18
Reinhard Zellner
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
Non-Arrhenius Behavior in Bimolecular Reactions of the OH Radical Reinhard Zellner Institut fur Physikalische Chemie der Universitat Gottingen, 3400 Gotfingen, West Germany (Received June 28, 1978)
Evidence is presented that rate measurements over a very wide temperature range (30G2000 K) for some reactions involving the OH radical show distinct non-Arrhenius behavior. An attempt to fit the available data empirically leads to k , = 106191'213exp(-1233 K I T ) cm3/mol.s and k z = 10s.0T16exp(-1660K/T) cm3/mol.s for the reactions with a finite activation barrier around room temperature ((1)OH + CH, CH3 + HzO and (2) OH Hz H20 + H) and k3 = exp(24.98 + 9.2 X 10-4T)cm3/mol.s and k4 = exp(27.1 + 1.5 X 10-3T) cm3/mol.s for reactions without substantial temperature dependence around 300 K ((3) OH + CO C02 + H and (4) OH OH HzO+ 0). The extent of non-Arrhenius behavior may plausibly be explained by means of the total reactive cross section, for which various forms are derived. A more direct explanation, however, is provided from measurements of the vibrational rate enhancement. It is shown that for OH + H2,vibrational excitation of hydrogen produces sufficient enhancement of the rate constant k O H + H 2 ( u = l ) / k O H + H 2 ( ~ = o=~ 1.5 (!,,~)102 at 298 K to account for the Arrhenius graph curvature of the overall thermal rate constant.
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Introduction
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0022-3654/79/2083-00 18$01.OO/O
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until recently been very scarce. A number of papers have been published in which non-Arrhenius behavior, or the temperature dependence of the Arrhenius activation energy E A ( T ) is , discussed in general terms of classical theories. These include EA(T ) expressions as given by the classical limits of transition state theory5Sz0as well as extensions of the TolmanZ2 approach to collision theory assuming various forms of excitation f ~ n c t i o n s The . ~ ~conclusive ~~~ result from these papers is that both activated complex and collision theory can produce non-Arrhenius behavior (even to an extent beyond what is required from recent experiments) by the temperature dependence of the partition function ratio (QIAB/QAQB) and the detailed form of the energy dependent total reactive cross section a ( E ) , respectively. Unfortunately, both approaches remain of qualitative nature. Neither a set of activated complex parameters, nor the shape of the excitation function (in the vicinity of the threshold energy for a reaction occurring a t thermal energies) can as yet be tested directly.38 However, collision theory provides the dynamical concept to account for internal excitation of the reactants which can be tested by measurement of the state specific rate constant hi. In terms of Arrhenius graph curvature such measurements can help to decide whether in a thermal system at elevated temperatures (where significant populations of vibrationally excited states occur) an important contribution of the total reactive flux occurs via the excited states. Such experiments circumvent the problem of having to know the energy dependence of the state specific cross sections and therefore provide a direct test of non-Arrhenius behavior in a Maxwell-Boltzmann system. The present paper is divided into three sections. First, the evidence for non-Arrhenius behavior of some bimolecular reactions involving the OH radical (OH + CH,, Hp, CO, OH) is given. The temperature range is above 300 K, where t h e contribution of tunneling to the observed curvature is expected to be small. Second, the experimental temperature dependence k (7') is transformed into the energy dependent total reactive cross section a(E). Although this step only provides a plausible interpretation, it is included in order to show what extent of deviation from the simple "line-of-centers" model may be operative and whether simple dynamical features of the reaction might be deduced. In the last section, results of an investigation of the vibrational enhancement in one of the above reactions (OH + H2(u = 1))are presented and the implications of these data to Arrhenius graph curvature of the thermal OH H2 reaction are derived.
Recent experiments covering very wide temperature ranges have shown that the rate constants of elementary bimolecular reactions can often no longer be represented by a simple Arrhenius expression. Instead, three parameter expressions of the form k ( T ) = AT" exp(-Eo/RT) with a positive T exponent n, Le., with concave upward curvature of the Arrhenius graph, have to be devised as empirical fits for the experimental data. Non-Arrhenius behavior is expected in theory; experimental evidence, however, has for a long time been obscured by the error limits of earlier gas kinetic measurements. Mainly as a consequence of improved techniques and sufficient extension of the temperature range, Arrhenius graph curvature has been detected. Elementary gas reactions that have now been identified t o exhibit non-Arrhenius behavior involve the reactants H (H(D) + Hp,lH + CH,, C2H6)j20 (0 + Hz),3OH (OH C0,4OH Hz,4C,5 OH CH4,6,7OH OH),',' CH3 (CH3 + HZ,"," CH3 + C2H6,2,12 CH3 + C3H8,C4HI0),l3c1 (c1+ CH414),and metal atoms (A1 + C0215). All of these reactions, with the exception of OH + CO and A1 + C02, are H-atom transfers. It is premature to say, however, whether this phenomenological statement is indicative to the observation of curved Arrhenius plots. The nature of H-atom transfer is certainly largely responsible for the Arrhenius graph curvature observed at lower t e m p e r a t u r e ~ ' ~where ~ J ~ large proportions of the reactive flux can be caused by tunneling through the potential barrier. l6 The present work is concerned with non-Arrhenius behavior of some bimolecular reactions of the OH radical, which have been the subject of study in this laboratory in recent years. The stimulus for this work came from two different sources. First, existing rate data in the low and high temperature range can usually not be encompassed by one single Arrhenius expression, and therefore the extent of non-Arrhenius behavior has t o be determined. This is of great importance to the modeling of combustion systems, in particular when rate data are needed in a temperature range where they have not been directly measured, but must be obtained by extrapolation. Second, non-Arrhenius behavior provides a tool to test elementary bimolecular rate theory. Although deviation from a simple Arrhenius behavior is predicted by all theoretical approaches, whether they are based on activated complex17 or collision theory18-20 concepts, t h e experimentally available information on which a "test" can be based has
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G
1979 American Chemical
Society
The Journal of Physical Chemistry, Vol. 83,No. 1, 1979 19
Non-Arrhenius 13ehavior in Bimolecular Reactions
( i ) Experimental Ehidence for Arrhenius Graph Curvature Some reactions of the OH radical have now been studied over a sufficiently wide range on the 1 / T scale that the extent of non-Arrhenius behavior may be determined. Progress in this direction has largely been achieved by the application of more and more refined kinetic techniques, in particular at temperatures between 300 and 500 K where flow system and static flash photolysis studies with sufficient OH detection sensitivity have provided the bulk of accurate rate data. IRate measurements at high temperatures are notoriously more difficult, and rate data derived from shock tube experiments or from flames may contain larger inaccuracies, which are mainly due to the difficulties of isolating an individual process in a complex reaction system and also to a generally strongly reduced detection sensitivity for free radicals. In view of this we have attempted in recent experiments on OH reactions to extend the flash photolysis/resonance absorption (FPRA) technique to higher temperatures. Measurements under static conditions (using a furnace to heat the gas mixture) can conveniently be made up to 900 K (OH C0;4e OH + CH:). To obtain even higher temperatures (-1500 K), Ernst et al.' have recently combined the FPRA technique with a conventional shock tube. In this system initiation of the reaction is only by flash photolys,is of H,O; thermal dissociation of either H 2 0 or the reactant (i.e., CH,) does not yet occur and the system remains kinetically simpler than one initiated by branched chain explosion. A high degree of isolation of a certain reaction can also be obtained by thermal initiation using specific radical sources. In this way, Ernst et al.9 have recently studied the disproportionation reaction OH OH H,O t 0 between 1200 and 1800 K using HNOBas an OH source. This work has confirmed the previously suggesteds non-Arrhenius behavior for this reaction, although its full extent is still a matter of speculation due to the surprisingly badly defined rate constant a t room temperature, where the lowest and highest value differ Iby a factor of 3, and a complete absence of data between 300 and 1200 K. Figures 1-4 are Arrhenius graph presentations of available rate data fiDr the reactions
+
+
10
o L
I
+ CH4
-
+ H20 HzO + H OH + Hz OH + CO +CO2 + H CH,
+
OH
+ OH +HzO + 0
(1)
+ CH,)
hJOH
+ H,)
4
[K-'I
1
..
L
JVS DL 52 WH GMO EHW
BBB
0
knkins eta1 1967
Drpdole, Lloyd 1970 Smith. Zellwr 1973 Weslsnbwg, d e b s 1973 Gardiner el a1 1974 Eberlus el 01 1971 Brabbs el 01, 1971 accepted d a b 01 3W K
LEEDS Baulch el 01 (e~oIulion)1972
X
-
v
01
10 -
9-
(2) (3)
(4)
including recent results from our laboratory. In no case can the low and high temperature data be encompassed by one single Arrhenius expression. Rate constants around 1000 K fall above the line extrapolated from the low temperature data by at least a factor of 2, and increasing to a factor of 5-7 in the 2000-K region, which even in unfavorable cases is most likely to be beyond experimental error. We have therefore attempted to represent these reactions by concavely curved "Arrhenius" lines as smoothed fits to the available data. For those reactions which have a finite iirrhenius activation energy around room temperature (01H + CH,, H2) we have chosen a AT,. exp(-E/RT) fit with the results
hl(OH
3
lo3/1
Figure 1. Arrhenius graph for reaction 1. References to the literature data may be found in ref 7. The heavy line is the smoothed interpolation corresponding to the expression k , = l o 6 " T 2 l3exp(-1233KIT) cm3/mo~-s.
-
OH
2
= 10"l 9 T 2 . I 3 exp(-1233K/T) cm3/mol.s = 10s.0T1.6 exp(-1660 K / T ) cm3/mol.s
Figure 2. Arrhenius graph for reaction 2. Cited literature references may be found in ref 25 and 26, from where additional references to low temperature data may also be obtained. LEEDS (ref 26) is the earlier consensus expression. The heavy line corresponds to the present evaluation k , = 108.'T1' exp(-1660KIT) cm3/mol.s.
The three parameters A , n, and E are of course strongly coupled and therefore the error limits of these expressions are better given for h itself. They are estimated to be for the CH,(H2) reaction zt20(20)% at room temperature (here all recent measurlements for both reactions overlap within the error limits stated for each reported experiment), *30(40)% a t 500 K, *50(60)% a t 1000 K, and *70(100)% a t 2000 K. For the reactions 3 and 4, a representation of k ( T ) in terms of the above three parameter expression failed to give a satisfactory interpolation of all available data. This is because for these reactions, which show a very small temperature dependence below 500 K, a Tn term provides tolo much temperature dependence below 500 K if n is adjusted to fit the high temperature variation and vice versa. A.n exponential temperature dependence
20
The Journal
of Physical Chemistry,
Vol. 83, No. 1, 1979 c . -
2000 loo0 12.5
I
500 400 I
,
T/K
300 I
,
"
.
k = e x p ( 2 4 98
+
92
T ) crn3 /mol s
Y
v
E'
J S d 110-
DFS
EHW n SN
1
2
4
1 0 ~ [K-'] 1 ~
Figure 3. Arrhenius graph for reaction 3. References to the literature may be found in ref 4d. The heavy line is the smoothed interpolation corresponding to the expression k , = exp(24.98 9.2 X lO-,T)
+
cm3/mo~.s.
(vibrational) ground state reaction. It may be implied, mainly from the low preexponential factors as observed in temperature dependent studies for reactions 1 and 2 around room temperature, that tunneling occurs. There is, however, no direct experimental evidence for this even down to temperatures of 240 K6,2sand its effect will be excluded from further consideration. The increase of k ( T ) at high temperatures above the low temperature Arrhenius line could then be understood as an increase of total reactive flux caused by reaction occurring from the excited states. (ii) k(T ) - u ( E )Considerations
SZ
3
Reinhard Zellner
In collision theory the thermal rate constant k ( T ) is connected with the energy dependent total reactive cross section g(E) vialg
h ( T ) = ( ~ p ) - O ~ ( 2 / h r i p )Jl m ~ E u ( E ) exp(-E/hT) d E (I) 0
where E and p are the relative translational energy and the reduced mass, respectively. Only for the so-called "line-of-centers'' model ( u ( E ) [l - ( E O / E ) ]does ) the above equation yield h ( T ) = A P exp(-E,/RT), with a preexponential temperature dependence according to the collision number. It has therefore been s u g g e ~ t e d ~that ~J~*~~ a behavior near threshold (E,) of u ( E ) other than linear (i.e., concave upward) may be responsible for strong Arrhenius graph curvature in thermal systems. Although this consideration assumes one single excitation function for all internal slates and only provides a plausible explanation (since, as has been pointed out above, there is no direct experimental proof) it shall still be presented briefly. The effect of vibrational excitation of the reactants is dealt with in section iii. The total reactive cross section a ( E ) may be obtained from eq I using an inverse Laplace transform.39 This can easily be done for special forms of h ( T )using transform tables.29The general case, however, employing the method of steepest descent, has been treated by Eyring and cow o r k e r ~ Following .~~ their approach, a rate constant of the form k ( T ) = AT" exp(-E,/RT) with A in units of cm3/ mol-s transforms to (hB is the Boltzmann constant and N is Avogadro's number):
-
I
13.0
1
RG
Rowlins, t a r d i n s 1976
I
0
20
I O
3X
103/T [K-']
Figure 4. Arrhenius graph for reaction 4. References are as follows RG,' EWZ,' AHWW," LEEDS consensus expression as evaluated by Baulch et a1.26 Further references for room temperature data points may be found in ref 9. The heavy line corresponds to the present evaluation k, = exp(27.1 1.5 X 10'3T) cm3/mol.s. The alternative representation k , = lOgi7T' cm3/mol.s is indicated by the dashed curve.
+
of the form k = exp(a + b T ) is more appropriate, and we deduce
h3(OH + CO) = exp(24.98 + 9.2
X 10-4T) cm3/mol-s
h4(0H + OH) = exp(27.1 + 1.5 X 10-3T) cm3/mol-s
+
For OH CO this expression has previously been derived by Baulch and D r y ~ d a l eand ~ ~by Steinert and Ze1lnerde and there is little doubt as to its reliability, even up to very high temperatures. For OH OH, however, the above expressionz7should be used with caution, due to the scatter of data at room temperature and the lack of a temperature dependence. If in the data evaluation the result of Albers et aLZ8on the reverse 0 + H 2 0 reaction is also included, then the change of the temperature dependence is discm3/mol-s tinctly smoother and h = AT" with A = and n = 1.14 is an equally appropriate representation. Both expressions are included in Figure 4. Clearly, more data are greatly needed. Measurements of the temperature dependence in the range 250-400 K are presently under way in this laboratory. The Arrhenius graphs of Figures 1-4 suggest that h ( T ) up to about 800 K would be quite well represented by one single Arrhenius expression. This could be (but not necessarily has to be) interpreted corresponding to the fully
+
u(E) =
(-i>~',~ 4NhBn
exp(n + 1.5) 1 -(E (n 1.5)f*+1E
+
-
(11)
This function is concave upward near the threshold (E,) for all n > 0.5. Application of this equation to OH + CH4, for example, where h ( T ) = 106.191"?.13exp(k1233 K / T ) cm3/mol.s, yields g(~)OH+CH,=
1
9.33 x 1 0 1 9 4 ~ 1.7 x 10-13)263 E
in units of A2 and E in ergs. This relation is presented graphically in Figure 5, which shows that o(E) indeed differs strongly from the simple "line-of-centers" model. Although the form of u(E) may represent nature reasonable well in the vicinity of Eo (which is most important in the present context of non-Arrhenius behavior) it certainly fails a t high collision energies where a(E) tends toward infinity. This is of course a consequence of the functional form of h ( T ) AT" which itself goes toward a. A correction for this artifact can be infiqity for T introduced by keeping the functional form of k ( T ) itself finite, which is most easily done by representing k ( T ) by means of a sum of exponential terms h ( T ) = LA,
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The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 21
Non-Arrhenius Behavior in Bimolecular Reactions
/
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103/T [ K - ' ]
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Figure 5. k ( T)-u(E) correlation for OH CH4 CH3 H,O. Two forms of representation are given: (a) a direct transform of k = AT". exp(-EIRT); and (b) a Iransform of a "sum of exponential terms" substitute expression (see text).
exp(-EJRT). The effect of this representation on r ( E ) is shown on the right-hand side of Figure 5 . We obtain a kind of a step function, each step being the result of the transform of an individual term of the exponential h ( T ) representation, r.,(E) ( l / E ) ( E- E J o 5and therefore r ( E ) = x u L ( E ) .The result could easily be smoothed out by introducing a larger number of terms. Certainly the step form behavior is not meant to be physically real, Le., to represent the energy dependence of detailed specific cross sections corresponding to specific internal states of the reactants. The a ( E )representation will qualitatively be similar for all reactions having the same h ( T )functional form, Le., OH + H2 will be similar to OH CH4. Other forms of k ( T ) , however, will have different a ( E ) representations. As a further example, the reaction OH CO COS H (3) shall be presented. A rate constant of the form h ( T ) = exp(a bT) cm3/ mo1.s transforms toz9
+
+
-
+
+
+
-
E /ergs, mo~ecu~e-l
+
Figure 6. k( T)-a(E) correlation for OH CO C02 H, Two forms of representation are given: (a) a direct transform of k = exp(a 6T); and (b) a transform of a sum of exponential terms. (The third term corresponding to line 3 on the left-hand side is not contained on the right-hand side since its energy region is off the scale.)
the ground state reactants, upward curvature of the Arrhenius graph may result. An account of the participation of internal states in terms of state specific cross sections was given by F1'iason and Hir~chfe1der.l~A still simpler formulation, which circumvents the energy dependence of specific cross sections, is provided in terms of specific rate constants. We may write for a reaction A + I3 at thermal equilibrium
, 1
htherm
= -[kU=OAB + QAQB
C kuABexp(-EuAB/RT)1
(iii) Non-Arrhenius Behavior as a Result of Vibrational Rate Enhancement (OH H,( v = l ) )
+
The aforementioned cross section considerations have neglected the internal structure of molecules. An increasing amount of experimental evidence on measurements of state specific rate constants k:' suggests, however, that these may vary strongly (usually increase) upon internal (usually vibrational) excitation. If therefore, in a thermalized system, and under conditions where vibrational relaxation is fast, the specific rates of reactants in vibrationally excited states may be larger than those for
(IV)
till
Here hL=OAB is the rate constant for the fully ground state reaction and the sum is to be taken over all Boltzmann weighted rate constants from specific vibrational states for either reactant A and Usually only the first excited vibrational states will be important. For the rate constant h,=l we may assume a vibrational enhancement according
to h,=l =: ALZlexp[-(Eo - aE,=,)/RT]
where f = b / k B . For OH + CO with h ( T ) = exp(24.98 + 9.2 x 10-4T) cm3/mol.s one obtains correspondingly 1 sinh (5.16 X 106E05) ( T ( E ) ~=~6.7 + ,X ~ ~ E in units of A2 and E: in ergs. This relation is presented graphically in Figurie 6. It represents a a(E) curve with a shallow minimum around 1 X ergs (h 1.5 kcal/mol) relative kinetic energy. Its unrealistic feature of going to infinity as E tends t o zero or infinity (as a consequence of h ( T ) = exp(a + bT)) can again be removed by an exponential sum representation of h ( T )with the first term having a very low (0.2 kcal/mol) activation energy. It is important to note that this representation also predicts a decline of t r ( E ) over the energy region 0.5-4 kcal/mol, which is the range of relative kinetic energy of the reactants up to 1500 K.
+
(V)
where CY is the fraction of the vibrational energy E,=l that is consumed selectively to overcome the barrier Eo. For the effect that eq IV and V have on Arrhenius graph curvature of hthprmit is important to determine whether the vibrational rate enhancement ( h ( u = l ) > h(u=O)) is primarily an effiect of an enlarged preexponential factor (A,=,)or of a reduced barrier.j Only in the first case can substantial curvature be expected. In the latter case the reduced barrier is exactly (CY = 1) or partly ( a < 1) cancelled by the Boltzmann factor.33 Rate constant measurements for reactions of vibrationally excited hydroxyl radicals with H,, CH,, and CO have recently been carried out by Spencer et al.34 The observed enhancements were small (hL=l/hL=o < 2, < 4, and < 2 for reactions of OH(u = 1) with H2, CH4, and CO, respectively) and are clearly insufficient to account for the observed non-Arrhenius behavior. In simple terms this could be interpreted as having excited the "wrong" bond. Clearly, in reactions in which OH abstracts, it is the bond to be broken and not the 0-H bond which changes strongly during the course of the reaction. Therefore, excitation of the reactant -H bond is expected to produce larger enhancements. This suggestion is strongly supported by result s from a trajectory c a l ~ u l a t i o nof~ the ~ ABC + D. four-atom exchange reaction AB + CD In view of thilg we have decided to investigate the reaction between OH and vibrationally excited hydrogen (H,(u=l)). First results36from this study as well as their implication regarding non-Arrhenius behavior of the thermal OH + €I2reaction shall be briefly presented. The experimeints were performed in a fast flow system
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22
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979
2000 1000
non-Arrhenius behavior for the thermal rate constant is predicted. The best agreement with the empirical curve through the experimental points is obtained for the larger values of A,=l. The gap around 1000 K could easily be removed by allowing a slightly larger activation energy for the ground state reaction (i.e., assuming that the low temperature activation energy is slightly reduced due to tunneling.)
300 -T/K
500
Reinhard Zellner
Conclusion
1
3
2
4
l@/T[K-']
Figure 7. Arrhenius graph of thermal and state specific rate constant for OH H, (H,(v=l)). The heavy curve is the empirical interpolation of all experimental results (cf. Figure 2). Light curves represent the extent of Arrhenius graph curvature as deduced from measurements on k (OH H2(v=1)) at 300 K and various assumptions for the preexponential factor ( A , = , ) or the fractional use ( a )of the excitation energy to reduce the ground state reaction barrier (see text).
+
+
using excess H,(u=l) and detection of OH by resonance fluorescence. Vibrationally excited hydrogen is produced by passing "cold" H2 over a heated tungsten filament and subsequent "freezing" of the vibrational population. Its concentration is determined by energy transfer to HC1 (added in excess of H2(u=1))and subsequent IR chemiluminescence. Details of the experiment will be given elsewhere.36 From the decay of OH in the presence of H,(u=l) we deduce k,(OH
+ H,(u=l)) = 6(?$)1011cm3/mol.s
a t 298 K, which corresponds to an increase in rate relative t o the ground state reaction of k2(u=l)/k2(u=0) = 1.5(f:t)1O2. Although we do not as yet have any direct proof, the magnitude of the rate constant suggests that it corresponds to the reactive OH + H,(u=l) H20 + H channel rather than to the V-V energy transfer process OH H,(u=l) OH(u=l) H, AE = 589 cm-l, for which a lower rate constant would be e ~ p e c t e d . ~ ; , ~ ~ The temperature dependence of the OH + H2(u=l) reaction is presently under investigation. In the absence of this information we may assume, however, that the total enhancement is partly due to a reduced threshold energy. (That the barrier is not totally removed, although the excitation energy of 11.9 kcal/mol exceeds by far the activation energy of the ground state reaction (EAO 4.4 kcal/mol), is suggested by the fact that h ( u = l ) is smaller than the preexponential factor for the ground state reaction.) For the discussion of non-Arrhenius behavior we therefore assume the following: (1) a likely upper limit of A,=l = 5 X cm3/mol.s (in this case a = 0.03 in eq V); (2) A,=l = 2 X 1014cm3/mol.s ( a = 0.07); and (3) A,=l = 5 X 1013cm3/mol.s ( a = 0.14). The resulting effect on kthermas calculated from eq IV is presented graphically in Figure 7 . It can be seen that for either choice of A,=l and a, the fractional use of the excitation energy, substantial
+
-
-
+
+
-
Evidence has been presented that a number of reactions involving the hydroxyl radical show distinct non-Arrhenius behavior. This can be explained plausibly through special forms of the energy dependent total cross section, in particular in the vicinity of the threshold or a t low total energies. However, studies of the vibrational rate enhancement seem to provide a more direct explanation. If a reaction is strongly enhanced upon internal (vibrational) excitation, then, in a thermalized system a t elevated temperatures and under conditions where vibrational relaxation is fast enough to maintain the vibrational equilibrium, a substantial proportion of the total reactive flux does occur via the excited states and this may result in non-Arrhenius behavior. I t is expected that this proposition will be tested for many more reactions, in particular for reactions between diatomics, where selective vibrational excitation is easily possible. Acknowledgment. The author is indebted to Professor
H. Gg. Wagner for stimulation, support, and valuable discussion of this work.
References and Notes (1) W. R. Schulz and D. J. Le Roy, J . Chem. Phys., 42, 3869 (1965); B. A. Ridley, W. R. Schulz, and D. J. Le Roy, ibid., 44, 3344 (1966). (2) T. C. Clark and J. E. Dove, Can. J . Chem., 51, 2147 (1973). (3) G. C. Light, J . Chem. Phys., 68, 2831 (1978). (4) (a) F. Dryer, D. Naegeli, and I.Glassman, Combust. Flame, 17, 270 Faraday (1970); (b) I. W. M.Smith and R. Zellner, J . Chem. SOC., Trans. 2 , 89, 1617 (1973); (c) A. A. Westenberg and N. de Haas, J. Chem. Phys., 58, 4061 (1973); (d) D. L. Baulch, and D. D. Drysdale, Combust. Flame, 23, 215 (1974); (e) W. Steinert and R. Zellner, Proc. Europ. Symp. Combust., 31 (1975). (5) W. C. Gardiner, Jr., Acc. Chem. Res., 10, 326 (1977). (6) R. Zellner and W. Steinert, Int. J . Chem. Kinet., 8, 397 (1976). (7) J. Ernst, H. Gg. Wagner, and R. Zellner, Ber. Bunsenges. Phys. Chem., 82, 409 (1978). (8) W. T. Rawlins, and W. C. Gardiner, Jr., J . Chem. Phys., 60, 4676 (1974). (9) J. Ernst, H. Gg. Wagner, and R. Zellner, Ber. Bunsenges. Phys. Chem., 81, 1270 (1977). (10) T. C. Clark and J. E. Dove, Can. J . Chem., 51, 2155 (1973). (11) P. C. Kobrinsky and P. D. Pacey, Can. J . Chem., 52, 3665 (1974). (12) P. D.Pacey and J. H. Purneil, J. Chem. SOC.,Faraday Trans. 1 , 68, 1462 (1972). (13) P. Camilleri, R. M. Marshall, and J. H. Purnell, J. Chem. Soc., Faraday Trans. 1 , 71, 1491 (1975); P. D. Pacey and J. H. Purnell, I n t . J . Chem. Kinet., 4, 657 (1972). (14) (a) D. A. Whytock, J. H. Lee, J. V. Michael, W. A. Payne, and L. J. Stief, J . Chem. Pbys., 66, 2690 (1977); (b) R. G. Manning and M. J. Kurylo, J . Phys. Chem., 81, 291 (1977); (c) M. S. Zahniser, 8. M. Berquist, and F. Kaufman, Int. J . Chem. Kinet., IO, 15 (1978). (15) A. Fontijn and W. Felder, J . Chem. Phys., 67, 1561 (1977). (16) M. J. Stern and R. E. Weston, Jr., J. Chem. Phys., 60, 2803 (1974); H. S. Johnston, "Gas Phase Reaction Rate Theory", Ronald Press, New York, 1966; R. P. Bell, "The Proton in Chemistry", Chapman and Hail, London, 1973. (17) S. Glasstone, K. J. Laidier, and H. Eyring, "Theory of Rate Processes", McGraw-Hill, New York, 1941. (18) L. S. Kassel, "The Kinetics of Homogeneous Gas Reactions", Chemical Catalog Company, New York, 1932. (19) M. A. Eiiason and J. 0. Hirschfelder, J. Chem. Phys.,30, 1426 (1959). (20) K. Shuler, J. Ross, and J. C. Light, "Kinetic Processes in Gases and Plasmas", A. R. Hochstim, Ed., Academic Press, New York. (21) T. C. Clark, J. E. Dove, and M. Finkelmann, paper presented at the VIth International Colloquium on Gasdynamics of Explosions and Reactive Systems, Stockholm, 1977. (22) R. C. Tolman, J . Am. Chem. SOC., 42, 2506 (1920). (23) R. L. Le Roy, J . Phys. Chem., 73, 4338 (1969).
Non-Arrhenius Behavior in Bimolecular Reactions (24) (a)M. Menzinger arid R. L. Wolfgang, Angew. Chem., 81, 446 (1969); (b) B. Perlmutter-Hayman, Prog. Inorg. Chem., 20 (1976). (25) W. C. Gardiner, Jr., W. G. Mallard, and J. H. Owen, J . Chem. Phys., 60, 2290 (1974). (26) D. L. Baulch, D. D. Dtysdaie, D. G. Horne, and A. C. Lloyd "Evaluated Data for High Temperature Reactions", Butterwotihs, London, 1972. (27) The Arrhenius curve represented by this expression was already drawn
in the original paper of Ernst et aL9 There, however, this curve was represented by k = 6.6 X 108T123cm3/mol s in an attempt to fit in the AT' form. The expression derived here should be given preference. (28) E. A. Albers, K. Hoyermann, H. G. Wagner, and J. Wolfrum, Symp. (rntl.) Combust., roc.], 73th (1971). (29) A. Erdglyi et al., Ed., "Tables of Integral Transforms", McGraw-Hill, New York, 1954. (30) (a)S. H. Lin and H. Eyring, Proc. Natl. Acad. Sci., U.S.A ., 68, 402 (1971); (b) S. H. Lin K. H. Lau, and H. Eyring, J. Chem. Phys., 55, 5657 (1971); 58, 12!61 (1973). For a recent review :jee J. H. Birely and J. L. Lyman, J. Photochem., 4, 269 (1975). M. Karplus, R. N. Porter, and R. D. Sharma, J . Chem. Phys., 43, 3259 (1965); A. Persky, ibid., 68, 241 1 (1978); R. Schinke and W. Lester, Jr., paper presented at the 1978 Molecular Collisions Conference, Asilomar, Calif., 1978. For cases where a is close to 1 or E, is very small, curvature in km is expected, even when A,=, is not enhanced. This follows from eq IV and V. The maximum increase of ktherm, however, does not extend beyond a factor of 2 . J. E. Spencer, H. Endo, and G. P. Glass, Symp. (Intl.) Combust., [ Proc.], 76th (1977). L. M. Raff, J . Chem. Phys., 44, 1212 (1966). R. Zellner, W. Steinert, and H. Gg. Wagner, manuscript in preparation. R. G. Miller and J. K. Hancock, J . Chem. Phys., 66, 5150 (1977).
It is important to note that cross section measurements in molecular beam and nuclear and photochemical recoil studies usually provide u(€)functions at high collision energies. (See, for example, J. Dubrin, Annu. Rev. Phys. Chem., 24, 97 (1973).) Measurements in the critical near-threshold region are comparatively inaccurate (J. G. Pruett, F. R. Grabiner, P. R. Brooks, J . Chem. Phys., 63, 1173 (1975).) It should be emphasized that this technique does not produce excitation functions of stringent physical significance. Within the scope of qualitative trends of u(€),however, its derivation from k( r ) is useflak It cannot be excluded that the state specific rate constants k, depend also on the reagent rotational excitation, as is suggested from trajectory calculations for a number of atom-diatomic reactions.32 In the absence of such information for the reactions considered here, we will neglect this effect and consider our approach as being accurate to a first approximaiion. Our detection method follows [OH(v=O)]and therefore can only distinguish between the two processes if vibrational relaxation of OH were fast. A study is presently under way to determine directly the relative amount of the reactive channel.
Discussion S. H. BAUER(Cornel1 University). When one begins a kinetic discussion with the statement that the bimolecular rate constant is proportional to the factor
Cexp (-hu,/ k r )[u
u1
exp (-Et/ k 2') d (Et/ k 2')
(where Et is the relative translational energy of the colliding species and IT,&) is the cross section for reaction for the internal state (i)l, he has already accepted two significant assumptions. First, he assumed that the vibrational state populations are Boltzmannian; this is not always the case, but such departures are not easy to demonstrate. More important, he allocates the entire burden of the observed deviation from an Arrhenius temperature dependence onto the shapes of the overlapping reactive cross sections, a t and immediately past their respective thresholds. One may question whether this procedure provides insight into the dynamics of reaction, since the required resolution for the u,(Et) functions is clearly not unique, were the function kbi(T) known with very high precision. Unfortunately, even that primary requirement is rarely fulfilled. R. ZELLNER. Regarding the assumption of a vibrational equilibrium, it has been demonstrated' that departures from the Boltzmann vibrational distribution in bimolecular reactions with important vibrational rate enhancement, and hence nonequilibrium contributions to non-Arrhenius behavior, are not significant.
The Journal of Physical Chemistry, Vol. 83, No. 1, 1979 2 3 The shape of the excitation functions G(E)a t and immediately past the threshold is certainly not unique. In the present evaluation it should primarily be understood as a plausible explanation of non-Arrhenius temperature dependence, in a situation where direct measurements of u(E) are unavailable. (1)T. C. Clark,
