5964
J. Phys. Chem. 1996, 100, 5964-5969
Thermodynamics of Electron Trapping and of Some Reversible Attachment-Detachment Reactions in Liquid Hydrocarbons A. Mozumder Radiation Laboratory† and the Department of Chemistry and Biochemistry, UniVersity of Notre Dame, Notre Dame, Indiana 46556 ReceiVed: September 11, 1995; In Final Form: NoVember 21, 1995X
Thermodynamic parameters for electron trapping from the quasi-free state have been calculated from equilibrium trapping and detrapping rates in n-hexane, cyclohexane, n-pentane, propane, 3-methypentane, and isooctane, in consistence with the variation of effective mobility with temperature. From these, the derived free energy and entropy of solution show systematic variation with V0, the energy of the lowest conducting state in the liquid. For several reversible attachment-detachment processes, the thermodynamic parameters of the reaction have been evaluated by referring to the quasi-free state of the electron in which the solute reaction is believed to take place. The calculated encounter reaction efficiency (η) of the attachment process in the quasi-free state generally decreases with the effective mobility. Only in n-hexane and cyclohexane the attachment reaction is nearly or partially diffusion-controlled. In isooctane, neopentane, and tetramethylsilane the reaction is inefficient (η j 0.1). Electron trapping is speculated within the context of Anderson localization.
Introduction It has recently been demonstrated1,2 that for a large class of liquid hydrocarbons the quasi-ballistic model provides an adequate description of the variation of effective electron mobility (µeff) with temperature (T). The model has also been extended to electron scavenging.3 In it the effective mobility is given by
µeff-1 ) 〈µ〉T-1 + 〈µ〉F-1
(1)
where 〈µ〉T ) (e/m)(ktf/kft)/(kft + ktf) is the ballistic mobility and 〈µ〉F ) µqf ktf/(kft + ktf) is the usual trap-controlled mobility. Here µqf is the mobility in the quasi-free state, and kft and ktf are respectively the rates of trapping and detrapping. While the equilibrium ratio kft/ktf can be given in terms of the trap density (nt) and the binding energy in the trap (0, considered positive) using a detailed balance argument,4 a specific model is required to give the individual rates. It has been shown2 that, for most hydrocarbons in which the measured mobility is η > 0.2 and not diffusion-controlled if η < 0.2. In Table 4 these are designated respectively by *, †, and ‡. In Table 4 we have generally used experimental data at the lowest temperature where reversible reaction can be seen. Of course, similar calculations can be made at higher temperatures. It should be noted that many attachment reaction rates are insensitive to temperature, and yet the reaction efficiency is low. This may happen if the reaction is nonadiabatic involving a small activation energy whence the temperature dependences of diffusion and reaction nearly cancel each other. The reverse (detachment) reaction is of course strongly activated. Within the limited scope of comparison we see from Table 4 that attachment reaction efficiency η generally falls with electron mobility; it is very low for all the reactions in tetramethylsilane. A similar phenomenon has been reported for irreversible electron scavenging reactions.3
TABLE 3: Thermodynamic Functions for Reversible Attachment-Detachment Reactions in Hydrocarbon Liquidsa solute
∆G°reac(qf)
∆H°reac(qf)
∆S°reac(qf)
styrene R-methylstyrene p-C6H4F2 p-C6H4F2 naphthalene styrene R-methylstyrene CO2 CO2
-67 -63 -46 -43 -57 -49 -44 -56 -35
-127 -129 -88 -104 -90 -103 -104 -112 -98
-199 -220 -142 -202 -111 -179 -202 -188 -213
solvent n-hexane cyclohexane isooctane neopentane
a All energies in kJ/mol. Entropy is in unit of J/(mol K). Overall uncertainty in the determination of ∆G° and ∆H° is ∼20% and that in ∆S° is ∼30% (see text).
Thus, we ignore trapping in this liquid and identify the quasifree state as the solvated state in this liquid. For this reason ∆G°soln for this liquid is taken from solute reaction cycle,8 rather than using the fixed value of ∆S°(qf). Since ∆H°soln and ∆G°soln are very nearly the same in this liquid, we can say that there is very little entropy change in solution. Comparing Tables 1 and 2, it is seen that although entropy change in trapping from the quasi-free state is negative, the overall entropy change in electron solvation is positive. Thus, the solvation process is driven both by enthalpy and entropy. Figure 3 generally shows that both ∆H°soln and ∆G°soln increase with V0 (i.e., becomes less negative), although there are local variations. However, the peaks around V0 ≈ 0 appear real. For some reversible reactions the thermodynamic parameters involving reaction in the quasi-free state are shown in Table 3 following the scheme of eq I and using eq 2. ∆X°reac (X ) G, H, or S) are taken from refs 6, 8, and 9 while Table 1 provides the data for ∆X°tr, except for neopentane in which case the thermodynamic trapping data are taken from reference 5. It is noteworthy that ∆X°reac(qf) (X ) G, H, or S) are negative for all the reversible reactions in the same way as for the overall reaction ∆X°reac. The entropy change is relatively large and the same factors responsible for large ∆S°reac seem to be operative for ∆S°reac (qf) as well.8 The main uncertainty in determining the thermodynamic parameters of solvation of the electron originates from the determination of V0, which is conservatively estimated to be ≈10 kJ/mol. Taking ∆S°(qf) a constant at 56 J/(mol K) also entails an error ≈8-10 J/(mol K). The total uncertainty increases with the absolute value of the thermodynamic parameters. We estimate the uncertainties in ∆G° and ∆H° in Tables 2 and 3 to be ∼20% and those in ∆S° to be ∼30%.
Conclusions (1) Electron solvation in hydrocarbon liquids from the vacuum state is accompanied by negative enthalpy and positive entropy changes, thereby both factors having considerable contribution to the free energy of solvation. In contrast, the entropy change in trapping from the quasi-free state is negative, the process being enthalpy driven. (2) The negative entropy change on trapping may be consistent with the Anderson model of localization, requiring no special trapping mechanism except disorder. (3) In low- and intermediate-mobility liquids the
TABLE 4: Electron Attachment Reaction in Liquid Hydrocarbons: Reaction Rate and Efficiency in the Quasi-Free Statea solvent n-hexane cyclohexane isooctane
neopentane tetramethylsilanec
a
solute styrene R-methylstyrene p-C6H4F2 p-C6H4F2 naphthalene styrene R-methylstyrene CO2 CO2 triphenylene phenanthrene naphthalene styrene R-methylstyrene
T (K) 329.4 329.6 296 298 361.3 321.0 299.1 306.0 297 322.9 307.7 273.1 237.7 219.5
kf1
k1 12
3.04 x 1014 1.9 x 10 1.4 × 1012 2.26 × 1014 9.1 × 1011 3.14 × 1014 2.5 × 1012 2.21 × 1014 9.0 × 1012 2.6 × 1013 1.2 × 1013 4.4 × 1013 1.5 × 1013 6.3 × 1013 6.94 × 1012 2.84 × 1013 5.5 × 1011 7.86 × 1011 b 2.0 × 1013 1.4 × 1013 1.0 × 1013 1.4 × 1013 1.5 × 1013
All specific rates in M-1 s-1. b Taken as kf1 ) k1 µqf/µeff with µeff ) 70 cm2
V-1 s-1. c
kdiff
η
remark
4.68 × 1014 4.68 × 1014 4.38 × 1014 4.40 × 1014 4.95 × 1014 4.61 × 1014 4.42 × 1014 4.47 × 1014 4.38 × 1014 4.63 × 1014 4.50 × 1014 4.17 × 1014 3.82 × 1014 3.64 × 1014
0.650 0.483 0.717 0.502 0.0525 0.0954 0.1425 0.0635 1.8 × 10-3 0.0432 0.0311 0.0240 0.0366 0.0412
* † * † ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡
In tetramethylsilane k1 ) kf1 in the absence of trapping.
Thermodynamics of Electron Trapping attachment rate in the quasi-free state greatly exceeds the overall attachment rate, while the detachment rates are nearly equal. (4) The efficiency of attachment reaction generally falls with electron mobility. Nearly diffusion-controlled reactions can only be seen in the liquids of lowest mobility. Acknowledgment. Dr. G. Ferraudi is thanked for reading the manuscript and Dr. G. Hug for sporadic discussions. References and Notes (1) Mozumder, A. Chem. Phys. Lett. 1993, 207, 245. (2) Mozumder, A. Chem. Phys. Lett. 1995, 233, 167. (3) Mozumder, A. J. Phys. Chem. 1995, 99, 6557. (4) Ascarelli, G.; Brown, S. C. Phys. ReV. 1960, 120, 1615. (5) Baird, J. K.; Rehfeld, R. H. J. Chem. Phys. 1987, 86, 4090. (6) Holroyd, R. A.; Gangwer, T. E.; Allen, A. O. Chem. Phys. Lett. 1975, 31, 520. (7) Warman, J. M.; de Haas, M. P.; Za´dor, E.; Hummel, A., Chem. Phys. Lett. 1975, 35, 383. (8) Holroyd, R. A. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 298.
J. Phys. Chem., Vol. 100, No. 14, 1996 5969 (9) Holroyd, R. A.; McCreary, R. D.; Bakale, G. J. Phys. Chem. 1979, 83, 435. (10) Allen, A. O.; Holroyd, R. A. J. Phys. Chem. 1974, 78, 796. (11) Allen, A. O.; Gangwer, T. E.; Holroyd, R. A. J. Phys. Chem. 1975, 79, 25. (12) Takayasu, H. J. Phys. Soc. Jpn. 1982, 51, 3057. (13) Tsurumi, S.; Takayasu, H. Phys. Lett. 1986, 113A, 449. (14) Rappaport, D. C. Phys. ReV. Lett. 1984, 53, 1965. (15) Anderson, P. W. Phys. ReV. 1958, 109, 1492. (16) Chandler, D.; Leung, K. Annu. ReV. Phys. Chem. 1994, 45, 557. (17) In all cases except isooctane where an experimental Hall mobility, 22 cm2 V-1 s-1, as measured by: Itoh, K.; Munoz, R. C.; Holroyd, R. A. (J. Chem. Phys. 1989, 90, 1128) is used for µqf. (18) Holroyd, R. A.; Tames, S.; Kennedy, A. J. Phys. Chem. 1975, 79, 2857. (19) Hamill, W. H. J. Phys. Chem. 1981, 85, 2071. (20) All V0 values are taken from: Handbook of Radiation Chemistry; Tabata, Y., Ito, Y., Tagawa, S., Eds.; CRC Press: Boca Raton, FL, 1991; Chapter VII. (21) Cf. eq 15 of ref 8.
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