1864
J. Phys. Chem. 1980, 84, 1864-1867
Electron Transfer in Reactions of Positronium Ions in Liquidst V. M. Byakov," V. I. Grafutin, 0. V. Koldaeva, and E.
V. Minaichev
Institute of Theoretical and Experimental Physics, Moscow, USSR (Received July 17, 1979)
One of the remarkable effects observed with a change in temperature is the appearance of a maximum in the positronium (Ps) rate constant K&sd in the reaction with an acceptor (Ac). The analysis of these dependences in liquids of different polarity has been made. It has been found that the temperature T(K,,) corresponding to the maximum value of rises when the polarity increases. The interpretation of the experimental data based on certain assumptions has been proposed. These assumptions are that (1)in the positronium reactions with the acceptor the electron transfer from Ps to Ac takes place according to Ps + Ac e,+ + Ac; and (2) the possibility of the electron transfer is due to thermal fluctuations in the solvent. At high temperatures the Kobsdvalue is determined by the probability of the electron tunneling from Ps to Ac. With the increasing temperature the probability value tends to zero because of the following dependence of the reorganization energy Ereorg on temperature 5": Ereox F (n> 1). In contrast, at low temperatures Kobd is controlled by diffusion. The rise of T(K,,,) with the increase of the solvent polarity is explained by the decrease of energy required to separate the reaction products to infinity.
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Introduction The investigation of the temperature and medium effects on the course of the positronium (Ps) reactions has attracted little attention so far, and that is why these effects have not yet received a close study. One of the most remarkable effects induced by the temperature changes is the appearance of a maximum in the observed reaction rate constant (Kobsd) of the positronium reaction with an acceptor (Ac) T o account for this fact the authors in ref 2 and 4 suggested that the positronium chemical reaction with an acceptor proceeds through an intermediate complex stage (Ps-Ac), and the positronium complex formation process in a solution may be described as follows
A, is the rate of pick-off annihilation, A, is the annihilation constant for positrons in the complex (Ps-~Ac),and K , and K2are the rate constants of the complex formation and the decay, respectively. One may deduce the following expression for the observed reaction rate constant:
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Assuming K , e-EIIRT,K2 e-E,IRT,and A, e-EcIRT, one can see from eq 1that a maximum in the Kobsd-temperature curve will take place if the correlation E , > El E, for the activation energies is realized. In this paper using the results of the positronium reaction constant measurements in weakly polar liquids, we would like to suggest another interpretation of the experimental results explaining both the extreme Kobsd temperature curve and the correlation between the temperatures corresponding to K,,,, T(K,,), and the properties of the medium. The emphasis is on the application of the ideas on the nonradiative electron transfer mechanism which takes into account thermal fluctuations in the solvent (see ref 5-7) and the interpretation of experimental data on positronium chemical reactions in weakly polar liquids.
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+Thispaper was presented at Colloque Weyl V. For a complete listing of the papers given at this conference see the May 15, 1980 issue of The Journal of Physical Chemistry. 0022-3654/80/2084-1864$01 .OO/O
Such a discussion seems to be necessary both for specifying some details of the electron-transfer theory (in particular the limitations imposed by the linear continuous medium liquid model) and for a clearer understanding of the positronium chemical reaction mechanism in liquids. Experimental Section The positronium lifetime spectra were measured by means of the standard fast-slow coincidence system described earlier.8 The temperature constancy in the ampule with the investigated solution was ensured by a thermostat, and the uncertainty in T did not exceed fl "C. Discussion In Figures 1 and 2 are shown the data on the temperature dependence of the Ps-Ac rate constants measured by us and by some other authors for various solvents. One can see that T(K-) for each acceptor essentially depends on the medium polarity. For example, T(K,,,) for Psnitrobenzene reaction (Figure 1)in benzene is about 80 "C higher than in hexane. The explanation of the maximum on the Kobsd temperature curve suggested in this paper is based on the following assumptions: (1) In the positronium reactions with the acceptor, electron transfer from Ps to Ac takes place. This assumption is based on the rather high values for these acceptors both of their electron affinity (0.5-1.2 eV) and of their reaction rate constants with the solvated electron (-lo1' M-' s-'), The possibility of the electron transfer from positronium to acceptor is suggested a t least in aqueous solutions by similar dependences of the rate constants of Ps and e, reactions with solutes on values of Red-Ox potentials. B The reaction of the H atom, which is to some extent a positronium analogue, with oxygen (the solutes in Figure 2) indicates the possibility of the electron transfer from Ps to Ac, too. (2) Electron transfer is possible through thermal fluctuations in the solvent according to the ideas developed in ref 5-7. The observed reaction rate constant may be represented as in eq 2, where W is the probability of an 4 T D P W P )v (2) = 4aDp + W(p)V electron tunneling from Ps to Ac at the reaction radius p , 0 1980 American Chemical Society
The Journal of Physical Chemistry, Vol. 84, No. 74, 7980 1865
Electron Transfer in Reactions of Positronium Ions
AG = -G,(Ps) - Ips - G,(Ac)
3Ot
+ EA(Ac) + G,(Ac-) + G,(e+) (6)
Flgure 1. Temperature dependences of the positronium-nitrobenzene reaction rate coristant in various solvents: (0)hexane: ((3)isooctane; (V) h e ~ t a n e(+) ; ~ t ~ l u e n e (A) ; ~ b e n ~ e n e (A) ; ~ water;3 (0)l-pentanok4 (0) octan01.~
Gs(A-) - G,(A) = 2a
I
-[
2 I
200
. 250
300
350
T(K1 Flgure 2. Temperature dependences of Ps-Ac reaction rate constant: (A) O2 in hexane); (0)O2 in isooctane.
V = 4ap2L anti L are the volume and the thickness of the reaction layer, respectively, and D is the coefficient of the relative diffusion of Ps and Ac. In accordance with ref 5-7 the charge-transfer probability in a time unit at rather high temperatures is r
Here x is the transmission coefficient, oeff= 1012-10*3s-l (ref 7) is the effective cyclic frequency of the solvent dielectric relaxation, and E,, and Gf are the reorganization (repolarization) energy of medium and the energy required to separate the reaction products to infinity, respectively. Approximately (see ref 10)
Ereorg - e2
-
( :)[ -- -
-
G=
'[
1 +1 -1 - ___ a3+b3 -
2a
2b
r
2,-4
p-]
1 - u3 + b3
cr
where Copt = n2!and E are the optical and static dielectric constants of the solvent, respectively, n is the refractive index, a and b are the radii of the reagents, r is the distance between their centers, e is the electron charge (the reorganization energy of the solvent depends on the quantity of the charge transferred only but not on the charges of each reagent separately), and AG is the difference of the electron levels lbetween which the electron transfer occurs (if we take into account dielectric energy charge). This difference corresponds to the reaction free energy
Ps -t- Ac
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The second term (Ip,) in the right part of eq 6 is equal to the positronium ionization potential in the gas phase; the fourth term (EA) is equal to the electron affinity of the acceptor; and the other terms are the solvation energies of the reagents. The difference G,(A-) - G,(A) between the anion and the neutral molecule solvation free energies is the charge of free energy resulting from the electron capture by the acceptor molecule in the solution, leading to the additional stabilization of the anion as a consequence of the dielectric polarization. A simplified expression of this difference is given by the well-known Born equation
es+ + Ac;
of electron transfer from the positronium to the acceptor
(7)
where a is calculated from the molar volume data. Positronium solvation free energy is estimated from the positronium ionization potential under vacuum and its energy in the condensed medium. The calculationsll demonstrate that the bond energy of the electron in the muonium in Ge or Si is reduced to of its vacuum value. Similar calculations for the positronium indicate that the ionization energy of Ps in the condensed phase is also notably lowered (to I 1 eV in Ge and Si) in comparison with the vacuum value (6.8 eV), the effective positronium radius being increased.12 Experimental evidence for this conclusion may be drawn from measurements of hyperfine splitting of muonium energy levels and from the positronium quenching by the external magnetic field. These effects are characterized by the parameter a = 13/(0)12/13/(O)lYac2 where 13/(0)12is the probability density for the localization of the electron on the muon (positron). For the muonium a is 0.44,0.58, and = 1 for Ge, Si, and H20, respectively.ll For the positronium, a is 0.34 (ref 13) and 0.85 (ref 14) in Si and H20, respectively, the lower value of a for Ps being induced by its greater Bohr radius. These circumstances account for the fact that the difference between thresholds of oxidation of Ps and ea? is only 2.5 eVa9 Taking this into account, one gets
where rB is the positronium Bohr radius, and teff is the effective dielectric constant whose value is determined mainly by the squared refractive index. Since teff is determined mainly by the values of n2and considerably less by that of 6 , it is seen from eq 6 and 8 that the positronium solvation energy is a weaker function of temperature than that of the anion, because dn/dT C dE/dT. (For example, in hexane dt/dT = 16 x W4;dn/dT = 5 x Using the experimental data on temperature dependences of n2 and E , one can make sure that the reorganization energy decreases more rapidly than I/ T when the temperature is high enough. This leads to the tunneling probability decreasing. As a result it turns out that 4aDp >> WV and the reaction rate becomes not diffusion controlled; i.e., in this temperature region Kobsd = WV 0. On the contrary, at low temperatures 4aDp
