J. M. HALE
3196
Electrode Reactions Involving Electronically Excited States of Molecules by J. M. Hale'" (Received December 8, 1968)
Cyanamid European Research Institute, Cologny, Geneva, Switzerland
Calculations are presented concerning two aspects of the electrode reactions of molecules M* in electronically excited states. Firstly, the current which would flow through an electrode is computed for the experimental arrangement in which M* is generated by irradiation of a solution of the ground-state molecules M through the electrode, which is assumed to be potentiostatically controlled in a region of potential where only M* is electroactive. Secondly, the rate of electrogenerationof M* is calculated, assuming the potential of the electrode to be switched instantaneouslyfrom one at which an anion M- is formed at a diffusion-controlledrate to one at which oxidation of M - to M* is feasible. In each calculation the quenching of M* t o M by electron exchange with the electrode is allowed for by using for this purpose arecently derived for mulation of an electrochemical rate constant valid at high overpotentials.
+
-
Introduction We shall consider, in the following, the electrode processes which can occur when an electronically excited molecule M* in a solution reacts a t a metallic electrode. The problem has been studied because it may provide a means of experimental verification of formulas recently derived for electrochemical rate constants a t very high overpotentials. lb A theoretical examination is made of the consequences to be expected from two kinds of experiment. Firstly, it is supposed that excited-state species, It*, are generated by irradiation of a solution of M and that these are detected a t an electrode. We assume that the bulk of the solution contains only the ground and excited states of the chosen solute molecule, together with an excess of indifferent electrolyte, and that M is not electroactive a t the electrode. This is an easily realizable state of affairs in practice, since it is necessary only to control the electrode potential to be somewhere between E+ and E-, the oxidation and reduction potentials of M, respectively. We allow for the possibility that :VI* may either be reduced or oxidized at the electrode and that the anions and cations formed may become immediately discharged at the electrode producing the ground-state moleculesZ
-
+ e "i.M- standard potential E-* M- -% M + e standard potential E E M* &+ I+ e standard potential E+* M + + e "L, M standard potential E+ M*
(la)
e
--t
(lb) (IC) (Id)
A forward-directed arrow over a rate constant IC denotes a reduction reaction, a backward-directed arrow an oxidation reaction. We recall that E+* The Journal of Physical Chemistry
= E+ ex/e and E-" = Ec / e , where ex is the excitation energy and e is the elementary charge. The first section, then, is concerned particularly with the dependence of the electrode current in such a system upon potential and the extinction coefficient of the solution a t the wavelength of the incident radiation. Secondly, we consider the efficiency of electrogeneration of excited states, simulating an experiment in which the electrode potential is switched instantaneously from a point cathodic of E- to one anodic of E-* and cathodic of E+ (that is, 6, < e(E+ - E-)). I n this case no incident radiation is present to cause formation of M* by other means, and our calculation therefore yields the likelihood that an excited molecule can be formed at, and escape from, the electrode before becoming quenched by a pathway involving the cation c
+ e standard potential E-* f MM + e standard potential E E M* -+ M + + e standard potential E+* z M standard potential E+ M+ + e
M-
-%
R/I*
--j
----f
(2a) (2b) (2c) (2d)
It is assumed in this section that E-* is less anodic than E+ in order that anion-cation annihilation may be ignored. a * 4 Radiative decay of M* produced according to reaction 2a was suggested by Maricle, Zweig, and (1) (a) American Cyanamid Co., Central Research Laboratories, Stamford, Conn. 06904. (b) J. M. Hale, J.Electroanal. Chem., 19, 316 (1968). (2) E. A. Chandross and R. E. Visco, J . Phvs. Chem., 72, 378 (1968). (3) R. E. Visco and E. A. Chandross, Electrochim. Acta, 13, 1187 (1968). (4) J. Chang, D. M. Hercules, and D. K. Roe, ibid., 13, 1197 (1968).
ELECTRONICALLY EXCITED STATES OF MOLECULES coworker^^^^ to be the source of luminescence observed when an electrode underwent the above-mentioned potential-time program when in contact with a solution containing phenanthrene.
The Rate Constants I n the reaction schemes considered here it is particularly important to describe correctly the behavior of a rate constant when the energy separation of the initial and final states of the electrode reaction is large. Consider, for example, the oxidation of an anion M- a t very high overpotential. The energy level of the electrode having the same energy as the excess electron in the anion a t equilibrium with its solvation environment might be several electron volts above the Fermi level.' I n this energy range of the conduction band of a metal the levels are virtually unoccupied, and the number of vacant levels of the metal having an energy near that of the electron in the anion becomes only weakly dependent on applied potential. This effect, then, would be expected to lead to "saturation" of the reaction rate suggesting that the conventional formula* for the rate constant, always increasing exponentially with applied overpotential, might not be applicable in this case. Moreover, the formula previously derived from electrontransfer theoryg
k
=
km exp{ - ( x / ~ ~ T ) [f I e(E - E , " ) / A ] ~ J(3)
(the positive sign is used for an oxidation reaction and the negative one for a reduction), where X is the reorganization energy parameter,'O predicts that the rate constant passes through a maximum k m at an overpotential of x/e volts. E,O is the standard potential, and E the electrode potential. This behavior is also unrealistic, as was pointed out by Marcus,11and is due to the inadequate account of the distribution of electron states in the electrode. A reexamination1 of the derivation of the rate constant from electron-transfer theory shows, however, that saturation of the rate constant a t high overpotential is accounted for in the basic equations. The property is lost in the course of the treatments published by Marcus,9 D ~ g o n a d z e , ' ~and ~ ' ~the authors of similar work^,'^'^^ because of an approximation made to derive an analytic form of the result valid at low overpotentials. It has been shown elsewhere' that the saturation property may be retained, in an approximation which makes use of a tabulated function in place of the analytic expression, namely
IC = O.51cm[l- erf(a)]
(4)
where a = [A
e(E - E C 0 ) ] / ( 2 2 / m )
j=
and erf represents the error function. Since erf(-x) = -erf(x) we have k(a) = k, - k ( - a ) , which leads to the sigmoidal property of the rate constant. I n
3197
the expression for a, the positive sign is taken for a reduction reaction and the negative sign for an oxidation. Unfortunately, the relationship between the rates of the reduction and oxidation directions of an oxidation-reduction reaction
is not quite valid in this approximation, but the consequences of this are not serious in the high overpotential regions of interest here. According to eq 4 a rate constant becomes independent of potential at 4 2 / ~ k T ) / e volts. overpotentials in excess of (A Choosing a typical value for X, 1 eV, we may compare the estimates of a t this overpotential made by the three approximations mentioned; we find N k, N lo4 cm + sec-'from eq 4, k N 1.83 x lo2 cm sec-l decreasing with overpotential from eq 3, and 6 x 1014 cm sec-l from the conventional formulation of electrode kinetics. I n choosing lo4 cm sec-' for IC, we are making the reasonable assumption that the maximum rate is collision determined and that the electron transfer is adiabatic. There is no reason to expect that X should be significantly different in the electrode reactions of groundor excited-state molecules. The only extra contribution could come from interactions of a dipole moment in the excited molecule. This, however, is likely to represent a very small reorganization energy under experimentally used conditions since the field of the dipole would penetrate very little into the electrolytic medium.
+
z
x
The Electrode Current Due to Reaction of Excited Species Employing diffusion and electrode kinetic equations we calculate the current due to the reaction system 1. The electrode potential is supposed to be controlled potentiostatically within the range E- < E < E+ so that the rate constants IC1 to k4 are time independent and backward reactions may be neglected. The radiation which generates M* in the solution is taken to (5) A. Zweig, D. L. Maricle, J. S. Brinen, and A. H. Maurer, J. Amer. Chem. SOC.,89,473 (1967). (6) D . L. Maricle, A. Zweig, A . H . Maurer, and J. S. Brinen, Electroehim. Acta, 13, 1209 (1968). (7) J. M. Hale and H. Bassler, Trans. Faraday Soc., 64, 2452 (1968). (8) P. Delahay, "New Instrumental Methods in Electrochemistry," Interscience Publishers, Inc., New Yorlr, N. Y., 1954, p 34. (9) R. A. Marcus, Ann. Rev. Phys. Chem., 15, 155 (1964). (10) R. A. Marcus, J. Chem. Phys., 24, 966 (1956); 43, 679 (1965). (11) R. A. Marcus, ibid., 43, 2654 (1965). (12) R. R. Dogonadre and Yu. A. Chizmadzhev, Proc. Russ. Aead. Sci., Ser. Chim., English Transl., 145,663 (1962). (13) R. R. Dogonadre, A. &I. Kuznetsov, and A. A. Chernenko, Russ. Chem. Rev., English Trans., 34, 759 (1965). (14) X. de Hemptinne, Bull. SOC.Chim. Fr., 2328 (1964). (15) X. de Hemptinne, Bull. SOC.Chim. Belges, 77, 21 (1968). Volume 73, Number 10 October 1060
J. M. HALE
3198 pass through and perpendicular to the plane of the electrode; hence, the calculation applies to an experiment in which monochromatic light is absorbed by the molecule in solution, but for which the electrode is transparent. The set of equations and boundary condiOions requiring solution is
c* + e l exp(-ex)
bC* b2C* -=D--dt ax2
el
i(1
+ &)(& + + lx3
8) X
(5)
7
b2CdC- = D-
Furthermore, we notice that near the potential E , = (E+ E-)/2 both L2 and 2 4 can be considered large and that it is therefore quite reasonable to investigate the asymptotic behavior of i for T , t >> D / h ~ , D / % ~The . appropriate expansion for f(x) in this region is 16b
+
6x2
bt
To simplify this result for the purpose of discussion of the shape of the current-voltage curve, we suppose first of all that t >> T . This enables us to reach the following formula for the current
= D-@C+ at bX2
(7)
At t = 0 , x 2 0, a n d a t x +
00,
t
20
c* = c- = c+= 0 Atx = O , t > O
of which we retain only the first term. This yields, then, a formula which can be used to discuss the properties of the current-voltage relationship
DbC+ ____bX
- k4C+ - Lac* t
II
The diffusion coefficients of all species are assumed to have an identical magnitude, D. T represents the lifetime of &I*in solution. e = e’Cb where e’ is the extinction coefficient of the solution at the appropriate wavelength, c b is the concentration of M in the bulk of the solution, and I is the intensity of the exciting radiation. x is the distance from the electrode along a normal to the surface, and t is the time measured from the instant of switching on the radiation. The equations were solved by the method of Laplace transformation. An expression is thereby obtained for the net observable current density over the illuminated area, arising from all possible reactions of the system
i = eIJeQ.
x
- Q 3 f ( Q 4 2 / 8 ) B l )exp(-e’)de’ j
{&4(&d8-)
The dimensionless symbols are QI
=Ad-,Qz
=
e
= t / r , &, = = Z32/7/D,
L1/7/D1Q3
The current is predicted to be zero at the potential which satisfies the condition
If the A’s for each of the oxidation-reduction couples involved were identical, then this potential should be near Em. The dependence of the current on electrode potential is sensitive t o the relative positions of the various standard potentials. A particularly distinctive kind of behavior results if the excitation energy of the molecule ex is smaller than P(E+ - E-) so that E+* and E-* are separated by a few tenths of a volt from E- and E+, respectively. I n this case there may exist regions of potential within which %2 and 5, are collision determined and both equal to km, while on one side of Em > k l . The expression for the current-voltage curve within these potential ranges then reduces to
E ~ Z , Q4
=
2 4 4 r / D . Also f(x) = exp(x2)erfc(x) where erfc represents the complimentary error function. lBa The
JOUTnal
of Physical Chemistry
(16) (a) W. Gautschi in “Handbook of Mathematical Functions,” M . Abramowitz and J. A. Stegun, Ed., Dover Publications, Inc., New York, N. Y., 1965, p 297; (b) p 298; (0) p 374; (d) p 377.
ELECTRONICALLY EXCITED STATESOF MOLECULES This condition, therefore, is characterized by the appearance of limiting currents on the anodic and cathodic sides of E,. The interpretation of the formula for i l i m is quite l/edE) would be the magnitransparent. e l / ( l tude of the current if every excited molecule arriving at the electrode were either reduced or oxidized and the product ion then diffused away. dDT is the average velocity of diffusion of the ion away from the electrode, where it is created, into the bulk of the solution, and k , is the velocity of removal of the ion by a collision-determined electrode reaction. Thus the ratio of these velocities represents the probability that an ion may escape from the electrode, or it is the fraction of the number of excited molecules arriving a t the electrode in unit time which cause a net current to flow. Figure 1 illustrates the shape of the i ( E ) relationship for the following choice of potentials and parameters: -0.5 V, E+* = E+ 1.5 V, E-* = 0.2 V, E , -1.2 V, E- = -2.5 V, and X = 1 eV. The quantity plotted is log ( i / i 1 l m ) and eq 4 was used for the k’s. I n magnitude, ilim is made small by the quenching reaction, that is, the successive oxidation and reduction described in eq l a and l b or ICand Id. Typically we may choose the following set of numerical values: B N 102 cm-l, 7 N 10-7 sec, D = 10-5 cm2 sec-l, I N 10’5 photons cm-2 sec-’, k , N lo4 cm sec-l, and t N 10-6 sec. These yield ili, N 10-l2 A cm-2. Evidently, detection of such a small current would be made virtually impossible because of the inevitable presence of impurities in an experimental system. A more favorable situation exists, however, when the potential is controlled near E- or E+ though within these extrema, for the oxidation rate of anions or the reduction rate of cations is slowed in this region. This is the cause of the nearly exponential increase of the current, revealed in Figure 1, between E+* and E-. The current might be detectable experimentally, therefore, provided that other quenching influences do not interfere. We may mention, for example, the possibility of resonance energy transfer from the excited molecules to the electrode or to the anions or cations, which could make the lifetime r very short in the neighborhood of the electrode. There is another situation in which observable current flow might be achieved, although the theoretical model needs to be modified in order to describe it. This arises when the ionic products of electron transfer to or from the excited molecule can react very rapidly with some species in solution, so that the fraction of ions which escape from the electrode-discharge reaction is increased.” It is also necessary that neither the homogeneous reaction partner present in solution nor the products of this reaction should be electroactive. A theoretical treatment of this problem and its application to experiment will be described in a forthcoming publication. l8
3199
6
c
+
I
I
I
1
I
I
I
=I
I
-.2
I
-A
I
I
I
I
I
-.6 -.8 -1.0 -1.2 -1.4
Figure 2. Current-voltage curve for reduction of electronically excited molecules a t a metal electrode; ex
> e(E+ - E-).
It is likely that no limiting current should be observ- E-) ; rather, the current increases able when ex > @(E+ steadily with overpotential as illustrated in Figure 2. This curve wa8 computed with the following choice of potentials: E-* = 1.03 V, E+ = 0.83 V, E, = -0.36 V, E- = - 1.55 V, E+* = - 1.75 V, X = 1eV. The photocurrent is predicted, by eq 5, to depend on wavelength parametrically through the extinction coefficient E. Indeed, when it is true that EZ/Z
