1792
I. RUFF
Photochemical studies of halogen molecules14indicate caging of halogen atoms to be quite effective. This accounts for the strong phase dependence found in their reaction as hot atoms. Conversely, the early finding of Bodenstein,lj that there is no gas-liquid phase effect in the photolysis of HI, supports our conclusion that caging of hydrogen atoms tends to be unimportant.
Acknowledgment. This work was supported by the U. S. Atomic Energy Commission. (12) J. C. (1953).
W.Chien and J. E. Willard, J . Am. Chem. Soc.,
75, 6160
(13) J. E. Willard, “Effects of Nuclear Transformations,” v01. I, AEC, Vienna, 1961. (14) J. Franck and E. Rabinowitsch, Trans. Faraday SOC.,30, 120 (1934). (15) As quoted by G. Kornfeld in the discussion to ref 14.
,
Extension of the “Band Model” to the Inner-Sphere Mechanism
of Electron-Transfer Reactions by I. Ruff L. Eotoos University, Institute of Inorganic and Analytical Chemistry, Budapest, Hungary Accepted and Transmitted by The Faraday Society
(December 1 , 1966)
The band model, suggested for interpreting the kinetic behavior of outer-sphere electron-transfer reactions, has been modified for the inner-sphere mechanism. The bridging ligand of the binuclear activated complex can be a favorized electron transporter, reducing the height of the energy barrier by its oxidation-reduction potential and the barrier width by its complex-formation ability. Because of these phenomena, the entropy of activation increases, resulting in a catalytic effect. I n addition to this, the bridging ligand can reduce the activation energy, if the formation enthalpy of the binuclear complex is larger than the energy of electrostatic repulsion between the reactant ions. Such types of catalytic effects of halogenide ions have been interpreted and the entropies of activation have been calculated. Good agreement has been shown between the theoretical and experimental results.
1. Introduction In earlier p a p e r ~ l -a~ new model has been suggested to calculate the kinetic parameters of oxidation-reduction and electron-exchange reactions, assuming an indirect electron-transfer mechanism. For the reactions considered in those works, it could be expected that the reactants approach each other, at most, up to the first coordination sphere in the activated complex. This requirement is satisfied at first by so-called “substitution inert” ions having a regular or nearly regular octahedral coordination sphere.5 Namely, the ions of “substitution labile” complexes are able to form temporarily a binuclear complex without an appreciable increase of the activation energy : Le., an activated complex can be formed in which the first coordination spheres of the two ions penetrate. It has been proved experimentally that electrontransfer reactions proceed by these two types of mechanisms (cf. r e v i e r n ~ ~ ~They ~ ) . are usually quoted as the “outer-sphere” and the “inner-sphere” mechanism. The Journal of Physical Chemistry
2. TheModel The model suggested, which has been shown to be in good agreement with the experimental results for reactions proceeding by the outer-sphere mechanism, is the following. It has been assumed that the reaction takes place either by electron transfer, in a narrower sense, with the aid of the intermediate reaction l s 2
H Z H + + e as follows
(1)
+ Hf+ + H H + oxz -+-redz + H +
redl
0x1
(1) I. Ruff, J. Phys. Chem., 69, 3183 (1965). (2) I. Ruff, Acta Chim. Acad. Sci. Hung., 47, 241 (1966). (3) I. Ruff, ibid., 47, 255 (1966). (4) I. Ruff, ibid., 52, 364 (1967). (5) D. R. Stranks and R. G. Wilkins, Chem. Reo., 57, 743 (1957). (6) H. Taube, Proc. R. A. Welch Found. Conf. Chem. Res., 6 , 7 (1962). (7) I. Ruff, K e m . Kozl., 26, 73 (1966).
INNER-SPHERE MECHANISM OF ELECTRON-TRANSFER REACTIOKS
or by hole transfer
+ OHOH + redz ox1
-+-+-
+ OH OH- + ox2 redl
Le., the intermediate process is OH-
E OH + e-
(2)
The Bohr radius of the electron belonging to the hydroxonium radical (H30) and that of the hole in the hydroxyl radical (OH) has been estimated ky a simple appr~ximation.~This value is about 40 A. Such a greatly extended wave function gives the possibility of considering the electron or hole to be a practically delocalized “free” particle like the charge carriers of semiconductors. It has been shown that the energy of the electron of the H30 radical, OH- ion, and the reactants can be calculated using the corresponding normal oxidationreduction potentials. Taking such energy conditions into account, the energy barrier of the electron-transfer reaction can be given as shown in Figure 1. Here the delocalized levels of H 3 0 and OH- are marked by the long horizontal lines. I n the equilibrium state, the upper level is practically unoccupied and the lower one is occupied. The energy ranges, lying below and above these, are unimportant for us, hence unoccupied and occupied states, respectively, cannot exist here, since their existence would result in the decomposition of the water because of the spontaneous transitions. The levels of reactants (electron donor and acceptor) may exist only between these two energy limits. The electron should overcome the rectangular potential barrier (dotted line) to accomplish one step of an electron transfer. This can happen either by ac-
1793
cepting higher energy values than the height of the barrier and forming an H30, or by the tunnel effect possessing lower energy than the barrier height. The activation energy is determined by the Franck-Condon restriction and, in the first case, by the barrier height. The hole transfer is quite similar, but everything happens in just the opposite direction and the barrier height is given by the lower energy limit and the unoccupied reactant level (hole donor). Thus the reaction rate observed macroscopically is one of the four possible paths: namely, which has the smallest free-energy change of activation, i e . , the fastest one. Figure 1 is very similar to the well-known band system of the semiconductors (conduction, forbidden, and valence band) both by its shape and meaning. Due to this similarity, this model will be called the “band model.” (There is a marked difference, of course: the Ha0 and OH levels are imperfection states too, not like the conduction and valence bands in semiconductors.)
3. Theoretical Equations
It has been shown4that the entropy of activation can be calculated by the equations
AS*
AS*, 4- AS’,
+ Ah’”, - R
(3)
where
and AS*t
=
-2R In c*
(6)
Here p , is the transition probability of the electron, p i is the probability of satisfaction of the ith configuration requirement of the activated complex ( A S * i represents what part of the collisions takes the place of a good orientation), and c* is the so-called normalized concentration : i.e., at this concentration the average distance between the reactants in solution is equal to the distance d between the reactants in the activated complex c* =
( d w A x 10-9-1
(7)
where N A is the Avogadro number. AS*, represents the entropy due to the change in the degrees of freedom of translation. In the case of transfer above the barrier, the activation energy is
P
,d Porition coordimtc Figure 1.
I
AE
3
U
(8)
where U is the barrier height (see Figure l), but then we have pe = 1 Volume Y9,iVumber 6
(9) M a y 1968
I. RGFF
1794 if the reactants situate inside of the effectivity space of one H30 or OH radical. Considering their mentioned Bohr radii, this means that the reactants should approach each other at least to about 80 '1. Thus on the basis of eq 6
Ab'*:, = $ 2 3 et1
(10)
Since we are speaking of the outer-sphere mechanism, the reactants can be considered to be of spherical symmetry, i.e., for the collision there is no configurational restriction. Thus we have
0
AS'i
(11)
(Here the term "collision" means that the two species are just a t the distance, d, required by the activated complex.) Thus using eq 3 , it can be written
AS*
=
21 eu
(12)
I n the case of the tunnel effect1,* the fastest path corresponds to the minimal d. For the outer-sphere mechanism, this occurs probably by the penetration of the second coordination spheres. Thus d is about 9.4 8, from which it follows that
AXit
0
(13)
On the basis of the Gamow equation
(T
- AE)
p , = exp - - d d 2 m ( i 7
)
(14)
where h is the Planck constant and m is the mass of the electron. Thus the total entropy of activation is
4a
- AE) - R
AS* = - R - - d d 2 m ( U h
(15)
since eq 11 is valid in this case too. I n the case of transition metal ions, the activation energy is given by
AE
= (1ODq)ox
- (1oDq)red
(16)
where the terms of the right-hand side are the crystalfield splittings of the (electron or hole) donor ion in its oxidized and reduced state, respectively. Equation 16 arises from approaching the ligand-electron repulsion energy by Van Vleck's This energy is needed t o preform the environment of the ion to be left by the electron or hole, as it is required by the Franck-Condon principle, i.e., to "push" the ligands toward the central ion. 4. Energy Barrier for the Inner-Sphere Mechanism I n the case of the inner-sphere mechanism, the electron transfer is preceded by the formation of binuclear complex of the configuration
nlI1%+- L -
1\12"+
Unlike the outer-sphere mechanism, now the energy barrier is determined by the properties of the ligand L T h e Journal of Physical Chemistry
and metal ions RL"+ and AIz"+. Though the wave function of L is much more localized than that of the hydrogen or hydroxyl ion, it will still be able to transport the electron, since here the distance between the initial and final position of the electron is smaller. Applying the usual expression, L functions as a bridge for the electron. If L is a halide ion, the binuclear complexes before and after the excitation of the hole, respectively, are nip+
-
T&C"-l,+
x- - ; \ ~ p + - x o - pp1.
-
Thus the intermediate reaction is
-X0
+ e-
(17)
Similar to the hydroxyl ion, the halide ion can play a part in the hole transfer, since this later intermediates the transfer by an electron-donation process followed by an electron-capture process. Using the approximations of the previous paper^,^-^ the relative energy of the hole of Xo could be taken into consideration by the normal oxidation-reduction potential with respect to eq 17, and that of the electron and hole of the metal ions can be calculated also by the corresponding normal oxidation-reduction potentials. Thus the energy barrier is altered, as seen in Figure 2 . It can be seen that the same possibilities can be taken into account as for outer-sphere electrontransfer: i.e., (i) electron and hole transfer is possible; (ii) both can take place either by tunneling or by the transfer above the barrier. The essential difference is that in this case the barrier height of the hole-transfer reaction is not determined by the energy limit E-, but by E,, if
E,
> E-
(18)
If E , < E-, the hydroxyl ions intermediate the reaction, since they can affect, even when they are not situating just between the reactants, being their wave functions much more extended. Similai-ly, it is possible for a suitable bridging ligand to catalyze the electron-transfer path reducing the actual barrier height. I n this case, it acts as an acceptor for an electron from the reducing agent. It must be a stronger oxidizing agent than the hydroxonium ion is. The bridging ligand was usually a halide ion in the investigated cases; later the treatment will be limited only to these. This does not mean, however, that the electrontransfer path has been excluded because the halides are catalyzers for the hole transfer. If the reactants are able to form a binuclear activated complex, the (8) F. H. Van Vleck, J . Chem. Phys., 3 , 803, 807 (1935).
1795
INNER-SPHERE MECHANISM OF ELECTROWTRANSFER REACTIONS
and
U
1
1
d
0
I
- x
Figure 2.
electron transfer through the Ha0 radical is favored as well, since the barrier width is decreased.
5. The Entropy of Activation The entropy of activation can be calculated making use of eq 3-6 in this case too. This requires the determination of d and U . (i) The distance between the reactants in the activated complex is d = r,
+ r, + 2rL
(19)
where r, and are the ionic radii of M"+ and hfm+, respectively, while rL is that of the ligand. The radius of the metal ion is markedly influenced by complex formation; no large error is made, however, when about 0.5 .& is assumlad. In Table I the calculated d values are shown using the ionic radii of the halide ions being in octahedral ;pm
Table I
1
X-
rLI
F-
1.36 1.81 1.95 2.16
c1Br-
I-
10,
v
-4.0 -2.3 -1.8 -1.2
d,
A
3.72 4.62 4.90 5.32
A s *t,
eu
-9.8 -7.2 -6.5 -5.5
coordination. I n the fifth column the values of AS *$ are given which have been calculated by applying eq 6 and 7. (ii) The probability of tunnelling can be determined with respect to eq 14 by the former d values and by one of the equation8
=
E+ - BO
(22)
where Ea is the energy corresponding to the normal oxidation-reduction potential of the appropriate donor. The suitable equation should be chosen among eq 20-22 depending upon whether the reaction is assumed to be hole transfer with E , > E- or E , < E-, or electron transfer. Among these entropy values, the favored one is that which results in the smallest AF *, since the reaction proceeds via this pathway. The application of eq 20 requires the knowledge of the normal oxidation-reduction potential due to the equilibrium described by eq 17. AS this contains halogen atoms, the potential cannot be observed experimentally. It can be calculated, however, on the basis of standard free-energy value^,^ assuming the hydration energy of halogen atoms to be zero. This seems to be a close approximation because the hydration energies of Xz molecules are only 3-5 kcal/mol (0.1-0.2 ev), too. The calculated normal oxidation-reduction potentials are given in the third column of Table I. If the composition of the donor ion is hlLEX, or RIL,X2, where L is MHOor HzO, the normal potential due to MLe has been used which does not essentially differ from the value corresponding to the mixed complexes. lo The values of A 8 *e, calculated in this way, have been summarized in the third column of Table I1 for some reactions studied experimentally. I n these calculations the observed activation energy has been applied for the value of AE. Namely, the correctness of the relations, obtained for the entropy of activation, can be controlled without using the theoretical AE values which could not easily be calculated. In addition, no new approximations and errors are introduced by this treatment. (iii) For the outer-sphere mechanism, the AS *i part of activation entropy, arising from the oriented collision, should not be taken into account, since the reactants can be considered, in close approximation, to be of spherical symmetry. That is not the case, however, for' the binuclear activated complex, because the bridging ligand occupies usually only one or two places in the coordination sphere of the reactants. At the other places there are other ligands unable t o form a bridge. Thus the probability should be also considered which is due to the suitable spacial orientation. The ideal orientation in the activated complex is the linear arrangement of the two metal ions and the halide (9) W. M. Latimer, "Oxidation Potentials," Prentice-Hall, Inc., New York, N. Y., 1952. (10) F. Basolo and R. G. Pearson, "Mechanism of Inorganic Reactions," John Wiley and Sons, Inc., New York, IT.Y . , 1958, p 318. Volume 73, Number 6
M a y 1968
1796
I. RUFF
Table I1
A s *e, No.
1 2 3 4 5 6 7 8 9 10 11
Activated complexa
Electron-Transfer Cr(II)-F-Cr(III)(NH3)5 Cr(I1)-Cl-Cr(III)(NH& Cr(II)-C1-Co(III)(N’H3)s Eu(11)-Cl-Co( III)( “3)s Eu(II)-Br-Co(III)(NH3)j V( II)-Cl-Co( III)(“3)s
eu
*calodq AS*obsd,
eu
Reactions -8.0 -23 -10.3 -9.4 - 11 . 3 -12.1 -10.3
Hole-Transfer Reactions \r(II)-Br-Co(III)(NH,)6 -11.4 Fe(I1)-Cl-Fe(II1) -10.1 Fe(I1)-C1-Fe(II1)Cl -9.8 Fe( 11)-F-Fe( 111) -10.1 Fe(I1)-F-Fe(II1)F -9.8
eu
-23 -22 - 24 -24 - 23
-30b -2ab -25“ - 30d -32d - 20‘
-23 -23 -21 -25 -24
-22d -24e -20’ -21’ -22’
* H. Taube, Proc. a The HzO ligands have been omitted. R. A . Welch Found. Conf.Chem. Res., 6, 7 (1962). A. Zwinkel and H. Taube, J . Amer. Chem. SOC.,83, 793 (1961). J. P. Candlin, J. Halpern, and D. L. Trimm, ibid., 86, 1019 (1984). e J. Silverman and R. W. Dodson, J . Phys. Chem., 56, 846 J. Hudis and A. C. Wahl, J . Amer. Chem. Soc., 75, (1952).
’
4153 (1953).
ion. It cannot be determined a priori how this configurational restriction is rigorous: Le., to what extent the metal-halide-metal angle might deviate from 180”. It seems to be acceptable, however, that the configuration is “good” as far as the metal-metal line passes nearer to the nucleus of the halide than to the center of another ligand. For reactant ions NL6X such positions take place of a probability of while for ML4X2ions this value is 1/3. This gives -3.6 and -2.2 eu, respectively, for A s *i* I n the fourth and fifth columns of Table I1 the observed values of the activation entropy are compared with the calculated over-all ones using eq 3. Considering the approximations and the average errors of the observed entropy of activation, an agreement within f5 eu can be regarded as a good one. On the basis of the good fit obtained in most cases, it can be concluded that the basic assumption of the theory seems to be right: Le., the band model is able to explain the entropy of activation of inner-sphere electron-transfer reactions. Thus the catalytic effect of the halide ions arises from two reasons with respect to the entropy of activation: (i) the barrier width i s decreased by the formation of the binuclear complex; (ii) the barrier height i s decreased for the hole when the halide can be oxidized more easily than the hydroxyl ion. It can be seen in Tables 1-11 that the first effect appears for every halide ion, while the second is expected only for C1-, Br-, and I- and, at first, is expected for systems favoring the hole-transfer path (Eo < -0.1 V). The Journal of Physical Chemistry
The data of Table I1 are interesting, from the point of view that the calculated as well as the observed values do not show any trend ; all of them are about - 20 eu. I n the case of the theoretical data this is the result owing to an inner compensation. Namely, the larger d value causes an increase of AS*, and a decrease of AS*e and vice versa. This confirms the applicability of the band model, since the same assumed energy conditions predicted properly a change of activation entropy from -40 up to +20 eu for the outer-sphere mechanism.1-4
6. The Activation Energy In the case of the outer-sphere mechanism, the relatively large distance between the reactants gave the possibility of neglecting the electrostatic interactions, I n this way the total activation energy could be explained by the Franck-Condon restriction. For the inner-sphere mechanism, however, the formation free energy of the binuclear complex and the electrostatic interactions should be taken into account, in addition to this. I t can be expected, in general, that the latter one enhances the activation energy, while the former one can enhance as well as reduce it, depending on the individual properties of the reactants. It can be seen that the problem is rather complicated to endeavor to obtain quantitative theoretical results. I t is probable, however, that the main part of the activation energy is the result of the Franck-Condon restriction. This can be seen in Table 111, where the activation energies of the inner-sphere transfers listed in Table I1 are compared with those of the outersphere transfers having the same donor. A catalytic Table I11
_--No.
1 2 3 4 5 6
7 8 9 10 11
Donor process
Calcd
AE, kcal/mol----
Outer sphere’
Cr(I1) + Cr(II1)
9.5
...
En(I1) + En(II1)
...
9.0“
V(I1) -* V(II1) Co(III)(NHs)s + Co(II)(”a)s Fe(II1) -*. Fe(I1)
13.2
...
13.2b 13.3
9.5
9.gd
Inner sphere
13.4 11.1 14.7 5.0 4.7 14.1 9.1 8.8 10.0 9.1 9.5
Obtained for electrode reaction: J. E. B. Randles and K. W. Somerton, Trans. Faraday Soc., 48, 937 (1952). * K. V. Krishnamurty and A. C. Wahl, J. Amer. Chem. SOC.,80, 5921 N. S. Biradar, D. R. Stranks, 31. S. Vaidhya, G. J. (1958). Weston, and D. J. Simpson, Trans. Faraday SOC., 55, 1268 J. Silverman and R. W. Dodson, J . Phys. Chem., 56, (1959). 846 (1952). The data in this column refer to isotopic electronexchange reactions.
1797
LIGHT-INDUCED PROTON EJECTION AND ELECTRON TRANSFER effect owing to the decrease in activation energy can be observed only in the case of reactions 4, 5 , 7, 8, and 10, but for the other reactions no catalysis of this type appears. Thus, in the latter case, the acceleration arises from the increase in the entropy of activation only: L e . , either the electrostatic repulsion and the free-energy change of the complex formation have no marked effect or they are opposite and compensate each other. As can be concluded from the present treatment, the electrostatic and polarization interactions are not the most important effects, unlike in the theories of Eyring and coworkers,ll Libby,12 and Marcus,13which, there-
fore, do not seem t o be easily extendable to the innersphere mechanism. On the other hand, this model does not contain such complicated parameters and computing processes as the conjugation theoryl47l6 does. (11) R.J. Marcus, R. J. Zwolinsky, and H. Eyring, J . P h y s . Chem., 58, 432 (1954). (12)W.F. Libby, ibid., 56, 863 (1952). (13) R. A. Marcus, J . Chem. Phys., 24, 966 (1956); 26, 867 (1957); Disc. Faraday Soc., 29, 21 (1960); J . Phys. Chem., 67,853 (1963). (14) J. Halpern and L. E. Orgel, Discussions Faraday SOC.,29, 32 (1960). (15) P.V. Manning, R. C. Jarnagin, and M. Silver, J . Phys. Chem., 68, 265 (1964).
Light-Indulced Proton Ejection and Electron Transfer in the Zinc Tetraphenylporphin-Benzoquinone System by Kenneth P. Quinlan Photochemistry Section, Energetics Branch, Space Physics Laboratory, A . F . Cambridge Research Laboratories, L. G. Hanscom Field, Bedford, Massachusetts 01730 (Received December 6 , 1967)
Excitation of zinc tetraphenylporphin in the presence of p-benzoquinone activates a reversible proton ejection and a n electron transfer attended by the formation of the benzosemiquinone ion radical. The reaction is slightly quenched by air and inhibited by large concentrations of benzoquinone. The results are similar to those found with the chlorophyll-benzoquinone system.
Introduction I n a recent communication, the chlorophylls were shown to eject, a proton during the light-activated single-electron transfer with p-benzoquinone. Possible sources of the origin of the proton from chlorophyll based upon previous work2p3 are the methine and the C-10 positions. A study of the zinc tetraphenylporphin (ZnTPP)-benzolquinone system offered an opportunity to see whether the presence of hydrogens at these particular sites is necessary for proton ejection. Two additional significant observations were also noted in the study of the chlorophyll-benzoquinone system. Proton ejection was in the same apparent proton-activity range whether air was present or not,' and high concentrations of benzoquinone inhibited proton ejection." Results are presented in the present paper to show that the light-induced reaction between zinc tetraphenylporphin and benzoquinone is slightly quenched by air and inhibited by large concentrations of benzoquinone.
Experimental Section The zinc a,P,y,G-tetraphenylporphin was kindly supplied by G. Sherman of Brandeis University. It was further purified by chromatography on a column of a mixture of magnesia and Hy-Flo Super Cel using the method of Seely and Calvin.6 Benzoquinone was purified by sublimation. Methanol was Baker Analyzed reagent spectrophotometric grade. Dimethylformamide was purified by distillation from calcium hydride under vacuum. The method of measuring the apparent pH values of the solutions has recently been described.6 I n the (1) K. P. Quinlan and E. Fujimori, J . P h y s . Chem., 71, 4154 (1967). (2) R. C. Dougherty, H. H. Strain, and J. J. Katz, J . A m e r . Chem. Soc., 8 7 , 104 (1965). (3) R. B. Woodward and V. Skaric, ibid., 83, 4676 (1961). (4) K. P. Quinlan, unpublished results. Electron spin resonance studies of the same system in dimethylformamide showed a large decrease in the esr signal with increasing amounts of pbenaoquinone. (5) G.R. Seely and M. Calvin, J . Chem. Phys., 23, 1068 (1955). (6) K. P. Quinlan and E. Fujimori, Photochem. Photobiol., 6, 665 (1967). Volume 78, Number 6 M a y 1968
