J. Phys. Chem. 1992, 96, 2512-2515
2572
It is interesting that, while both the activation energy and the preexponential factor for the CION02 reactionI2are quite different than those reported here for the ClNO, reaction, they are similar to those for the combined channels in the OH + ClNO reaction.I0 The similarity in the kinetic parameters for the OH ClONO, reaction to those for the combination of channels l a and l b of the O H ClNO reaction suggests that there are channels other than direct abstraction occurring in the OH + ClONO, reaction as well, such as those shown in Table IV. It would therefore be worthwhile to identify and measure the products of the O H + CIONO, reaction to confirm the presence of more than one reaction channel. In the troposphere, assuming a temperature of 298 K and a peak O H c ~ n c e n t r a t i o nof~ 1~ X lo7 radicals ~ m - the ~ , lifetimeI6 of ClNO, with respect to the reaction with O H is 7 = (k3[OH])-' = 33 days. Using absorption cross sections determined recently in this laboratory30 and published actinic flux value^^'*^^ and assuming a quantum yield of unity33over the actinic region, the
+
+
(29) Eisele, F. L.; Tanner, D. J. J . Geophys. Res. 1991, 96, 9295. (30) Ganske, J. A.; Berko, H. N.; Finlayson-Pitts, B. J. J . Geophys. Res.,
in press. (31) Demerjian, K . L.; Schere, K. L.; Peterson, J. T. Ado. Enoiron. Sci. Technol. 1980, 10, 369. (32) Peterson, J . J . US. Environmental Protection Agency Report No. EPA-600/4-76-025; U S . Government Printing Office: Washington, D.C., 1976. (33) Nelson, H. H.; Johnston, H. S . J . Phys. Chem. 1981, 85, 3891.
lifetime of CINO, with respect to photolysis ranges from 44 min to 31 h at solar zenith angles of 0-86', corresponding to noon and dusk, respectively. Thus, the O H reaction is too slow to compete with photolysis. Conclusions
The Arrhenius parameters and products of the gas-phase reaction of O H with ClN02 have been determined using a fast flow discharge system with two different wall coatings in the temperature range 259-348 K. The preexponential factor, activation energy, and formation of HOC1 as the product are consistent with chlorine abstraction being the only significant reaction channel occurring over this temperature range. This is in contrast to the O H ClNO reaction where both direct abstraction and a more complex reaction path are comparable at room temperature. The similarities between the preexponential factors and activation energies for the reaction of O H with ClNO, where two reaction channels are known to exist, and with C10N02, where product studies have not been reported, suggest that there are at least two significant reaction paths in the ClON0, reaction as well. Product studies of the O H + CIONO, reaction would shed some light on this possibility.
+
Acknowledgment. We are grateful to the National Science Foundation (Grant ATM-9005321) and especially to Dr. Richard Carrigan and Dr. Jarvis Moyers for support of this work and to T. Jayaweera for assistance with some of the experiments.
Semiclassical Calculations of Transport Coefficients and Rotational Relaxation of Nitrogen at High Temperatures Gert D. Billing* and Lichang Wang Department of Chemistry, H . C. 0rsted Institute, University of Copenhagen, DK 2100 Copenhagen 0, Denmark (Received: September 13, 1991; In Final Form: November 20, 1991)
Rate coefficients for rotational relaxation and transport phenomena have been calculated for nitrogen at high temperatures by using a semiclassical method, in which the translational and rotational motion of the two molecules are treated classically whereas the vibrational motion is quantized. The results are in much better agreement with high-temperature experimental data than those obtained with a classical treatment of the vibrational motion.
1. Introduction Transport coefficients and rotational relaxation of nitrogen have been calculated using a completely classical mechanical At low temperatures these results are in good agreement with experimental data, but the agreement becomes poorer at higher temperatures. This indicates whether higher order terms in the Wang Chang-Uhlenbeck theory, which has been used in the calculations of transport coefficients and rotational relaxation, and/or a quantum mechanical treatment of the vibrational motion should be considered at high temperatures. In the present calculation, we still introduce the first-order WCU theory, but consider a semiclassical treatment of the inelastic diatom-diatom collisions instead of the classical rigid rotor model or vibrating-rotor model used in ref 3. In the present paper the translational and rotational motion of the two diatomic molecules are treated classically; the vibrational motion is quantized. As
a result, a set of coupled first-order differential equations (equations of motion) for the classical part of the system and for the quantum amplitudes for the vibrational motion are solved. A Monte Carlo averaging procedure over initial variables is used in order to calculate average cross sections so as to obtain the transport coefficients and rotational relaxation in the temperature range from 200 to 2000 K. This is done by considering 10 values of the total energy, Le., the sum of the rotational energy of both molecules and the kinetic energy of the relative motion. At each energy about 250 trajectories are used. In section 2 the semiclassical theory and the first-order WCU results for the rate constants are given. In these calculations, an extended sitesite potential of Ling and Rigbf is used. The calculations of collision integrals and rate constants for nitrogen are presented in section 3. In the final section, we compare the results with experimental data and previous calculations. 2. Theory
( I ) Wang Chang, C . S.; Uhlenbeck, G. E.; de Boer, J . Sfudies in Sfotistical Mechanics; de Boer, J., Uhlenbeck, G. E., Eds.; North Holland Publishing Co.: Amsterdam, 1964; Vol. 2. (2) Nyeland, C.; Poulsen, L. L.; Billing, G. D. J . Phys. Chem. 1984, 88, 1216. (3) Nyeland, C.; Billing, G. D. J . Phys. Chem. 1988, 92, 1752.
0022-3654/92/2096-2S12$03.00/0
Here we shall give a brief description of the semiclassical method in which the rotational and translational motions are (4) Ling, M . S. H.; Rigby, M. Mol. Phys. 1984, 51, 8 5 5 .
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 6, I992 2573
N2 Rotational Relaxation and Transport Phenomena treated classically whereas the vibrational motion is quantized. For further details, the reader is referred to refs 5 and 6. The Hamiltonian for the collision of two nitrogen molecules is
H = - 1( P i 2r
plus corresponding equations for the Y and 2 components. We notice that the Lagrange multiplier is given by5
+ P$ + P i ) + m'+ HY' + HVib + HVib + 01
02
V(~,~l,YI,dl,~2,YZ,d2) (1) where is the reduced mass of the two molecules, Px, Py, Pz are the X, Y, 2 components of the momentum vector P for the relative motion, R the center of mass distance between the two molecules and ri ( i = 1, 2 ) vibrational coordinates, yi and & are spherical angles specifying the orientation of the two molecules in a body fixed system with R axis as the z axis. The Hamiltonian for the rotational motion is then expanded around the equilibrium vibrational distances pi.
The second term in eq 2 is the rotational-vibrational coupling term, which is included by introducing the rotationally disturbed vibrational wave functions
where the zeroth order wave function is given as
For the quantum degrees of freedom, we get from the time-dependent Schriidinger equation the following set of coupled equations for the amplitudes:
ih6,,,, = C ~ ~ ~ f l ~ l ~ f l ~ z I V ( ~+~ ~ ~ 2 , ~ ) I ~ f l l ~ f l z ~ nn ;;
i p,,
i h ~dfl~,dfl~,I~foldn,dn,~l~n,nz exp( + Ed, - E,, - E,$
1
(15)
where
is a Coriolis (centrifugal stretch) coupling term. For more details, the reader is referred to ref 7. From the above equations we obtain the information needed for a Monte Carlo evaluation of the average over the initial Boltmann distributions of the rotational and translational energy, i.e., to evaluate the collision integrals:
and
*,$, = - j k 1 ~ 3 ( & , I r-1F,I&,)
(5)
The total wave function for the vibrational degrees of freedom is now expanded as
* = C ~ f l l f l z ~ ~ ~ ~ ef xl lp~j -~ip1, ,~ +d En$) f l z ~ ~ (6) 2~ nlnz
where the collision cross section .",' dQ = 2r%'(b)bdb, Pff!(b)is the transition probability, and ti = Ei/kBT,y2 = 1/4mv2/kBT, and QA = C i exp(-ti). The following five collision integrals were considered for the calculation of transport coefficients, the rotational relaxation, and cross section for depolarized Rayleigh spectral line widths:
(KA421)
where E, =
+
(7)
We now consider the equations of motion for the classical degrees of freedom. This is done by introducing the effective potential, which depends only upon the classical coordinates: V ~ ~ ( R , Y I , ~ I , Y= Z(*lV*) *~Z~~)
+
+ +
+
i=1,2
where c, = cos y,, sc, = sin y, cos $,, ss, = sin y, sin +,, and A, is a Lagrange multiplier. The following 18 equations of motion are now obtained from eq 9:
aR
ax +
y4 sin2 5
x = px/r (10) ava, ac, av,, asc, av,, ass, --+--+-ac, ax asc, ax ass, ax
1 2
- -(At)2 sin2 $,
+
Ur2- YY' cos Fl)
(8)
In this manner, the effective Hamiltonian for the classical degrees of freedom in Cartesian coordinates is obtained. 1 1 HI= - ( P i P$ P;) -(p:, p;, Pi,) 2r ;=1,2 2mj V~,(R,~I,CZ,SCI,SC*,SSI,SSZ,~) + C ii(F? - xf - y f - zf) (9)
+ +
(18)
({(ti
- z)[(ti
- tj)Y2- (tk
- ~ / ) Y Y 'COS
= PJml
(22)
where tin, is the internal heat capacity. (b) The cross section for depolarized Rayleigh (DPR) scattering is defined as
(12) (c) The shear viscosity 4 is defined as 7-1
(5) Billing, G . D. Cbem. Phys. 1975, 9, 359. (6) Billing, G. D. G e m . Phys. 1974, 5, 244; Comput. Phys. Rep. 1984, 1 , 237: Compuf. Phys. Commun. 1987, 44, 121.
FII)
where 6 is the scattering angle, At = c k + t, - ti - t j , and 6 is the angle with which the direction of the rotational angular momentum of one of the molecules changes due to the collision. Using the first-order Sonine expansion of the Wang-Chang-Uhlenbeck theory, one can get the rate constants as defined below: (a) The relaxation collision number p is defined as
(11) XI
(21)
=
"5keT ( 17. sin2 E -
(7) Billing, G . D. Comput. Phys. Rep. 1984, I , 237
2574 The Journal of Physical Chemistry, Vol. 96, No. 6,1992
(d) The self-diffusion coefficient D is defined as
Billing and Wang TABLE I: Collision Integrals" as a Function of Temperature ~ _ _ _ T, K eq 18 eq 19 eq20 eq21 eq22 200 500 800 1100 1400 1700 2000
(e) The thermal conductivity is defined as c
0.441 0.381 0.353 0.335 0.322 0.313 0.304
" In units of where
0.348 0.356 0.360 0.363 0.365 0.366 0.368
1.60 1.85 1.99 2.09 2.17 2.24 2.30
0.696 0.842 0.929 0.993 1.04 1.09 1.12
0.814 0.906 0.956 0.992 1.02 1.04 1.06
cm3 s-I.
TABLE 11: R o h t i o ~ Relaxation l and Transport Properties as a Function of Temperature
3. Results We have in the present calculations used the extended site-site potential given by Ling and Rigby4 for the short-range part of the interaction, i.e. V,, = A exp(-aR - OR2)
(29)
where A = 22.568 au, a = 1.186 au, P = 0.09 au. R denotes the atom-atom distance between atoms in different molecules. The long-range part of the potential was taken as a quadrupolequadrupole interaction added to a R-6 dispersion potential. The C, dispersion constant was set to 24.4 au? and for the quadrupole moment Q the theoretical result -1.136 au was used. For vibrating molecules we need the quadrupole derivative, which was set to the experimental value9 (aQ/dr), = 0.933 au. This potential was also used in our previous rigid rotor and vibrating rotor calculations in which the vibrational motion was treated classically.3 A Monte Carlo procedure is used in order to average over the molecular collisions. The collision integral of eq 17 can be rewritten in the form
({I)= J d(BWF(I(U) exp(-PCr)
z d E ( k B V '
T, K 200 500 800 1100 1400 1700 2000
P
~DPR''
Ab
qC
od
3.7 4.9 5.7 6.4 6.9 7.3 7.7
50.6 32.8 26.2 22.5 20.1 18.3 16.9
15.1 33.4 50.1 66.0 81.2 96.0 110.4
10.8 23.3 34.7 45.7 55.7 65.5 75.0
0.0871 0.450 1.04 1.85 2.86 4.02 5.42
aIn A. units of cm2 s-' at 1 atm. P
I
W / m K . CInunits of 10" P a s . I
I
I
I
I
1
-1
0
SC
units of
-I
el-
(30)
1dj, dj2 dl dj', dj', N-:jJ21() ( 3 1 )
I is the orbital angular momentum, Ji, j'i are the initial and final rotational angular momenta, and Nu are the number of trajectories at a given value of the total 'classical" energy U,where -
60k I
In the present calculations, 10 energies (U) in the range 15010000 cm-' and about 250 trajectories at each energy are used. This gives a standard deviation of about 5% on the transport properties and one slightly larger on the rotational relaxation collision number. Using an analytical fit to the function FI,(U), it is possible to reduce the uncertainty slightly (see ref 2). We have in Table I given the collision integrals and in Table I1 the (8) Berns, R. M.; Van der Avoird, A. J . Chem. Phys. 1980, 72, 6107. (9) Reuter, D.; Jennings, D. E. J. Mol. Spectrosc. 1986, 115, 294. (IO) Prangsma, G.J.; Alberga, A. H.; Bennaker, J. J. M. Physico (Amsterdam) 1973, 64, 278. ( I 1 ) Carnevale, E. H.; Corey, C.; Larsen, G.J . Chem. Phys. 1967, 47, 2829. (12) Keijser, R. A. J.; Van den Hout, K. D.; deGroot, M.; Knaap, H. F. P.Physico (Amsterdam) 1974, 75, 51. (13) Vogd, E. Ber. Bunsen-Ges. Phys. Chem. 1984, 88, 997. (14) Touloukain, Y. S., Ed. Thermophysical Properties ojMatter, Plenum: New York, 1970; Vol. 3, 1933; Vol. 1 1 . (15) Shashkov, A. G.; Abrameko, T. N.; Aleinikova, V. I. J . Eng. Phys. 1985, 49, 83.
401 30-
'4
I
\
i
1010 0
400
800
1200 T/ K
1600
2000
Figure 2. Same as Figure 1 for the DPR cross sections (in A*) as a function of temperature. Experimental data at 300 K (ref 12) indicated by A. Values obtained with a classical mechanical treatment of the vibration are indicated at 1200 and 1600 K by 0.
The Journal of Physical Chemistry, Vol. 96, No. 6,1992 2575
N2 Rotational Relaxation and Transport Phenomena
6
7
70
F
LO
301
i
20
0
400
I
0
800
1200 TIK
1600
2000
Figure 3. Same as Figure 1 for the viscosity. Experimental data from refs 13 and 14 are indicated by A and 0 , respectively. 0 indicate values obtained by using a classical mechanical treatment of the vibrational motion. I
140
130
11 0 100 I
y
0
90 I
801 *:0 7 0 1
3200 1 &
.
1
I
I
I
I
0
400
800
1200
1600
2000
1
T/K
Figure 4. Same as Figure 1 for the thermal conductivity. Experimental data from refs 14 and 15 are indicated by 0 and A, respectively. 0
indicate values obtained by using a classical mechanical treatment of the vibrational motion. transport properties as a function of temperature obtained by running approximately 250 trajectories at 10 energies. 4. Discussion Figures 1-5 show the results of the present calculations and the experimental data, previous rigid rotor and vibrating rotor (VR) calculations for the rotational relaxation, depolarized Rayleigh spectral line widths, viscosity, thermal conductivity, and diffusion coefficient. For the rotational relaxation, the results of the present semiclassical calculations are smaller than those obtained with the rigid rotor classical mechanical method. However, the error bars on each of the calculations are about 5-10% for the rotational relaxation and about half as much for the other transport properties. Thus, at low temperatures, the rigid rotor and the present semiclassical calculations are within the error bars of each other. We also notice that there is a large scatter in the
LOO
800
1200 TIK
I
1
1600 2000
Figure 5. Diffusion coefficient as a function of temperature. The solid line indicates the present result. Rigid rotor results from ref 3 art indicated by a dashed line. The vibrating rotor results obtained using a classical mechanical treatment of the vibrational motion are shown at 1200 and 1600 K (0).
experimental data for the rotational relaxation number. What is more important is that the high-temperature data obtained with a vibrating rotor model in ref 3, where the vibration is treated classically, gives rotational relaxation numbers and transport properties which seem to be inconsistent with the low-temperature data. For the DPR cross sections, the present calculations are slightly higher (within the error bars of the calculation) than those obtained previously. The most interesting results however are those obtained for the viscosity and thermal conductivity. In both situations, the present calculations have given a satisfactory agreement with the experimental data and it is shown that the quantum corrections introduced in the semiclassical treatment do improve the agreement at higher temperatures. At lower temperatures the two calculations agree within the error bars of each. For the diffusion coefficients, the present and the previous rigid rotor calculations give the same results. In order to treat high-temperature transport properties correctly, it is necessary to include the vibrational degrees of freedom in the dynamical calculations. We have shown that if the vibrational motion is included in the dynamics it should be treated quantally rather than classically. A classical mechanical treatment of the vibrational motion fails due to both the quantum nature of this degree of freedom and/or problems with the zero point energy of the vibrational degree of freedom as mentioned in ref 3. Although methods for dealing with the zero point energy problem has recently been suggested,I6 we have not used these here, the reason being that the present semiclassical calculations are not much more expensive to carry out than the complete classical calculations. The reason for this is first, that in the latter case the vibrational motion has to be followed accurately around the turning points; i.e., the integration step length becomes usually very small. Second, the semiclassical calculations converge rapidly as far as the expansion of the vibrational part of the problem is concerned. Thus,only a small number of vibrational product states have to be included in the expansion of the total wave function.
Acknowledgment. The present research was supported by the Danish Research Academy and the Danish Natural Science Research Council. (16) Miller, W. H.; Hase, W. L.; Darling, C. L. J . Chem. Phys. 1989, 91, 2863.
