficients kf and kd for NO3 Formation and Decay" m 2 1 /
atm 2 4 12 25 50 100
150 200
W02)0/ torr
[Ole/ [Nolo/ 10"kfl 1012kd/ [NO210 [NO210 [NO210 [NO210 1012k2 2 0.2 0.4 0.4 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.07 0.04
0.1 1.o 0.5 0.4 1.0 1.0 1.0 1.o 1.o 1.o 1.o 1.o 2.0 3.0 6.0
0.03 0.015 0.025 0.02 0.02 0.02 0.015 0.1 0.2 0.4 0.015 0.015 0.015 0.015 0.015
5 3 4 4 3 5 7 8
8 7 8
6 2.4 4 3 2.3 2.5 2.3 3.4 4.5 8 2.4 2.3 2.3 3.0 2.5
1.5 1.3 1.3 1.8 1.8
TABLE 11: Rate Coefficients from Evaluation in Ref 15, Used for Numerical Simulations
reaction
--
+
Discussion The interpretation of our experimental rate coefficients kf and kd will be based on the kinetic properties of the mechanism 0 N02-NO + 0 2 (4)
+
+ M +NO3 + M + M NzOs + M
+ + + - + + - + NO
NO,
(3) (2)
2N0,
(5) As long as the oxygen atoms are not yet consumed, also the side reactions 0 N O M NO2 M (6)
0
NO3
NO2
+
k, = 1.6 X 10-l'
-
-
0 + NO,
O2
(7) can contribute. The reaction of oxygen atoms with N205,thermal decomposition of N 2 0 5(eq l), and ozone formation by reaction of oxygen atoms with O2can be neglected under our condition^.'^ The N 2 0 / N 2 ratios used in our experiments were always small enough to ensure a complete electronic deactivation of O('D) (15) Baulch, D.L.; Cox, R. A.; Hampson, R. F.; Kerr, J. A,; Troe, J.; Watson, R. T. J . Phys. Chem. Ref. Data, in press. See also: J . Phys. Chem. Ref. Data 1980, 9, 295; 1982, 1 1 , 327.
SKI)
X lo-"
x lo-" x 10-11 x 10-12 ko = [N2]9X lo-)* k , = 2.2 x lo-" F, = 0.8 ko = [Ni]3.7 X
F, = 0.34 2 x 10-11
NO + NO, 2N02 0 + NO N2 NO2 + N2
2.1 2.0
in the form kf/[N0210and kd/[NO2],-,. Table I summarizes our results for a variety of experimental conditions. Since single-flash experiments in a short path length high-pressure photolysis cell had to be employed, a relatively large noise of the signals like those in Figure 1 could not be avoided. Averaging of several experiments, therefore, was necessary. Within the scatter, the simple time laws appeared adequate for all concentrations illustrated in Table I. (Deviations from these time laws became apparent only in the simulations; see below.) It was found that kf was always at least a factor of 10 larger than kd. An effect of bath gas pressure on kd was not directly visible from the measurements. However, experiments with an excess of initial oxygen atom concentrations over NO2 always showed much larger kd values than experiments with an excess of NOz over 0. Addition of N O to the N20-N02-N2 mixtures resulted in an increase of kd. This increase was more pronouncd for mixtures with NO2 excess over 0 than for mixtures with oxygen excess over NO2. The given observations are all consistent with the kinetic properties of the relevant reaction mechanism of the system as shown below. In order to decide which data provide the most direct access to k2, the reaction mechanism is simulated in the following on the basis of the best available set of rate coefficient~.]~A comparison of our data with these simulations then will allow for a "fine tuning" of the input rate coefficients under conditions where the observables are most sensitive to individual reactions.
0 + NO2
2.6 7.2 4.4 9.3
2.0 2.0
" k2 evaluated from kd by using numerical simulation (k,/[NO2lO, kd/[NO2I0,and k2 in cm3 molecule-' d).
NO3 + NO2
k/(cm3 molecule-'
--
O('D) + N2 0 + N2 O('D) + NzO 2 N 0 O('D) + N20 N2 + 0, 0 + NO2 NO + 0 2 0 + NO, + N, NO, N2
ko = [Nj]1.0 X lo-" k, = 3 X lo-" F, = 0.85
4
NO,
+ 0,
1 x 10-11
1.0
NO,
x
I
2. 0.5 U
I
0
Figure 2. Calculated concentration profiles (100 atm of N2, [0]o/[N02]o = 0.2, [NO],/[NO,], = 0.014, rate coefficients from Table 11).
1.0
Cn
U I
0 I 0.5 0
5
lk,+k,)
IO INOzlot
15
20
Figure 3. Calculated concentration profiles (100 atm of N2, [O]o/[NOz]o = 2.0, [NO],/[NO,], = 0.0, rate coefficients from Table 11).
atoms without NO formation by reaction of O(lD) with N20. On the other hand, a small fraction of NO2 besides N 2 0 is also photolyzed by the laser flash at 193 nm via NO2 hv+ NO 0
+
+
The NO yield produced by this reaction15 was typically between 1% and 3% of the initial NO2. Since kd is sensitive to the N O / N 0 2 ratio, this small initial NO concentration was always included in our simulations. Table I1 shows the set of rate coefficients from ref 15 which were used for a first simulation of reactions 2-7. For typical conditions of our work, with an excess of NO, over oxygen atoms of the order of [0],/[N02], = 0.2, the simulation predicts a conversion of oxygen atoms to NO, of the order of [N0,],/[0], 0.5k3/(k3 k4). Under these conditions one should also expect k, e 2(k3 + k4) and kd E 2k2 at 2 atm, or kd 1.2k2 at 100 atm. With added NO, reaction 5 leads to an increase of NO3 consumption. On the other hand, with an oxygen atom excess over NO,, NO, consumption via reaction 7 and the combined effect of reactions 6 and 2 lead to a pronounced increase of kd. In order to illustrate these kinetic properties of the reaction system, we show in Figures 2 and 3 concentration profiles for experiments with NO2 excess, and with oxygen atom excess, such as calculated with the rate coefficients from Table 11. For small [O],/[NO,], concentration ratios, the ratios of kf/( [NO,],(k, + k4)) and kd/( [NO&k2) are relatively insensitive to the exact rate coefficients of other reactions. For this reason,
+
5086 The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
Croce de Cobos et al.
c
m c
-3 W
0
aJ
0
E r?
5
\ Y
-
+
Figure 4. Falloff curve of recombination reaction NO2 NO, + M N2O5 4- M ( T = 295 k 3 K 0 , this work; 0, ref 5; A, ref 1; M, ref 2; dissociation data from ref 1 and 2 converted with equilibrium constant from ref 4; full curve: falloff expression given in the text).
we have used mostly the experiments with [O],/[NO,], = 0.2 to derive improved values of k2 by the use of the kd/( [NO,],k2) ratios from the simulation. The resulting k2 values are included in Table I (last column). They represent the final result of our work. A final simulation with the k2 data obtained did not alter this evaluation. The error limits of the derived k2 values, due to the scatter of the kd data and the uncertainties of the simulations, are estimated to be about f20%. The experiments a t 4 and 12 atm used relatively high [0],/[N02], ratios such that the accuracy of the corresponding k2 values was only f40%. The error limits for the measured kf values are much larger than those of kd, being close to a factor of 2. Figure 4 shows our high-pressure data for kz in comparison with earlier low-pressure data from ref 1-5. The agreement is very good. However, the present direct measurement leads to a slightly higher value of the limiting high-pressure rate coefficient of
k2,, = (2.2 f 0.5) X
cm3 molecule-’
s-I
Together with the low-pressure value
kzo = [N2](3.0f 1.5)
cm3 molecule-’ s-’
X
a broadening factor of the falloff curve of3 F, = 0.36 and a width of the falloff curve of N = 0.75 expressionI6
k2 k2,-
k20/k2’m
- log F, in the
F~1+[l0ll(k2o/k2,.)/lVI2~-’
1 + k2o/k2,-
of k , by a factor of 3 was observed when the pressure was raised from 1 to 10 atm. k l from ref 7 at 10 atm is converted into k2 5 X 10-l2 cm3 molecule-’ s-’ such that mechanistic complications have to be responsible for the disagreement. By reduction of the measured k f / [NO,], values from experiments with [O]o/[NO,]o= 0.2 on the basis of the present simulation, absolute values for k3 k4 can also be derived. Subtracting the presumably pressure-independent literature value for k4 from Table 11, one obtains absolute values of k3 over the falloff range. These can be compared with the results from relative rate measurements from ref 17 over the same pressure range. The good agreement of measured and simulated k f /[NO,], values demonstrates the agreement between the present absolute measurements and the relative rate data from ref 17. Therefore, the older data given in Table I1 are confirmed by our work. The most direct access to reaction 5 is provided by experiments with added NO under conditions with NOz excess over oxygen atoms, e.g., the series of experiments at 100 atm shown in Table I. The increase of kd/[N0210with increasing [NO]o/[N02]ois very pronounced. It is fully consistent with the simulation based on the literature value of k5 given in Table 11. It should be noted representation that the slope of the kd/[N02],vs. [NO]O/[N02]0 to a major extent is governed by k5,but to a minor extent also depends on the other reactions. From the good agreement between simulated and measured slopes, we therefore conclude that our experiments confirm a value of
+
k5 = (2 f 1) X lo-’’ cm3 molecule-’ s-’ This value can now be assigned a smaller error limit than that given in ref 15 (A log k5 = f0.5).
the falloff curve of Figure 4 can very well be represented such as shown in the figure. A theoretical interpretation of the derived ko, k,, and F, values may follow the analysis given previ~usly.~ The agreement in Figure 4 apparently rules out the high values from ref 6 . Although in ref 6 the bath gas O2was used, near to the high-pressure limit a discrepancy by a factor of 3 cannot be attributed to different collision efficiencies. The old data from ref 7 apparently also have to be discarded. In ref 7 an increase
Acknowledgment. Financial support of this work by the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 93 “Photochemie mit Lasern”) is gratefully acknowledged. A.E.C.C. also thanks the Consejo Nacional de Investigaciones Cientificas Y Tdcnicas de la Republica Argentina for a foreign exchange fellowship.
(16) Troe, J. J . Phys. Chem. 1979,83, 114. Gilbert, R. G.; Luther, K.; Troe, J. Ber. Bunsenges. Phys. Chem. 1983, 87, 169.
(17) Troe, J. Ber. Bunsenges. Phys. Chem. 1969, 73, 906. Hippler, H.; Schippert, C.; Troe, J. Symp. Int. J . Chem. Kinet. 1975, 1, 27.
Registry No. N2, 7727-37-9; N20, 10024-97-2; NO, 10102-43-9; NO,, 10102-44-0; NO,, 12033-49-7; N205, 10102-03-1.
