ESTIMATION OF THE ISOMERIZATION RATEOF NITROUSACID
269
Estimation of the Isomerization Rate of Nitrous Acid'
by I. C. Hisatsune Department of Chemistry, Whitnwre Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16806 (Received July 84, 1067)
The rate constant for the unimolecular isomerization of cis-nitrous acid to the trans acid has been calculated by using the transition-state model. The one-dimensional potentialenergy function for the barrier to internal rotation of the HO group and the necessary force constants and structural parameters of the transition complex were obtained from an earlier infrared study of isotopic nitrous acids. Quantum correction arising from the penetration of the potential barrier does not appear to be important. Over the temperature range from 0 to 50", the calculated rate constants fit the Arrhenius form 1.33 X 1013exp( - 10.9 kcal/RT) sec-l and 1.21 X l O l 3 exp( - 11.0 kcal/RT) sec-' for HN02 and DN02,respectively. The isomerization half-life is about three orders of magnitude shorter than that of the formation of the acids from NO, NOz, and H20. Introduction A quantitative kinetic study of the rapid gas-phase NO2 H20 = 2HN02, has not equilibrium, ?NO been reported, but Wayne and Yost2 have found that the forward reaction at about 24" has a half-life as short as 14 msec. Since the infrared spectra of all species in this equilibrium, including those of the cisand trans-nitrous acids as well as of other oxides like which may be present in the N203, N204,and "Os, equilibrium system, are well known, this reaction may be suitable for study by the technique of rapid-scan infrared spectroscopy reported recently by Jensen and Pimentel.3 However, during the temperature dependence study of the infrared bands of isomeric nitrous it was noted that the isomerization of the acid also appears to be very rapid. If this isomerization rate is indeed rapid and comparable to the rate of formation of the acid, then the interpretation of the kinetic data for the equilibrium may not be possible at the present time, since one can conceive of three possible competing equilibria in addition to those involving N208,1\T~O4,and "Os. (Equilibria for the latter three oxides can be minimized by using higher temperatures and lower partial pressures of the reactants.) Thus, an a priori estimation of the isomerization rate of nitrous acid will be helpful before one undertakes the experimental study of the formation equilibrium. I n the present paper, the unimolecular rate constant for the isomerization of cis-nitrous acid to the trans molecule has been calculated. A transition state model6 was employed from which the rate constant is given by
+
k
=
+
I'*(kT/h) (F'/F) exp( - AEo/RT)
(1)
where the various symbols have the usual meanings. Structural parameters, vibrational frequencies, and
the potential-energy function separating the two isomers were taken or estimated from our earlier work on nitrous acid.4 The transmission correction, I'*, was calculated by fitting an unsymmetrical Eckart potentialenergy function6 to the rotational barrier. Johnston and Heicklen' have transformed the Eckart potential into a convenient form to calculate tunneling corrections, and their results were used here. Results The fundamental vibrational frequencies of the isotopic cis-nitrous acids observed in our earlier study4 are listed in Table I. The structure of the cis isomer is not known, but the structure of the trans acid from microwave spectroscopy* is reported to be planar with r(H0) = 0.954 A, r(N-0) = 1.433 A, r(N=O) = 1.177 A, LHON = 102' 3', and L O N O = 110" 39'. The principal moments of inertia given in the table for our molecules were derived by assuming the same cis structure as was used before: namely, r(H0) = 0.96 A, r ( N - 0 ) = 1.46 A, r(N=O) = 1.20 A, LHON = 104",and LONO = 114". The rotational barrier potential function between (1) Work supported by Grant AP-18 from the National Center for Air Pollution Control, U. S.Public Health Service. (2) L. G. Wayne and D. M. Yost, J. Chem. Phys., 19, 41 (1951). (3) R. J. Jensen and G. C. Pimentel, J . Phys. Chem., 71, 1803 (1967). (4) G. E. McGraw, D. L. Bernitt, and I. C. Hisatsune, J . Chem. Phys., 45, 1392 (1966). (5) S.Glasstone, K. J. Laidler, and H. Eyring, "The Theory of Rate Processes," McGraw-Hill Book Co., Inc., New York, N. Y., 1941. (6) C. Eckart, Phys. Rev., 35, 1303 (1930). (7) H. S. Johnston and J. Heicklen, J. Phys. Chem., 66, 532 (1962); in the reference, the term in brackets in eq 15 should be [al(f 1) a2]'/2. (8) A. P. Cox and R. L. Kuczkowski, J . A m . Chem. Soc., 88, 5071 (1966).
+
Volume 7.2, Number 1 January 1968
270
I. C. HISATSUNE
Table I: Vibrational Frequencies (cm-1) and Principal Moments of Inertia (amu A2)
YI
~2
~3 ~4
YS
Y6
(OH) (N=O) (H-0-N) (N-0) (0-N-0) (torsion)
cis-
cis-
HNOi
DNOi
3424 1640 1261 853 608 638 6.158 41.15 47.31
2525 1625 1008 814 601 504 7.244 42.08 49.33
HNOi*
3544 1628 1263 849 634 (611i)a (705i)b 6.134 41.57 46.10
DNOa*
2581 1628 1001 797 626 (466i)" (538i)b 6.952 43.96 48.00
a Torsion force constant = -0.1604 mdyne A/radian2; see text. b Torsion force constant = -0.2134 mdyne A/radian2; see text.
the trans and the cis isomers was deduced previously from the infrared spectra4 to be 2V(kcal/mole) = 1.17(1
- cos e)
+
where e = 0" corresponds to the trans isomer minimum potential energy, which is 0.389 kcal/mole below the cis minimum. Since the barrier maximum was a t 0 = 86" from the cis minimum, this angle was taken for the nonplanarity of the transition complex, i.e., the angle between the planes defined by O N 0 and HON in the complex. The remaining structural parameters of the complex were taken to be essentially intermediate between those of the cis and trans acids: r(H0) = 0.96 A, r(N-0) = 1.45A,r(N=O) = 1.20A, LHON = 103", and L O N 0 = 112". The principal moments of inertia of the complex, listed in Table I, were calculated with these parameters. Once the structure of the transition complex is fixed, its vibrational frequencies can be calculated from estimated force constants. I n the present case, the cis "overlay" force constants, derived earlier from leastsquares fitting of both the cis and trans acid fundamental frequencies, were used (See Table I11 of ref 4). The calculated frequencies for the normal and the heavy-acid transition complex are given in Table I. Two sets of imaginary torsional frequencies are also listed here, and these were calculated from two different negative torsional force constants whose origin is described below. The cosine rotational barrier given by eq 2 is illustrated in Figure l by the solid curve. Here, the potential energy is in units of V* = 11.15 kcal/mole and the torsional angle x = e is in radians. These notations, as well as the subsequent ones in this section, are the same as those used by Johnston and Heicklen.' In order to estimate the tunneling corrections for this The Journal of Physical Chemistry
-2.0
-I,O X (radian)
Figure 1. Rotational potential-energy barrier in nitrous acid: the solid curve is the cosine barrier, and the dashed and dotted curves are Eckart potential functions with force constants of -0.1604 and -0.2134 mdyne A/radianZ, respectively.
rotational barrier, an unsymmetrical Eckart potentia1 function
v
= --Ay(l
- y)-1 - By(1 - y)-2
(3) for which the transmission coefficient, K , is given by a closed form was fitted to the cosine barrier. Here, A and B are functions of barrier heights from the cis and trans minima, respectively, and y is an exponential function of x, the barrier height, and the force constant at the barrier maximum. The second derivative at the maximum of the cosine barrier gives a value of -0.1604 mdyne A/radian2 for the latter force constant. The resulting Eckart function is shown in Figure 1 by the dashed curve. Since this function is broader than the cosine barrier, a second Eckart potential function was derived by fitting the barrier maximum, the inflection point on the cis side of the cosine barrier, and the two barrier minima. For this function, which is illustrated by the dotted curve in Figure 1, the force constant at the barrier maximum was -0.2134 mdyne A/radian2. The quantum-mechanical transmission coefficient, derived from the two force constants described above, is shown as a function of scaled initial energies in Figure 2. The classical transmission coefficient would be a step-function at E/V* = 1.0. It is apparent that loo-- *,*.--------- $'
-- /,
0.8
--;
0.6 -I
/-0,4
-,/ --0.2 -I
I
I
I
p'
..3'
1
1
1
,
ESTIMATION OF 'THE ISOMERIZATION RATEOF NITROUSACID Table 11: Calcuhted Rate Constants T,"K
HNOi
X 10-4,aec-1--DNO2
273 283 293 298 303 313 323
2.46 4.99 9.67 13.2 17.9 32.1 55.2
2.00 4.09 7.94 10.9 14.8 26.5 45.7
-k
kh/kd
1.23 1.22 1.22 1.21 1.21 1.21 1.21
tunneling is significant only when the energy of the reactant molecule is within about 10% of the barrier maximum. Johnston and Heicklen' have also tabulated quantum correction factoirs I'*, obtained by integrating the transmission coefficients over a Boltzmann distribution of incident molecules. These correction factors are expressed in terma of a's and u*, which are proportional, respectively, to the ratio of barrier height to the imaginary frequency and to the ratio of the imaginary frequency to the absolute temperature. Examination of the tabulatedl results indicates that for a given set of a's, the correction factor increases as u* becomes larger, Le., larger imaginary frequencies and lower temperatures. I n our case, the largest imaginary frequency given in Table :I: corresponds to u* = 3.4 at 25". For this value of u*, the published table extends only to a = 20, while lour a's are near 35 and 41 for the two Eckart potential functions. However, the numerical trend in the table suggests that the quantum correction factor in the present case must be at most about two and more probably less. Since the variation, a t a given temperature, of experimental rate constants for complex reactions in of this order, eq 1 can be used without the r*to get a reasonable estimate of the isomerization rate constants. Results calculated on this basis are presented in Table 11. Over the temperature range from 0 to 50", which appears to be a convenient range to study experimentally the nitrous acid formation and isomerization reactions, the calculated isomerization rate constants listed in Table I1 for HN02 and DNO2 fit well the Arrhenius
27 1
form of 1.33 X 1013 exp(-10.9 kcal/RT) sec-1 and 1.21 X 1013exp(- 11.0 kcal/RT) sec-', respectively. The only experimental result with which our calculated rate constants can be compared appears to be that from a study of the decomposition of chloroformic acid by Jensen and Pimentel.3 These authors found this reaction to be first order and interpreted their results on the basis of a rate-determining step involving the isomerization of the cis-chloroformic acid to the trans acid. The rate constant they obtained over the temperature range from 15 to 70" was 5 X 1013 exp (-14 kcal/RT) sec-l. Since the rotational barrier in cis-chloroformic acid is probably very similar to that in cis-formic acid, which in turn has a barrier9 2.2 kcal/ mole greater than that in cis-nitrous acid, the result obtained by Jensen and Pimentel suggests that our calculated rate constants are reasonable. As was stated in the Introduction, there has been no quantitative kinetic study of the formation of nitrous acid from NO, NO2, and HzO. However, Wayne and Yost2 have reported that the formation half-life at about 24" can be as short as 14 msec, and that the reaction rate a t the same temperature is approximately 10-2 atm/sec when the partial pressures of the reactants are all about atm. The calculated isomerization half-life of cis-nitrous acid, on the other hand, is about 5 psec at 25", and if the partiaI pressure of the cis acid is about atm, then the isomerization rate will be about lo3 atm/sec. Thus, even though the results of Wayne and Yost may be opened to question because their experimental points show considerable scatter, it appears, at the present time, that the isomerization reaction of nitrous acid is faster by about five orders of magnitude than the rate of formation of the acid from NO, N02, and H2O.
Acknowledgments. I am grateful to Dr. D. L. Bernitt of our Computation Center for calculating the vibrational frequencies of the transition complex, and to Professor Julian Heicklen for helpful discussions. It is a pleasure to acknowledge the continued financial support of our studies by the National Center for Air Pollution Control. (9) L, Bernitt, K. o. Hartman, and I. c. Hisatsune, J , Chem.
,,
Phya., 42, 3653 (1965).
Volume 72,Number 1 January 1968
