J. Phys. Chem. 1989, 93, 3141-3145
3141
Theoretical and Experimental Determination of the Lithium and Sodium Superoxide Bond Dissociation Energies John M. C. Plane,* B. Rajasekhar, Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, Florida 33149
and Libero Bartolotti Department of Chemistry, University of Miami, Coral Gables, Florida 331 24 (Received: July 18, 1988; In Final Form: October 19, 1988)
The bond energies of the alkali-metal superoxides are poorly known. In this paper, ab initio calculations using the unrestricted Hartree-Fock and fourth-order Mdler-Plesset perturbation theories have been employed to study the structure and electronic energies of Li02 and NaO,. Several standard large split-valence Gaussian basis sets were used in the calculations. Comparison of the Hartree-Fock geometries and vibrational frequencies with experimental measurements indicates that the 6-3 11G basis set is optimal. The calculated bond energies are Do(Li-02) = 296 kJ mol-' and Do(Na+) = 185 kJ mol-'. Experimental estimates of the lower limits of these bond energies are then derived from a kinetic study in the gas phase of the reactions Li,Na + O2 + N2 at about 1100 K, yielding DO(Li-O2) 1. 306 kJ mol-' and Do(Na-02) 1 202 kJ mol-'. The very good agreement between the experimental lower limits and theoretical bond energies (3% for Do(Li-02) and 9% for Do(Na-0,)) implies that these lower limits are in fact very close to the actual bond energies. The recommended values of Do(Li-02) = 306 kJ mol-' and Do(Na-02) = 202 kJ mol-' are both about 40% higher than currently accepted values derived from studies of alkali metals in flames.
Introduction The alkali-metal superoxides Li02 and N a 0 2 are of theoretical interest and are important in the atmospheric and combustion systems described below. Both superoxides have been observed by infrared, Raman, and ESR spectroscopy of the molecule formed when the alkali-metal vapor and O2 are coadsorbed onto an inert gas matrix at low temperature^.'-^ Their formation in the gas phase has been inferred from the observation that the reaction between the alkali-metal atoms and O2is termolecular, implying a recombination process.610 In the solid phase, N a 0 2 can be obtained only by reacting N a 2 0 zwith O2at 300 atm and 800 K, while Li02 cannot be isolated." The superoxides are known from matrix isolation studies,'-s as well as the semiempirical calculations of AlexanderI2 and a number of ab initio calculations on LiO2,'*lS (1) Andrews, L. J. Chem. Phys. 1969, 50, 4288. Hatzenbuhler, D. A.; Andrews, L. Ibid. 1972, 56, 3398. Smardzewski, R. R.; Andrews, L. Ibid. 1972, 57, 1327. Smardzewski, R. R.; Andrews, L. Ibid. 1973, 58, 2258. Andrews, L. J. Phys. Chem. 1969,73,3922. Smardzewski, R. R.;Andrews, L. Ibid. 1973, 77, 801. Andrews, L.; Hwang, J.-T.; Trindle, C. Ibid. 1973, 77, 1065. (2) Jacox, M. E.; Milligan, D. E. Chem. Phys. Lett. 1972, 14, 518. (3) Huber, H.; Ozin, G. A. J. Mol. Spectrosc. 1972, 41, 595. (4) Adrian, F. J.; Cochran, E. L.; Bowers, V. A.; J. Chem. Phys. 1973,59, 56. (5) Lindsay, D. M.; Herschbach, D. R.; Kwiram, A. L. Chem. Phys. Lett. 1974,25, 175; J . Phys. Chem. 1983,87,2113. Lindsay, D. M.; Garland, D. A. J. Phys. Chem. 1987, 91, 6158. (6) Husain, D.; Plane, J. M. C. J. Chem. Soc., Faraday Trans. 2 1982, 78, 163. Husain, D.; Marshall, P.; Plane, J. M. C. Ibid. 1985, 81, 301; J . Photochem. 1986, 32, 1. (7) Silver, J. A.; Zahniser, M. S.; Stanton, A. C.; Kolb, C. E. Proceedings
of the 20th International Symposium on Combustion; The Combustion Institute: Pittsburgh, PA, 1984; p 605. (8) Kramer, S. D.; Lehmann, B. E.; Hurst, G. S.; Payne, M. G.; Young, J. P. J. Chem. Phys. 1982, 76, 3614. (9) Plane, J. M. C.;Rajasekhar, B. J. Phys. Chem. 1988, 92, 3884. (10) Plane, J. M. C.; Rajasekhar, B. J. Phys. Chem., accompanying paper in this issue. (1 1) Cotton, F. A,; Wilkinson, G . Aduanced Inorganic Chemistry: A Comprehensiue Text; Interscience: New York, 1972. (12) Alexander, M. H. J. Chem. Phys. 1978, 69, 3502. (13) Billingsky, F. P.; Trindle, C. J. Phys. Chem. 1972, 76, 2995. (14) ONeill, S. V.; Schaefer, H. F.; Bender, C. F. J. Chem. Phys. 1973, 59, 3608. (15) Grow, D. T.; Pitzer, R. M. J. Chem. Phys. 1977, 67, 4019.
0022-3654/89/2093-3141$01.50/0
to exist as an ion pair A+(02)-, in an isosceles-triangular (C,) configuration. In the ground 2A2state of the molecule the alkali-metal outer s electron is almost completely transferred into The ring structure of the the 3a antibonding orbital of the 02.12 molecules makes them particularly interesting from a theoretical point of view. The geometries and vibrational frequencies of both molecules have been measured accurately in the matrix isolation studies referred to above.'-5 By contrast, the bond dissociation energies are poorly known and have mostly been estimated from studies on Li and N a seeded into oxygen-rich flames. In the case of Li02, Phillips and co-workersI6 obtained a value of Do(Li-02) = 222 f 25 kJ mol-' from flame photometric experiments. This was in good agreement with an estimated value of 220 kJ mol-' for the bond energy by Alexander,I2 based on the Hartree-Fock calculation of L i 0 2 by Grow and Pitzer.Is Phillips and co~ o r k e r s ' ~ Jalso ' determined a value for Do(Na-02) = 234 f 13 kJ mol-'. In this case, however Alexanderlz argued that the bond energy would be only about 150 kJ mol-', by making assumptions about the ionic nature of the Na+02- bond. This considerably lower value has since been supported by further work on sodium in oxygen-rich flames, following the demonstration by Husain and Plane6 that the reaction Na
+ O2 + N2
-
Na02
+ N2
(1)
was about 3 orders of magnitude faster than had been deduced in an early flame experiment by Carabetta and Kaskan.I8 JensenIg reinterpreted this flame data'* using the revised kl and obtained Do(Na-02) = 170 f 25 kJ mol-'. Hynes et aLzoobtained a value of Do(Na-0,) = 146 f 21 kJ mol-' from a new set of flame measurements. Finally, an approximate upper limit to Do(Na-02) of 184 kJ mol-' was deduced by Figger et aL2' from a molecular beam experiment. (16) Dougherty, G. J.; McEwan, M. J.; Phillips, L. F. Combust. Ffame 1973, 21, 253. (17) McEwan, M. J.; Phillips, L. F. Trans Faraday Soc. 1966, 62, 1717. (18) Carabetta, R.; Kaskan, W. E. J. Phys. Chem. 1968, 72, 2483. (19) Jensen, D. E. J. Chem. Soc., Faraday Tram. 1 1982, 78, 2835. (20) Hynes, A. J.; Steinberg, M.; Schofield, K. J . Chem. Phys. 1984,80, 2585. (21) Figger, H.; Schrepp, W.; Zhu, X . J. Chem. Phys. 1983, 79, 1320.
0 1989 American Chemical Society
3142
The Journal of Physical Chemistry, Vol. 93, No. 8,1989
An important reason for knowing the bond dissociation energies of these molecules accurately is in order to characterize satisfactorily the flame chemistries of the alkali metals under many common flame conditions. The superoxides are formed rapidly in oxygen-rich flames through recombination reactions such as reaction The ability of the superoxides to act as metal sinks in the afterburn region of a flame will depend on their unimolecular dissociation rates at high temperatures and their rates of reaction with abundant flame radicals such as 0, H, and OH; the rates of both types of reactions will be affected significantly by the superoxide bond strengths. In particular, the Li/Na ratio technique has been established as an important method for determining the hydrogen atom concentration in the afterburn region of a flame.22-24 The method is based on the assumption that Li and LiOH are in equilibrium in the flame. We have discussed elsewhereg how the Li/LiOH balance may be perturbed through the rapid formation of Li02 and subsequent production of LiOH through reaction with H atoms, perhaps invalidating this technique in certain types of flames. A second system in which the alkali-metal superoxides play a major role is in the chemistry of the alkali metals in the mesosphere. The major source of the alkali metal is believed to be meteoritic,25and the free metal atoms have been observed in layers at about 90 km altitude.26 A number of recent atmospheric model^^'-^^ have indicated that the principal sinks for the metal atoms immediately beneath this layer are the superoxides, formed by recombination reactions with 02.As in the case of flames, the bond strengths of the superoxides will substantially affect their rates of reaction with abundant mesospheric species such as 0 and H atoms, and hence their effectiveness as sinks of the metal atoms. In this paper, a set of ab initio calculations on the ground states of both molecules is presented first, using a series of standard split-valence Gaussian basis sets. Equilibrium geometries and vibrational frequencies are computed at the self-consistent-field (SCF) level and compared with the experimental The optimal basis sets are then used to calculate the bond energies of both molecules with respect to dissociation to the ions, including correlation energy corrections. The bond energies with respect to neutrals are then obtained by use of accurately known ionization energies and electron affinities. The second part of the paper consists of an experimental derivation of the lower limits to these bond energies from a series of high-temperature kinetic studies of reaction 1 and the analogous reaction Li
+ O2 + N2
-
L i 0 2 + N2
(2)
The third part of the paper compares and discusses the results of the theory and experiment and then concludes with recommended bond energies for both superoxides.
Theoretical Determination of Do(Na-02) and Do(Li-02) The major features of the alkali-metal superoxide potential surfaces have been discussed in detail by Alexander,I2 who also reviewed the earlier ab initio investigations of LiO2.I3-l5 The surface contains two minima, corresponding to a linear A-0-0 molecule (Cmu)and to an isosceles-triangulr geometry ( C 2 J . All the ~ t u d i e s ' ~ agree - ' ~ that the absolute energy minimum corresponds to the 2A2surface in CZugeometry.
Plane et al. TABLE I: Comparison of ab Initio and Experimental Geometries of LiOj NaO,, and 0, A%"
rAd/A
r w / A
A%a
2.2 0.3
1.33 1.36 1.34
1.9 0.9
2.8 3.3
1.33 1.35 1.34
1.5 0.9
1.33 1.35 1.34
1.5 0.6
Li02 experiment[ UHFi6-3 1G UHF/6-311G
1.77 1.81 1.78
experiment I UHF/6-31G UHFi6-311G
2.07 2.13 2.14
NaO,
02-
experimenP UHFi6-31G UHF/6-311G
Percentage difference from the experimental value.
Although some covalent character must be present in the ground-state alkali-metal superoxides, since the purely ionic and covalent diabatic curves cross at only 2.95 and 3.06 8, for Li02 and Na02, respectively,I2 there is strong e~perimentall-~ and theoreti~all~-'~ evidence that they are predominantly ionic. Hence, the energy required to dissociate to the ions is expected to correlate well with the equilibrium molecular geometries. The overall dissociation energy to the neutral metal atom A and O2is then given by Do(A-02) = E(A+) + E(O,-) - E(A02)
- IP(A)
+ EA(02) + ZPE
(3)
where E( ) indicates the absolute electronic energy; IP(A) is the ionization potential of the metal and EA(02) is the electron affinity of oxygen, both of which are accurately known from experiment;30 ZPE is the difference between the zero-point vibrational energies of 02-and AOz. For eq 3 to produce accurate bond energies, the basis sets must approach the Hartree-Fock (HF) limit. We used a number of the standard split-valence basis sets available within the Gaussian 86 set of programs.31 Langhoff and B a ~ s c h l i c h e rhave ~ ~ pointed out that, in bond energy calculations on molecules of this kind, the atoms approach the H F limit much more quickly than the molecular systems, and so the SCF values of Do almost invariably increase monotonically with improvements in basis set quality. We found a satisfactory convergence of Do values (within 0.08 eV) for both Li02 and N a 0 2 using the 6-31G and 6-31 1G basis sets (see below). These basis sets comprise inner-shell functions each written in terms of a linear combination of six Gaussians and valence shells represented by numbers of Gaussian primitive~.~~~~' The equilibrium geometries of both molecules were first calculated from an unrestricted Hartree-Fock (UHF) S C F optimization which determined that the ground states of both molecules are 'A2 in C2, symmetry. The corresponding electron configurations are Li02 1b, (a)4aI(a)4al (P)5a1( a V a ~ ( f l1)a2(a) 1b l ( W ~ d a ) 3 b 2 ( P )
NaOl
(22) Bulewicz, E.; James, C. G.; Sugden, T. M. Proc. R . SOC.London, A 1956, 235, 89.
(23) McEwan, M. J.; Phillips, L. F. Combust. Flame 1965, 9, 420; Ibid.
1967, 11, 63.
(24) Muller, C. H.; Schofield, K.; Steinerg, M. J . Chem. Phys. 1980, 72, 6620. (25) Goldberg, R. A.; Aikin, A. C. Science 1973, 180, 294. (26) Sandford, M. C. W.; Gibson, A. J. J. Atmos. Terr. Phys. 1970, 32, 1423. Jegou, J.-P.; Chanin, M.-L.; Megie, G.; Blamont, J. E. Geophys. Res. Lett. 1980, 7 , 995. (27) Liu, S. C.; Reid, G. C. Geophys. Res. Lett. 1979, 6 , 283. (28) Thomas, L.; Isherwood, M. C.; Bowman, M. R. J . Atmos. Terr. Phys. 1983, 45, 587. (29) Swider, W. J. Geophys. Res. 1987, 92, 5621.
(30) JANAF Thermochemical Tables, 3rd ed.; Chase, Jr., M. W., Davies, C. A,, Downey, J. R., Jr., Frurip, D. J., McDonald, R. A,, Syverud, A. N., Eds.; J . Phys. Chem. Re5 Data 1985, 14. (31) Gaussian 86; Frisch, M. J.; Binkley, J. S.; Schlegel, H. B.;Raghavachari, K.;Melius, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, C. M.; Kahn, L. R.; Defrees, D. J.; Seeger, R.; Whiteside, R. A,; Fox, D. J.; Fleuder, E. M.; Pople, J. A. Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh PA, 1984. (32) Langhoff, S. R.; Bauschlicher, C. W. J. Chem. Phys. 1986,84,4474. (33) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. A b Inirio
Molecular Orbital Theory; Wiley-Interscience: New York, 1986.
L i 0 2 and NaO, Bond Dissociation Energies
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 3143 this accounts for most of the decrease in D0(Na-02). A minor change in Do(Na-02) from 185 kJ mol-' is therefore likely on increasing the flexibility of the basis set beyond the 6-3 11G level.
TABLE 11: Ab Initio Calculations of the Mulliken Electron Populations and Dipole Moments of Li02 and NaOl
40
kDID
-0.3812 -0.4136
6.7415 8.7759
-0.3704 -0.4203
6.4388 8.8141
qA
Li02 UHF/6-3 1G UHF/6-31 IG
0.7623 0.8272
Experimental Determination of the Bond Energies Method. Both reactions 1 and 2 were investigated by the
Na02 UHF/6-31G UHF/6-31 IG
0.7409 0.8407
TABLE III: Comparison of ab Initio and Experimental Vibrational Freauencies for LiO, and NaOl B2/cm-l A%. A,/cm-l A%. A,/cm-I A%'
Li02 experiment' UHF/6-31G UHF/6-311G
494.4 281.0 362.4
experiment I
332.8 244.3 274.8
UHF/6-31G UHF/6-311G a
-43.1 -26.7
698.8 771.0 771.7
10.3 10.4
1097 1215.5 1269.0
10.8 15.7
13.8 12.3
1080 1231.6 1280.6
14.0 18.6
Na02 -26.6 -17.4
390.7 444.8 438.8
Percentage difference from the experimental value.
Table I compares the U H F equilibrium geometries using the 6-3 1 G and 6-3 11G basis sets with the experimental geometries for Li02, NaO,, and O T . ' ~ ~There ' is very good overall agreement, particularly for the 6-31G basis set calculation of Li02 and 02-, as expected because of the increased flexibility of the repreen tat ion.^^ We also performed U H F optimizations using a basis set with single first polarization functions (6-3 1G*). Although the energies of both the superoxide molecules and the 02-anion were improved, the (0-0)-bond lengths in the optimized geometries were significantly shorter than the e~perimental.',~~ Finally, a basis set containing diffuse functions (6-31SG) was employed. energy but This produced an expected i m ~ r o v e m e n in t ~ the ~ 02no significant improvement in the superoxide energy, leading to Do values that were much lower than with the 6-31G or 6-311G sets. The Mulliken charge distributions and the dipole moments of both superoxides are listed in Table 11, confirming their highly ionic nature.'-5 Table I11 compares the U H F vibrational frequencies using both basis sets with experiment. The U H F calculations with the 6-3 11G basis set produce better agreement for both low-frequency vibrations. The two A2 frequencies are consistently higher than the experimental by 10-18%. The size of this discrepancy is commonly observed in comparisons of H F frequencies with experimental measurement^^^ and would be reduced significantly if the measured frequencies's2were corrected for anharmonicity. By contrast, the B2 frequencies are uniformly lower than the experimental values' by 17-44%, which is unusual in H F c a l c ~ 1 a t i o n s . l ~ ~ ~ ~ Fourth-order Moller-Plesset (UMP4) perturbation theory34was then used to determine the correlation energy correction at the UHF geometries. Table IV illustrates the calculations of the bond dissociation energies from eq 3 at the UHF and UMP4 levels using the 6-3 1G and 6-3 11G basis sets. The correlation energy correction is seen to increase Do by 15-30 kJ mol-'. In the case of Li02 there is a very close convergence (0.9 kJ mol-' at the UMP4 level). This arises because the Li+ energies are very close to the H F limit, and there is an almost identical lowering of E(Li02) and E(OT) on going to the more flexible 6-31 1G basis set. Thus we obtain Do(Li-02) = 296 kJ mol-'. By contrast, in the case of N a 0 2 the bond energy Do(Na-02) from the 6-3 1 1 G basis set is 14 kJ mol-' smaller than from the 6-31G set at the UMP4 level. Once again, the lowering of E(Na02) and E(0,) are very similar on going to the 6-31 1G set. However, E(Na+) is also lowered since the 6-31G and 6-31 1G basis sets have not approached the H F limit for this species, and (34) Moller, C.; Plesset, M. S. Phys. Reu. 1934, 46, 618.
technique of time-resolved laser induced fluorescence spectroscopy of the alkali-metal atoms following the pulsed photolysis of the corresponding alkali-metal iodide vapor in an excess of O2and N2. The purpose of these experiments was to examine whether the reactions reached an equilibrium in the experimental time scale of =lo ms which is governed by the rate of diffusion of the metal atoms out of the beam of the probe laser. A measured equilibrium constant would then yield Do through the application of statistical mechanics. The experiments were performed at the highest possible experimental temperature of 1100 K so that the equilibrium would be furthest toward the alkali-metal atom and 02. A detailed experimental description has been given and only a brief description follows. The reactions were studied in a stainless steel reactor enclosed in a furnace. LiI and NaI molecules are suitable photolytic precursors for studying reactions of Li or Na because they possess large photolysis cross sections below 390 11111,~~ and the high S / N ratios that are obtained by using them in this experimental system are prerequisite to the analysis described below. Other precursors, such as the hydroxides and superoxides, were not used here because they photolyze less r e a d i l ~although ,~ they have the advantage that there is no potential for production of 12,which can be a serious c ~ n t a m i n a n t ~ ~ ' ~ (see below). For each reaction, the corresponding powdered iodide was placed in a tantalum boat in a heat pipe coupled to the reactor and then heated to a suitable temperature to generate a vapor pressure of about lOI3 molecule cm-3 in equilibrium above the molten salt. The vapor was then entrained in a flow of N2 and carried into the central chamber where it was mixed with a flow of O2 in N2 and photolyzed by a small flash lamp. N a atoms were detected by selectively pumping one of the D and Li atoms at lines at h = 589.0 nm (Na(32P3/2)-Na(32S1/2)), X = 670.7 nm (Li(22PJ)-Li(22S112),using a nitrogen-pumped dye laser (laser dyes Rhodamine-6G or DCM, respectively; bandwidth = 0.01 nm). Fluorescence was then measured with a gated integrator after passing through an interference filter centered at 589 nm or 670 nm. The accurate determination of the temperature in these experiments is important with regard to extracting thermochemical information. The temperature of the gas mixture flowing through the reactor was measured in situ by a shielded Chromel/Alumel thermocouple permanently inserted through the reactor all.^*'^ Since it was demonstrated that the gas was in thermal equilibrium with the reactor walls and that the conductive transfer of heat by the gas was more important than radiative transfer from the reactor walls to the shielded thermocouple, we estimate that the temperature of the gas could be measured absolutely to within about 10 K at 1100 K. This uncertainty does not produce a significant error in the estimates of bond dissociation energies which follow. Materials. Oxygen (99.995%, Liquid Carbonic) was trapped at 77 K before use. Nitrogen (99.999%, Liquid Carbonic) was used without further purification. NaI and LiI (99%, Aldrich, anhydrous) were each refluxed in the heat pipe at 700-850 K for several hours prior to kinetic experiments to remove traces of 12. Results. Figure 1 illustrates an example of the time-resolved decay of the LIF signal at h = 589 nm, generated by the pulsed photolysis of NaI in an excess of O2 and N2 at 1 1 0 0 K. The recording of the LIF decay was delayed for 0.1 ms after the flash lamp fired, in order to avoid any interference from long-lived luminescence from the walls of the central chamber induced by the flash lamp. The signal-to-noise ratio of these decays is more than 200:l. There is no evidence of the Na atom decay in Figure 1 reaching equilibrium. In fact, the set of eight such decays used (35) Brodhead, D. C.; Davidovits, P.; Edelstein, S. A. J . Chem. Phys. 1969, 51, 3601.
The Journal of Physical Chemistry, Vol. 93, No. 8, 1989
3144
Plane et al.
-
TABLE IV: Calculation of ab Initio Bond Dissociation Enereies ~
~~
E(A02)"
E(O23"
D,(AO, A+ OF)*
D,(A-02)b,c
Do(A-O,)b*d
-149.5232565 -149.7724646 -149.5743572 -149.847 402 5
750.1 780.2 751.9 779.3
273.5 303.6 275.4 302.7
266.3 296.4 268.2 295.5
635.8 656.3 626.6 642.2
183.2 203.7 174.0 189.6
178.9 199.4 169.7 185.3
E(A+)"
+
LiO, UHF/6-31G UMP4/6-3 IG UHF/6-311G UMP4/6-311G
-157.044997 6 -1 57.306 646 8 -157.097 2300 -157.380 585 8
-7.2354800 -7.235 4800 -7.2358390 -7.235 8390
UHF/6-31G UMP4/6-31G UHF/6-311G UMP4/6-311G
-3 1 1.425 059 9 -311.682 1540 -31 1.477 639 6 -311.756691 6
-161.6592765 -161.6592165 -161.664 232 9 -1 61.664 232 9
NaO, -149.5232565 -149.7724646 -149.574 357 2 -149.847 402 5
"Units: hartrees. bunits: kJ mol-'. cCalculated using IE(Li) = 5.390 eV, IE(Na) = 5.138 eV, and EA(0,) = 0.44 eV (ref 31). dDifferencein zero-point energies is calculated from ref 1 and ref 31.
! 10
5
TABLE V: Derivation of the Experimental Bond Dissociation Energies [021/1015 Kwa/10-'5 cm3 Doa/kJ
1.85 3.70 5.55 7.40 5.0 9.9 19.8 24.7
2
I
400
600
800
LIF, = LIFOexp(-k't)
1000
k' = kdiff + kl,Z[O,l
(4)
[N21
(5)
where kdia describes the diffusion of metal atoms out of the volume created by the intersection of the flash lamp and the dye laser beams.36 kdincan be determined by measuring k'in the absence of 02.LIFOis the LIF signal extrapolated back from 0.1 ms to t = 0. The solid line in Figure 1 illustrates such a fit. There is no evidence in Figure 1 for the unimolecular dissociation of N a 0 2
+ O2 (+N2)
(-1)
being sufficiently fast for reaction 1 to approach an observable equilibrium, which we have clearly been able to observe in other reaction^.^'*^* Nevertheless, such decays can be used to derive lower limits for the equilibrium constants K , for reactions 1 and 2 at 1100 K, assuming that the superoxide molecules do not undergo further reaction. The lower limits to K , are obtained (36) Plane, J. M. C. J . Phys. Chem. 1987, 91, 6552. (37) Plane, J. M. C.; Saltzman, E. S. J . Chem. Phys. 1987, 87, 4606. (38) Plane, J. M. C.; Rajasekhar, B. J . Chem. SOC.,Faraday Trans. 2 1988, 84, 273.
23.7 11.6 11.5 11.5
213 206 206 206
(LIFO - LIF,)
where LIF, is the measured LIF signal a t time t after the flash lamp fires. k'is the pseudo-first-order rate constant given by
Na
+ O2 + N2
0.10 0.04 0.02 0.01
psecs
in the present calculations are all well fitted by a single-exponential decay down to the base line of the form
-
327 326 320 314
by inspecting each decay to determine a value of the LIF signal close to the base line but still clear above the noise. If this value of the LIF signal is termed LIF, (see Figure l ) , then
1
I
Figure 1. Time-resolved decay of the LIF signal from Na atoms at X = 589 nm (Na(32P,)-Na(32S,,,)) following the pulsed photolysis of NaI vapor at 1100 K [O,] = 2.47 X 10I6 molecule ~ m - [N,] ~ ; = 4.36 X lo" molecule cm-). The solid line is a best fit of the form A exp(-k't) through the data, extrapolated back to t = 0. The broken line indicates the decay that would be observed if Do(Na-02) = 185 kJ mol-'.
N a 0 2 (+N2)
6.16 4.3 1 2.67 1.49
mol-'
" Lower limit; see text for discussion of diffusion corrections.
0
200
0.45 0.28 0.12 0.09
Na
4
LL -
0
kdirr/k' molecule-' Li + 0, + N,
molecule cm-3
Ke, 2
LIF,[021
(6)
The lower limit to the A 0 2 bond energy is then extracted from a statistical mechanical calculation of K , using the experimental geometries and vibrational frequencieslJ?n the partition functions. Although the selection of LIF, is somewhat arbitrary, the final estimate of Dois not very sensitive to the choice of this parameter. For example, changing LIF, by a factor of two results in a 2% variation of Do. The contribution of kdiffto k'(eq 5) has to be accounted for, particularly when the contributions to k' from chemical reaction and diffusion are of similar size. Table V records the results of analyzing LIF decays of reactions 1 and 2 according to eq 6. For both reactions, the 0, concentration was varied by about a factor of 5 . In the case of the Li reaction, the ratio of kdiff/k'ranged from 0.09 to 0.45, leading to a corresponding increase in K,.by a factor of 6. A plot of K , vs kdiff/k'yields, from extrapolation to kdiff = 0, a value for Kq of 1.0 X lo-'' cm3 molecule-'. This corresponds to D0(Li-02) I310 kJ mol-'. For the Na reaction the kdiff/k'was much smaller (0.01-0.10) and the estimates of K , become tightly grouped, corresponding to Do(Na02) I 206 kJ mol-'. Finally, a remaining uncertainty in deriving these lower limits is the possible role of I, reactions. We have discussed e l ~ e w h e r e ~ , ' ~ the production of I2 from the heterogeneous reaction between 0, and adsorbed alkali-metal iodides on the hot walls of the reactor. The reactions Na,Li I, are extremely fast39and thus only a trace of I2 (12/02= 0.02%) would approximately double the measured k'and convert half the Na atoms formed in the flash to NaI rather than NaOz, leading in turn to an overestimation of K However, I, is overwhelmingly dissociated to I atoms at 1100 I?;30 and thus about 5% of the 0, flowing through the reactor would have to
+
(39) Edelstein, S . A.; Davidovits, P. J . Chem. Phys. 1971, 55, 5164
J . Phys. Chem. 1989, 93, 3 145-3 151 react on the walls to form sufficient I, to double k'. This seems unlikely because the ratio of alkali-metal iodide to 0, entering the reactor is probably less than 0.2%. Furthermore, in our determination of k2 we also used LiOH and LiO, as photolytic precursors of Li atoms and measured k2 within 30% of the rate constant measured using LiI.9 Allowing for the possibility that 30% of k' in decays such as that pictured in Figure 1 is due to the reactions between akali-metal atoms and I, yields the marginally lower limits Do(Li-0,) L 306 kJ mol-' and Do(Na-02) 1 202 kJ mol-' which we adopt as our final lower limits.
Discussion We have thus obtained estimates of the bond energies of Li02 and NaO, from ab initio calculations and their lower limits from experiment. When attempting to compare between theory and experiment, we make reference to the work of Bauschlicher and L a n g h ~ f f ,who ~ ~ ,have ~ carried out an extensive intercomparison of the ab initio and experimental bond energies for most of the alkali-metal monoxides and hydroxides. Those a u t h o r ~con~~,~ clude that agreement within 5% can generally be achieved when the ab initio calculations use a sufficiently flexible basis set. In the case of our study of Li02, the ab initio value and the experimental lower limit differ by only 10 kJ mol-', or 3%. We believe that this indicates that the experimental lower limit must be close to the true value and recommend a value of Do(Li-0,) = 306 kJ mol-'. This value is substantially higher (40%) than the current value of 222 f 25 kJ mol-' obtained from flame measurementsI6 and theory.', These flame experiments by Dougherty et a1.l6 probably need to be reanalyzed now that k,(Li + 0, + N,) has been determined directlyg and found to be 2-3 orders of magnitude faster than was assumed16 by analogy with O2 + Nz. The semiempirical calculation of the reaction H Alexanderi2 is based on the ab initio study of LiO, by Grow and Pitzer.ls They15 employed a contracted gaussian basis set with
+
~~
(40) Bauschlicher, C. W.; Langhoff, S. R. J . Chem. Phys. 1988,88,6431.
3145
polarization functions which led to a caculated (0-0)- bond distance which was significantly smaller than experiment,' as found in the present study with the 6-31G* basis set. We consider that the excellent agreement in this work between the ab initio and experimental geometries (Table I), and the convergence in Do(Li-0,) between basis sets (Table IV), has produced a more accurate calculation of the bond energy. In the case of NaO,, the agreement between the ab initio and experimental determinations is less satisfactory, though with a difference of only 17 kJ mol-' (9%). This is illustrated in Figure 1, where a broken line indicates the decay that would have been observed if reaction 1 approached equilibrium with Do(Na-0,) = 185 kJ mol-' and assuming an absence of I2 in the reactor and no other removal of NaO, besides dissociation (reaction -1). The observed LIF decay in Figure 1 clearly does not support the theoretical value of Do(Na-0,) = 185 kJ mol-'. However, as discussed above, the theoretical bond energy decreases by 14 kJ mol-' on going from the 6-31G to the more flexible 6-31 1G basis set. A tight convergence between the basis sets, as seen in the case of LiO,, has not been reached and the theoretical Do(Na-02) can be expected to change further on increasing the flexibility of the basis set, although the direction is difficult to predict. We therefore recommend Do(Na-0,) = 202 kJ mol-'. This value is close to the molecular beam study of Figger et al.,l However, it is significantly higher than the values of the bond energy obtained in the recent flame studies of Jensenigand Hynes et al.,,O by 19% and 39%, respectively, indicating that the chemistry of the alkali metals in oxygen-rich flames is still not completely understood.
Acknowledgment. This work was supported under Grant ATM-8616338 from the National Science Foundation. We thank F. Miller0 for supporting the purchase and running costs of the Gaussian 86 program and S. Mroueh for assisting with the calculations. Registry No. Li02, 12136-56-0; N a 0 2 , 12034-12-7.
New Approach to Weak-Collision Factor ,8 in Thermal Unimolecular Reactions Wendell F o r d DPpartement de Chimie Physique des RPactions, UA 328 CNRS, INPL-ENSIC et UniuersitP de Nancy I , I Rue Granduille, 54042 Nancy Cedex, France (Received: July 28, 1988: In Final Form: October 10, 1988)
A new expression for the weak-collision correction factor @, suitable for rapid computation, is proposed, based on a matrix approach and related to the energy diffusion version of the Fokker-Planck equation (eq 19 in the text). It makes use of a transition matrix constructed for a specified transition probability model but does not require the actual solution of the master equation; thus the number of matrix operations is minimized, with consequent saving in machine time. This approach is useful for a closer identification of the weak-collision falloff with a particular transition probability model. The new approach is applied to the decomposition of ethane as test case and compared with results obtained by actual solution of the weak-collision master equation and with other definitions of @.
1. Introduction The pressure dependence of k , , ~the , unimolecular rate constant for dissociation at some finite pressure in a gas-phase thermal system, gives rise to the well-known "falloff" that generally serves for the comparison of theory with experiment. It is usually represented by the well-known RRKM form of kd [see eq 7 below and accompanying text] which incorporates the assumption that collisions are "strong", Le., that every collision deactivates an energized molecule with unit efficiency. It is agreed that the strong-collision version of kuniis apt to be poor in many instances,
but it has the virtue of making the falloff calculations relatively quick and easy. This is a nontrivial consideration if large amounts of data on many different molecular systems are to be treated, considering that full treatment of "weak" collisions, while feasible, is more demanding. It is therefore not surprising that over the years there have been many proposals for shortcuts that would conserve the simplicity of the strong-collision calculation and at the same time correct, at least approximately, for weak-collision effects. While they may differ in detail, in essence these proposals'-' boil down to mul-
Present address: Laboratoire de Physicochimie ThBorique, UA 503 CNRS, Universite de Bordeaux I, 33405 Talence Cedex, France.
(1) Tardy, D. C.; Rabinovitch, B. S. J . Chem. Phys. 1966, 45, 3720; J. Chem. Phys. 1968, 48, 1282.
0022-3654/89/2093-3 l45$01.50/0
0 1989 American Chemical Society
