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
6442
Lamb et al.
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984
n \
1.2 A\ REACTION COORDINATE
\ \
\
_f
Figure 1. Potential energy surfaces for a reaction proceeding through an intermediate with no angular momentum and with large angular mo-
mentum. It has been proposed2g4that these effects are due to the formation of an intermediate complex which can dissociate into either reactants or products. In the first paper of this series8 we have presented a theory for calculating the rate constants of reactions proceeding via an excited intermediate. The presence of an intermediate allows us to have a tight transition state with a small or negative threshold energy. A negative threshold energy allows a reduction in the activation energy; values as negative as -1.8 kea1 mol-’ may be achieved for reactions of this type.8 In calculating the rate constants, it is necessary to be careful in treating angular momentum effects. In this paper we apply this theory to the reactions of OH with HONO, and H 0 2 N 0 2 . In section I1 of this paper we outline the method used in our calculation of rate constants. In section I11 we describe the models we have used for the OH H N 0 3 reaction, and section IV we discuss the results for this reaction. The model used for the OH HN04 reaction and the results of those calculations are given in section V. In section VI we address the question of what pressure effects might be expected.
+
+
11. Method of Calculation The type of reaction that we are considering may be described by the scheme A
+ B &Y* k
k-1
k2
products
(3)
For any given energy, E, and angular momentum, J , we may use RRKM theory to calculate the rate constant, k,(E,J), for dissociation to products. Then, if we can calculate the concentration of intermediate, [Y(E,J)],as a function of E and J , we can integrate over all E and J to get the experimental rate constant, kexptl.
Referring to Figure 1, we note that the minimum energy required for reaction to take place will depend on J. This minimum energy may be determined by either the first transition state, TS1 (as is the case for J = 0 in the figure), or the second transition state, TS2 (large J in Figure 1). For energies above the minimum, we assume that the intermediate is in steady state. The concentration of Y can then be expressed in terms of k,(E,J), k 1 ( E , J ) , and the equilibrium constant between the intermediate and reactants. All of these quantities can be calculated from RRKM theory and the usual methods of statistical mechanics. The details of this procedure were presented in the first paper of this series.s In the final expression for the rate constant all terms dependent on the intermediate cancel out. The rate constants for dissociation of the intermediate depend on the amount of energy available for redistribution among the various degrees of freedom of the transition states. Due to conservation of angular momentum, the rotational energy cannot be redistributed. Since the rotational energies of the two transition states may be quite different, we take account of the angular momentum as well as the total energy in computing the rate constant. This results in a double integration, over E and J , which requires a great deal of computer time. To avoid this laborious procedure, we begin with the transition-state theory rate constant for TS2 and then calculate two corrections. The first correction is due to the fact that there is a minimum energy with which the intermediate may be formed.
\
1.2i A ’0
Figure 2. Geometry of the second transition state for OH + HNO, (model I). For the intermediate the hydrogen bend is asymmetric.
The second is for the difference between the steady-state and equilibrium concentratins of the intermediate. As long as these corrections are not too large, this method is faster and more accurate than the direct integration. Because the intermediate is formed with energies that exceed the barrier for H atom transfer, we do not include any corrections for tunneling. Both the hydroxyl radical and nitric acid (and, by analogy, peroxynitric acid) have substantial dipole moments. We therefore expect that they may experience a long-range, attractive interaction. Thus, the transition state for the formation of the intermediate will be very loose with no threshold energy; the intermediate will be formed at essentially collision frequency (cm3 molecule-’ s-l). Since the experimental rate constant (1.3 X for OH HNOJ is much smaller than this, we conclude that the difference between the steady-state and equilibrium concentrations of intermediate will be small. Most collision complexes dissociate to reactants rather than products. Because the first step is so fast, the principal effect that TS1 has on the rate is that, for low J , it will determine the minimum energy required for reaction. For low values of J the maximum of the centrifugal potential will occur at large separations of the two reactants. Thus, the moment of inertia will be large and the rotational energy at the centrifugal barrier will be very low. As a result, we might expect that TS1 will have essentially no effect on the minimum energy required for reaction; at low J the barrier at TS1 is negligible, and at high J the centrifugal barrier at TS2, which is tight, determines the threshold energy. This conclusion is verified by actual calculations. We begin by assigning to T S l a moment of inertia appropriate for a loose transition states9 Since this moment is an average over a thermal distribution, it should be too small; it is only for low values of J that TS1 is important. For the reactions considered here, increasing this moment has only a small effect on the rate constant.
+
+
111. Transition-State Models for OH HN03 In order to calculate the rate constants, we must estimate the vibrational frequencies and moments of inertia of the transition states. V,, the potential energy at TS2, is adjusted to fit the data. Properties of the intermediate are only needed in considering the pressure dependence of the reaction; they have no effect on the low-pressure rate constant. For the first transition state we use a hindered Gorin modeL9 All frequencies are assumed to be the same as in the reactants. The nitric acid molecule is assumed to rotate freely, but the hydroxyl radical is restricted to a solid angle of 27r steradians. This is because the dipole-dipole interaction will be attractive for only half of all possible orientations. If we assume a side-by-side dipole interaction, the maximum of the centrifugal barrier occurs at 5 A. This yields a moment of inertia of 335 amu A2 for the
(8) Mozurkewich, M.; Benson, S.W., submitted for publication in J . Phys.
Chem.
(9) Benson, S . W. “Thermochemical Kinetics’’; Wiley: New York, 1976.
Negative Activation Energies and Curved Arrhenius Plots TABLE I: Vibrational Frequencies (in c d ) for the Reactants, Second Transition State, and Products for OH "0,' mode reactants model I model I1 products NO2 asym str 1708 1600 1600 1480 NO2 sym str 1325 1200 1200 1060 &NO2 str 647 1060 1060 1480 N O z scis 879 630 630 380 0 - N O 2 rock 579 480 480 380 NO2 wag 762 760 760 380 HO str 3550 2000 2000 3825 HON bend 1331 1200 1200 HO torsion 456 1200 1200 HOH bend 1200 1200 1654 OH str 3735 3800 3800 3936 800 HON bend 340 HO torsion 400 O N 0 bend 600 ONO' bend OON bend 400 OH0 torsion 185
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6443
+
~
a
For assignments see text.
adiabatic rotations, As we shall see, the rate constant is fairly insensitive to the details of TS1. In model I we assume that the intermediate contains a planar, six-membered, hydrogen-bonded ring (see Figure 2). The stronger H bond should be between the hydroxyl 0 and the OH group of the nitric: acid. When the distance between these two oxygen atoms becomes sufficiently short (-2.4 A), the hydrogen bond should be symmetrical;10i.e., when the H atom is midway between the two oxygens, it is at a minimum in the potential energy. In this transition-state geometry, shown in Figure 2, there is essentially a three-center OH0 bond. The largest moment of inertia is 195.5 amu A2; we take this as the adiabatic moment.* The moment of the active external rotation is 30.5 amu A2. At the temperatures considered here only vibrational frequencies of less thafi 1000 cm-' will have a significant effect on the rate constant. In assigning these frequencies, we have used the largest values which seem reasonable. The reason for this is that higher frequencies produce a more negative activation energy. As shown below, even with the highest reasonable frequencies this model does not produce as negative an activation energy as is observed experimentally. The lowest frequencies are those associated with the hydroxyl 0 atom. Other than the reaction coordinate these low frequencies are an O N 0 bending motion and a torsion about the N O bond. For the bend we use a typical heavy-atom bending frequency of 400 ~ m - ' the ; ~ actual frequency is probably lower than this since the O H 0 bond is so long. The torsional barrier is due to the developing ?r character of the NO bond. For a barrier of 30 kcal mol-', which is almost certainly high, the frequency is 185 cm-'. The frequencies associated with the NO3 group have been measured for both HNO3I1and NO3;'* we have used the average for the transition state. There is a low-frequency torsional motion of the hydroxyl H about the OH0 bond. Assuming that the barrier is due to a 5 kcal hydrogen bond, we calculate a frequency of 340 cm-'. The remaining modes are all relatively high-frequency hydrogen motions. For these we assign typical values of O H stretching (two modes) and bending (three modes) freq~encies.~ The bridging H motion is lowered to 2000 cm-I since it moves in a somewhat broader well than a typical OH stretch. The frequencies are listed in Table I along with those of the reactants and products. As we shall see, model I does not reproduce the observed temperature dependence. As a result, we have also considered a second route which yields a tighter TS2. In model I1 we assume that the hydroxyl radical attacks from above (or below) the plane (10) Pauling, L. "The Nature of the Chemical Bond";Cornell University Press: New York, 1960. (1 1 ) McGraw, G. E.; Bernitt, D. L.; Hisatsune, I. C. J . Chem. Phys. 1965, 42, 237. (12) Ishiwata, T.; Fujiwara, I.; Naruge, Y . ;Obi, K.; Tanaka, I. J . Phys. Chem. 1983.87, 1349.
L7A
1 I
I I
I
1i.38A I I
Figure 3. Geometry of the second transition state for OH f HNO, (model 11). TABLE 11: Experimental and Calculated Values of the Rate Constants for OH + HNO," 1oi3k, cm3 molecule-' s-' 225 K 300 K 400 K I,, at TSl, amu A2 1.3 0.7 3.0 experimental model I 335 1000 3000 339 TST
2.0 2.1 2.1 2.1 2.5
1.3 1.4 1.4 1.4 1.5
1.o 1.o 1.o 1.o 1.o
model I1 335 1000 3000 339 TST
2.4 2.7 2.7 2.7 5.0
1.3 1.4 1.4 1.4 1.9
0.8 0.8 0.8 0.8 0.9
Experimental values from DeMore et ai." For model I, V2 = -0.6 kcal mol-'; For model 11, V2 = -1.0 kcal mol-'. bTreated as active rotations.
of the nitric acid. As it does so, bonding interaction develops between the radical and the N atom. This would cause the N-OH bond to lose what ?r character it possesses and permit the hydroxyl group in nitric acid to rotate freely. We could then form the four-center transition state shown in Figure 3. The 0-N distance of 1.7 A corresponds to a bond order of 'I2; this distance is taken as the reaction coordinate. The frequencies for the NO3group and the hydrogen motions are the same as in model I, with one exception. The low-frequency torsion of the hydroxyl H is replaced by a HON bend of the hydroxyl radical. Because of the long, weak ON bond, this frequency should be lower than for the other hydrogen bends; we have used 800 cm-'. The hydroxyl group oxygen motions include the reaction coordinate and two bending motions relative to the plane of the NO3 group. A typical heavy-atom bending frequency is 400 cm-I. Bending in the plane of the ring will encounter additional resistance due to the bridging H; we use 600 cm-I for this mode. The frequencies are summarized in Table I. Finally, we note two additional features of this model. First, since the hydroxyl radical can attack from either above or below the NO3 plane, there will be a reaction path degeneracy of two. Also, the transition state is very nearly a spherical top. Hence, we set the adiabatic moment equal to 78.3 amu AZ,which is the cube root of the product of moments. All three external rotations are adiabatic; there are no active rotations.8
IV. Results and Discussion for OH + HNOB The bwt fits to the experimental data were obtained with V, = -0.7 kcal mol-' for model I and -1 .O kcal mol-' for model 11. The results are presented in Table 11. In section I1 we observed that the appropriate adiabatic moment for TS1 should be much larger than the thermal average moment of 335 amu A2. From the results in Table I1 we see that increasing the moment to 1000 amu A2 has a significant effect only at the lowest temperatures.
The Journal of Physical Chemistry, Vol, 88, No. 25, 1984
6444
io00
10.0
1
0
Y
x
t
TfKl 50040b""300
OH+HN03
250
200
-1
I
I
Figure 4. Arrhenius plots of experimental and calculated rate constants for OH + HNOp. The adiabatic moment for TSl was set equal to 1000 amu A2. Increasing the moment to 3000 amu A2 has virtually no further effect. Hence, we conclude that at 1000 amu AZthe moment has become effectively infinite. In Figure 4 we present Arrhenius plots of rate constants calculated from models I and I1 along with the line recommended by De More et a1.13 Values of the activation energy for the range 225-300 K are the following: model I, -760 cal mol-'; model 11, -1140 cal mol-'; NASA panel,I3 -1550 cal mol-I. Even with the tighter model I1 we see that the calculated activation energy is not as negative as the experimental value. However, when we compare the actual values of the rate constants given in Table 11, we see that the discrepancies are on the order of 10%. The experimental uncertainty is larger than this; reported activation energies range from -1.3 kcal mol-' to -1.7 kcal m0l-'.~9~Also, most experiments have found rate constants at 400 K which are somewhat larger than the values calculated form the NASA panel's Arrhenius fit.13 This is because the NASA panel only fit data below 300 K, if all the data were included, a smaller activation energy would result. The NASA panel ascribes the high values of the rate constant at 400 K to curvature of the Arrhenius plot. Such curvature is to be expected at high temperatures.* To determine the extent of curvature predicted by our models, we have carried out calculations up to 1000 K; these results are shown in Figure 4. For neither model is there a significant deviation from linearity at 400 K. The rate constant for model I passes through a minimum between 600 and 700 K. For model I1 the rate constant does not level off until nearly 1000 K. Although Kurylo et aL3 report a definite curvature above 300 K, other worker^,,^-^ find no evidence of curvature. Considering the reported uncertainties in the data, experiments at higher temperatures will be necessary in order to determine the extent of curvature. Another possible source of curvature is that a second reaction path may become available at high temperatures. This could produce different products, such as H 2 0 2and NO,, or the same products by a different mechanism, such as direct abstraction of H by OH. The A factor for direct abstraction should be about cm3 molecule-' s-'. If this route has an activation energy 4X (13) DeMore, W. B.; Molina, M. J.; Watson, R. T.; Golden, D. M.; Harnpson, R. F.; Kurylo, M.; Howard, C. J.; Ravishankara, A. R. Jet Propulsion Laboratory,California Institute of Technology, Pasadena, CA, 1983, JPL Publ. 83-62.
Lamb et al. TABLE III: Calculated Activation Energies (in cal mol-') as a Function of Temperature for OH + HN03 model I model I1 temp range, K 200-225 290-310 400-450 600-700
E,,, -720 -780 -600 0.0
-E,tIRT 1.7 1.3
Ea,, -970
-E,,,IRT 2.3
-1270
2.1
0.7 0.0
-1490 -1 220
1.8 0.9
of 2 kcal mol-', the rate constant a t 300 K will be an order of magnitude less than the measured rate constant. However, at SO0 K, the rate constant will be about the same as that calculated from model 11. Thus, we would have substantial curvature above 400 K. The negative activation energy is due to the average energy of TS2 being less than the average energy of the reactants. In the center of mass system the average energy of TS2 is, at low temperatures, RT.' The average energy of the reactants is 4RT (three translational and five rotational degrees of freedom). Thus, a lower limit on the activation energy is -3RT. As the temperature increases and vibrational contributions to the energy become more important, the activation energy will increase and eventually become positive. In Table I11 we show activation energies for various temperature ranges for each of the two models. We see that, as expected, the ratio of E,,, to R T becomes less negative as T increases. However, since RT is getting larger, E,,, stays nearly constant over the range 200-400 K for model I and actually becomes more negative for model 11. Since the calculations are fairly laborious, we have examined an approximation in which all rotations are treated as being active. This eliminates the need to integrate over J as well as E; the results are presented in Table 11. We see that this gives essentially the same results as the more rigorous calculations with large moments of inertia for TS1. This is because the rapid equilibrillm between reactants and intermediate causes the intermediate to "forget" whether angular momentum was conserved in forming it. This is then reflected in the equilibrium between the intermediate and TS2. The situation is analogous to the high-pressure (transition-state theory) limit in unimolecular decomposition; in that case there is no difference between the adiabatic and active rotations. This does not mean that we may account for adiabatic rotations by including, as is sometimes done in RRKM calculation^,^^ a factor of the thermal partition function for these rotations. When V, is negative, no degrees of freedom will have a thermal distribution. As discussed in section 11, we carry out these calculations by beginning with the transition-state theory (TST) rate constant and calculating corrections. The TST rate constant corresponds to the case where the intermediate undergoes frequent collisions so that it is in thermal equilibrium with the reactants. For comparison, the TST rate constants are presented in Table 11. The first correction to the TST rate constant accounts for the fact that it is V I ,and not V,, that determines the minimum energy required for reaction to take place. This lowers the rate constant by about SO% at 225 K and 10% at 400 K. The second correction accounts for the fact that the first step proceeds at a finite rate so that the intermediate is in steady state rather than equilibrium. The TST rate constant is 3 orders of magnitude larger for TS1 than for TS2. Nevertheless, this correction amounts to nearly 10% at 200 K and about 2% at 400 K. This is a consequence of V, being less than V I ;for sufficiently low energies (a few tenths of a kilocalorie) the density of states will be less at TS1 than at TS2. For these energies the first step becomes rate limiting. These low energies contribute a disproportionate share of the rate constant; this is why the temperature dependence is negative.' As a result, the effect of TS1 is magnified, especially at low temperatures. Finally, we comment on the calculations for this reaction carried out by Marinelli and J o h s t ~ n .They ~ assumed that the reaction (14) Robinson, P. J.; Holbrook, K. A. "UnimolecularReactions"; Wiley: New York, 1972.
Negative Activation Energies and Curved Arrhenius Plots
The Journal of Physical Chemistry, Vol, 88, No. 25, 1984 6445
TABLE I V Vibrational Frequencies (in cm-I) and Assignments for HNOAAs Determined from F H R Data and Analogous Molecules
obsd frea" 1728 1304 803
922 3540 94 1 1397
freq of analogous molecules CH3CFONOqb (0)OOH' ., 1759 1301 804 709 633 455 928 860 303 3310 940 1440 152 I
freq used in current
calculation 1728 1304 803 709 633 45s 922 303 3540 941 1440 152
mode NOz asym str NO, svm str NO; Lis NO2 wag O-NOz str NO2 rock 0-0 str
N-0-OH bend OH str OH torsion HOO bend O-NOz torsion
"Reference 15. bReference 17. "Reference 16. proceeds via a direct mechanism through a transition state resembling that of our model I and with no threshold energy. They found that even when the all vibrational frequencies were 1000 cm-' or greater they were unable to get an activation energy of less than -3RT/2. With these frequencies they found an absolute value of the rate constant that is too low. In order to account for these discrepancies, they propose a rather unusual treatment of the reaction coordinate. Our results support an alternate explanation. First, the presence of an intermediate permits V, to be negative. This allows the average energy at TS2 to be as small as R T whereas in a direct mechanism the energy must be a t least 5RT/2 ( R T for the reaction coordinate, 3RT/2 for the overall rotations.* Thus, we can achieve activation energies as much as 3RT/2 more negative than with a direct mechanism. Second, our model I1 provides justification of the high vibrational frequencies needed to keep the average energy of TS2 near its minimum value.
V. Models and Results for OH + HN04 The calculations of rate constants for OH H N 0 4 is very similar to that for OH HNO,; hence less detail will be presented here. As in the case of O H + HNO,, we need to assess the vibtational frequencies and moments of inertia for TS1 and TS2 (Figure 1). However, in the case of peroxynitric acid there are several uncertainties in the structural properties of the reactants, transition states, and products of the reaction. While there are spectroscopic data and frequency assignments available on HNO,, such data are sparse and incomplete on H N 0 4 . Niki et al.15 have measured the infrared gas-phase spectrum of peroxynitric acid the vibrational frequencies that they observed are listed in Table IV, along with frequencies for the analogous molecules FONO, and CH,C(O)OOH. On the basis of on these frequency assignments, we choose the values of the frequencies for H N 0 4 as listed in Table IV, along with suggested mode assignments. Peroxynitric acid is probably planar, or very nearly so, due to the internal hydrogen bond formation. Peroxyacetic acid displays this same hydrogen bond formation and is planar.I6 The products of this reaction have not been identified experimentally. We follow the general assumption they are the thermodynamically most favorable ones, H 2 0 , NO,, and 0,. For the first transition state we use the hindered Gorin model described in section 111. The dipole moment for H N 0 4 is not known; we assume that it is similar to that for H N 0 3 . Applying the same arguments as in section 111, we find that the rate constant is only slightly sensitive to TSI. This will be discussed in more detail later.
+
+
(15) Niki, H.; Maker, P. D.; Savage, C. M.; Breitenbach, L. P. Chem. Phys. Lett. 1977, 45, 564. (16) Swern, D. "Organic Peroxides"; Wiley: New York, 1970; Vol. 1. (17) Shimanouchi, T. J . Phys. Chem. Ref. Data 1973, 2, 239.
I
\ !.70A'\
oog*:----; j0Yf L41A
\
1.73A
,300(
Figure 5. Geometry of the second transition state for OH
+ "0,.
TABLE V: Vibrational Frequencies (in cm-') for the Second Transition State for OH + HNO, mode assignt mode assignt NO, asym str 1700 OH torsion 1200 NO, sym str 1300 HOHbend 1200 NO, scis 780 HONbend 1200 NO, wag 710 HO str 2000 NO, rock 455 HON bend 800 NO, twist 250 (hydroxyl) OH str 3500 0 - O s t r 1100 (hydroxyl) other ring modes 1-250 {3-210
We have considered two models for the intermediate, analogous to models I and I1 of the O H H N 0 3 reaction. One is a planar, hydrogen-bonded, seven-membered ring. This model suffers from the same deficiencies as model I in the nitric acid case, Le., the second transition state is too loose to account for the observed temperature dependence. Also, the geometry of the ring is unfavorable; several bond angles are too large. The second model is analogous to model I1 in the O H + HNO, case (Figure 5). In this model we assume that the hydroxyl radical attack is from out of the plane of the peroxynitric acid. During the attack a bonding interaction develops between the radical and the N atom. The formation of products then takes place via a concerted process. The reaction coordinate is taken as one of the ring modes. The vibrational frequencies and moments of inertia used for the second transition state are listed in Table V. The NO2 frequencies in the intermediate were assigned from a combination of H N 0 4 and NO, frequencies.18 H and O H frequencies were assigned in the same manner as model I1 for OH HNO,. The ring modes are somewhat difficult to quantify. We have removed the light H atom in the consideration of these modes, leaving four heavy atoms with six associated frequencies. Four of these frequencies are stretches and two are deformations. One of the stretches is the reaction coordinate. Another ring mode, the 0-0 stretch, has been assigned at 1100 cm-' by using Badger's rule for a bond order of The other four modes were chosen to give the experimentally observed temperature dependence. This results in three 210-cm-' modes and one 250-cm-' mode. These values are only slightly lower than the low-frequency modes observed in loose-ring compounds and therefore seem to be reasonable assignments. The moments of inertia for TS2 are 64.0, 158.0, and 175.8 amu A'. In separating the adiabatic and active degrees of freedom, we must treat the molecule as either a spherical or a symmetric top.* Since our case is intermediate between these two cases, we must treat each of these as approximations. To test, them, we tried three different cases. With all three moments treated as adiabatic we have case 1 (spherical top) where the moment of inertia (121.1 amu A2) is the cube root of the product moment. Case 2 is the symmetric top case where two moments are adiabatic (175.8 amu A*) and one is active (57.33 amu A,). In addition,
+
+
(18) Herzberg, G. "Molecular Spectra and Molelcular Structure"; Van Nostrand Reinhold: New York, 1945; Vol. 2.
2or[
6446 The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 TABLE VI: Experimental and Calculated Values of the Rate Constants for OH HNO,"
+
lad
case
at T S l ,
amu A2
10'2k, cm3 molecule-' s-I 241 K 295 K 336K 8.8 5.3 4.1 4.4 3.3 6.8 5.0 3.6 8.3 8.9 5.2 3.7 9.1 3.1 5.3 37.0 11.6 6.3 9.4 5.4 3.8 7.8 4.9 3.6 ~~~~
exptl 1
TST 2 3
350 1000 3000 10000 3000
3000 3000
T (K) 2O :
3yO
2f5
2 7
OH + HNO,
--
,
Lamb et al.
THIS WORK
- - - - SMITH
'In
ET AL.
n
/
Y-0
"Experimental values from Smith et al.' V2 = -2.9 kcal (case 1, all TS2 rotations adiabatic; case 2, two TS2 rotations adiabatic; case 3, all TS2 rotations active). in case 3 we looked for the effect of treating all of the moments as active ( 121.1 amu A2), The calculation of the rate constants was carried out using the direct double-integration method of the first paper of this series.* It was necessary to use this method because the correction method, which was used in the nitric acid calculations, resulted in corrections to the TST rate constant that were too large, yielding an erroneous rate constant. The results for model 11are presented in Table VI. The best fit was achieved with V2= -2.9 kcal mol-'. In section I1 we observed that the appropriate adiabatic moment for TS1 should be much larger that the calculated thermal average moment of 350 amu A2. The rate constant is most sensitive to this when all rotations are adiabatic (case 1). For this case we examined the effect of varying the moment from 350 to 10000 amu A2. From the results shown in Table VI, we can see a similar situation to nitric acid in the trend of increasing the moment of TSl. All other calculations were run with a moment of 3000 amu A2.
Examination of the effect of changing the number of adiabatic modes (cases 1-3 above) shows that there is little difference between cases 1 and 2. Treating all rotations as active is a worse approximation than in the nitric acid case. As in the case of nitric acid, we consider two possibilities for the first step in the reaction scheme between reactants and intermediate. First, we make the approximating assumption that the rate of this step is very fast, i.e., that equilibrium is established. Second, we take into account the fact that this step actually proceeds at a finite rate such that the intermediate is in steady state rather than in equilibrium. Using the steady-state case rather than the equilibrium approximation lowers the rate constant by about 20% at 241 K and about 10% at 336 K. We use the steady-state rate constant in our results. In Figure 6 we present Arrhenius plots of rate constants calculated from model 11, along with the data of Smith et al.7 and Trevor et Trevor et al. measured the rate at pressures less than 1 torr and found a temperature-independent rate constant. Possible reasons for the discrepancy between the two experimental results, suggested by Smith et al., include the pressure difference between the two studies and the indirect technique employed by Trevor et al. in measuring HN04 concentrations. Values of the activation energy are -1.4 kcal mol-' for our calculation and -1.3 kcal mol-' for the work of Smith et al. The experimental Arrhenius plots show no evidence of curvature; the calculated plot has slight curvature. The experimental and calculated values of the activation energy are above the lower limit of -3RT, as in the case of nitric acid. It is significant to note that in both the H N 0 3 and H N 0 4 reactions the nonplanar model can explain the negative temperature dependence. This gives further credence to its applicability. VI. Pressure Dependence If these reactions do proceed via intermediate complexes, then their rates should depend on the pressure. This is because col(19)
Trevor, P. L.; Black, G.;Barker, J. R. J. Phys. Chem. 1982,86, 1661.
1
I
I 4
3
I
I
5
I
4000/~ ( K - ' ) Figure 6. Arrhenius plots of experimental and calculated rate constants for OH HN04 The adiabatic moment for TSl was set equal to 3000 amu A2. All moments for TS2 are adiabatic (case 1).
+
lisional deexcitation may remove enough energy from the intermediate so that it can no longer dissociate to reactants. Since the concentration of the intermediate will be greater at low temperatures, the pressure effect should be most pronounced at low temperature. Margitan and Watson4 have reported such a pressure dependence for O H HN03. At 238 K they find that the rate constant in 100 torr of H e is 35% larger than in 20 torr. At 298 K the enhancement is about 10%. They estimate that their measurements are accurate to 10%-15% of the total rate constant. However, Smith et aL7 report that, within lo%, there is no difference between the rate constants in 50 and 760 torr of He a t both 240 and 300 K. Robertshaw and SmithZoreport that at 300 K the rate constant in 3.8 atm of Ar is 3 times the low-pressure rate constant. They also report a low value of 8.4 X cm3 molecule-' s-' for the low-pressure rate constant. Finally, preliminary results from Molina et are indicative of up to a 15% increase in the rate constant between 50 and 760 torr bf H e and about a 30% increase between 10 and 50 torr of He at low temperature. Overall, these results are inconclusive, but they do indicate that the pressure enhancement in 1 atm of He is probably in the range of 10%-50%. If the well depth is less than 10 kcal mol-', the equilibrium constant for the reaction
+
OH
+ HN03 * H,N04
will be less than, roughly, cm3 molecule-'. In measurements of these rate constants, the concentrations of H N 0 3 is of the order of 1015molecules ~ m - thus, ~ ; less than one OH radical in lo7will be consumed in establishing this equilibrium. Consequently, the steady-state concentration of the intermediate will be quickly achieved and the pressure-dependent rate will be due to subsequent reactions of the intermediate. Since we have V, < 0, the deexcited intermediate can undergo unimolecular decomposition to products without having sufficient energy to be able to decompose to reactants. If the well is shallow, this process will be much faster than any bimolecular reaction of the intermediate. We therefore propose that in addition to reaction 3 the following sequence takes place:
OH + HNO,
-
3 -
H2N04*
H2N0,
k'
H 2 0 + NO3 (4)
H 2 N 0 4indicates those intermediate molecules which do not have (20) Robertshaw, J. S.; Smith, I. W. M. J . Phys. Chem. 1982, 86, 785. (21) Molina, M. J., private communication, 1984.
The Journal of Physical Chemistry, Vol. 88, No. 25, 1984 6447
Negative Activation Energies and Curved Arrhenius Plots
TABLE VII: Parameters Describing the Pressure Dependence of OH + HNOl in He for Various Values of the Well Depth. P 225 K 300 K V, kcal mol-' 3 5 7 3 5
om3 niolecule-' 1033kc,cm6 molecule-2s-I iOi3k,, cm3 molecule-'^-^ 10I9[M] I,2, molecule cm-3
0.57 0.38 2.4 63
3.0 2.0 2.4 12
13 8.7 2.4 2.8
0.23 0.13 0.50 37
k, is the low-pressure, third-order rate constant, k, is the difference between the low- and high-pressure limits, and of He at which half of k , is achieved. sufficient energy to dissociate to reactants. The two-step forming the stabilized intermediate has an overall termolecular rate constant k,. The overall rate constant, k+, of reaction 4 will add to that of reaction 1; it may be approximated as
where
k , is the difference between the high-pressure and zero-pressure limits of the total rate constant. Both of these quentities were calculated in section 111; the high-pressure limit is simply the transition-state theory rate constant. Equation 5 provides us with the falloff curve for reaction 4. As noted in section 111, the difference between the TST and low-pressure rate constant is greatest at low temperature. Thus, the high-pressure rate constant will have a more negative activation energy. Referring to Table 11, we see that the TST rate constant for model I has an activation energy of about -0.9 kcal mol-'. Thus, pressure effects cannot bring this model into agreement with the experimental activation energy. In the following we examine the pressure dependence of model 11. To determine how rapidly the high-pressure limit is reached, we wish to calculate [MI llz the concentration of M at which k+ = k, 12. This is given by k , may be calculated from collision theory and the equilibrium constant for formation of the excited intermediate. Letting X be the collision efficiency and Z be the collision frequency, we have k, = XZK,*
(6)
where K,* is the equlibrium constant for formation of excited intermediate from the reactants. In the first paper of this seriess we presented a method of calculating Kq* that is appropriate when the intermediate well is shallow and TSl is very loose. In this case, we treat all rotational degrees of freedom as being able to exchange energy freely. The validity of this approximation is supported by the rate constant calculations of section 111. From the results in part 1, we write the rate constant for collisional deactivation as k, =
w / Q * Q ~ ) S , ' ~+ EV) ~ ~ P ( - E / R T )d~
(7)
In order to evaluate eq 7 we need to specify the properties of the intermediate. One possibility is that the intermediate, H2N04, is the radical of the nitrogen analogue of phosphoric acid. If this is so, then the two OH groups are equivalent. Thus, if the hydroxyl radical is isotopically labeled, scrambling of the label will occur since the intermediate usually dissociates to reactants. However, in the case of O H HN04 the symmetrical intermediate could dissociate to "0, and H 0 2 . Since this overall reaction is 25 kcal mol-' exothermic, we expect that the intermediate will generally dissociate to products. Thus, the reaction should proceed at near collision frequency with zero activation energy if this were the mechanism. We therefore propose that in the intermediate the nitric acid (or peroxynitric acid) is similar to the unbound species. The O H is bound to this by a dipole-dipole interaction (- 3 kcal mol-') plus, perhaps, a weak covalent interaction with the nitrogen atom.
+
[M]l/2
is the concentration
The frequencies are essentially those of the reactants plus a weak N O stretch and O N 0 bends at 300 cm-', an H O N bend at 800 cm-l, and a free rotation of the O H group. The external rotations have moments of about 80 amu A2. We estimate N ( E V) using the Whitten-Rabinovitch approximation,22disregarding the two high frequencies. The result is
+
N(E
k , = k{k,/k,
7 5.1 2.9 0.50 1.7
1.2 0.68 0.50 7.3
+ V) = 1.4 x 10-'(E + V + 11113
with E in kcal mol-' and N in mol kcal-I. For He as a collision partner we use X = 0.1 and a collision radius of 4.1 8,in evaluating eq 7 numerically. The results at 225 and 300 K for several values of the well depth, V, are presented in Table VII. The results for [MI 1/2 are highly uncertain since neither the well depth nor the collision efficiency is known. However, certain general conclusions can be drawn. First, from the results of section 111, the high-pressure limiting rate will be about double the lowpressure limit a t 225 K and will be about 35% larger at 300 K. Half of this increase will be achieved at a pressure somewhere in the range of a few hundred torr to 10 atm of He. As the temperature decreases from 300 to 225 K, k, will increase by a factor of 3 and [MI ljz will increase by about 70%. These results are consistent with the results of Margitan and Watson and of Molina. If the well is somewhat deeper and the collision efficiency is larger, then it is possible that this reaction is actually near the high-pressure limit. This is consistent with the result of Molina, which indicate that the pressure dependence is much stronger below 50 torr than above 50 torr. The low rate constant and activation found by Connell and Howardz3provide support for this possibility, but the low-pressure results of Jourdain et aL6 contradict this. To test this possibility, we have carried out calculations on model I1 with V, = -0.8 kcal mol-'. In the high-pressure limit, we find kTST = 1.01 x IO-', exp(281/T) cm3 molecule-' s-l in good agreement with the experimental results cited earlier. In the low-pressue limit, ko = 7.17 X 10-9T1.94 cm3 rnolecule-'s-', At 225 K, with a well depth of 10 kcal mol-' and X = 0.25 for He, cm6 molecule-2 s-I, k, = 1.25 X lo-', we have k , = 1.5 X cm3 molecule-' s-', and [MI,/, corresponds to about 20 torr. Therefore, our model may be consistent with the reported rate constant for this reaction being in with the low-pressure or high-pressure limit. Experiments on the isotopic reaction, OD DNO,, may be able to resolve this question of pressure dependence. For this reaction, zero-point energy changes will make V,, in model 11, more negative by 0.8 kcal mol-'. We have carried out rate constant calculations using reported frequencies for DNO, (McGraw et al.), dividing all H atom frequencies at TS2 by 2'/* and letting Vz = -1.8 kcal mol-'. We find, at low pressure, k = 6.85 X 10-8T2,23 and, at high pressure, kTsT = 5.93 x 1 4 exp(1250/T). At 225 K, the difference between the low- and high-pressure limits is nearly a factor of 4; thus, it should be much easier to detect the pressure dependence for this reaction. For O H + HNO, the TST rate constant is much larger than the low-pressure rate constant (see Table VI). Since the TST rate constants are so large, the finite rate of the first step will probably
+
(22) Forst. W. "Theory of Unimolecular Reactions"; Academic Press: New York, 1973. (23) Connell, P.; Howard, C. J., unpublished results taken from ref 7.
6448
J. Phys. Chem. 1984, 88, 6448-6453
significantly reduce the high-pressure limit. The well depth for hence, this reaction is probably similar to that for O H “0,; the termolecular rate constant at low pressure will also be similar. This means that the relative pressure dependence for H N 0 4 will be much less than for HNO,. The high-pressure limit will only be approached at extremely high pressures.
water, the experimental results can be fit quite well. This model predicts that the activation energy will be negative up to a very high temperature (-900 K) unless a second mechanism, such as direct abstraction of H, becomes important. The high-pressure limit of the reaction O H HNO, M will be characterized by the establishing of thermal equilibrium between the reactants and the intermediate followed by unimolecular decomposition of the intermediate. This high-pressure limit will not be much higher than the low-pressure limit. The pressure dependence will be very weak but will be strongest at low temperatures. The present results cannot distinguish whether the reaction is near the low- or high-pressure limits. For OD DNO, the pressure dependence should be more pronounced. For OH HN04 a mechanism similar to that for O H H N 0 3 successfully predicts the rate constants as a function of temperature. The pressure dependence of this reaction is also expected to be very weak.
+
+
VII. Summary The rate constants and temperature dependences of the reactions of OH with HNO, and H N 0 4 may be calculated by assuming that the reactions proceed via an excited intermediate. In applying the method of calculation presented in the first paper of this series, it is appropriate to use a very large moment for the adiabatic rotations of the first transition state. Treating all rotations as active is an excellent approximation for the reaction OH + HNO,. For OH + H N 0 4 , where the rate of decomposition of the intermediate to reactants is not much faster than the rate of decomposition to products, this approximation is not as good. To produce a large negative activation energy, the second transition state must be quite tight. This permits both a very negative value of V2and a low internal energy. For OH HNO, a planar, six-membered, hydrogen-bonded ring cannot be made tight enough. However, if the reaction proceeds via attack of the O H at the nitrogen atom followed by four-center elimination of
+
+
+ +
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. We thank Dr. Mario Molina for the discussion and use of his data prior to their publication. Registry No. OH, 3352-57-6; “OS, 7697-37-2; HN04, 26404-66-0.
+
Quasi-Liquid Crystals of Thermochromic Spiropyrans. A Material Intermediate between Supercooled Liquids and Mesophases Felix P. Shvartsman and Valeri A. Krongauz* Department of Structural Chemistry, The Weizmann Institute of Science, Rehovot, 76100 Israel (Received: June 8, 1984; In Final Form: August 24, 1984)
Crystals of thermochromic spiropyrans containing mesogenic groups give isotropic melts. However, these materials form metastable films with mesogenic properties below the crystal melting points. A strong birefringence of the films appears concommitantly with the thermoconversion of 5% of the spiropyran into the colored merocyanine form. The mesophase is stabilized on alignment in an electrostatic field. Measurements of the optical anisotropy of the aligned films show that only the merocyanine molecules are oriented along the field, while the spiropyran molecules do not exhibit linear dichroism. Polar and nonpolar dyes introduced to the films exhibit behavior similar to that of the merocyanine molecules. It is concluded that the spiropyran molecules form small ordered domains similar to lyotropic micelles; the interaction between the domains endows the bulk material with “quasi-liquid-crystalline” (QLC) properties. The location of the merocyanine molecules in QLC structures is discussed.
-
Introduction The photo- and thermochromic conversions of spiropyrans into merocyanine dyes are among the most extensively investigated reversible reactions in solutions.’ The color changes are associated with the following reactions: NO.
SPIROPYRAN
MEROCYANINE
We have reported several new types of molecular organizates formed on photo- and thermochromic conversion of spiropyrans into m e r ~ c y a n i n e s , ~and - ~ based on the capability of the mero(1) Bertelson, R. C. In ‘Photochromism”; Brown, G.H., Ed.; Wiley: New York, 1971; Chapter 3. (2) Krongauz, V. A,; Goldburt, E. S . Nature (London) 1978, 271, 43. (3) Knogauz, V. A.; Fishman, S. N.; Goldburt, E. S . J. Phys. Chem. 1978, 82, 2469.
0022-3654/84/2088-6448$01.50/0
cyanine dyes to give “giant” molecular aggregates.8 One type is the so-called quasi-crystals, which is formed on irradiation of solutions of spiropyrans in an electrostatic field and consists of submicron-size globules aligned in a “string-of-beads” s t r u ~ t u r e . ~ ~ ~ The globules are composed of highly dipolar crystalline cores (assemblies of dipolar molecular stacks) covered by amorphous envelopes. The cores and envelopes both consist of different spiropyran-merocyanine complexes. Quasi-crystals exhibit optical nonlinearity and generate the second h a r m ~ n i c . ~ A second type of organizate occurs in atactic polyvinyl macromolecules bearing spiropyran side groupse5 These macromolecules crystallize by virtue of intermoleculear stacking of mero(4) Meredith, G. R.; Williams, D. J.; Fishman, S. M.; Goldburt, E. S.; Krongauz, V. A. J . Phys. Chem. 1983, 87, 1697. ( 5 ) Krongauz, V. A,; Goldburt, E. S . Macromolecules 1981, 14, 1382. (6) Goldburt, E. S.; Shvartsman, F. P.; Krongauz, V. A. Macromolecules, in nrea ._. r-
(7) Goldburt, E. S.; Shvartsman, F. P.; Fishman, S. N.; Krongauz, V. A. Macromolecules 1984, 17, 1225. (8) Sturmer, D. M.; Haeltine, D. W. In “The Theory of the Photographic Process”; Jammes, H., Ed.; Macmillan: New York, 1977; Chapters 7 and 8.
0 1984 American Chemical Society
