J. Phys. Chem. 1995,99, 12529-12535
12529
Falloff Behavior in the Thermal Dissociation Rate of N20 W. Dale Breshears Chemical Reactions, Kinetics and Dynamics Group, MS J567, Los Alamos National Laboratory,
Los Alamos, New Mexico 87545 Received: March 6, 1995; In Final Form: May 25, 1 9 9 9
The striking difference in falloff behavior (pressure dependence) observed between N2O and Ar as collision
-
partners in the thermal dissociation rate of N20 is noted. A mechanism proposed by Lindars and Hinshelwood [Proc.R . SOC.London, Ser. A 1955, A231, 1781 involves transitions, both unimolecular and collision-induced, from the ground singlet electronic state to a dissociating triplet state. The mechanism was fit to a wide range of experimental data for neat N20 and N20-Ar mixtures (273 individual data points), accounting quantitatively for the observed behavior [0.006 average rms deviation in ln(k)]. Results derived for the limiting high- and low-pressure (M = Ar) rate constants exceed current recommended values by factors of -1.4 and -2, respectively, over the temperature range 925-2000 K. An alternative mechanistic interpretation involving a finite rate for intramolecular vibrational relaxation is discussed.
1. Introduction rX10'2
The rate of unimolecular dissociation of N20,
is of considerable interest due to the spin-forbidden nature of the reaction, Le., singlet reactant going to triplet plus singlet products. It has been the subject of numerous experimental studies; with a few exceptions, these fall into two general categories. The first, spanning the period 1924-1985, comprises measurements on static gas in heated cells in the temperature range 876-1227 K, with both neat (undiluted) N20 and various N2O-diluent mixtures. The second, spanning the period 1961- 1992, comprises measurements in shock tubes in the temperature range -1400-3500 K, all of which were performed on mixtures of N20 dilute (a few hundered parts per million to a few mole percent) in an "inert" gas such as Ar or N2 (see the reviews in 1973 by Baulch et a1.,I3 in 1984 by Hanson and Salimian,I4 and the 1992 paper by Michael and LimI5). Johnsson et al. recently reported measurements in a flow reactor in the temperature range 1000-1400 K using gas mixtures of -200 ppm N20 in various diluents at atmospheric pressure. They also review other recent works of similar nature. A significant feature of the available data is the remarkable difference in pressure dependence, or falloff behavior, of the first-order rate constants for reaction R1 between collision partners M = N2O and M = Ar. This difference is illustrated in Figure 1, which shows the data of Lindars and Hinshelwood* at 993 K and of Huntefi at 997 K, both measured in static, neat N20; the data of Lindars and Hinshelwood9 at 993 K for addition of Ar to 50,100, and 200 Torr of N20; and those of Olschewski, Troe, and Wagner (0TW)l8 at 1700 K, measured in shock waves in mixtures of -0.1% N20 in Ar. The data at 1700 K have been muliplied by 1 x in Figure 1. The 1700 K data, with the predominant collision partner M = Ar, exhibit a loglog slope near unity (Le.,k proportional to concentration, [MI) for [MI I -1 x moVcm3, rapidly falling off to nearconstant values for [MI ? -1 x moVcm3. In contrast, the data for neat N20 exhibit a nearly constant log-log slope of -0.6 over a comparable span in log[M]. The N20-Ar l 6 3 l 7
@
Abstract published in Advance ACS Absrructs, July 15, 1995.
-
1x10'~
-
c
v)
x
1x10'4
1x i o 5 1x10'7
1x10'6
1x105
1x104
1x10~
i~io.*
[M] (mole/cm3)
Figure 1. First-order rate constant, k, for N20 thermal dissociation vs total concentration, [MI. Data for neat (undiluted) N20: Lindars and Hinshelwood: T = 993 K (0);Hunter,6 T = 997 K (0). Data for N20-Ar mixtures: Lindars and Hinshelwood? T = 993 K, initial N20 pressure 50 Torr (A), 100 Torr (e), 200 Torr (V); Olschewski et ul.,'* T = 1700 K ( k x is plotted) (0).The solid lines result from the
least-squares fit described in the text. mixture data at 993 K show only a very slight dependence on Ar concentration. In 1955, before the existence of any shock tube data, Lindars and Hinshelwood9 noted this striking difference in falloff behavior between their own measurements for neat N208 and various N2O-diluent mixture^.^ They proposed a simple mechanism9 to account for the qualitative features of the data but did not evaluate any rate constants quantitatively. ForstI9 also noted this rather dramatic difference in falloff behavior and suggested that the data for OTWi8 may be suspect. In the present paper, rate data for neat N20 and for N20-Ar mixtures are analyzed quantitatively in terms of the mechanism proposed by Lindars and Hin~helwood.~ The results satisfactorily account for the observed differences in falloff behavior between collision partners M = N20 and M = Ar. In section 2, the available data are reviewed briefly, and in section 3, the Lindars-Hinshelwood9 mechanism is discussed. Section 4
This article not subject to U S . Copyright. Published 1995 by the American Chemical Society
12530 J. Phys. Chem., Vol. 99, No. 33, 1995
Breshears
describes the procedure and results of the analysis. The results are discussed and compared with those of other workers, both experimental and theoretical, in section 5, and an altemative physical interpretation of the Lindars-Hinshelwood mechanism is suggested. In all subsequent discussions, the notation of Troe20 is adopted, wherein k, is defined as the true first-order rate constant in the high-pressure limit and as the pseudo-firstorder rate constant in the low-pressure limit. The second-order, bimolecular rate constant in the low-pressure limit is then given by W[M], where [MI is the total concentration of collision partners.
2. Review of Experimental Data Measurements in category one, i.e., on static gas in heated cells at 876-1227 K, are considered first. In all such studies except that of Loirat et al. (LCFS),', rates were determined by measuring the pressure change at one or more times during the course of the reaction. Some investigators also measured NO p r o d u ~ t i o n . ~ ~Although ~ - ~ . ' ~ the earliest of these measurements dates back more than 70 years, descriptions of experimental apparatus and procedures given by the various authors suggest that accurate pressure-time measurements were well within the scope of then-current technology. In the most recent study, Loirat et al. (LCFS)l2 measured time-dependent concentrations of N20, N2,02, and NO and derived rate constants for reaction R1 by matching computed concentration profiles to the data. The overall thermal decomposition reaction is assumed to be 2 N 2 0 +f(2N2
+ 0,) + (1 - j)(N2 + 2NO)
(R2)
with 0 If I 1. From the stoichiometry of reaction R2, the rates of change of total pressure P and of N2O concentration are related at constant temperature and volume by 1"d dt
-2-
dp
dt
Some authors report half-times, t1/2, for the reaction of N20,'35-7 some report "integral" rate constants derived from measurements at a single reaction time with the assumption of first-order disappearance of N20?33 and others report "initial" rate constants derived from measurements at very early reaction times.8-'0 Analysis of all reported temporal profiles for pressure or N2O concentration (the table on p 286 of Hinshelwood and Burk,' Table I of Hunter,6 Table I of Lewis and Hinshelwood,' Table I11 of Lindars and Hinshelwood,8 Figure 2 of Kaufman et a1.,I0 and Figure 2 of LCFSI2) indicates that in all cases the decay of [N20] is exponential within experimental scatter over at least the first half of the reaction course. Thus, integral and initial rate constants may be compared directly, and reaction half-times can be converted to first-order rate constants by the relation
Some uncertainty exists concerning the contributions of the fast reactions
+ 0 - N, + 0, N 2 0 + 0 -2 N 0
N20
(R3) (R4)
to the measurements in category one (see the discussion by Baulch et al.,I3 pp 85-86). With the exception of LCFS,', none
of the reported rate constant has been corrected for such contributions. All reported rate constants used in the present analysis were arbitrarily divided by two. Some uncertainty also exists conceming possible heterogeneous contributions. Hinshelwood and Burk' and Musgrave and Hinshelwood5 report experimental observations indicating that such contributions to their measurements are probably small. Nevertheless, in an analysis of the data available in 1951, Johnston2' assumed a small heterogeneous contribution at low pressures. However, an extensive study of heterogeneous effects by Lindars and Hinshelwood* in 1955 casts some doubt on this assumption. Their findings conceming the effect of heterogeneous reaction on NO production rates are in accord with those of Kaufman et al.'O From the findings of Lindars and Hinshelwood,8summarized in their Figures 1-6, we conclude that the first-order rate constants for initial N20 pressures in the range 9-500 Torr at 993 K, presented in their Figure 1, are demonstrably free of any significant heterogeneous contribution. For the remainder of the data in category one, such contributions,to varying extent, cannot be ruled out. Several measurements in category one on mixtures of N20 with various diluent gases have been reported. The focus here is on those in N20-Ar mixtures. Lewis and Hinshelwood7 measured rates at 925 K in mixtures of 20% and 33% NzO, with N20 partial pressures in the range -20-560 Torr. Lindars and Hinshelwood9 report measurements at 993 K with up to -2000 Torr of Ar added to 50, 100, and 200 Torr of N20. Bell et ul.l1 measured rates at 981 K with up to -350 Torr of Ar added to 99 Torr of N20. As will be shown below, the data of Volmer and Froehlich3 for neat N20 are in disagreement with the majority of other such data; their measurements reported for N20-Ar mixtures4 were not included in the present analysis. It should be noted that none of the reported measurements in category one span a range in total concentration sufficient to define unambiguously either the high-pressure or the lowpressure limit. In their 1973 review, Baulch et ~ 1 .noted ' ~ considerable scatter in reported shock tube measurements of W[M] (mostly with M = Ar) in the range 1400 IT I 3500 K (see their figure, p 70). They recommended the values reported by OTW,'8
k d [ M ] = 5.0 x lOI4 exp(-29200/T) cm3/(mol s) ( M = Ar) (3) k, = 1.3 x 10" exp(-299OO/T) s-'
(4)
derived from measurements in dilute mixtures (0.02- 1.O %) with Ar,in the approximate temperature range 1400-2500 K. OTW covered a wide range of total concentration, 5 x I [MI I2 x lov3mol/cm3, apparently sufficient to define both the low- and high-pressure limits (see Figure 2 of 0TWl8 and Figure 1 of the present paper). Theirs remains the only reported study to date to accomplish this feat. Their results, eqs 3 and 4, were corrected to account for the observed effects of the fast follow-up reactions R3 and R4 (see Figure 1 of O W ) . In a 1984 review, Hanson and SalimianI4 noted significantly improved agreement in more recent shock tube values reported for W[M] (again, mostly with M = Ar) (see their Figure 4). Their recommendation for W[M] (M = Ar,Kr) in the range 1500-3600 K differs significantly from eq 3 only at temperatures above -2500 K. Michael and LimI5 recently reported shock tube measurements of W[M] with M = Ar and Kr in the temperature range 1540-2500 K, which they note agree well with other recent
Falloff Behavior in N20
J. Phys. Chem., Vol. 99, No. 33, 1995 12531
measurements, with the recommendation of Hanson and Salimian,I4 and, thus, with the results of O W , eq 3. The conclusion we draw from this summary is that the results of O W ' 8 for W[M] with M = Ar, derived from their measurements at [MI I6 x moYcm3, have stood the test of time and that, by implication, their data at higher concentrations are also valid. O W reported measured first-order rate constants as a function of [MI only at 1700 and 2000 K, and these (values from Figure 2 of divided by two) are the only shock tube data used in the present analysis.
where the function fl[M]) is given by
(6) and the limiting low- and high-pressure first-order rate constants are
3. The Lindars and Hinshelwood Mechanism In 1955, prior to the existence of any shock tube data, Lindars and Hinshelwood9 proposed the following simple mechanism to account for the extended falloff behavior observed in measurements on neat, static N20 and for their own measurements with added gases (Ar, N2, C02, and CF3: 1. 2.
-
N20*
4.
N203*
-
-
N203*
N2
+0
They defined N20* as singlet molecules energized sufficiently to be capable of transitions, both bimolecular (step 2) and unimolecular (step 3), to a triplet state, N203*,which dissociates irreversibly (step 4). They concluded that description of the highest-pressure data of Huntefi requires including a second triplet state, N203**, with the following additional steps:
6.
k4
fo=l+-
+ M -N20* + M N,O* + M N ~ o ~+* M N20
3.
5.
respectively. The function fl[M]) has low- and high-pressure limiting values
-
+M N20** + M N2*
7.
N20**
8.
N,o~**
+M N203** + M
N20**
-
N 203**
N,
+o
Lindars and Hinshelwood recognized that this mechanism is overly simplified: "This must obviously do less than complete justice to reality, since each triplet state will have associated with it different energies corresponding to different vibrational and rotational quantum levels. Each will thus in fact have a range of values for all the constants of any collisional or spontaneous processes in which it is involved They presented an analysis of the qualitative features of a model based on this mechanism but did not evaluate any rate constants quantitatively. They note that the essential feature of the model is the inclusion of both bimolecular and unimolecular transitions to the triplet state(s) (steps 2 and 3, 6 and 7). As will be shown below, a model based on steps 1-4 provides a satisfactory quantitative description of the data analyzed. Application of the steady-state approximation to [N20*] and [N203*] and use of the fact that detailed balance requires k3lk-3 = kdk-2 leads to the expression
(9)
k-3
respectively. Several features of eq 5 are worthy of note. The algebraic form was chosen to illustrate clearly that forf= 1, eq 5 reduces to the expression for the simple Lindemann mechanism (see, e.g., Robinson and Holbrook,22 pp l5ff). In fact, for any range of [MI over which f is constant, corresponding to fo or f-, eq 5 will be indistinguishable from the Lindemann form. Thus, a pseudo-high-pressure limit may exist with rate constant
Equation 5 departs from the Lindemann form in the range of [MI over whichfvaries between the limits of eqs 9 and 10 only ifthe productfl[M])(kdk) is of order unity or greater over this range. For gas mixtures, the first-order rate constants for steps 1 and 2 and their reverse are given by
k,[M] = c k , [ M j ] ,
i = 1, -1,2, -2
(12)
j
where the sum is over all components in the mixture and k, is the bimolecular rate constant for collisions with component j .
4. Data Analysis: Procedures and Results Data for neat N20 included in the analysis are those of Hunter: of Nagasako and Volmer? of Lewis and Hinshelwood,' and of Lindars and Hinshelwood.8 The motivation for this selection will be apparent from the results. Data for N20-Ar mixtures included in the analysis are those of Lewis and Hinshelwood,' of Lindars and Hinshelwood? of Bell et al.," and of OTW.'8 This combined data set for both neat N20 and N20-Ar mixtures contains 273 individual points. The data of Johnsson et aZ.,I6comprising measurements at various temperatures, each at a single pressure (one atmosphere), provide no direct information about the fall off behavior and were not included in the analysis. Equation 5 was fit to the total combined data set by the method of least squares. The quantities k,~ I M k, - 2 ~ l k - 3 ,and
12532 J. Phys. Chem., Vol. 99, No. 33, 1995
Breshears
TABLE 1: Least-Squares Parameter Values quantity
k, ki (M = NzO) kl (M = Ar) k-ztk-3 (M = N 2 0 ) k-dk-3 (M = Ar) kdk-3
A , (mol/cm3; s) 2.53 4.26 3.54 5.18 9.86 6.26
x 10" x 1015 x 1014 x 107 x 1015 x lo3
1x l 0'
E , kcaUmol 60.66 58.55 56.17 11.96 70.07 12.16
1x100
I
1
kdk-3, with M = N20 and Ar, were each represented by the simple Arrhenius form; e.g.,
Q 1x10'
1x102
For each such quantity, the parameters Ai and Ei/R were adjusted to minimize the sum of squared residuals in In(@. With two quantities, ~ I and M k - z ~ l k - 3 ,assuming different values for M = N 2 0 and Ar, a total of 12 parameters were adjusted in the fitting process, which was performed using the numerical Solver routine in version 4.0 of the Microsoft Excel software package on a Macintosh Quadra 800 personal computer. The parameter values obtained from the fitting procedure .are shown in Table 1. The average root-mean-square deviation for the fit is 0.006. With the reduced variablez0
1x103
1 ~ 1 0 ~ 1x10"
1x10~
1x10~
I X I ~
X
Figure 2. Results (solid line) of the least-squares fit described in the text, plotted in the form of eq 15. Individual points are for neat (undiluted) N 2 0 data included in the fit: Nagasako and Volmer2 (0); HunteP (0);Lewis and Hinshelwood' (A); Lindars and Hinshelwood8
(0). 1x100
the function
2 1x101 1
derived from eq 5, takes the Lindemann form and is "universal" in the sense that all data describable by the assumed mechanism, at all temperatures and for all collision partners M, should fall on a single curve. The results of the fit are compared with the neat N 2 0 data, plotted in the form of eq 15, in Figure 2 and with the N20-Ar mixture data in Figure 3. It should be emphasized that the solid curves in Figures 2 and 3 result from a single fit to the combined data for neat N 2 0 and N 2 0 - A r mixtures; the individual data points are plotted separately in Figures 2 and 3 simply for ease of visualization. It is apparent from Figures 2 and 3 that the fit to the data is excellent, with no indication of systematic deviations. The solid lines in Figure 1 are the results of the fit for the data displayed there. In Figure 4, the results of the fit are compared with those data for neat N20 not included in the analysis. The data of Hinshelwood and Burk' and of Musgrave and Hinshelwood5 are systematically low, while those of Volmer and Froehlich3 and of Loirat et ~ 1 . are ' ~ systematically high. The results of Kaufman et a1.I0 are in better agreement but exhibit a systematically lower slope. The data of Johnsson et a1.I6 for N20-AI (pseudo-first-order rate constants from their Figure 5 ) lie above the results of the fit by factors of about 4. 5. Discussion
Values of k, calculated from the parameters in Table 1 are shown in Figure 5, along with the results of O W ' * from eq 4 above. The present result exceeds that of OTW by about 40% over the temperature range 1400-2000 K. From Table 1, the activation energy is 60.7 kcal/mol, while eq 4 yields 59.5 kcal/ mol. Values of W[M] for M = N20 and Ar, calculated from the parameters in Table 1, are shown in Figure 6 , along with the
'
1x102 1x102
I
I
I
1x10'
1x100
1x l 0'
lxld
X
Figure 3. Results (solid line) of the least-squares fit described in the text, plotted in the form of eq 15. Individual points are for N20-Ar mixture data included in the fit: Lewis and Hinshelwood' (0);Lindars and Hinshelwood9 (0);Bell et al." (A); Olschewski et al.Is (0).
results of 0TWi8from eq 3 above, for M = Ar. The present result for M = AI exceeds that of OTW by factors of about 2 over the temperature range 1400-2000 K. From Table 1, the activation energy for M = Ar is 56.2 kcdmol, while eq 3 yields 58.0 kcdmol. The activation energy for M = N20 from Table 1 is 58.6 kcdmol, and the values of W[M] exceed those obtained here for M = Ar by a factor of -3.5 over the temperature range 925-1020 K. The factor is comparable to that reported by Dove et aLZ3for the relative efficiencies of N20 and AI in the vibrational relaxation of N20 (-2.25-4.0). Values of k-2/k4 for M = N20 and Ar, obtained from the parameters in Table 1 using the relation k - ~ / k 4= (k-2/k-3)/(kd k-3), are shown in Figure 7. The strikingly different character of k-2/k-4 for these two collision partners accounts for the differences in falloff behavior illustrated in Figure 1. Equation 15 can be rearranged to yield
k = (-)F(x) X k, I+x
Falloff Behavior in N20
lXIOO
J. Phys. Chem., Vol. 99, No. 33, 1995 12533
I %“/e I
1XlO.l
--$
1x10~
-
1x106
-
35
1x10~
-
0
-F
25 %
,\
‘
\\.\
\ \ \ \ \ \
I
A
1x i o 3
\\
5 1 x 1 0 ~-
0
1x10’2
‘
I
I
1xlo.2
1x101
1x103
’
1x102
’
1UlOl I
1x10.~
1x100
0.4
\
I
1
I
I
I
I
I
0.5
0.6
0.7
0.8
0.9
1.0
1.1
X
1.2
1ooorr (K’)
Figure 4. Results (solid line) of the least-squares fit described in the text, plotted in the form of eq 15. Individual points are for neat (undiluted) N20 data not included in the fit: Hinshelwood and Burkl (0);Volmer and Froehlich3 (0);Musgrave and HinshelwoodS (A); Kaufman et d.l0(0); Loirat et ~ 1 . (v). ’ ~ 1x10~
Figure 6. Values of the limiting low-pressure bimolecular rate constant, k d [ M ] ,obtained from the least-squares fit described in the text for M = N20 (-) and M = Ar (- -) and by Olschewski et dl*for M = Ar, eq 3 (- -). 1x106
I
I
I
t
1x10~
1x102
-
1 x 1 0 . ~-
I
I
0.4
0.5
1x104
I
0.6
I
I
I
0.7
0.8
0.9
\
g
1x10~
35
-
1x102
3
1x10’
* t 1x100
1x10‘
I
1
I
1.0
1.1
1.2
1OOOrT ( K I )
Figure 5. Values of the limiting high-pressure rate constant, k,, obtained from the least-squares fit described in the text (-) and by Olschewski et a1.,I8 eq 4 (- -).
in which
is analogous to the “broadening” factors defined by Troe and c o - w ~ r k e r s .Since ~ ~ ~ fl[M]) ~ ~ and, thus, g(x) approach a finite limit at low pressure (low x), eq 9, and unity at high pressure (high x), eq 10, F(x) approaches unit at both low and high pressures. Figure 8 shows broadening factors, computed from the parameters in Table 1, for the range of experimental conditions covered by the data shown in Figure 1. For M = Ar at 1700 K, F 7 0.9 at all concentrations, and the experimental fall-off curves are virtually indistinguishable from the Lindemann form. For the N20-Ar mixtures at 993 K, F is roughly constant over the range 1 x < [MI < 1 x 10-1 mol/cm3, corresponding approximately to the pseudo-high-pressure limit given by eq 11. The physical interpretation of the mechanism proposed by Lindars and Hinshelwoodgis worthy of some discussion. Their assumption of a finite rate for the unimolecular dissociation of
1
c I
I
l*l”
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1ooorT ( K ’ )
Figure 7. Values of k - 2 ~ / kobtained ~ from the least-squares fit described in the text for M = N2O (-) and M = Ar (- -).
N203*, step 4 above, would seem to imply that N203* corresponds to a bound triplet state. However, it has been assumed in theoretical treatments that dissociation occurs via transitions from the bound singlet ground state to a repulsive triplet state (see, e.g., discussions by Lorquet et a1.25and by Yau and F’ritchard?6 and references cited therein), which then dissociates with a rate comparable to vibrational frequencies, s-I. Figure 7 shows that for M = N20 and for M = Ar at T 1. -1500 K, k-2M/k4 2 1 x lo4 cm3/mol. For k4 -1 x loi3s-’, values of k-2M 1 -1 x 10’’ cm3/(mol s) are obtained, 2-3 orders of magnitude higher than “gas kinetic” collision frequencies. Conversely, setting k-2M = 1 x lOI4 cm3/(mol s), a value typical of gas kinetic collision frequencies, yields k4 5 -1 x 1O’O s-l, apparently much too low for dissociation from a purely repulsive triplet state. The arguments presented above suggest that the physical interpretation Lindars and Hinshelwood9 attached to their mechanism may be inconsistent with the quantitative results of the present analysis. The fact remains, however, that kinetically the mechanism successfully accounts for the observed complex falloff behavior over a wide range of experimental conditions,
Breshears
12534 J. Phys. Chem., Vol. 99, No. 33, 1995
0.3
0.1 o'2
c
1
0.0 I
1x10"
I
I
I
IXIO-~
I
1x10'~
1x10'~
I
1x10.~
I IXIO'*
[MI(mole/cm3)
Figure 8. Broadening factors, F , defined by eq 17, calculated from the results of the least-squares fit described in the text, for conditions corresponding to the experimental data plotted in Figure 1: Neat (undiluted) NzO, T = 993 K (-); N20-Ar mixtures, T = 993 K, initial N20 pressure 50 Torr (- -), 100 Torr (- - -), 200 Torr (- - - -); N20 infinitely dilute in Ar, T = 1700 K (- -).
including the very different effects of N20 and Ar as collision partners. This success provides motivation to explore possible alternative physical interpretations of the Lindars and Hinshelwood mechanism. One such alternative is suggested by the results of classical trajectory calculations reported by Marks and Thompson?' who employed potential energy surfaces with the lowest energy singlet-triplet crossing at -63 kcal/mol, corresponding to the linear configuration. They found surface crossing (and hence, dissociation) to be enhanced by localization of vibrational energy in the N20 stretching modes and depressed by localization in the bending mode. A time scale L -1 x lo-'' s was estimated for unimolecular statistical redistribution of energy between the bending and stretching modes. We replace steps 1-4 of the Lindars and Hinshelwood9 mechanism by
1a. 2a.
+M N20t + M N20
-
+M N20* + M
N20t
3a.
N20t
N20*
4a.
N~O* N ~ o ~ *
-
and assume that the triplet N203* dissociates essentially instantaneously. Here N20t and N20* are singlet ground-state molecules with the same intemal (vibrational and rotational) energy, N20t representing essentially all such molecules and N20* representing a subset with distributions of intemal energy favorable for unimolecular transition to the triplet state, step 4a. Step l a represents collisional activation to levels in the singlet state from which the triplet is energetically accessible. The reverse of steps 2a and 3a is collisional and unimolecular channels, respectively, for statistical redistribution of a fixed total intemal energy. In the usual statistical treatments of unimolecular reaction rate theory, step 3a is assumed to be infinitely fast (see, e.g., Robinson and Holbrook,22pp 99-108). If, however, the rate of step 3a is finite, as the results of Marks and Thompson27 suggest, collisional redistribution, step 2a, becomes significant at sufficiently high pressures. The result
estimated above for k4, now applied to kda, i.e., k4a 5 -1 x 1O'O SKI,is in general agreement with values obtained by Marks and Thompson27at the lowest energies (see their Tables I1 and 111). From a direct calculation of the surfaces of intersection of ground-state N20 and the three triplet states correlating with the products of reaction R1, Chang and Yarkony28 report a minimum crossing energy of 58 kcal/mol, corresponding to a linear configuration. This result contradicts the 1982 deduction by Lorquet et a1.25of -40 kcal/mol, corresponding to a bent configuration at 130". Further, the body of available experimental data, taken as a whole, provides no firm evidence to support the prediction by Forst,19 based on the lower value, that the activation energy for reaction R1 decreases from -60 kcal/ mol at the high-pressure limit to -40 kcal/mol at the true lowpressure limit. The only previous study in which w[N20] was determined quantitatively is the analysis by Johnston2' of the data available in 1951 for neat N20. Values were taken as the slope of a linear fit to the lowest concentration data, the intercept being interpreted as the contribution due to heterogeneous reaction. While the results of the present analysis agree with those of Johnston to within 20% on the average over the temperature range 989-1053 K, the agreement is to some extent fortuitous, given the differences in approach. Endo et al.29 reported measurements of W[M] in the approximate temperature range 1700-2400 K for M = He, Ne, Ar, Kr, Xe, N2, and CF4. Values relative to those for M = Ar range from 0.7 for M = Xe to -6 for M = CF4. The value of -3.5 for M = N20 near 1000 K from the present analysis is consistent with these results. One remaining question of interest is the striking difference in the temperature dependence of k - 2 ~between M = N20 and Ar, illustrated in Figure 7. It is important to note that, within the strict context of the mechanism suggested above, step 2a involves no transfer of internal energy to or from the N20 molecule, only redistribution of the existing intemal energy. The rates of such collisional processes will be sensitive to the details of the N20-M interaction potential, which obviously will be considerably more complex for M = N20 than for M = Ar. Small, long-range features may well be important and could result in rate constants considerably larger than those for "normal" gas kinetic collisions.
Acknowledgment. This work was supported by the US. Department of Energy. I thank Drs. David Funk, John Lyman, and Richard Oldenborg for stimulating and fruitful discussions. References and Notes (1) Hinshelwood, C. N.; Burk,R. E. Proc. R. SOC.London, Ser. A 1924, A106, 284-291.
(2) Nagasako, N.; Volmer, M. Z . Phys. Chem. 1930, BIO, 414-418. (3) Volmer, M.; Froehlich, H. Z . Phys. Chem. 1932, BZ9, 85-88. (4) Volmer, M.; Froehlich, H. Z . Phys. Chem. 1932, BZ9, 89-96. ( 5 ) Musgrave, F. F.; Hinshelwood, C. N. Proc. R. SOC.London, Ser. A 1932, A135, 23-39. ( 6 ) Hunter, E. Pro. R. SOC.London, Ser. A 1934, A144, 386-412. (7) Lewis, R. M.; Hinshelwood, C. N. Proc. R. SOC. London, Ser. A 1938, A168, 441-454. (8) Lindars, F. J.; Hinshelwood, C. Proc. R. SOC.London, Ser. A 1955, A231, 162-178. (9) Lindars, F. J.; Hinshelwood, C. Proc. R.SOC.London, Ser. A 1955, A231, 178-197. (10) Kaufman, F.; Gem, N. J.; Bowman, R. E. J . Chem. Phys. 1956, 25, 106-115. (11) Bell, T. N.; Robinson, P. L.; Trenwith, A. B. J . G e m . SOC.1957, 1957, 1474-1480. (12) Loirat, H.; Caralp, F.; Forst, W.; Schoenenberger, C. J. Phys. Chem. 1985, 89,4586-4591.
Falloff Behavior in N20 (13) Baulch, D. L.; Drysdale, D. D.; Home, D. G. Evaluated kinetic data for high temperature reactions; Buttenvorths: London, 1973; Vol. 2. Homogeneous gas-phase reactions of the H2-Nz-02 system, pp 69-92. (14) Hanson, R. K.; Salimian, S. In Combustion Chemistry; Gardiner, W. C., Jr., Ed.; Springer-Verlag: New York, 1984; pp 351-421. (15) Michael, J. V.; Lim, K. P. J. Chem. Phys. 1992, 97, 3228-3234. (16) Johnsson, J. E.; Glarborg, P.; Dam-Johansen, K. Twenty-Fourth Symposium (International) on Combustion; The Combustion Institute: Pittsburgh, PA, 1992; pp 917-923. (17) Glarborg, P.; Johnsson, J. E.;Dam-Johansen, K. Combust. Flame 1994, 99, 523-532. (18) Olschewski, H. A.; Troe, J.; Wagner, H. G. Ber. Bunsenges. Phys. Chem. 1966, 70, 450-459. (19) Forst, W. J . Phys. Chem. 1982, 86, 1776-1781. (20) Troe, J. Ber. Bunsenges. Phys. Chem. 1983, 87, 161-169. (21) Johnston, H. S. J . Chem. Phys. 1951, 19, 663-667. (22) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley-Interscience: London, 1972.
J. Phys. Chem., Vol. 99, No. 33, I995 12535 (23) Dove, J. E.; Nip, W. S.; Teitelbaum, H. Fifteenth Symposium (International) on Combustion, The Combustion Institute: Pittsburgh, PA, 1975; pp 903-916. (24) Gilbert, R. G.; Luther, K.; Troe, J. Ber. Bunsenges. Phys. Chem. 1983, 87, 169-177. (25) Lorquet, A. J.; Lorquet, J. C.; Forst, W. Chem. Phys. 1980, 51, 253-260. (26) Yau, A. W.; Pritchard, H. 0. Can. J . Chem. 1979, 57, 17311742. (27) Marks, A. J.; Thompson, D. L. J . Chem. Phys. 1991, 95, 80568064. (28) Chang, H. H.; Yarkony, D. R. J . Chem. Phys. 1993, 99, 6824683 1. (29) Endo, H.; Glanzer, K.; Troe, J. J . Phys. Chem. 1979, 83, 2083 -2090. JP950632L
