10
The Journal of Physical Chemistry,
Vol. 83, No. 7, 1979
J. V. Michael and J. H. Lee
Rate Constant Measurements at Constant Temperature by the Flash Photolysis-Resonance Fluorescence Technique and Recombination-Dissociation Theory for NO2 and NOCl J. V. Michael" and J. H. Leet Astrochemistry Branch, Laboratory for Extraterrestrial Physics, NASAlGoddard Space flight Center, Greenbelt, Maryland 20771 (Received June 28, 1978) Publication costs assisted by NASAIGoddard Space Flight Center
The flash photolysis-resonance fluorescence (FP-RF) technique for directly measuring thermal rate constants is discussed. Specific consideration is given to experimental accuracy with due regard to systematic and random errors. Data for the recombination and dissociation reactions, 0 + NO + M * NOz + M and C1 + NO M * NOCl + M, are reviewed. Theoretical calculations, which are based on the recent theory of Troe, are presented for these two systems. A modification of this theory is presented and tested with the same two systems.
+
I. Rate Constant Measurements at Constant Temperature by the FP-RF Technique The first absolute rate constants that were measured by the flash photolysis-resonance fluorescence technique were reported by Braun and Len2i.l" Since this 1967 work, facilities have become available in several other laborat ~ r i e s . l -Between ~ 100 and 500 rate constants have been measured, thereby making this method a major contributing technique to the literature of gas phase chemical kinetics. The technique is simple in concept. Transient atom or radical species are produced flash photolytically on a time scale which is short compared to their decay. Generally, continuous wave radiation, which is in resonance with lower and upper electronic states of the atom or radical, is absorbed by the transient species. If the upper state has a sufficiently short lifetime for emission compared to that for pressure quenching or dissociation, then a fraction of the resonance radiation is isotropically scattered. The scattered radiation is detected by single photon counting techniques at right angles to both photolytic and resonance radiation sources in a well-defined intersection volume. The decay of scattered photons is generally recorded by a multichannel analyzer operating in the multiscaling mode. Controlled experimental conditions allow for flash repetition with a subsequent increase in signal to noise. The scattered photon count will always be directly proportional to the lower state concentration if two conditions are met. First, the transient concentration cannot be so large that absorption occurs in a nonlinear portion of the curve of growth. Nonlinearity is caused by a mismatch of resonance source and absorber line shapes.l0 A further complication occurs if the concentration is even higher. Then radiation imprisonment by the absorber is possible. The second condition is related to the first. The line shape of the resonance source should be as close to nonreversed as is practical. This point, and additionally, intensity, have been the basis for using tunable dye lasers as sources for resonance fluorescence studies.3b However, if a reversed resonance source is used one should be aware that nonlinearities in curves of growth set in at lower *Author to whom correspondence should be addressed. Visiting Professor of Chemistry, Catholic University of America, Washington, D.C. 20017. 'Research Associate, Catholic University of America, Washington, D.C. 20017; permanent address, Department of Energy and Environment, Brookhaven National Laboratory, Upton, L.I., N.Y. 11973. 0022-3654/79/2083-0010$01 .OO/O
absorptions than with nonreversed sources. A nonreversed source is also desirable from the standpoint of absolute sensitivity for detection, the Doppler line shape of the nonreversed source being in closer resonance with that of the absorber. The limiting factor affecting the absolute sensitivity is the oscillator strength for the transition. I t is proportional to the absorption cross section and is obviously a property of the transition. The sources of uncertainty in rate constant measurements are both systematic and random. Systematic uncertainty can be further broken down into two categories, systematic errors and systematic factors. In the discussion to follow we will consider each source of error with specific reference to the apparatus and procedures which are used in our laboratory. The major source of systematic error is in absolute pressure measurement. We measure absolute pressure with an MKS Baratron Model 77 capacitance manometer. Three separate pressure measurements enter into the calculation of reactant concentration for a kinetic experiment. Since premixed experimental mixtures are used, the first measurement is reactant pressure, and the second is total gas pressure. The third measurement is the pressure of the flowing experimental mixture through the reaction cell. The manufacturer claims an absolute pressure accuracy of -0.5% for the dc output. Thus, cumulative errors could give an uncertainty of -1.5% in reactant concentrations. Because the device is mechanical with electronic processing and readout, we periodically check the calibration by the more direct but less accurate method of oil manometry with cathetometer. The systematic factors are not apparatus dependent but are instead dependent on the reaction to be studied. These are as follows: (a) the possible perturbation of the time profile of the observed species by reactions of other primary photolysis products,11J2(b) similar perturbation by secondary reactions of the products from the observed species-reactant interaction,13 (c) possible loss of the reactant by homogeneous or heterogeneous reactions in the reaction cell,11J2J4and (d) possible loss of reactant in the mixtures while in storage.14 The quantity observed in these experiments is scattered photon count as a function of channel. The multichannel analyzer time base is controlled by a highly accurate crystal oscillator. Thus, the temporal profile of photon counts is obtained with high accuracy. Since photon counts are proportional to the lower state concentration, given the qualifications noted above, the experiment yields a con-
0 1979 American Chemical Society
Recombination-Dissociation Theory for NOp and NOCl
centration profile. Generally this profile is logarithmic with time because both diffusional loss rates and depletion by reactant (if reactant is much greater than observed species concentration) are fiirst-order processes. Therefore, the only diagnostic possibility in these experiments is the quality of the first-order signal decay for an experimental mixture with some easily variable experimental conditions. These conditions are (1) flow rate of the experimental mixture through the cell, ( 2 ) photoflash intensity, (3) photoflash wavelength, and (4) use of more than one source molecule whiich gives the same observed species on photolysis. The general procedures which are adopted are to first observe whether the signal decay is truly logarithmic under some given initial experimental conditions. If curvature in such plots is noted then complications such as (a) or (b) above are indicated. Systematic experiments with the four diagnostic tools are then designed in order to eliminate the complication. When the set of conditions is found which gives first-order decay, experiments with variable reactant concentration are then carried out with the allowed variations of flow rate, photoflash intensity, photoflash wavelength, and oft entimes, source molecule. If decay constants are not linearly proportional to reactant concentration then complications (c) and/or (d) above may be indicated. Usually decomposition in the reaction cell can be diagnosed by flow rate variation. When loss of reactant in storage or in the reaction cell is indicated, we have relied on both chemical analysis14 or spectrophotometric analy&11J2 of the gas mixture both before and after it flows through the cell. Such procedures decrease the precision in reactant concentration determination, and we have accordingly increased quoted error in a fairly subjective fashion. The most important errors, however, arise from random sources. Signal decay is always observed to decrease to a background level which results from scattered resonance radiation from the walls of the reaction cell. Thus, the significant quantity in any channel is the accumulated counts, N , minus the accumulated counts at long times, NB (background). I f the count rate is not too large then Gaussian statistics hold,15 and the standard deviation is the square root of the accumulated counts; i.e., fN1I2and fNB1/’. It is obvious that for small values of N - NBnear to the end of species decay that errors can be very large since the error in the difference is approximately f(W2 + NB”’). We accordingly use a weighted linear leastsquares analysis of the logarithm of N - NBvs. channel. We randomize these effects even further by analyzing data from a given experiment four to five times from different starting and stopping channels. Standard deviations from the mean of these analyses are either similar to those for each analysis or greater, and we always choose the greater value. We additionally generate a standard deviation from the mean from repeat experiments under similar experimental conditions. Depending on the system, these values range from f 2 to 3:10%, and therefore dominate in the error analysis. Rale constants are calculated from differences in decay constants with and without reactant. Since the diffusional decay constants (without reactant) are typically less than 10% of those obtained when reactant is present and have similar uncertainties, the errors in the differences are only slightly increased. The evaluation of the accuracy of FP-RF rate constant determinations can certainly be carried out objectively. If systematic and random errors are considered, and all systematic factors have been explored and show no complications, then the objective conclusion is that rate constants can be measured with standard deviations
The Journal of P h y s m l Chemistry, Vol. 83, No. 1, 7979
11
ranging from &3to A1270 from the mean. We reach this conclusion essentially from arguments based on experimental repeatibility. Of course the number of repeat determinations is never sufficient to indicate Gaussian statistics so that the usual confidence limit on one, two, or three standard deviations has to be accordingly decreased, and because of the limited set, confidence evaluation becomes quite subjective. Another subjective consideration is the assumption that premi,xtures are indefinitely stable. One can envision that the heterogeneous processes of physical and chemical adsorption could alter the gas cornposition of premixtures. Even though there is an extensive literature on both the time16 and ~ t e a d y - s t a t e dependence l~ of such processes, these considerations are still subjective since there are little or no data available for the ternary systems which have been used. Our procedural choice is to trust the general conclusions of the theory. We therefore always use pressures well below the saturation pressure, and we use storage bulbs with low surface-to-volume ratios. When chemical intuition suggests a potential complication, we make provision to analyze gaseous mixtures by independent methods. Lastly, we again rely on repeatibility of results with the decreas,ing pressure of the storage gas during a set of experiments as the final criterion. In conclusion subjective estimates of error are often unquantitative. ‘We suggest that scientific reporting should have objectivity as the most important goal. We have accordingly chosen to emphasize only objective estimates of error in past reports from this laboratory, and this emphasis will be continued in the future. In part I1 of this paper, results for the recombination reactions 0 + N O + M and C1+ NO M are given and are discussed from a theoretical point of view. These reactions have been studied in this laboratory by the FP-RF technique as a function of temperature. No emphasis has been given in part I to temperature measurement since t he techniques are well known. We point out that even though the flash photolysis technique is used, the extent of phlotolysis in this experiment is small, and local heating effects are entirely negligible.
+
11. Recombinaition-Dissociation Theory for NOz
and NOCl Recombination Results. Results for the 0 + NO + M and C1 + NO -k M reactions18-20were obtained in our laboratory by the FP-RF technique. A t a given total pressure and temperature, decay constants of atoms were measured as a function of [NO]. Linear least-squares analysis yielded an apparent bimolecular rate constant as the slope and the first-order diffusional rate constant as the positive intercept. The set of experiments was repeated as a function of total pressure but still at constant temperature. The resultant apparent bimolecular rate constants a t each pressure were then analyzed by linear least-squares methods to yield the termolecular rate constant a t the given temperature. Linear least-squares values of intercepts from such analysis were nearly always within one standard deviation and were always within two standard deviations of zero. The temperature was changed so that approximately equal increments in T1were covered over the available range, and the procedure was repeated giving ultimately the temperature dependence of the termolecular rate constant. Separate determinations were then made with various heat bath gases. The experimlental results are summarized as follows: 0
+ NO + R4
-+
NO2
+M
(217 IT I500 K)
12
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
J. V. Michael and J. H. Lee
TABLE I: Input Data for t h e Recombination-Dissociation Calculationa AH;,
molecule NO NO2
kcal mol-’ 21.46 8.59
0 NOCl
58.989 12.83
c1
28.52
c/k,
mol wt,
w,
K
g mol-’
4.55
190
30.008 46.008
3.57
668
16.000 65.465
cm-’ 1903.6 1357.8 1665.5 756.8
u, A
2.58 3.41 3.85 NZ Data from ref 28 and 29.
htHe = 0.772 h?’ = 2.77
X
htNz= 3.80
X
C1 + NO
X
10-z’T1.s6or (9.01 k 1.16) X exp(594 f 3 3 / T ) 10-27T1.szor (15.5 f 2.0) x exp(584 f 35/77
-
NOCl
htHe = (4.11 f 0.20)
+M
(200 5 T 5 400 K)
htAr= (4.40 =k 0.30) X 10-32 (T= 298 K)
X
All data are expressed in cm6 molecule-z s-l. Dissociation Results. TroeZ1has reviewed earlier shock tube data for the NOz dissociation by Huffman and Davidson,22 Fishburne et al.,23 and Zimetz4 and has combined these data and those obtained in his laboratory. His analysis is accepted by Baulch et aLZ5whose recommendation is used here:
+M
-
NO
+ 0 + M (1400 5 T 5 2400 K)
kdA’ = 1.83 x lo-’ exp(-32‘iOo/T) f 2570 Dissociation experiments with NOCI in N2 have been carried out by Ashmore and BurnettZ6between 473 and 573 K. Shock tube experiments in Ar have also been reported between 890 and 1193 K by Maloney and Palmer.27 Ashmore and Burnett’s results and their estimate of error are accepted here. In order to assess the errors in the Maloney and Palmer results, their data have been analyzed by linear least-squares methods: NOCl M NO + C1+ 1LI
+
hdNz = 3.02
2(1 +. 2 5
2.279(-115)
1
e-hc(lzl.l)/kT)
3 g h c ( i s a . s ) / k T $. e - h ~ ( 2 z 6 . j ) / k T
1
Parentheses denote the power of ten.
htNz = 3.85 X 10-27T-1.91 or (11.8 f 1.0) X exp(532 f 20/T)
NO2
1 2
10.22 119.5 47.6
10-27T1.63 or (10.8 f 1.2) X exp(523 f 3 0 / T )
+M
g
us
35.457
He Ar
a
1800 594.9 331.5
I or molecular units 1.64(-39) 15.40(-117)
X
ZLJ in eq 1 is the Lennard-Jones rate constant for collisions of the vibrationally-rotationally excited species (either NOz* or NOCl*) with heat bath gas, M.31 I t is calculated from t / k and values for the excited species and heat bath gas (Table I) with the combining rules, t I 2 = (t1e2)I/* and gI2= (al a2)/2where T* = hT/clz. Then
+
ZLj =
*
CTI~~Q 2)(T*)(8.~hT/pLjl *(~
Q*(2,2)(T*) are tabulated integrals.29 Though Troe30 has given an analytic expression for Q*(z,2)(Tv), it is more accurate and no more difficult to use interpolated values from the tabulation. The other terms in eq 2 have their usual meanings. p in eq 1 is the vibrational density of states for the molecule a t the threshold energy, Eo. The product, pRT, results from integration of the sum of states of vibrational states (which are assumed to be in a Boltzmann distribution) over the available energy from Eo to infinity. Thus, it is approximate, but for triatomics this approximation is quite accurate. p is calculated by methods given by Whitten and R a b i n ~ v i t c h . ~ ~ Qblb is the vibrational partition function for either NOz or NOC1. It is calculated from the vibration frequencies given in Table I by usual methods. The remaining terms have been derived and discussed by T r ~ e For . ~ ~the present triatomic cases the specific formulae are Fanh
(s F*ot
= (s
-
= [is
-
11/(~ - 3/21]”
(3)
l)!
+ 1/2)! [ ( E ,+ a E , ) / k T ] 3 1 zX 2.1~(E~/hT)~/~
+
exp(-18006/T)
hdA’ = (1,47_+O2$t) x
f
25%
exp(-17510/T)
All data are expressed in cm3 molecule-’ s-’. Theory. The theoretical calculations for the recombination-dissociation reactions follow those recently presented by Troe. The input data from spectroscopic, thermochemical, and transport property sources are given in Table I.28;29The limiting strong collision low pressure rate constant for dissociation is given by Troe30 as
(2)
2.15(E0/hT)”~- 1
[ ( E Of
UE,)/(S
1/2)h?’]
(5)
Fanh is the ratio of anharmonic to harmonic state densities and is a function of the total number of oscillators, s , and the number of Morse oscillators, m, which are coupled to s - m harmonic oscillators. For NOz, s = 3 and m = 3, but for NOCl, s = 3 and m = 2. FE is the energy dependence of the harmonic oscillator density of states, and E‘,,, is the factor which takes into account the rotational contributions. Fintand Fco,, are both unity for the present cases. The terms E, and a in eq 4 and 5 result from the Whitten and R a b i n o v i t ~ hmethod ~~ for evaluating state densities. E , is the zero point energy, and a is given explicitly by these authors. Inspection of eq 2-5 shows that the significant temperature dependence in the preexponential
The Journal o f Physical Chemistry, Vol, 83, No. 1, 1979
Recombination-Dissociation Theory for NO, and NOCl
factor of eq 1 arises from eq 2, eq 5, Qvlb, and T . The strong collision termolecular rate constant, ktM, is given by the product, hdMKeq, where K, is the equilibrium constant for 0 NO = NOz or C1 + N b = NOC1. K,, for both equilibria is tabulated at various temperatures in the JANAF tables.z8 Since values at specific temperatures are necessary in the present calculations, we have used the data in Table I in statistical mechanical calculations, and the comparison of calculated values with the JANAF values for NO2 agrela within 3% for 217 IT I500 K. However, a siimilar comparison for NOCl shows that the calculated values are lower a t 200 and 239 K ( 1/10 and -1/5, respectively) than those from the JANAF tables. We therefore suggest that the tabulation is in error at these low temperatures and have used the present statistical mechanical values here. Equation 1 is then evaluated for the two cases over the same range of temperatures as used for the experimental determinations. The ratio of the experimental value (as taken from analytical representations given in the earlier section) to the calculated value gives P,, the collisional deactivation efficiency factor. The results are presented in Table 11. h," =: kdMKeqis then evaluated at each experimental temperature, and similar ratios are calculated yielding p, from the recombination results. These are also presented in Table [I. Troe has additionally derived a theoretical expression for the temperature dependence of a,
+
N
-(S) is the average energy transferred in the collision
13
system, 0 + NO = NOz, is one of the best known such systems in thermal gas phase kinetics. Inspection of the data for C1 + NO = NOCl shows that they are not as extensive; however there is no experimental reason for considering them to be inaccurate. Specific consideration of FP-RF results, in general, show good agreement on many reactions between laboratories where the same or other techniques have been used. Thus, since the answers are known, any disagreement between theoretical analysis and experiment lhas to be due to inadequacies in theory. p, values greater than one and variability of -(a) may not be due to theoretical inadequacies. However, use of the theory in its present stage of development requires scaling factors on both B, and - ( L E ) . This need for parameterization introduces an unfortunate arbitrariness that requires additional, and perhaps new, experiments to be designed. Also additional theoretical development may be needed. This latter point has supplied the motivation for the modification in Troe's theory that follows. Modification of Troe's Theory f o r Nonlinear Triatomics. The energy transfer mechanism for atom A addition to diatom B is
14 + B e AB* AB*
+M
-
AB
h1,h-1
+M
h2
where h2 is identified with the Lennard-Jones collision rate constant. In the low pressure limit, htM = ZLJ(hl/kl) where hl/h-l is an equilibrium constant, K,,*, between reactants and excited adduct. Consideration of the back reaction and insistence on microscopic reversibility shows that ktM = hdMKeqwhere K, is the equilibrium constant for A B = AB. Thus, hdd = ZLJ(Keq*/Keq). Both K,,* and K,, can be evaluated by usual statistical mechanical metlhods with
+
of vibrationaily excited species with the heat bath gas. For large values of -(X), p, is close to unity since FE is generally also close to one. Inspection of the B, values for 0 + NO in Table I1 shows some unexpected features. P C K 2 exceeds unity a t the three lowest temperatures. This particular case has been more directly studied by llippler et al. at room t e m p e r a t ~ r e . ~ ~ Quenching experiments have suggested a value of PcN2= and 0.5 to 0.6 at 295 K, and, therefore, the calculated value does not agree with this experimental value. Both PCh and PcHe are less than unity, and PCHeapproximately obeys eq 6 with - ( A E ) considered as a constant. However, the low temperature values of P$ can only be reconciled with eq 6 if - ( S i is considered to decrease from 3.1 to 1.3 All terms have their usual meanings in eq 7 and 8. Qvr* is the partition function for the vibrationally-rotationally kcal/mol as temperature increases from 217 to 500 E(. Inclusion of the high temperature dissociation results excited triatomic and is specifically the sum over all vishows the opposite effect, namely, that -( AE) increases bration-rotation states of products of energy dependent from 1.0 to 1.3 kcal/mol as temperature increases. Indegeneracy factors times a Boltzmann factor. The spection of the p, values for C1+ NO in Table 11, though summation (or integration when state densities are large less than unity in all cases, still requires variation of -( 9) enough to be considered continuous) is from the threshold for reconciliation with eq 6. Also note that when Fro,does energy to infinity, and if there is no energy barrier for not change appreciably with temperature; Le., for the high recombination then the threshold energy is Eo as shown temperature dissociation data, that p, becomes essentially in eq 7 and 8. As pointed out by T r ~ ethe , ~vibrational ~ constant. state density, p, varies slowly enough so that the vibrational Whether 0,values in excess of unity and the apparent contribution generates p(Eo)hT exp(-Eo/hT) to a good variability of - ( A E ) are the result of variability in the approximation, especially for triatomics. Thus, h d M here experimental results, is a valid question. Experimental is closely similar to eq 1. The concept is the same, namely, accuracy has to be carefully considered before the present that hdMis dependent on collisions and on the statistical subtle theoretical effects can be properly assessed. The fraction of vibr,stion-rotation phase space as evaluated results for 0 + NO + Ar have recently been reproduced from the properties of the triatomic alone. Troe then within experimental error by Anderson using the flash treats anharmonic and rotational effects separately; photolysis-chemiluniinescent technique.34 Other recomhowever, we propose to calculate Qvr* by the Marcusbination results h a w been recently reviewed34and are in Whitten-Rabinovitch method of vibration-rotation state substantial agreement with those considered here. The summation with all vibrations and rotations contributNOz dissociation results are also in good agreement with ing.35~36 With the same assumption as Troe, Qvr* = pvrone another. I t therefore appears to us that the reversible (Eo)hTexp(-Eo/hT), where
14
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
J. V. Michael and J. H. Lee
R
hl
h
hl
*
W
t
d.
d
w
t0
5
w
G
E
G z"
4
E
0
a a
e 0"
z
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
Recombination--Dissociation Theory for NO2 and NOCl
TABLE 111: Theorebical Results from t h e Modification
molecule NO,
!T,
molecule-*
:K 217 252 298 355 418 500
35.9 27.5 20.7 15.5 12.0 9.1
He
217 252 298 355 418 !SO0
46.4 36.3 27.7 21.0 16.4 12.6
.4r
!117 252 298 :355 418
34.5 26.1 19.2 14.2 10.8 8.1
.M Ig2
!SO0
s-*
2500
NOCl
He N,
298 200 239 ‘298 400
Ar
a
298 r390 ‘350 11000 1050 1100 1150 1193
s-
39.8 63.4 46.6 32.3 20.2
473 500 560 620
5.63(-24) 4.45(-23) 2.11(-21) 4.62(-20) 31.2 2.14(-16) 7.19(-16) 1.75(-15) 3.89(-15) 7.99(-15) 1.54(-14) 2.57(-14)
Pr
0.59 0.59 0.58 0.56 0.54 0.52 (0.56) 0.24 0.24 0.24 0.24 0.24 0.23 (0.24) 0.36 0.36 0.36 0.35 0.34 0.32 (0.351 0.24 0.23 0.23 0.23 0.25 0.26 (0.24) 0.10 0.24 0.24 0.22 0.20 (0.23) 0.16 0.16 0.16 0.16 (0.16) 0.14 0.20 0.20 0.21 0.22 0.23 0.24 0.24 (0.22)
+ (1pw)E,]s-l+(l./Z) [l + /3(0.605)(E,/E0)~/~ X e x p ( - 2 . 4 1 9 ( ~ , / ~ , ) ~ /[r(s ’ ~ ) ] + r/2)FIwLl}(9)
= (5
1
All terms have their usual meanings or have been discussed elsewhere. With the good assumption that the principal moment of inertia product in the excited molecule is equal to that of thie ground state, gAB* = gA,, and uAB* = uAB, k d M and ktM are
ZLJ {[Eo-I- UEz]S-1+(”2)[1 + p(0.605) X (hT)1/2 ( E , / E o ) ~ ex~(-2.419(E~/E,)”~)l ’~ /[QVlbABr(S + r/2)nw,II exp(-Eo/kT) (10)
kdM =
and
--
f /
/
2.0
3.0
5.0
4.0 1O3K/T
FTT-
Figure 1. Comparison of k? for 0 -t NO -t M calculated from eq 11 = 0.56; with experimental values: (0)data for M = N, line is for ( 0 )data for M = He, line is for = 0.24. Data are from ref 18 and 19, p, values are from Table 111.
6,
6,
20
30
1O4K/T
O + NO + A r
e N02+ A r
I\\
0 15
-32
I
\
I
01
I
20
,
I 30
T
1
I
40
‘ I
(-40
50
IO3 K / T
+
+
Flgure 2. Comparison of calculated k? and kdA‘for 0 NO Ar NO2 Ar with experimental data: ( 0 )k Ar is from ref 19, line is from eq 11 with 6 3 = 0.35 (Table 111); (I) k,,A range of experimental data (f25%) from ref 25, line is from eq 10 with the same p, value; Le., 0.35 from the recombination result in Table 111.
*
Parentheses indicate the power of ten.
&(Eo)
I
molecule-’
2.56(-17) 3.50(-16) 2.67(-15) 1.34(-14) 4.94(-14) 1.45(-13)
1500 1’700 1900 2100 2300
I
15
+
htM = ZLJ(172AB/ImAmB)3’2(gAB/gAgB)((111213)1’2cBh2)/ (1BcAB(kT)“’2)([Eo+ U E , ] ~ - ~ + 1 ( ~+/ ~p(0.605) )[ X (E,/JE,)~’~ exp(-2.419(EO/E,)ll4)]/ [Qvlb$(s + r/2)naLl\(11)
The data in Table I have been used in eq 10 and 11 to predict for both the NOz and NOCl systems the strong collision values of k d M and htM. In both systems the values of s, the number of oscillators, and r, the number of rotations, are three. The results are shown in Table I11 along with 0,values dlerived from the analytical representations of experimental data given earlier. Where temperature has been varied, the inferred 0,values are well represented by constants for both recombination and dissociation data.
16
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
and therefore their meaning may be simplified, (b) the temperature dependences for both recombination and dissociation in both systems are accurately predicted, and (c) the absolute BcN2 room temperature value for 0 NO by Hippler et al.33 is consistent with the modification. Lastly, it should be realized that P, values are parameters which bring predictions into correspondence with experimental measurements. Troe30 has documented an empirical correlation with the number of atoms in the heat bath gas which is remarkably invariant for a variety of activated molecules, and this correlation obviously increases the possibility for accurate predictions. Nevertheless, there is as yet no satisfactory theory for their absolute estimation.
103K/T 10
00
-+-30
T
1
+
--35
1 .NOtMLNOCI+ M
V'
/I'
,
I\
'{-
Ar
--
N2'\
- 55 O
L
-
-
-
20
.~
30
40
IO3K / T
+ -+
1
As pointed out previously from the recombination results,*O both systems give relative 3, values of 1,6;1,0:0,7 for Nz:Ar:He, and this apparent uniformity of behavior suggests that the p, values from the recombination results are to be preferred. We have accordingly used the recombination results to illustrate the agreement with actual data in Figures 1-3 and to predict the dissociation rate constants in Figures 2 and 3. The representations of dissociation rate constants in these figures are not actual data but instead are the range of experimental errors quoted earlier. If these errors are realistic then the prediction for NOz Ar is uniformly high by 15% from the maximum allowable value while that for NOCl f N2 is high by -10%. The NOCl Ar prediction is well within the error limits. Several points can be made which are common to both the Troe theory and the modification. High experimental accuracy is needed in order to test either method. Specifically with regard to the NOz and NOCl systems, either method shows that the systems are microscopically reversible. If p, values are arbitrarily taken to be 1.0, the maximum, then rate constant predictions for recombination and dissociation will generally be overestimates by as much as 10 for any bath gas where the energy transfer mechanism is dominating. Thus, any experimental result which indicates 3, 1 3 almost certainly indicates the predominance of a bound complex nechanism. Another common feature in recombination predictions is that a significant part of the temperature dependence arises from electronic degeneracy factors. The two methods do not predict the same behavior of B, with temperature P, being T dependent with Troe's method and being constant in the modification. To be sure, the modification is not nearly as detailed as Troe's theory and includes no anharmonicity or other correction factors (eq 3 and 4). We point out that these factors are essentially constants and would tend to cause /3, overestimates in the modification. However, the modification has some attractive features: (a) 8, values are constants,
-
+
+
-
Acknowledgment. We acknowledge support by NASA under Grant No. NSG 5173 to Catholic University of America. We also express gratitude to Dr. L. J. Stief for helpful discussions and for support.
50
Figure 3. Comparison of calculated ktMand kdMfor CI NO M * NOCl M with experimental data ( 0 )kY2 is from ref 20, line is from eq 11 with 8,= 0.23 (Table 111). (I) kdNz range of experimental data (f25%) is fro i ref 26, line IS from eq 10 with the s a q e2:3/ value; i.e , 0 23 from [he recombination result in Table 111; (I) Ar is from ref 20 and infers P / = 0 14 (Table 111). k: range of exp. rimental data is from linear least-squares analysis of data in ref 27, line IS from eq 10 with p, = 0.14
+
J. V. Michael and J. H. Lee
References and Notes (a) W. Braun and M. Lenzi, Discuss. Faraday Sac., 44, 252 (1967); (b) M. J. Kuryio, N. C. Peterson, and W. Braun, J . Chem. Phys., 53, 2776 (1970);(c) M. J. Kuryio, Chem. Phys. Lett., 49, 467 (1977): (d) R. G. Manning and M. J. Kurylo, J . Phys. Chem., E l , , 291 (1977). R. T. Watson, G. Machado, S ,Fischer, and D. D. Davis, J , Chem. D. D. Davis, G. Machado, B. Conaway, Y. Phys., 65, 2126 (1976); Oh, and R. Watson, ibid., 65, 1268 (1976). (a) J. E. Davenport, B. A. Ridley, L. J. Stief, and K. H. Weige, d . Chem. Phys., 57, 520 (1972);(b) G. Hancock, W. Lange, M. Lenzi, and K. H. Welge, Chem. Phys. Lett., 33, 168 (1975). F. Stuhi and H. Niki, J . Chem. Phys., 57, 3671, 3677 (1972). R. B. Klemm and L. J. Stief, J . Chem. Phys., 61, 4900 (1974). T. G. Slanger and G. Black, J. Chem. Phys., 53, 3717,3722 (1970). Heavy atoms: (a) R. J. Donovan, H. M. Gillespie, and R . H. Strain, J . Chem. Soc., Faraday Trans. 2 , 73, 1553 (1977);(b) D. Husain, L. Krause, and N. K. H. Slater, ibid., 73, 1668 (1977). R. A. Perry, R. Atkinson, and J. N. Pitts, Jr., J . Chem. Phys., 64. 1618,3237 (1976);J . Phys. Chem., 81, 296 (1977). R. B. Klemm, paper presented at the Fall Meeting, Eastern States Section, The Combustion Institute, Hartford, Conn., Nov, 1977. A. C. G. Mitchell and M. W. Zemansky, "Resonance Radiation and Excited Atoms", Cambridge University Press, Cambridge, 1934. J. H. Lee, J. V. Michael, W. A. Payne, and L. J. Stief, J. Chem. Phys., 69, 350 (1978). J. V. Michael, J. H. Lee, W. A. Payne, and L. J. Stief, J. Chem. Phys., 68, 4093 (1978). D. A. Whytock, R. B. Timmons, J. H. Lee, J. V. Michael, W. A. Payne, and L. J. Stief. J . Chem. Phvs.. 65. 2052 (19761. J. V. Michael, D. A. Whytock, 2, H. Lee, W. A: Payne, and L. J. Stief, J . Chem. Phys., 67, 3533 (1977). R L. Kiobuchar, J. J. Ahumada, J. V. Michael, and P. J. Karol, Rev. Sci, Instrum., 45, 1073 (1974). J. Crank. "The Mathematics of Diffusion", Oxford University Press, London, 1956. See, for example, K. J. Laidier, "Chemical Kinetics", McGraw-Hili, New York, N.Y., 1965. D. A. Whytock, J, V. Michael, and W. A. Payne, Chem. Phys. Lett., 42, 466 (1976). J. V. Michael, D. A. Whytock, and W. A. Payne, J . Chem. Phys., 65, 4830 (1976). J. H. Lee, J. V. Michael, W. A. Payne, and L. J. Stief, J. Chem. Phys., 68, 5410 (1978). J. Troe, Ber. Bunsenges. Phys. Chem., 73, 144 (1969). R. E. Huffman and N. Davidson, J. Am. Chem. Sac., 81,231 1 (1959). E. S.Fishburne, D. M. Bergbauer, and R. Edse, J. Chem. Phys., 43,
1847 (1965). E. Zimet, J . Chem. Phys., 53, 515 (1970). D. L. Baulch, D. D. Drysdale, D. G. Horne, and A. C. Lloyd, "Evaluated Kinetic Data for High Temperature Reactions", Vol. 2,Butterworths, London, 1973. P. G. Ashmore and M. G. Burnett, Trans. faraday Sac., 58, 1801
(1962). K. K. Maloney and H. B. Palmer, Int. J. Chem. Kinet., 5, 1023 (1973). "JANAF Thermochemical Tables", Natl. Stand. Ref. Data Ser., Natl. Bur. Stand., No. 37 (1971). J. 0.Hirschfelder, C. F. Curtiss, and R. B. Bird, "Molecular Theory of Gases and Liquids", Wiley, New York, N.Y., 1964. J. Troe, J . Chem. Phys., 66, 4758 (1977);M. Quack and J. Troe, Gas Kinet. Energy Transfer, 2, 175 (1977). S.C.Chan, J. T. Bryant, L. D. Spicer, and B. S.Rabinovitch, J . Phys. Chem., 74, 2058 (1970).
Recombination-Dissociation Theory for NOp and NOCl (32) G. Z.Whitten and B. S. Rabinovitch, J. Chem. Phys., 38, 2466 (1963). (33) H. Hippler, C. Schippert, and J. Troe, Int. J . Chem. Kinet. Symp 7 , 27 (1975). (34) L. Anderson, paper presented at the 13th Informal Photochemistry Conference, Clearwater Beach, Fla., Jan, 1978; R. F. Hampson, Jr., and D. Garvin, NaN. Bur. Stand. Spec. Pub/., No. 513 (1978). (35) G. Z. Whitten and B. S. Rabinovitch, J. Chem. Phys., 41, 1883 (1964). (36) R. A. Marcus, J . Chem. Phys., 20, 359 (1952).
Discussion W. H. DUEWER(Lawrence Livermore Laboratory). In the theoretical estimation of low pressure dissociation or recombination rates, the density of molecular vibrational energy levels at threshold appears as a multiplicative factor. These energy level densities can be calculated in various ways. Using experimental values for the first and second anharmonic constants and a direct count procedure to evaluate the density of states for several triatomic molecules near threshold, I have found that the state densities from the calculations made using coupled anharmonic oscillators exceed those for harmonic oscillator models by a factor that often exceeds the correction factors that can be deduced from a Morse oscillator moclel (such as those estimated by Professor Troe) by an amount comparable to the correction factor derived from the Morse oscillator model. Use of a coupled Morse oscillator model leads t o state densities near threshold roughly 4 times as large as the harmonic oscillator state densities for HzO, and roughly 2 times as large as harmonic oscillator state densities for NOz. Use of such state densities would tend to reduce p values for NOz.
J. TROE(Institut fur Physikalische Chemie der Universitat Gottingen). These accurate low temperature measurements are beautiful and very important since they offer a test for the adequateness of our handling of centrifugal effects and hence of our understanding of the long-range potential. However, I think that the result of your study, namely, the finding of a temperature independent p,., is an artifact which is due to inadequate handling of rotational effects. My older rotational factors Fro,,which you compared with your rotational factors, were a first approximation designed for application to broad temperature ranges. At low temperatures one can do better by using modifications as described in my paper a t this meeting (see also J. Troe, J . Chem. Phqs., in press). Here Fro,is formulated for van der Waals or Morse potentials combined with polyatomic centrifugal potentials. The latter are particularly important. The results of these low temperature correction'$confirm the results of my earlier analysis, namely, that pCis a temperature-dependent function. For the H20system, e.g., in Ar p, decreases from values near 0.3-1 at 300 K to 0.01 near 6000 K. Also for the ClNO and NOz systems treated by you, p , turns out to decay with temperature in agreement with our knowledge of (LE). (It is important to note, as you pointed out, that the equilibrium constants for the C1 NO * ClNO equilibrium in the JANAF tables at 200- 400 K are in error by a factor up to 10 which can be very misleading and which was responsible for the apparently
+
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
17
low p, values in the table of Troe ( J . Chem. Phys., 66, 4758 (1977)).l (1) Editor's Note (D. Garvin): The error in the equilibrium constants for C1+ NO = ClNO mentioned by Dr. Troe is due to an incorrect value for H" - Hoz98at 200 K for ClNO in JANAF Thermochemical Tables (Natl. Stand. K e f . Datu Ser., Nutl. Bur. Stand., No. 37). ,4corrected set of JANAF thermal functions for ClNO has been published by Chase et al. (J.Phys. Chem. Ref. Data, 4, 1-175 (1975)).
J. V. MICHAEL.Both of the comments refer to techniques of state counting at the threshold energy. In the full paper, we chose to use the Marcus-Whitten-Rabinovitch method, and admittedly this method is by no means unique even though it is a popular method of choice. As pointed out in the full paper, inclusion of anharmonicity factors of the type noted by Dr. Duewer will tend to reduce p, values by a constant factor which is temperature independent. Thus, the predicted temperature dependences of the rate constants remain unaffected. The comment by Professor Troe properly calls attention to the differences in approach between his method and modification presented in the full paper. At issue is the treatment of rotation effects. Both Troe and we suppose that a common theory reversibly connects recombination with dissociation, and collisional deactivation effects should therefore be connected in both experiments. Troe, in his comment, suggests the importance of knowing with high accuracy the low temperature recombination rate constants in order to understand centrifugal effects in the NOz and NOCl thermal systems. If accurate recombination results are important in this demonstration then we conclude that centrifugal effects on these two systems near the threshold energy have not yet been properly demonstrated because the alternative modification yields constants for the p, factors. This conclusion may be true of all thermal systems near the threshold; however, we choose not to induce this general claim merely from the two systems of interest even though the data for both recombination and dissociation are quite accurate. As we point out, there is a critical need for directly measuring the temperature dependence of collisional deactivation efficiency factors for thermal systems near the threshold energy. In both of the formalisms given in the paper, 8, is in actuality an experimental quantity. We show explicitly that Troe's theory, as it now stands, will require at least two scaling parameters whereas the modification only requires one. It therefore appears to us that the real issue is where to place the unknownness in the theory, unknownness in our opinion being directly proportional to the number of parameters needed in order to bring theory into correspondence with experimental answers. It is obvious that we believe that little insight is gained by nesting the unknownness further into the theory in quantities that have no experimental basis. Questions of' adequacy of treatment are therefore irrelevant at the present time. However, they may become relevant in the future when either a first principles theory for the absolute calculation of p, and/or direct experimental measurements of the temperature dependence of p , in thermal systems near the threshold energy become available.
