NOTES
3183
agreement with values predicted by conventional equations for diffusion-controlled reactions. At least at 2 5 O , the measured diffusion coefficient is still somewhat larger than that predicted by the equation that treats an atom as a stationary sink in an isotropic continuum; however, the discrepancy is not as large as it previously appeared to be. Acknowledgment. The original experimental work was carried out at the University of Oregon and was supported in part by the U. S. Atomic Energy Commission. The opportunity to discuss this situation with Professor. Stokes was provided by a Fulbright award to visit the Victoria University of Wellington, New Zealand. The manuscript was prepared while on a National Science Foundation Senior Postdoctoral Fellowship at the Max Planck Institut fur physikalische Chemie. (2) R. M. Noyes, J . Am. Chem. SOC.,8 6 , 4529 (1964).
O n the Role of Water in Electron-Transfer Reactions. I
by I. Ruff Institute of Inorgan.ic and Anulytical Chemistry, L. E(ltv6s University, Budapest, Hungary (Received Novembsr 16, 1964)
The kinetic investigation of electron-transfer reactions revealed, in general, that the high rate of these reactions could not be accredited simply to collisions of the reaction partners.4 Making use of the number of collisions per unit time, good agreement could be reached between calculated and experimental reaction rates by assuming relatively large ionic sizes. Eyring and co-workersl and Marcusa explained electron transfer using a model which allowed electron transfer even when the reactants were separated from each other by a considerably larger distance than the sum of their respective ionic radii. Both theories have been based on the electrostatic properties of the more or less isolated activated complex. In the present paper a new model is suggested to explain the velocity of electron-transfer read’ions on the basis of tunnel effect, where more emphasis is laid on the properties of the solvent-particularly of water-than in the above-mentioned theories. According to experience, the electron-transfer reactions are first-order reactions in the involved ions.
Thus, the rate of reaction can be described in general by the equation
where v is the rate of exchange, k is the Boltzmann constant, T is the absolute temperature, h is the Planck constant, xe gives the probability of the electron’s transfer in activated state to the other ion (the transmission coefficient of the electron), E* is the energy of activation, R is the universal gas constant, and [ox] and [red] are the concentrations of the lower and higher charged ions, respectively. Hence, to evaluate the rate constant
K
= IcT -Xe
exp(
-&)
h
xeand E* is to be determined theoretically. If the xe electron transmission coefficient is supposed to be the probability of the transfer of an electron through a potential barrier lying between the ions, then for the determination of xe the shape of the barrier, its height, the distance of the two interacting ions, and the kinetic energy of the electron must be known. Some informations on these parameters can be gained through the following considerations. It is well known that only those ions are stable in solution for which the bonding energy of the attached electron falls between certain limits, determined by the nature of the solvent; e.g., in water an electron cannot be bonded to an ion with an energy lower than the bond strength between an electron and a hydrogen atom dissolved in water. On the other hand, however, it cannot be bonded with an energy higher than the bond strength of an electron in the HzO molecule; otherwise, the solvent would suffer chemical decomposition. Thus, in water (at equilibrium conditions) only such ions can exist which have oxidation-reduction potential values confined to the -2.4 v.
< eo < $2.10
v.
(3) range. The upper limit is given here by the thermodynamically calculated normal oxidation-reduction potential of the
H=H++e (1) process, while the lower limit is given by that of the
1
(1) R. J. Marcus, B. J. Zwolinski, and H. Eyring, J. Phys. Chem., 58,432 (1954),and referencestherein. (2) A. C. Wahl, 2.Elektrochem., 6 4 , 90 (1960). (3) R. A. Marcus, J . C h . ,Phys. 2 4 , 966 (1956); 26, 867 (1957); Can. J. Chem., 37, 155 (1959); Discussions Faraday SOC.,29, 21 (1960);J. Phys. Chem., 67, 853 (1963).
Volulne 80, Number 9 September 1966
NOTES
3184
H20zOH
+ H+ + e
(11)
process. In the calculation of the potential of equilibrium 11, the heat of hydratation of the OH radical was supposed to be zero4; thus this value was most probably too low. Using in the calculation the hydration energy of one hydrogen bridge, the mentioned oxidation-reduction potential will be about -2.4 v. (The origin of the potential scale is set by the normal potential corresponding to the
Hz
-
2
2H+
+ 2e
2.10
iI
..................
6
oi
I
.$or
GboP
-2.4
(111)
process.) Thus, the energy of the electron bonded to the ion may cover a range of 4.5 e.v. Its position related to the relative energy levels of equilibria I and I1 is expressed by the normal oxidation-reduction potential. These energy conditions axe plotted schematically in Figure 1 . The “donor” level represents here the state of the very electron which is attached by smallest energy to the ion with lower charge, Le., the state of the electron which is involved in the exchange. The “acceptor” level is, on the other hand, the lowest, yet unfilled energy level of the ion with higher charge. These two states axe energetically equivalent since they refer to two ioat of the same element, being in different states of oxidation. A onedirection exchange means the transfer of an electron from the donor level to the acceptor level. This can happen in one of the following ways. (1) A hydrogen atom is formed and it “transports” the electron to the acceptor level if the energy of activation is larger than (or equal to) the difference between the donor state and the bond energy of the electron attached to the hydrogen atom, which is repm sented by the upper horizontal line. This electron transport may run,however, faster than determined by the diffusion velocity of the hydrogen atom since the electron can go from hydrogen atom to hydrogen atom also by tunnel effect. This is so because the exchange reactions are carried out generally in strong acidic solutions, where the mutual distance of hydrogen ions is a few kgstr$ms. Hence, in this case, the electron moves “quasi-free” up to the acceptor ion in the nearly periodic potential field of the hydrogen ions. According to another theory: the “free” electron bound to a proton may move with the Grotthus mechanism. This mechanism is supported by the experimental fact that the Fe(I1)-Fe(II1) exchange p r o c d in ice too, and the process is not diffusion controlled.” (which is regarded approximately equal to the aCti-
Position ooordinate.
Figure 1.
vation energy) is not suflicient to overcome the barrier. In this case it can reach the acceptor ion only with the aid of the tunnel effect. Utilizing these ideas one can determine the parameters of the potential barrier opposing the exchange reaction. The shape of the barrier is in close approximation rectangular (dotted line in the figure); its height is given by
Eo = 2.10
- Eo e.v.
(4)
where EOis the energy corresponding to the oxidationreduction potential. Thus, to describe the passage of an electron through the rectangular potential, we can use the Gamow equation xe
= exp(-:dd2nz(Eo
- E*))
(5)
where m is the mats of the electron and d is the width of the barrier. Since the donor ion will be activated, quite independently of its distance from the other reactant, electron transfer is possible at all values of d. Therefore, in a 1 M solution (with respect to both ions) an average value ford (9.4 A.) may be used in the calculation of the resultant reaction rate. In order to calculate the xe transmission coefficient and in consequence to determine theoretically the R rate constant d e k e d by eq. 2, it would be necessary to calculate E*,in addition to the above parameters. E* is, according to the Frank-Condon principle, the t h d energy which is necessary for the rearrangement of water moleculw caused by the new charge distribution due to the electron transfer.‘ One can imagine this fact as follows: the electron can leave
(5) R.A. Home and E. H. Aselrod, J . Chena. Phgs., 40,1618 (1964).
NOTES
3185
Table I
System
V(I1)-V(II1) Ce(II1)-Ce(1V) Co(en)2 +-Co( en),*+ V(II1)-V(1V) Fe(I1)-Fe(II1) NP(V)-NP(VI 1 Tl(1)-Tl(II1) Fe(CN)&-Fe(CN)P Mn(V1)-Mn(VI1) Ce(111)-Ce( IV) Co(I1)-Co( LCI)
E*, kcsl./mole
13.2 7.7 14.3 10.7 9.9 8.3 12.1 4.7 10.5 23.0 21.6
Eo, v.(I
Electron-transfer reactions 0.25 20.5 -1.61 54.0 Hole-transfer reactions ~ 2 7 -0.33 26.0 -0.77 19.5 -1.15 14.6 -1.25 10.4 -0.36 18.2 -0.56 9.4 -3.0 -1.61 -1.81 -5.5
--O.l
8.1 5.1 -7 7.5 8.2 8.8 6.0 8.3 9.6
...
...
b
7.6 4.1
C
6.4 7.6 7.8 8.1 6.0 6.3e 10.0” 18.2 16.7
’
See ref. 4. K.V. Krishnamurty and A. C. Wahl, J. Am. C h .Soc., 80,5921 (1958). See ref. 1. D. Cohen, J. C. Sullivan, and J. C. Hindman, J. Am. C h . SOC.,76, 552 (1954). ‘ Recalculated to the pH of the kinetic measurements. See ref. 2. J. Shankar and B. C. de Souza, J. Inorg. Nucl. C h m . , 24, 693 (1963).
the ion with smaller charge only if the thermal excitation had already removed the ligands from its first coordination sphere to a distance corresponding to the complex of the ion with higher charge. Thus,an excitation of the ligands producing valence vibrations is the required condition for electron transfer. The calculation of E* is, however, in this way very problematic because, generally, the corresponding bond energies and force constants are unknown though the applicability of the described model can be controlled by comparing the observed xe-values with the data calculated by the combination of (5) and the measured E*values. Up to this point only the upper limits of the possible donor-energy values have been considered. The electron transfer, however, can be envisaged in another way, too. Thk happens namely if the electron is excited from the level corresponding to process I1 (represented in the figure by the lower horizontal line) to the acceptor state, whereas the appropriate amount of activation energy must be yielded. An HzO+ molecule, an electron deficiency (a hole), is created in this process, which can move comparatively fast t o the donor state in the periodic potential field of the water molecules. When axriving at the donor, the hole recombines with its electron. As in the example of electron transfer, the hole can reach the acceptor level only by tunnel effect if the excitation energy is smaller than the difference between the considered state and the energy corresponding to the process 11. (From the point of view of the electron,
’
the acceptor level is a donor level.) The circumstances are in this case completely analogous to that of the electron tunneling with the only deviation that now
E” = 2.4 - Eo e.v. (Taking into account the hole, the energy scale is now reversed, and hence the opposite sign of the original Eo- 2.4term.) According to the cited principles, two ways are available for every electron-transfer reaction: one by “electron transfer” and another by “hole transfer.” The two ways of reaction are parallel, and therefore the ratedetermining process is the faster one. One can thus admit that the electron-transfer reactions take place mainly by %ole transfer” if the ion pair has a smaller normal oxidatioD-reduction potential than about -0.1 v., and they occur mainly by “electron transfer” if the oxidation-reduction potential of the partners is higher than this value. If, however, it turns out that in the different modes of reaction the smaller probability of tunneling is coupled with lower energy of activation and the larger probability with the higher one, then it may happen that the two rates of reaction are commensurable (SW eq. 1)and so may be observed experimentally. This is the case, for example, in the Ce(II1)-Ce(1V) exchange reaction.’ The pH dependence of the reactions has two reasons. (1) The change in pH of the solution may result in a change of the coordination sphere of the reactants Volum 69, Numbm 9 Septmbm 1986
NOTES
3 186
(e.g., hydrolysis), thus causing a deviation in the Franck-Condon restriction by the change of the force constants; this means a change of the activation energy. Thus, the reaction order in hydrogen ion is governed by coordination chemical laws. These reasons are responsible for the pH dependence of the Ce(II1)Ce(1V) reaction, proceeding by hole transfer. (2) The limits of the potential in reactions I and I1 are also pH dependent, and thus another “slight” pH dependence can be observed6 (eq. 5). This later phenomenon was taken into account in calculating the A!fn042--3!h04- and Fe(CN)64--Fe(CN)63- systems. When both effects are significant, their separation is very difficult. In Table I the log (x,kT/h) values calculated with the aid of (5) according to the described method are compared with the measured values of the action constant. (For the calculation of the observed action constant, the experinientally obtained values of the entropy of activation were used.) In some exchange reactions, e.g., U(1V)--U(VI), the agreement between calculated and observed values seems to be poor. The explanation for it is that these ions preferably form polynuclear hydroxy complexes, and therefore the OH- bridges existing between the reactants are involved in the electron transfer. The Ce(II1)-Ce(1V) and Co(I1)-Co(II1) hole-transfer reactions unfold an interesting fact. According to experience here xe > 1. Up to Eyring’s theory about the rate of reaction
AS*
xe = exp-
R
(7)
where AS* is the entropy difference due to activation. Positive AS* means that the degrees of freedom of the activated complex grow larger. According to the model suggested in the present paper, this is obvious. The activation energy belonging to the mentioned exchange readion happening by hole transfer is namely higher than E”, which refers to a “free” hole. The increase in the degrees of freedom is thus reasonable. Regarding the figures collected in the table, we have to remark that, when a reaction proceeds by two-electron transfer, the probability of the simultaneous tunneling of two electrons is given by the square of the single electron’s transfer probability. The Tl(1)TI(II1)exchange was calculated on this basis.
(6) I. Ruff, Acta Chim. Acad. Sci. Hung., in press.
The J O U Tof ~Physical Chemistry
Electron Trapping in Rigid Ethanol-Methyl-2- tetrahydrofuran Mixtures
by L. Shields Department of Physical Chemistry, The Universitv of Leeds, Leeds 2, England (Received January 28, 1966)
Color centers produced by ionizing radiation in rigid organic solvents have been assigned to trapped or solvated electrons112although the latter term implies the orientation of solvent molecules to an unrealizable extent. The trapped electron bands are solvent dependent having less energy in methyl-2-tetrahydrofuran (MTHF) than in alcohols’; in this respect they resemble the optical absorption bands of solvated halide anions which are shifted to low energies in low polarity solvents. In mixed solvents the halide bands, ascribed to charge-transfer-to-solvent transitions,*J move continuously, but not uniformly, with mole fraction of either solvent between the limits set by the bands in each pure solvent. These observations are interpreted in terms of a continuous interchange of solvent molecules between the solvation shell and the bulk solvent, thus allowing for the continuous variation of the solvation shell with solvent composition. The nonuniformity of the shift is attributed to preferential solvation by the more polar solvents, particularly those solvents containing hydroxyl groups. The effect of solvent composition on the trapped electron band in relation to the solvation spectra of halide ions in mixed solvents should test the application of the term “solvated” electronsto these color centers. The nature of trapping sites in vitreous ethers and alcohols has been discussed.’ In the former, electrons are trapped in relatively large holes, and in vitreous alcohols the electric field of the electron is sufficient to align the hydroxyl dipole of the peripheral layer of molecules and thereby further confine and stabilize the electron. Following from a simplz description for the electron trapped in a spherical potential well, it can be shown that the optical band energy is governed by cavity size which is solvent dependent. The (1) M. J. Blandamer, L. Shields, and M. C. R. Symons, J . Chem. SOC.,1127 (1965). (2) W.H.Hamill, J. P. Guarino, and M. R. Ronayne, J. Am. Chem. SOC.,84,500 (1962); 84,4230 (1962); Radiation Res., 17,379 (1962). (3) M. J. Blandamer, T. R. Grifiiths, L. Shields, and M. C. R. Symons, Trans. Faraday Soc., 60, 1524 (1964). (4) M.Smith and M. C. R. Symons, Discussions Faraday SOC.,24, 206 (1956). (5) T. R. Griffithsand M. C. R. Symons, Trans. Faraday SOC.,56, 1125 (1960).
