J. Phys. Chem. 1984,88, 6435-6441
E,,, = -mRT/2
(A.ll)
When n is sufficiently large for both summations to converge to the exponential, (A.8) becomes E,,,
+
VTs - mRT/2
(A.12)
This is the same activation energy that we get in the TST limit with many collisions. This is because the large number of classical degrees of freedom serves as a heat bath just as collision partners would a t high density. The limit in eq A.10 is the one of interest for significant negative activation energies in reactions between small molecules. It states that the average energy of the transition state is just R T so that the activation energy is just RT minus the internal energy of the reactants. This has a simple physical explanation. At energies well above the zero pont the density of states changes slowly relative to the Boltzmann factor. As a result, the average energy is the same as for systems with a constant density of states, such as a classical oscillator or a two-dimensional rotor. Appendix B For a molecule with principal moments of inertia Z,., Zy, and I,, the kinetic energy is
6435
and the total angular momentum, K,is determined by LZ = L,z Lyz L,2
+
+
If we assume that I, IZy II,, then, for a particular value of L, the energy must lie in the range (LZ/12) 5 E,,, 5 (L2/Z,.)
(B.1) An amount of energy equal to the lower limit is unavailable for exchange with other degrees of freedom. We take this into account by assuming that we have an adiabatic two-dimensional rotor with moment of inertia I,. The energy associated with this rotor can change only through a change in geometry which alters 12.
The total rotational energy can vary over the range indicated in eq B.1. If we have I, >> I,., then the upper limit will be very large. In this case we may take the remaining degree of freedom as a one-dimensional rotor. The moment of inertia must be chosen so as to producq the correct overall partition function for the three rotational degrees of freedom. If I, is not very different from I,, there will be very little energy available for exchange with other degrees of freedom. In this case we should treat the molecule as a spherical top with moments (ZxZ,Jz)1/3.All three rotational degrees of freedom will be adiabatic. The degeneracy associated with J is ( 2 J 1)2 rather than (25 + 1) as in the case where I, >> I,.
+
Negative Activation Energies and Curved Arrhenlus Plots. 2. OH 4- CO Michael Mozurkewich, John J. Lamb, and Sidney W. Benson* Department of Chemistry, Donald P. and Katherine B. Loker Hydrocarbon Research Institute, University of Southern California, Los Angeles, California 90089 (Received: March 22, 1984)
In the first paper of this series we have presented a method for calculating the rate constants of reactions proceeding through an intermediate complex. Here we apply that method to the reactions OH + CO and OD + CO over a temperature range of 200-2000 K. A number of models of the transition state for decompositionof the intermediate to products were examined. The best results were achieved by using a bent transition state with the OCO bend as reaction coordinate. The pressure dependence of these reactions can be explained only if the transition state for formation of the intermediate is fairly tight with a significant threshold energy.
I. Introduction The gas-phase reaction of hydroxyl radical with carbon monoxide
OH
+ CO
-+
H
+ COZ
is an important component of the atmospheric HO, cycle. It also plays a critical role in combustion processes. There are two aspects of this reaction which suggest that it is not a simple elementary reaction. First, the Arrhenius plot is strongly curved;' the activation energy is near zero at 300 K and increases to about 7 kcal mol-' a t 2000 K. The second interesting aspect is that the rate constant increases with p r e s s ~ r e . ~ - ~ Dryer et aL6 showed that the temperature dependence of this reaction could be explained by a tight transition state with a small (1) Baulch, D. L.; Drysdale, D. D. Combust. Flame 1974, 23, 215.
D. M.; Hampson, R. F.; Kurylo, M. J.; Howard, C. J.; Ravishankara, A. R. Jet Propulsion Laboratory California Institute of Technology, Pasadena, CA, 1983, JPL Publ. 83-62. (3) Perry, R. A.; Atkinson, R.; Pitts, J. N. J. Chem. Phys. 1977,67, 5577. (4) Paraskevopoulos, G.; Irwin, R. S.J . Chem. Phys. 1984, 80, 259. (5) DeMore, W. B. Inr. J. Chem. Kind., in press. ( 6 ) Dryer, F.; Nageli, D.; Glassman, I. Combust. Flame 1971, 17, 270. (2) DeMore, W. B.; Molina, M. J.; Watson, R. T.; Golden,
0022-3654/84/2088-6435%01.50/0 , I
,
threshold energy. The pressure dependence can also be explained if a stable intermediate is O H C O =FHOCO* H C02
+
+
+
A potential energy surface for such a reaction is shown in Figure 1. As shown in the first paper of this series: the presence of an excited intermediate can explain both strongly curved Arrhenius plots and negative temperature dependences in what otherwise appear to be elementary reactions. We have developed a modified version of RRKM theory for calculating rate constants for reactions of this type. The theory allows for the possibility that the formation of the intermediate may not be fast relative to the decomposition to products and permits V, to be less than VI. In this paper we apply this theory to the OH CO reaction. Smith and Zellner7 carried out transition-state theory calculations for a mechanism involving an H O C 0 intermediate. They obtained reasonably good agreement with experiment but found too strong a temperature dependence and too low a rate constant
+
(7) Smith, I. W. M.; Zellner, R. J. Chem. SOC.,Faraday Trans. 2 1973, 69, 1617. (8) Smith, I. W. M. Chem. Phys. Lert. 1977, 49, 112. (9) Mozurkewich, M.; Benson, S. W. J. Phys. Chem., preceding paper in
this issue.
0 1984 American Chemical Society
6436 The Journal of Physical Chemistry, Vol. 88, No. 25, 1984
Mozurkewich et al. -H
0,961
0,
\2 ,free
a)
w
2.8H
HOCO REACTION COORDINATE
\
---+-
\
'c-0
Figure 1. Potential energy surface for the reaction of OH with CO. Heats of formation are in kcal mol-' at 300 K.
1.i3A
Figure 2. Assumed structure of the intermediate (trans form). 1.152\
TABLE I: Vibrational Frequencies (cm-I) and Moments of Inertia (amu A') for the Intermediate and Transition States Considered for the OH CO Reaction
+
TS 1 TS2 vibrational mode HOCO" contactb tightb,c S-Zbsd bendingb linearb' 0-H str 3456 3570 3500 rc 2400 rc C=Ostr 1833 2140 1900 2330 2100 1830 C-0 str 1077 rce rc 1310 1200 1300 HOC bend 1261 900 840 870 840 (2) OCO bend 615 425 660 (2) rc 635 (2) torsion 615 640 rotations adiabatic 50 (2) 130 (2) 63 (2) 51 (2) 51 (2) 54 (2) active 2.3 5.5 (2) 2.7 1.4 1.2 internal 0.68 (2) 0.87 Reference 1 1 * Reference 12. Reaction coordinate. I
e
Reference 10.
Figure 3. Assumed structures of the first transition state: (a) contact
transition state, (b) tight transition state. H \
\
a)
j.32,
sc a 0
1.258
4.18
Reference 7.
for the reaction OD + CO. We have carried out calculations using our theory in order to determine (1) what effect the choice of reaction coordinate has on the calculated rate constant, (2) what effect the first transition state has on the rate constant, (3) where the linear model proposed by Goldenlo can fit the experimental data, (4) whether a reasonable account of the observed pressure dependence can be obtained, and ( 5 ) whether we can explain the observed isotope effect. In section I1 we describe the models that we have used for the transition states, and in section I11 we outline the method used to calculate the rate constants. The results of these calculations are presented and discussed in section IV. In sections V and VI we examine the pressure dependence and isotope effect. 11. Transition-State Models
The HOCO intermediate has been observed in matrix isolation studies," and its IR spectrum has been assigned. The assumed structure is shown in Figure 2, and the vibrational frequencies and moments of inertia are given in Table I. As shown in the first paper of this series, the low-pressure rate constant depends only on the properties of the transition states and not at all on the intermediate. However, the structure and frequencies of the intermediate are of great value in estimating these properties for the transition states. Also, a knowledge of the intermediate is necessary for estimating the pressure dependence. In the following, the first transition state (TS1) refers to the decomposition of the intermediate to reactants. The second transition state (TS2) refers to decomposition to products. We have considered two models of the first transition state. These are shown in Figure 3 with frequencies and moments given in Table I. For the contact transition state we use a hindered Gorin model with the 0-C distance taken to be the van der Waals (10) Golden, D. M. J . Phys. Chem. 1979,83, 108. (11) Milligan, D. E.; Jacox, M. E. J . Chem. Phys. 1971, 54, 927.
H'
c)
H_ _ _ _ _ _ - 0 1.3%
0
C=
1.258
l.iSi
Figure 4. Assumed structures of the second transition state: (a) Smith-Zellner model, (b) OCO bending model (trans form), (c) linear model.
contact distance (2.8 A). The moments of inertia are effective moments which include factors to account for the limited solid angle available.'* In the tight transition state the OH torsion is assumed to be a free internal rotation. The vibrational frequencies are those used by Golden.lo These are based on the frequencies of the intermediate with the bending frequencies lowered by 30% to account for the weakening of the C-0 bond. For the second transition state we have considered three different models. These are shown in Figure 4. The first, with the heavy atoms linear, is essentially that of Smith and Zellner.' The vibrational frequencies are those of C 0 2 plus a HCO bending frequency which is taken from the tables given by Benson.lz We feel that the Smith-Zellner model is not very realistic. Using the force constants reported by Milligan and Jacox," we calculate that the 30 kcal mol-' released informing the intermediate would be insufficient to open the OCO angle to more than 160". Also, we prefer to think of the slow OCO bending motion as taking place in an effective potential due to the rapidly moving H atom. These arguments lead us to propose the "bending" model geometry shown in Figure 4 with the OCO bend taken as reaction coordinate. The bond angle of 160" for this transition state can also be arrived at by considering the addition of H to COzto form HOCO. Usually the addition of a free radical to a closed-shell molecule (12) Benson, S. W. "Thermochemical Kinetics"; Wiley: New York, 1976.
Negative Activation Energies and Curved Arrhenius Plots has an activation energy of no more than about 10 kcal mol-'. The models considered here require that this activation energy be about 25 kcal mol-'. The "extra" 15 kcal mol-' is just about what is needed to produce a bent, rotating C 0 2 molecule with an OCO angle of 160'. In this model the vibrational frequencies for the heavy atoms are assumed to be the average of the corresponding frequencies in the intermediate and in CO,. The OH stretching and bending frequencies are lowered by 30% from those of the intermediate. This corresponds to applying Badger's for a bond order of 1/2. The torsional frequency is calculated by using the small moment of the C 0 2 group (0.39 amu A2) and a barrier height of 7 kcal mol-' (1/2 the rotational barrier for the intermediate). A factor of 2 is included in the rate expression to account for cis-trans isomerization. We might consider a transition state with the same geometry as the bending model but with the OH stretch taken as the reaction coordinate. Such a transition state would be somewhat looser due to the replacement of a high-frequency OH stretch with a lowfrequency OCO bend. This would result in a larger threshold energy (chosen to fit the rate constant at 300 K) and a more strongly curved Arrhenius plot. Comparing the parameters listed in Table I, we see that such a model would be very similar to the Smith-Zellner model. It is very likely that the reaction coordinate is actually some mixture of the OH stretch and OCO k n d . This simply says that, at the top of the barrier, the direction in which there is no restoring force involves a simultaneous change of both coordinates. One of the vibrations in the transition state would also be a mixture of the stretching and bending motions. If we begin with the bending model, we must lower the OH stretching frequencies as we mix in the OCO bend. As we do this, we will get results that are more and more like those of the Smith-Zellner model. In this sense the bending and Smith-Zellner models provide two extreme cases. Finally, we consider the completely linear transition state proposed by Goldemlo The above critisms of the Smith-Zellner model apply to this model as well. Also, we do not feel that a hydrogen approaching along the C02axis would be very effective in producing the necessary bending of the OCO angle. Since this model is very tight, and consequently requires a negative value of V2,it is interesting to examine what sort of temperature dependence it produces. The parameters listed for this model in Table I are those given by Golden.lo All of these models are very nearly prolate tops. Hence, we take two rotational degrees of freedom as adiabatic and the third as active. For the adiabatic moment of inertia we use the largest of the three moment^.^ We treat the active rotation as a onedimentional rotor with the moment chosen to give the correct product of moments. All rotations are treated classically in computing sums and densities of states. 111. Method of Calculation
In order to compute the rate constants, we use a modified version of RRKM theory. The mathematical details are presented in the first paper of this ~ e r i e shere ; ~ we only outline the method. We are considering a reaction scheme of the form A
+B
Y* k-i
kl
products
The rate constants and the intermediate concentrations are functions of the total energy, E , and angular momentum, J. We can express the total rate as
For k,(E,J) we use the usual expression from RRKM theory. If we were to use V, (Figure 1) for Eo and the thermal equilibrium values for Y(E,J), then this expression would reduce to the (13) Robinson, P. J.; Holbrook, K. A. "Unimolecular Reactions"; Wiley: New York, 1972.
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6437
+
TABLE 11: Calculated Rate Constants for OH CO 1O13kerptl,cm3 molecule-' s-' VIa Vza 200 K 300 K 500 K 1000 K 2000 K
experiment Smith-Zellner bendingb bending bending linear
1.314
0.7 1.0 1.0 1.2
0.7 1.0
1.0 0.7 1.6 -5.8
1.3 1.2 0.96 0.90 0.56
1.5' 1.5 1.8 1.5 1.5 1.4
1.8' 1.9 2.3 2.1 2.2 2.7
2.8l 3.6 3.9 3.5 3.8 4.2
6.9' 10.9 9.3 9.3 9.7 6.7
"In kcal mol-'. bCalculated by using transition-state theory. All other calculations use the method described in section 111.
transition-state theory (TST) rate; Le., it will be the product of the equilibrium concentration of Y* and the TST rate constant for k,. There are two sources of deviation from the TST rate. First, the intermediate is in steady state rather than equilibrium. The concentration is then related to the equilibrium concentration by [Y(E,J)I = [Y(E,J)l,,/(l
+ k2(4J)/k-'(E,J))
If the first transition state is looser than the second and V2is not too negative compared to Vl then we expect to have k-, > k2 and this correction will be fairly small. The second correction is due to the fact that, if Vl > V2,the minimum energy, Eo, in (2) will be determined by the first transition state. Hence, not all states of Y which have sufficient energy to decompose to products will be populated. If the moments of inertia of the two transition states are very different, rotational effects become important. We have discussed these effects in detail in the previous paper; they have little effect on the reaction considered here. There are two methods that we have used to evaluate (2). The first is a straightforward numerical computation of the double integral. This method is very laborious, especially at high temperatures. For temperatures in excess of 1000 K the integral will not converge until energies of many tens of kcal mol-' are reached. The second method makes use of the fact that when V2is not much smaller than V,, the rate constant does not differ much from the TST rate constant. In this case we begin with the TST rate and calculate the corrections mentioned above. This method, where appropriate, is much faster than the direct integration of eq 2. In this paper we have used the second method for all calculations except those on the linear model at less than 1000 K. In these latter calculations both the TST rate and the corrections are large compared to the actual rate. Therefore, direct integration was necessary in order to obtain reasonable accuracy. Even though TS2 involves a hydrogen motion, we do not apply any corrections for tunneling or for a transmission coefficient less than one. This is because the intermediate is formed at very large energies (>30kcal mol-') so that these corrections should be small.
IV. Results and Discussion The calculations were carried out as described previo~sly.~ In addition to the frequencies and geometric parameters specified above, we need the potential energies, Vl and V2,for the first and second transition states. These are chosen in order to fit the experimental data. For the initial calculations we have assumed Vl = V2and used the tight model for the first transition state. That this is appropriate is indicated by the pressure dependence data which we discuss later in this paper. The resulting rates are only slightly lower than they would be if calculated from transition-state theory. Using a looser first transition state or a smaller value of VI would produce an even smaller deviation from TST but would not be consistent with the observed pressure dependence. The results of calculations on the Smith-Zellner and bending models are presented in Table I1 and Figure 5 along with the values =commended by Baulch and Drysdale' (above 300 K) and Davis et al.I4 (220-300 K). The recommended Baulch and (14) Davis, D. D.; Fischer, S.; Schiff, R. J . Chem. Phys. 1974, 61, 2213.
6438 The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 T (K) 20,0 2000 1000 I
l
500 400 300 250 I
l
I
LO
2.0
3.0
!OOo/T
(K-')
I
200
I
I
I
4.0
5.0
i0.0-
--
-
Y)
5.0I
E n 0
-
5
-
c P
I
i.0-
0.5-
Figure 5. Arrhenius plots of the experimental and calculated rate constants for OH + C O Baulch curve above 300 K, Davis et al. below 300 K.
+
Drysdale curve was calculated from log k = -12.95 3.94 X 10-4T, which best fits the experimental data. Suggested error limits for the rate constants calculated from this equation are f 2 0 % at 300 K,increasing to 430% a t 1000 K and above. Recent measurements by Ravishankara and Thompsonls (250-1040 K) are in general agreement with the earlier measurements. The threshold energies were chosen to fit the rate constant a t 300 K. For comparison, we have also presented the TST rate constants for the bending model. As expected, the difference is small and decreases with increasing temperature. The experimental rate constant at 2000 K may be in error by as much as 50%. Even if this is so, the Smith-Zellner model gives a value that is too high. Treating the OCO bend as reaction coordinate gives better agreement with the Baulch curve. There is a substantial difference between the bending and Smith-Zellner models a t temperatures below 300 K. Davis et al.14 have found that the rate constant at 200 K is about 10%3-0% less than at 300 K. As mentioned in section 11, we could take a mixture of the OH stretch and OCO bend as reaction coordinate in the bending model. This would produce results more like those of the Smith-Zellner model, i.e., closer agreement with the Davis data below room temperature and a greater deviation from the Baulch curve at high temperatures. The data are not certain enough in either case to determine the extent to which this should be done. As noted above, lowering VI will not affect these results significantly. To show the effect of increasing VI, we present, in Table 11, the results of one calculation on the bending model with Vl = V2 0.5 kcal mol-'. This increases the minimum energy for which reaction may take place and, as a result, produces a slightly stronger temperature dependence. thus, a change of this sort will not improve the agreement between the Smith-Zellner or bending models and the experimental data. Also, as discussed below, having V, > V2 is not consistent with the isotope effect. For the linear transition state, V2must be negative in order to achieve a sufficiently high rate constant. At low temperatures the TST rate constant will be very large. In this case the correction due to the fact that VI rather than V2determines the minimum energy required for reaction is nearly as large as the TST rate constant. As notes above, this requires that we compute the rate constant by direct integration of eq 2 for temperatures less than 1000 K. At 2000 K the rate constant is insensitive to Vl. With V2 = -5.8 kcal mol-' the 2000 K rate was fit fairly well. Using this
+
(15) Ravishankara, A. R.; Thompson, R. L. Chem. Phys. Lett. 1983,99, 317.
Mozurkewich et al. value of V2,we then adjusted VI to fit the 300 K rate; this was achieved with VI = 1.6 kcal mol-'. The results of the calculations with these parameters are presented in Table I1 and Figure 5. We see that the Arrhenius plot calculated from the linear model is much straighter than is observed experimentally. When hot H atoms, with energies of 1-2 eV, collide with thermal C 0 2 molecules, the probability of producing OH and CO is less than 0.1%.'6717From the A factor and collision frequency for H + C 0 2 we calculate that the probability of sufficiently energetic thermal collisions producing reaction is of the order of 10%. The hot H atom data therefore indicate that it is advantageous to have a substantial fraction of the activation energy in the form of internal excitation of COz. Vibrational excitation of the product C 0 2 is expected from both the Smith-Zellner and bending models due to the distortion of C 0 2in the transition state. Studies of the reaction rate of vibrationally excited OH or CO should provide a means to separate the effects of the two transition states. If energy is fully randomized in the intermediate, then the rate of quenching reactant vibrations should be equal to the rate of formation of the intermediate. Also, the reaction rate of vibrationally excited OH or C O should allow the determination of the energy dependence of the branching ratio for decomposition to products and reactants. This differs from the energy dependence of the overall reaction since, for the tight TS1, kl has a significant dependence on the nonvibrational energy of the reactants. Dreier and Wolfrum'* have measured the rate of reaction of OH with vibrationally hot CO. The data are only sufficiently accurate to show that the reaction rates for hot and thermalized CO are of the same order of magnitude. Spencer et al.I9 have reported that the rate of reaction of vibrationally excited OH with CO is, at most, twice as fast as thermal OH. Unfortunately, they do not appear to have accounted for the excitation of OH via the sequence
OH(v=O) H
+ NO2
+ CO -*
+
H
+ CO1
OH(Y>O)
+ NO
which should occur readily in this system. As a result, their rate constants are very likely too low.
V. Pressure Dependence Reactions proceeding via a relatively long-lived intermediate are expected to have rate coefficients which increase with increasing pressure. This is due to stabilization of the intermediate by collisions resulting in the removal of enough energy so that it can no longer decompose. Thus, we replace the above scheme with OH f CO
c HOCO*
-
H
COP
lM HOCO
+
The formation of the HOCO radical from OH CO is about 31 kcal mol-' exothermic.'2 Thus, at low concentrations we should be able to neglect collisional excitation of HOCO. Under these conditions the rate of reaction will be the sum of the rate of decomposition and the rate of collisional deexcitation. We can express the pressure dependence in terms of a thirdorder rate contant, klII,given by
where X is the collision efficiency, Z,, is the collision rate constant, and Kq* is the equilibrium constant between the excited inter(16) Oldershaw, G . A.; Porter, D. A. Nature (London) 1969, 223, 490. (17) Tomalesky, R. E.; Sturm, G. E. J . Chem. Soc., Faraday Trans. 2 1972, 68, 1241. (18) Dreier, T.; Wolfrum, J. Symp. (In?.) Combust., [Proc.],18, 1980 1981,aoi. Glass, G. P. Symp. (Int.) Combust., [Proc.], (19) Spencer,J. E.; Endo, H.; 16,1976 1977,829.
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6439
Negative Activation Energies and Curved Arrhenius Plots mediate and the reactants. The actual difference will be somewhat less than this since the intermediate will be at less than the equilibrium concentration. From the comparison between the TST and actual rate constants in Table 11, we see that this error is about 10%. In estimating the pressure effect, we neglect this. Of the quantities in eq 3 the one that we have the least knowledge of is A. Therefore, we rearrange (3) to
=
kIIIZcollKeq*
(4)
and see if the values of X we obtain are reasonable. In the preceding paper9 we derived an expression for Kq* that is suitable for a tight TS1 and a deep well. This is
where Q$ = RTfB, is the partition function for adiabatic rotations of the intermediate and By and Bl are the rotational constants for the intermediate and TS1. The density of states, N(Vl V y ) ,of the intermediate at the threshold for TS1 is calculated by using the Whitten-Rabinovitch approximation.20 The pressure dependence of the OH + C O reaction has been studied by a number of groups; most of these have used steadystate photolysis. Due to the possibility of reactions of product species, such as HOCO, with OH, we fell that these experiments are less reliable than ones using flash photolysis. For our calculations we have used the flash photolysis measurements of Perry et Extrapolating their data in SF6 to 1 atm gives a rate constant 2.5 times the low-pressure limit, corresponding to kIIr = 9.1 X cm6 molecule-z s-'. They also find that added Ar has no measurable effect on the rate constant as is consistent with the expectation that Ar should be a much less efficient collision partner than SFs. Using the tight and contact transition states, we have calculated the collision efficiencies, A, needed to fit the experimental data for SF6. The parameters from Table I were used for the transition state and intermediate. W e used a well depth of 31 kcal mol-'; the results are not sensitive to this parameter. We have assumed a collision radius of 2 A for HOCO and 2.5 A for SFsSz1 For the tight transition state we set Vl = 1.0 kcal mol-'. N(32) cm3 molecule-'. is 1 X lo5 states kcal-' and Keq* is 7.3 X Using eq 4, we find X = 0.48. Applying transition-state theory to the tight TS1 yields a high-pressure limiting rate constant of 1.6 X cm3 molecule-' s-'. Since the rate constant at 1 atm is about one-fourth of this, we should increase X to about 0.64 to account for the fact that we are in the falloff region rather than the linear, low-pressure region. This is a very reasonable value for the collision efficiency. With the contact TS1 we use VI = 0.0. N(31) is 9 X lo4 states kcal-I, Keq* is 7.3 X cm3 molecule-', and X is calculated to be 0.05. This is extremely low. For this transition state the high-pressure limit is 9.5 X lo-" cm3 molecule-' s-' so at 1 atm we are in the linear, low-pressure region. Hence, we see that a tight transition state with a significant barrier is needed to obtain a reasonable collision efficiency.
+
TABLE 111: Vibrational Frequencies (cm-I), Moments of Inertia (amu A*),and Changes in Zero-Point Energy (kcal mol-') for OD CO Transition States vibrational mode TS1 tight S-Z TS2 bending linear D-0 stretch 2500 rc' 1800 rc DOC bend 830 630 570 670 (2) moments adiabatic 67 55 56 59 active 3.1 2.6 1.8 internal 1.2
+
AV
-0.2
1.o
0.1
0.8
Reaction coordinate.
TABLE I V Calculated Pate Constants for OD + CO iOi3kexpt,,cm3 molecule-' s-I Via V," 200 K 300 K 500 K 1000 K 2000 K Smith-Zellner 0.5 1.7 0.08 0.23 0.58 1.9 7.1 bending 0.8 1.1 0.56 0.95 1.5 3.0 8.9 1.3 bending 1.0 0.8 linear 1.4 -5.0 0.52 1.1 1.8 2.7 4.2
In kcal mol-'.
be explained if the pressure dependence for the OD reaction is much stronger than for the OH reaction. Indeed, Paraskevopoulos and Irwinz3have reported that I atrn of He enhances the rate for OD C O by a factor of about 2.5 but does not increase the rate of O H CO. We will return to this question of pressure dependence below. There are two factors that can contribute to the decrease in rate upon isotopic substitution. First, replacing H with D will double the rotational partition function for the hydroxyl radical. The effect of isotopic substitution on the vibrational partition functions of the second transition state will be less than this. Hence, these entropy factors will lower the rate but by less than 50%. The second way that isotopic substitution can lower the rate is due to the fact that, for the models considered here, the hydrogen vibrational frequencies are lower in the transition state than in the reactants. As a result, the zero-point energy is lower and isotopic substitution will increase V2relative to the reactants. For this to affect the rate significantly, we must have V2 L Vl. We have carried out calculations for OD CO by modifying the parameters used for the OH + CO calculations. Geometries were kept the same. The reported frequencies for O D and DOCO" were used to assign the OD stretching frequencies in the same manner as was used for the OH + CO transition states. The DOC bending frequencies were chosen to satisfy the Teller-Redlich product rule.25 Vl and Vz were then obtained by applying the appropriate zero-point energy changes to the values used for OH + CO. The parameters which were changes from those used for OH C O are listed in Table 111, and the results of the calculations are listed in Table IV and plotted in Figure 6. If we accept the lower value (5 X cm3 molecule-' s-I) for the OD CO rate constant, then we see that the moddel of Smith and Zellner gives too strong an isotope effect and the bending and linear models give too weak an effect. The difference is due to the fact that for the Smith-Zellner model, in which the O H stretch becomes the reaction coordinate, isotopic substitution increases V2by more than for the bending model, in which the O H stretch is present in the transition state. For the linear model, changing V2has little effect since Vl determines the minimum energy for which reaction can take place. Also, we note that for the same reason choosing Vl > V2 produces a weaker isotope effect with the bending model. The results of Jonah et al. indicate that the in good agreement rate constant for OD + CO is about 1.1 X with both the bending and linear models. As in the O H + C O
+
+
+
+
+
VI. Isotope Effects Westenberg and Wilsonz2and Paraskevopoulos and Irwin23have measured the rate constant for OD C O at 300 K. They both find that this rate constant is about one-third of the rate constant for OH + CO. In contrast, Jonah et aLZ4have found that the rate constant for OD CO is about 25% lower than for OH CO at temperatures from 300 to 1000 K. Since Jonah et al.24 carried out their experiments in 1 atm of Ar, this discrepancy could
+
+
+
~~
(20) Forst, W. 'Theory of Unimolecular Reactions"; Academic Press: New York, 1973. (21) Chan, S. C.; Rabinovitch, B. S.; Bryant, J. T.; Spicer, L. P.; Fujimoto, T.; Lin, Y. N.; Pavlou, S. P. J . Phys. Chem. 1970, 74, 3160. (22) Westenberg, A. A.; Wilson, W. E. J. Chem. Phys. 1966, 45, 338. (23) Paraskevopoulos, G.; Irwin, R. S. Chem. Phys. Len. 1982, 93, 138. (24) Jonah, C. D.; Zeglinski, P.; Mulec, W. A,, personal communication.
( 2 5 ) Wilson, E. B.; Decius, J. C.; Cross, P. C. "Molecular Vibrations"; McGraw-Hill: New York, 1955.
6440
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984
TABLE V Parameters for Pressure Dependence of OD
Mozurkewich et al.
+ CO'
iOi3k1,cm3 molecule-I
10I9k3/kz,cm3 molecule-I
Wkz
exptl
S-1
exptl
bath gas
exptl curve
calcd
curve
line
calcd
curve
line
calcd
CF4
2.4 2.2
19.2 19.2 19.2
4.4 4.4
23 21 30
19 19 19
7.1 7.8 2.3
0.76 0.92 0.59
0.80 0.27 0.09
N2 He
'The experimental values are for the curve given by Paraskevopoulos and Irwinz1and for a straight line through their data. The calculated values are for the-bending model with a tight T S l . -
,o.o 2000
t
I
I
T
1000 1
(IO
500 400 I
I
300 250 I
I
200 I 4
OD + CO
.-.- -...Smith
8 Zellner Model Linear Model
fit by Paraskevolpoulos and Irwin, we would have to use a much tighter transition state and/or a larger value of V,. Neither of these changes seems physically reasonable, and neither would give agreement with the OH C O data. The parameter k 3 / k 2is the slope of rate constant vs. pressure in the low-pressure, linear region. This is obtained from
+
where kII is the third-order rate constant. We estimate k111from the OH C O pressure dependence. The ratio of collisional stabilization rates for OD CO to OH CO will be determined by two factors: the ratio of density of states and the change in V,. As long as the well is deep enough that we may treat all the vibrations classically the density of states for DOCO is 2(21/2) times the density for HOCO at a given energy. For the intermediate we treat vibrations classically and, in calculating the density of states, measure the energy from the minimum of the electronic potential. For the reactants, which are at low energy, we must measure the energy from the vibrational ground state. Thus, when N ( E ) is calculated from the Whitten-Rabinovitch approximation, the zero-point energy difference of 1.4 kcal mol-' between OH and O D has the effect of making the well shallower for OD CO. This reduces the density of states of DOCO at the top of the well by about 25%. For O D the density of rotational states will be twice that for OH. Hence, the overall effects of density of state changes is to increase the equilibrium constant (eq 3) by about 10% for DOCO. For the model we have used, V, will be smaller by about 0.2 kcal mo!-' for O D + CO. Thus, the collisional stabilization constant, kIIl,should be about 50% larger for O D CO than for O H CO. The relative increase in rate is greater since the low-pressure rate for O D is smaller. The values of k3/k2 listed in Table V are calculated by assuming that the ratios of collision efficiencies, SF6:CF4:N2:He = 0.5:0.3:0.1:0.02, are the same as for HOCOS4 We see that the reported pressure dependence is much stronger than we would expect from the O H CO data. In order to get such a large pressure dependence, we would need a looser first transition state and/or a lower value of VI. This is the opposite of what is needed to make the experimental and theoretical values of k , agree. An alternate explanation is to recognize that Paraskevopoulos and Irwin's curve fits their data far better than one would expect on the basis of the standard deviations, that they assign to the individual points. In fact, most of their points fall within one standard deviation of a straight line through their data (see Figure 7). Since their largest measured rate constant (1.8 X is less than one-tenth of the calculated high-pressure limit, we expect the rate constant to depend linearly on the data. The parameters resulting from this assumption are listed in Table V. These parameters are much closer to the calculated values than the ones resulting from Paraskevopoulos andd Irwin's fit. In particular, the low-pressure limiting rate constant is about 8 X cm3 molecule-' s-I. This is substantially higher than the value measured by Paraskevopoulos and Irwin and by Westenberg and Wilson but is lower than the value obtained by Jonah et al. With the linear fit the pressure dependence, k3/k2, is much closer to the values predicted from the HOCO data. Considering the differences resulting from the two methods of viewing the data,
+
1.0
2.0
3.0
4.0
5.0
(ooo/T (K-') Figure 6. Arrhenius plots of the experimental and calculated rate contants for OD + CO. reaction, if we include some O D stretching character in the reaction coordinate of the bending model, we will get results that are more like those of the Smith-Zellner model, Le., a stronger isotope effect. At 2000 K there is little difference between the absolute rate constants predicted by the Smith-Zellner and bending models. The Smith-Zellner model predicts that O D reacts about 30% slower than OH; the bending model predicts a lowering of 10%. Both of these are in reasonable agreement with the 25% reduction measured by Jonah et al.24 Paraskevopoulos and Irwin23have measured the rate of the reaction OD C O as a function of pressure with several collision partners. They find that the pressure dependence is about the same for N2and CF4 with a definite curvature to the plot of rate constant vs. pressure. For He the effect is somewhat weaker and the plot shows only a slight curvature. Paraskevopoulos and Irwin assume that the curvature is due to approaching the high-pressure limit where the rate constant is determined by k l . They fit their data to the equation
+
where k3 is the rate constant for collisional deexcitation. In Table V we list the parameters that they obtained along with those calculated from our model. We obtain the calculated value of k l from transition-state theory. The calculated value of k_,/k2 is chosen to agree with the low-pressure limiting rate constant, ko.
We used the value of ko of 9.5 X calculated from the bending model. We see that in order to match the values from the curve
+
+
+
+
+
+
J . Phys. Chem. 1984,88, 6441-6448
100
200
300
400
500
600
PRESSURE (torr)
Figure 7. Comparison of straight line and curve for fitting the pressure dependence of OD + CO in CF4. Data from Paraskevopoulos and Irwin;23 error bars are one standard deviation.
any conclusions must be regarded as very uncertain.
VII. Conclusions The strongly curved Arrhenius plot exhibited by the reaction O H CO can be explained by assuming the existence of an intermediate complex. We find that the low-pressure rate is determined almost entirely by the properties of the transition state for dissociation to products (TS2). A bent geometry for this transition state is indicated by the fact that hot H atoms have a very low probability for reacting with thermal CO,. Also, this enables us to understand the large barrier for addition of H to
+
coz.
Various models of the transition state were examined. The best results were achieved by using a slightly bent transition state with
6441
the OCO bending motion as reaction coordinate and a threshold energy, V,, of about 1.0 kcal mol-' above the ground state of the reactants. Even in this case the temperature dependence is somewhat stronger than for the curve recommended by Baulch and Drysdale.' If the reaction coordinate has some OH stretching character, then the temperature dependence will be weaker at low temperatures and stronger at high temperatures. The pressure dependence of the rate constant depends on the properties of both the intermediate and the transition state for dissociation to reactants (TS1). Since the proposed intermediate is fairly well-known, this permits us to determine the nature of the transition state. In order to fit the pressure dependence with reasonable collision efficiencies, we find that the transition state must be rather tight with a threshold of about 1.0 kcal mol-' relative to the reactants. For the reaction OD + C O the situation is less clear. The experimental data are sparse and not entirely consistent. Our model is not inconsistent with the available data when the uncertainties in that data are considered. Including OH stretching character in the reaction coordinate strengthens the isotope effect; the entire range of experimental values can be fit in this way. The pressure dependence of the O D C O reaction should be about 50%, stronger than for OH CO, and the high-pressure limit should be somewhat larger. A more precise determination of the pressure dependence is needed to determine if the strong curvature reported by Paraskevopoulos and Irwin is correct. If this is the case, the model proposed here will need substantial revision.
+
+
Acknowledgment. This work was supported by the National Science Foundation and the Army Research Office under Grants No. CHE-79-26623 and DAAG29-82-K-0043. Registry No. OH, 3352-57-6;CO, 630-08-0;OD, 13587-54-7.
Negative Activation Energies and Curved Arrhenius Plots. 3. OH -t HN03 and OH 4"04
John J. Lamb, Michael Mozurkewich, and Sidney W. Benson* Donald P. and Katherine B. Loker Hydrocarbon Research Institute, University of Southern California, Los Angeles, California 90089 (Received: April 17, 1984)
The method developed in the first paper of this series has been used to calculate the rate constants and activation energies for the reactions of OH with HNO, and HN04. The reactions are assumed to proceed via stable intermediates. For the reaction OH + HNO, a planar intermediate containing a six-membered hydrogen-bonded ring predicts a less negative activation energy than is observed experimentally. However, if the reaction proceeds via attack by OH at the nitrogen atom followed by four-center elimination of water, the experimental data can be fit very well. A similar model fits the temperature dependence of OH + "0,. Both reactions are predicted to have a weak pressure dependence. The reaction OD + DNO, is predicted to have a larger pressure dependence.
I. Introduction The dominant loss processes for HO, in the lower stratosphere have been shown to be the reactions of hydroxyl radical with nitric acid and with peroxynitric acid:' OH + HONO, H,O + NO, (1) OH
--
+ H02N02
products
(2) Recent investigation^^-^ have determined that reaction 1 has a (1) WMO Global Ozone Research and Monitoring Project, "The Stratosphere 1981: Theory and Measurements", Report 11; World Meterological Organization: Geneva, Switzerland, 1982. Cicerone, R J.; Walters, S.; Liu, S . C. JGR, J. Geophys. Res. 1983, 88, 3647. (2) Wine, P. H.; Ravishankara, A. R.; Kreutter, N. M. JGR, J . Geophys. Res. 1981, 86, 1105.
0022-3654/84/2088-6441$01.50/0
negative temperature dependence: Le., the reaction rate increases with decreasing temperature over the range 220-440 K. The activation energy is about -1.5 kcal mol-'. A similar temperature dependence has been reported7 for the reaction of hydroxyl radical with peroxynitric acid over the tempeature range 240-330 K; the activation energy is about -1.3 kcal mol-]. In addition to these negative activation energies both reactions have preexponential factors that are unusually low for atom abstraction reactions. (3) Kurylo, M. J.; Cornett, K. D.; Murphy, J. L. JGR, J. Geophys. Res., 1982,87, 3081. (4) Margitan, J. J.; Watson, R. T. J. Phys. Chem., 1982, 86, 3819. (5) Marinelli, W. J.; Johnston, H. S. J . Chem. Phys. 1982, 77, 1. (6) Jourdain, J. L.; Poulet, G.; LeBras, G. J . Chem. Phys. 1982, 76, 5827. (7) Smith, C. A.; Molina, L. T.; Lamb, J. J.; Molina, M. J. I n f . J . Chem. Kinet. 1984, 16, 41.
0 1984 American Chemical Society
