J. Phys. Chem. 1996, 100, 9385-9388
9385
Anomaly of the Ortho/Para Ratio of Hydrogen Molecules Generated by γ-Irradiation of Solid Alkanes Tsuneki Ichikawa* and Hiroshi Yoshida Department of Molecular Chemistry, Graduate School of Engineering, Hokkaido UniVersity, Kita-ku, Sapporo, 060 Japan ReceiVed: January 10, 1996X
The gas chromatographic measurement of the ortho ()triplet nuclear spin)/para ()singlet nuclear spin) ratio of hydrogen molecules generated by γ-irradiation of alkanes revealed that the ortho/para ratio depends on the physical phase of alkanes during the irradiation. The ortho/para ratio for liquid alkanes was equal to the high-temperature limit of 3, which implies that the rate of hydrogen atom abstraction from alkanes by hydrogen atoms, the main mechanism of the formation of hydrogen molecules, does not depend on the nuclear spin states of the abstracting and abstracted hydrogen atoms. The ortho/para ratio for solid alkanes was about 2.6, which indicates that hydrogen molecules are trapped for more than a few second adjacent to the alkyl radicals generated in pair with the hydrogen molecules, during which the lattice vibration of the alkyl radicals induces the simultaneous transition of the rotational and the nuclear spin states of the hydrogen molecules. A quantummechanical theory of the ortho-para transition has been given.
Introduction Abstraction of a hydrogen atom from a molecule by a hydrogen atom is one of the simplest chemical reactions, so that it has been extensively studied both experimentally and theoretically. One of the characteristic features of the hydrogen abstraction is its large quantum effects on the reaction rate. The activation energy for the abstraction is on the order of 1000 K, so that the abstraction is expected to be practically prohibited at temperatures below 77 K. However, this is not the case. The abstraction proceeds rather quickly due to quantummechanical tunneling of a C-H hydrogen atom through the potential barrier (for a recent review, see refs 1 and 2). Recently, another quantum-mechanical effect, the nuclear spin effect, was reported. Miyazaki et al. measured the rate of hydrogen atom abstraction from hydrogen molecules by hydrogen atoms at 4.2 K and found that the abstraction from parahydrogen with antiparallel nuclear spins is about 3 times faster than that from ortho-hydrogen with parallel nuclear spins.3,4 They suggested that this effect does not arise directly from the difference of the nuclear spin states but indirectly from the difference of the rotational states between ortho- and parahydrogen. Ortho-hydrogen is allowed to have odd rotational quantum numbers, whereas it is 0 or even numbers for parahydrogen. Although several mechanisms have been proposed for explaining the relation between the rotational quantum state and the rate of hydrogen atom abstraction,5,6 the details of the mechanism are still not clear. Very recently, Hase et al. reported a complementary experimental result for the nuclear spin effect. Using Raman spectrometry, they measured the ortho/para ratio of hydrogen molecules generated from solid ethanol at 77 K by the reaction
‚ CH3CH2OH + γ-ray f ‚H + CH3CHOH or CH3CH2O‚ ‚ CH3CH2OH + ‚H f H2 + CH3CHOH and found7 that the ratio is much less than the statistical value of 3/1, which is expected to be obtained if the rate of hydrogen X
Abstract published in AdVance ACS Abstracts, May 1, 1996.
S0022-3654(96)00122-0 CCC: $12.00
atom abstraction does not depend on the nuclear spin state or the rotational state of the product molecule. However, they later found8 that the observed ratio is not the original ratio just after the abstraction but is the ratio after thermodynamic equilibrium due to spin-phonon interactions mediated by concomitantly generated free radicals with a concentration more than 0.1 mol/ dm3. The above result indicates that the reduction of the concentration of the free radicals is necessary to observe the original ortho/para ratio. However, it is impossible in their experiment due to poor sensivity of the Raman spectroscopy. In the present study, we observed the original ortho/para ratio of hydrogen molecules generated by γ-irradiation of organic solid and liquid by applying gas chromatography for more sensitive detection of ortho- and para-hydrogen9,10 and found that the ortho/para ratio is different from the statistical value only when the molecules are generated in solids. We propose this to be the transient trapping of hydrogen molecules adjacent to free radical particles which are generated in pair with hydrogen molecules. Experimental Section Chemicals used were spectroscopic grade hexane and methylcyclohexane and reagent grade cyclohexane. These were used without further purification. The chemicals were sealed in glass tubes after degassing with freeze-pump-thaw cycles and were irradiated at constant temperatures with 60Co γ-rays. The dose rate for the samples was 15 kGy/h. The hydrogen molecules generated by the irradiation were collected at ambient temperature and were analyzed by gas chromatography at 77 K using a 250 × 3 mm diameter alumina-packed glass column (80100 mesh transition metal-free activation grade 1 alumina B from ICN Biomedicals, used after heating at 400 K in helium flow for 1 h). The flow rate of helium as the carrier gas was 50 cm3/min. The gas stream from the column was passed through a short combustion tube (40-60 mesh copper oxide, 600 K), and the resultant water vapor was detected with a TCD cell at 400 K. The sensitivity of the chromatography to the gaseous hydrogen was as low as 10-3 cm3. Results and Discussion Experimental Evidence of Ortho-Para Conversion after the Formation of Hydrogen. Figure 1 shows the gas chro© 1996 American Chemical Society
9386 J. Phys. Chem., Vol. 100, No. 22, 1996
Ichikawa and Yoshida TABLE 2: Ortho/Para Ratio of Hydrogen Molecules Generated by γ-Irradiation of Methylcyclohexane irradiation temperature/K
irradiation time/ha
storage temperature/K
storage time/h
ortho/para ratio
77 (solid) 77 (solid)b 77 (solid) 142 (solid) 163 (liquid) 163 (liquid) 179 (liquid) 295 (liquid)
0.3 1 1 1 1 1 1 1
77 77 77 142 163 77c 179 295
0.1 1 24 0.1 0.1 24 0.1 0.1
2.6 ( 0.1 2.6 ( 0.1 2.6 ( 0.1 2.6 ( 0.1 3.0 ( 0.1 3.0 ( 0.1 3.1 ( 0.1 3.0 ( 0.1
a The radiation dose was 15 kGy/h. b Addition of 2, 5, 10, and 100 vol % cyclohexane, a scavenger for thermalized hydrogen atoms, did not affect the ortho/para ratio. c Gaseous hydrogen remaining after freezing the sample at 77 K was evacuated just after the freezing.
Figure 1. Gas chromatograms of hydrogen equilibrated at 77 K (A) and at 295 K by passing the hydrogen gas through Fe3+-loaded zeolite X and those of hydrogen generated by γ-irradiation of solid hexane at 163 K (C) and of liquid hexane at 179 K (D) for 0.3 h. Column: 250 × 3 mm diameter glass column packed with 80-100 mesh activation grade 1 alumina B from ICN Biomedicals, 77 K. Carrier gas: helium, 50 cm3/min. Detection: TCD at 400 K of water vapor generated from hydrogen by passing through a 30 mm combustion tube packed with 40-60 mesh copper oxide, 600 K.
TABLE 1: Ortho/Para Ratio of Hydrogen Molecules Generated by γ-Irradiation of Hexane irradiation temperature/K
irradiation time/ha
storage temperature/K
storage time/s
ortho/para ratio
77 (solid) 77 (solid) 77 (solid) 163 (solid) 163 (solid) 179 (liquid) 295 (liquid)
0.3 1 1 1 1 1 1
77 77 163 163 77 179 295
0.1 24 24 0.1 24 0.1 0.1
2.5 ( 0.1 2.5 ( 0.1 2.5 ( 0.1 2.5 ( 0.1 2.5 ( 0.1 2.9 ( 0.1 2.9 ( 0.1
a
The radiation dose was 15 kGy/h.
matograms of hydrogen equilibrated at 77 K (A) and at 295 K (B) by passing the hydrogen gas through Fe3+-loaded zeolite X and those of hydrogen generated by γ-irradiation of solid hexane at 163 K (C) and of liquid hexane at 179 K (D) for 0.3 h. The theoretical ortho/para ratio of the equilibrated hydrogen is 0.97 and 3 at 77 and 295 K, respectively, which agrees with the peak area ratios of the chromatograms A and B. Table 1 summarizes the ortho/para ratios for γ-irradiated hexane. It is evident from Table 1 that the ortho/para ratio of hydrogen depends not directly on the irradiation temperature but on the physical phase of hexane during the irradiation. The ortho/para ratio for liquid hexane is close to the statistical value or the high-temperature limit of 3, irrespective of the irradiation temperature. On the other hand, the ortho/para ratio for crystalline hexane is about 2.5 and is independent of the irradiation temperature. Storage of the irradiated crystalline samples did not change the ortho/para ratio, which indictes that the ortho-para conversion does not take place after the irradiation. Table 2 summarizes the ortho/para ratios for γ-irradiated methylcyclohexane. Although the state of the solid methylcyclohexane is not crystalline but glassy, the result for methylcyclohexane is essentially the same as that for hexane; the ortho/ para ratio is about 3 and 2.6 for the liquid and the solid states, respectively. It has been established by many researchers11-13 that hydrogen molecules are generted mainly by hydrogen atom abstraction
of hot and thermalized hydrogen atoms, which in turn are generated from alkanes, as
RH2 + γ-ray f ‚RH + ‚H(hot) H‚(hot) + RH2 f ‚RH + H2 ‚H(hot) f ‚H(thermal) ‚H(thermal) + RH2 f ‚RH + H2 The thermalized hydrogen atoms are scavenged by alkenes such as cyclohexene,14 whereas the hot hydrogen atoms are not. For elucidating the effect of the initial energy of hydrogen atoms on the ortho/para ratio, we examined the effect of cyclohexene on the ortho/para ratio for solid methylcyclohexane. As shown also in Table 2, addition of cyclohexene did not change the ortho/para ratio, which indicates that the ortho/para ratio does not depend on the energy of the hydrogen atoms. One of the main differences between liquid and solid alkanes is the lifetime of alkyl radicals which are generated in pair with the hydrogen molecules. The alkyl radicals disappear very quickly in liquid alkanes due to radical-radical combination, whereas the lifetime of the radicals is more than a few days in solid alkanes. The difference of the ortho/para ratio between solid and liquid alkanes strongly suggests that the ortho to para conversion takes place in solid alkanes due to coexisting alkyl radicals. The ortho/para ratio of 3 for liquid alkanes suggests that there is no preferential formation of ortho or para hydrogen in hydrogen atom abstraction. The ortho/para ratio depends not on the temperature of the medium or the kinetic energy of the parent hydrogen atoms but simply on the spin degeneracy of hydrogen, because the hydrogen atom abstraction is a highly exothermic reaction. No ortho-para conversion takes place without alkyl radicals, so that the ortho/para ratio for liquid alkanes maintains its original value. In solid alkanes, on the other hand, the alkyl radicals mediate the ortho-para conversion of geminate hydrogen molecules. The ortho/para ratio therefore decreases toward the thermodynamical equilibrium value. Now the question is why the ortho-para conversion was not observed after γ-irradiation. We believe that the ortho-para conversion ceases shortly after the formation of hydrogen molecules due to the diffusion of the hydrogen molecules from the geminate alkyl radicals. In the following section we will try to estimate the lifetime of the radical-hydrogen molecule pair from the observed ortho/para ratio for solid alkanes. Quantum Theory of Ortho-Para Conversion. The explicit Hamiltonian for a hydrogen molecule interacting with an
Ortho/Para Ratio of H Generated by γ-Irradiation
J. Phys. Chem., Vol. 100, No. 22, 1996 9387
electron spin at R ˜ is given by
Fermi’s second golden rule, as
H ˜ )H ˜0 + H ˜d
(1)
A10 ) 2π|〈Ψ0|U|Ψ1〉/p|2/dω
where
) D22[2π|〈Q0|U|Q2〉/p|2/dω]
H ˜ 0 ) -[p2/(2mp)](∇12 + ∇22) + U(r˜1,r˜2) - p2γp2[3{I˜1(r˜2 -
= [3p2γeγpr/{R4pω20)}]2 ×
r˜1)}{I˜2(r˜2 - r˜1)}/|r˜2 - r˜1|5 - ˜I1˜I2/|r˜2 - r˜1|3)] (2)
≈ 2.5 × 10-12 (10-8 cm/R)8k20/s
H ˜ d ) p2γeγp[3{S˜ (R ˜ - r˜1)}{I˜1(R ˜ - r˜1)}/|R ˜ - r˜1|5 S˜ ˜I1/|R ˜ - r˜1|3] + p2γeγp[3{S˜ (R ˜ - r˜2)}{I˜2(R ˜ - r˜2)}/|R ˜˜ - r˜2| ] (3) r˜2| - S˜ ˜I2/|R 5
3
mp is the mass of the hydrogen atom, and r˜1 ) r˜ and r˜2 ) -r˜ are the locations of proton 1 and proton 2 in the hydrogen molecule. All the operators in Hamiltonians (2) and (3) have their usual meanings. As usual, the rotational angular momentum has been assumed here to be quenched. The wave functions for ortho- and para-hydrogen molecules under Hamiltonian H ˜ 0 are given by
O ) Q2J+1Tx, Q2J+1Ty, Q2J+1Tz P ) Q2JS
(4)
respectively, where QJ is a rotational wave function with a rotational quantum number J, and S and T are nuclear spin wave functions defined by
S ) (R1β2 - β1R2)/x2 Tx ) (β1β2 - R1R2)/x2, Ty ) (β1β2 + R1R2)/x2, Tz ) (R1β2 + β1R2)/x2 (5) A term in Hamiltonian H ˜ d causing the transition between the ortho and the para state must have an (I˜2 - ˜I1) term, which is approximately given by the Taylor expansion of H ˜ d with respect to r, as
˜ | )[{d(R ˜ + r˜)/dr}]r)0[S˜ {(I˜2 - ˜I1)R ˜} + H ˜ d′ = (3p γeγp/|R 2
[(ω10/ω20)3/{1 - exp(-pω10/kT)}]k20 (9)
where pωJK ) 59.3{J(J + 1) - K(K + 1)} cm-1 is the difference of the rotational energies with the rotational quantum numbers J and K, r ) 0.37 × 10-8 cm is half of the bond length of the hydrogen molecule, and k20 is the rate of the spin-allowed transition from J ) 2 to J ) 0 by a zero-point lattice vibration. The value of K20 is roughly estimated from the line width of the Raman spectrum of hydrogen molecules in organic solids (∼10 cm-1)8 to be 3 × 1011/s. Since the rotational state of more than 90% of the hydrogen molecules is either J ) 0 or J ) 1 below 200 K, the times for the ortho-para conversion, 1/A10, at 77 K are estimated from eq 9 to be 5 min, 2.1 h, 21 h, and 130 h at R/10-8 cm ) 2, 3, 4, and 5, respectively. The other mechanism of ortho-para transition is the simultaneous transition of the rotational and the nuclear spin states due to time fluctuation of H ˜ d′ or the phonon-induced fluctuation of the location of an unpaired electron with respect to a hydrogen molecule.15 The fluctuation is given by
χ ) χ0 sin ωt, y ) y0 sin(ωt), z ) R + z0 sin(ωR/Vo) sin(ωt) (10) where x0, y0, and z0 are distortions due to a phonon of angular velocity ω in a medium with density F (F = 0.9 g/cm3 for alkanes) and longitudinal and transverse sound velocity Vo and Vt (Vo = Vt = 200 000 cm/s for alkanes) and are defined by
x02 ) y02 = [{pω/(8π2FVt3)}/{exp(pω/kT) - 1}]dω z02 ) [{pω/(4π2FVo3)}/{exp(pω/kT) - 1}]dω
(11)
5
˜ ) - (5/R ˜ )(S˜ R ˜ ){(I˜2 - ˜I1)R ˜} + R ˜ {S˜ (I˜2 - ˜I1)}]r (I˜2 - ˜I1) (S˜ R (6) It is convenient to set the origin of the coordinate at the center of rotation of the hydrogen molecule and to put the unpaired electron at R on the z-axis. The electron spin is then quantized about the z-axis, since the electron-nuclear dipole interactions about the x- and y-axes are averaged out due to quick rotation of the hydrogen molecule. It is noted that J ) 0 does not mean a stopped rotor, since the potential about the hydrogen molecule is not perfectly spherical and the J ) 0 state has some kinetic energy which is usually much larger than the spin energy. There are two mechanisms for the ortho-para transition. One is the static mixing of the ortho and the para states. Assuming that the ground ortho and para states are mixed as
Ψ0 = Q0S + C1Q1Tx Ψ1 = Q1Tx + D0Q0S + D2Q2S
H ˜ d,t = (3p2γeγprS˜ z/R5)[4{cos θ (I˜2x - ˜I1x) + sin θ cos φ (I˜2z - ˜I1z)}χ + 4{cos θ (I˜2y - ˜I1y) + sin θ sin φ (I˜2z - ˜I1z)}y - 3{sin θ cos φ (I˜2x - ˜I1x) + sin θ sin φ (I˜2y - ˜I1y) - 2 cos θ (I˜2z - ˜I1z)}z] (12) Assuming Ψ0 ) Q0S and Ψ1 ) Q1Tx, the transition rate B10 from Ψ1 to Ψ0 is given by
B10 ) 2π|〈Ψ1|H ˜ d,t/sin ωt|Ψ0〉/p|2 exp(pω10/kT)/dω = (x3p2γ2γpr/R5)2{pω10/(2πp2FV3)}/ {1 - exp(-pω10/kT)} = 4.4[(pω10/cm-1)/{1 - exp(-pω10/kT)}]
(7)
(10-8 cm/R)10 (13)
(8)
The times for the ortho-para conversion at 77 K are estimated to be 1.8 s, 1.8 min, 29 min, and 4.6 h at R/10-8 cm ) 2, 3, 4, and 5, respectively. It is evident that the ortho-para transition due to the static mixing of the ortho and the para states is negligible in solid alkanes.
and the lattice vibration exerts a perturbation potential
U(r˜,t) ) U exp(-iω10t)
Since x, y, z , R, r, the time-dependent perturbation is approximately given by
the rate of transition A10 from Ψ1 to Ψ0 is estimated from the
9388 J. Phys. Chem., Vol. 100, No. 22, 1996
Ichikawa and Yoshida
The ratio of the rate of J ) 0 to J ) 1 conversion with respect to the rate of J ) 1 to J ) 0 conversion should be equal to the Boltzman factor at temperature T, so that the rate of J ) 0 to J ) 1 conversion is expressed by
B01 = 9B10 exp(-pω10/kT)
(14)
where a factor of 9 is a product of the rotational and the spin degeneracy for the J ) 1 state. Provided that the spin-allowed transitions from J > 1 to J ) 1 and J ) 0 states are much faster than the spin-forbidden transition from J ) 1 to J ) 0, the time for the change of the ortho/para ratio from the initial value of 3 to the final value of 2.6 at 77 K is estimated from eqs 13 and 14 to be 1 s, 1 min, 17 min, and 2.7 h at the electron-hydrogen separation distances of R/10-8 cm ) 2, 3, 4, and 5, respectively. It is evident from the estimation of the conversion time that only an alkyl radical adjacent to a hydrogen molecule mediates the ortho-para conversion. The rate of ortho-para conversion depends on the temperature. Increase of the temperature causes the increase of the conversion rate. Increase of the temperature, on the other hand, causes a decrease of the lifetime of the radical-hydrogen molecule pairs due to increase of the thermal diffusion rate of the hydrogen molecules. Since these two effects cancel each other out, the ortho/para ratio does not depend much on the irradiation temperature. Conclusion Hydrogen atom abstraction from alkanes by a hydrogen atom results in the pairwise formation of hydrogen molecules and alkyl radicals. The ortho/para ratio of the resultant hydrogen is equal to the statistical or the high-temperature limit of 3. In
solid alkanes the hydrogen molecules are transiently trapped adjacent to the geminate alkyl radicals, during which the orthopara conversion takes place. After a few seconds, the hydrogen molecules part from the alkyl radicals by diffusion, so that the ortho-para conversion does not take place after the irradiation. In liquid alkanes, on the other hand, both the hydrogen molecules and the alkyl radicals diffuse very quickly. No anomaly of the ortho/para ratio is therefore observed for liquid alkanes. Acknowledgment. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture, Japan. References and Notes (1) Ichikawa, T. J. Syn. Org. Chem. Jpn. 1994, 53, 442. (2) Tolkachev, V. A. Khim. Fiz. 1991, 10, 1207. (3) Fujitani, Y.; Miyazaki, T.; Masaki, N. M.; Aratono, Y.; Tachikawa, E. Chem. Phys. Lett. 1993, 214, 301. (4) Miyazaki, T.; Hiraku, T.; Fueki, K.; Tsuchihashi, Y. J. Phys. Chem. 1991, 95, 26. (5) Bowman. J. Phys. Chem. 1991, 95, 4921. (6) Takayanagi, T.; Masaki, N. J. Chem. Phys. 1991, 95, 4154. (7) Hase, H.; Ishioka, K. Radiat. Phys. Chem. 1992, 39, 329. (8) Ishioka, K.; Hase, H. Radiat. Phys. Chem. 1994, 44, 617. (9) Moore, W. R.; Ward, H. R. J. Am. Chem. Soc. 1958, 80, 2909. (10) Moore, W. R.; Ward, H. R. J. Phys. Chem. 1960, 64, 832. (11) Willard, J. E. In Radiation Chemistry; Farhataziz, Rodgers, Eds.; VCH Publishers: New York, 1987; Chapter 13, and references therein. (12) Miyazaki, T.; Hatano, Y.; Fujisaki, N. In Handbook of Radiation Physics and Chemistry; Tabata, Tagawa, Eds.; CRC Press: Boca Raton, 1991; Chapter X, and references therein. (13) Gyorgy, I. In Radiation Chemistry of Hydrocarbons; Foldiak, Ed.; Elsevier: New York, 1981; Chapter 2, and references therein. (14) Freeman, G. R. Can. J. Chem. 1960, 38, 1043. (15) Ichikawa, T.; Kurshev, V. J. Chem. Phys. 1993, 99, 5728.
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