OUTER AND INNER MECHANISM OF REACTIONS OF EXClTED MOLECULES BY ALBERT WELLER Labor f. physikal. Chemie, Wiederholdstr. 15, Stuttgart, Germany Received 16th February, 1959 The distinction between outer and inner mechanism (introduced by Forster 1) is out- lined in connection with the quenching of fluorescence. The same distinction is applied to acid-base reactions of excited molecules, where the inner mechanism is known in prin- ciple. The rate constants of these reactions exhibit a rough parallelism with the cor- responding equilibrium constants. Upper limiting values of the reaction rate constants are calculated using the equilibrium constants, the diffusion coefficients of the reactants, and steric factors (closely related to the statistical factors of the respective equilibria).Good agreement with the experimental rate constants proves that the " inner " proton transfer equilibrium is established in a time which is much shorter than the mean lifetime of the excited molecules. Exceptions occur with proton transfer reactions between nitrogen atoms. The influence of the environment on the inner mechanism is discussed. Reactions of excited molecules (A*) which lead either to quenching or to transformation 2-4 of fluorescence can be studied by fluorescence measurements, when the reaction rates are comparable with the reciprocal mean lifetime 701 of the excited molecules. For bimolecular reactions in solution the two reactants are required to form an encounter complex. Transfer of excitation energy to quenchers which absorb at longer wavelengths than the fluorescent molecules may occur over distances considerably larger than the encounter distance.5 This kind of reaction will not be treated in this paper.If, however, due to forbidden transitions, weak mutual coupling exists between the partners, the distance required for energy transfer will eventually become equal to the encounter distance. The different ways possible for the formation of the encounter complex (A*-Q) which have led to the distinction between diffusional (or dynamic) and static quenching may be represented in the following manner : diffusional static kl A* + Q +A*.Q k-1 f K" I A + Q + A.Q. This scheme includes only the outer mechanism which describes the intermolecular conditions of quenching. The additional reaction, k2 A*.Q + unstable intermediate or photoproduct (11) representing the proper quenching process, completes the reaction scheme for fluorescence quenching.The rate constant k 2 depends on the inner mechanism and determines to a certain extent the quenching effect. The influence of this rate constant k 2 on the overall reaction and its implications will be discussed below in connection with the acid-base reactions of excited molecules in water. Before this, however, the conditions for diffusional and static quenching will be considered. 28A . WELLER 29 The following expression for the concentration dependence of the relative quantum yieldflfo can be derived 6 from the complete reaction scheme (I) and (11) : where P is the probability of the occurrence of the inner mechanism.For P --f 1, the well-known expression : (3) f = 1 fo (1 + K"cQ)(l klCQTO) is obtained. The relaxation time of the association equilibrium in the ground state Teq depends on the sum of the equilibrium concentrations of the reactants which may be approximated by the total concentration cQ, because usuallycA < cQ : then The possible difference between the dissociation rate constants in the ground and excited state-due to a different amount of interaction in either state-is accounted for by the superscript O (indicating the ground state). On the other hand, the plausible assumption is made that the association rate constants in the ground and excited state are equal. The relative amounts of diffusional and static quench- ing will depend on the ratio of the two 7s.Static quenching occurs when req > TO. This condition implies that P -+ k2/[k2 + (l/ro)]. On the other hand, when Teq -S r o diffusional quenching occurs and P --f k2/(k2 + k-1). In either case, k2 must be at least comparable to the reciprocal mean lifetime for measurable quenching to occur. Since Teq decreases with decreasing viscosity, quenching will be essentially diffusional in solvents of low viscosity like water, except in such cases where k-i becomes small, because of strong electronic interaction between A and Q in the encounter complex. When the theory of diffusion-controlled reactions is applied to the diffusional type outer mechanism, it must be considered that non-stationary processes may be involved, due to the short time available for the reaction.In addition, the effect of electrostatic forces of ionic charges and the influence exerted on the overall reaction by the rate constant k2 must be taken into account. In the expression finally derived, ( 5 ) the exponential factor corrects for the non-stationary part of the reaction, and _ - f - exp 1- K~Q~flfro)'l fo 1 + kCQTO ' is the stationary overall reaction rate constant which for y = 1 becomes identical with the rate constant derived by Debye 8 for diffusion-controlled ionic reactions. These reactions depend on the relative diffusion coefficient, DA + DQ, the ratio, and the encounter distance a; E = dielectric constant. The number N' of par- ticles per millhole corrects for the dimensions (1. mole-1 sec-1) used for second- order rate constants.The factor y in eqn. (6) is the probability that the reaction ( k 2 ) will occur when the reactants have reached the distance a. Therefore y30 REACTIONS OF EXCITED MOLECULES furnishes information with regard to the inner mechanism. On the other hand, y2 appears in the exponential factor, so that with decreasing y the non-stationary part of the reaction decreases much more rapidly than the stationary part. It is in this respect that similar expressions which have been derived by Umberger and La Mer 9 and by Sveshnikov 10 differ from eqn. (5). In those equations the effect of the inon-stationary reaction decreases linearly with y. With diffusion-controlled reactions of more complex molecules, there may be involved a steric factor CJ which accounts for the possibility that no reaction will occur unless the reactants come into contact with their mutually reactive sites.In this case one can write Y = CJP, (8) where p is now the probability that the proper reaction (k2) takes place when a sterically favourable encounter of the reactants occurs. Since little is known about the inner mechanism of quenching, it seems worthwhile to apply the above formulae and considerations to other reactions of excited molecules, where the inner mechanism is known in principle. Such reactions may then serve as a model for quenching reactions. With acid-base reactions of the general type * kl * k2 * k3 * AHzA + B Z B + (AH-B)"Af"B + (A-HB)zA+ZB + A(ZA-1) + HB(ZB+l), (111) k-1 k-2 k-3 the excited acid, splitting off a proton, goes over into the excited conjugate base (and vice versa) ; the fluorescence of the species thus formed is an additional source of information. Expressions analogous to eqn.(6), derived 11 for the relative quantum yields flfo and f 'If '0 (where the dashed quantities refer to the species formed by the reaction), have been used to separate the stationary and non-stationary part of the reactions, so that values of the stationary overall rate constants can be no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TABLE ~.-ACID-BASE REACTIONS IN WATER AT 25" AH+ B H3O+ +OH- H3O+ + CH3COO- AcrH+ + OH- * H30+ + R b R ~ H + C~H~COO- RbH + C3H7COO- RbH + CH3COO- NH: + A& H3P04 + Rb- R ~ H + HCOO- RbH f H2P04- HCOOH + Rb- Ac:H++ NH3 CH3COOH + $0- K 5.0 x 1015 5.7 x 104 2.2 x 103 650 115 1 02 88 49 26 8.7 0,204 0.115 0-039 0.01 1 kexpt.1. mole-' sec-1 13 f 2 x 1010 4 5 & 0.5 X 1010 1.85 & 0.15 X 1010 4.8 f 0.3 X 1010 2-86 * 0.15 X l o 9 2.76 & 0.15 X 109 2.90 IJ, 0.15 X 109 2-91 A 0.2 x 109 0.57 f 0.03 X 109 2.40 j, 0.15 x lOQ 6.0 f 0-4 X 108 2.8 & 0.2 X 108 0.22 i 0.02 x 108 0.33 f 0.02 x 108 14.6 8.58 6.02 10.1 2.08 2.00 2.23 1.97 2.65 2.58 2-09 2.44 3.05 2.08 12 13 4 15 15 15 15 16 4 15 16 15 4 15 (20°C) AcrH+ = acridinium kation ; RO- = 8-naphtholate ion ; ROH = 8-naphthol. Acr = acridine;A. WELLER 31 obtained if the mean lifetimes TO and TO’ respectively are known.* A number of these acid-base reactions thus evaluated are given in table 1. They are arranged in order of decreasing equilibrium constant K and it is seen that the overall rate constant k decreases roughly in the same order. All values are extrapolated to zero ionic strength.Two reactions, (1) and (2), measured by Eigen and co- workers 12913 are included in table 1. The relative diffusion coefficients have been calculated from ionic conductances and, for uncharged molecules, taken from the literature or evaluated by means of a nomogram given by Othmer and Thakar.14 Values of y, quoted in table 2, have been calculated from the rate constants k with the aid of eqn. ( 6 ) using a = 7.5 A. This later assumption seems justified for two reasons. (i) 7.5 A corresponds to two layers of water molecules between the proton-exchanging groups. (ii) With reaction (l), a < 7.5 leads to y Z 1. So, if a common value of a is chosen, it must not be smaller than 7.5 A.no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Y 1.0 0.59 0.35 0.53 0.24 0-24 0-23 0.24 0.037 0.17 0.05 1 0.022 0.001 5 0-0028 TABLE 2 kcalc. 1. mole-1 sec-1 13 x 1010 3.8 x 1010 1.8 x 1010 4.5 x 1010 2.9 x 109 2.8 x 109 3.1 x 109 3.2 x 109 4.4 x 109 2.9 x 109 6.8 x 108 3.5 x 108 1.7 x 108 0.34 X 108 With regard to the probability p , two limiting cases may be imagined con- sidering reaction scheme (111) : (i) The equilibrium, determined by the rate constants k2 and k-2, has a re- laxation time, (k2 + k-2)-1, much shorter than the mean lifetime of the excited molecules. As a result, equilibrium is established at once when an encounter of the reactants occurs. In this case p + p f = 1, (9) wherep’ refers to the back reaction of (111). then (ii) k2 and k-2 are much smaller than any other rate constant of scheme 111, p + p l - = a .(10) This case is realized with the slow reaction of pseudo-acids and will therefore Using eqn, (9), upper limits of p and pl can be calculated : not be considered here. where p = cm + a; p’ = 1/(1 + C), (1 1) p (T‘ ( D , + D,)6’ (eb - 1) p’ (T (D, + 0,)s (ed’ - 1) C = - = - K In this equation, which follows simply from eqn. (6) and (S), (T’/(T is the statistical factor which removes contributions to K due to symmetry. As an example, the equilibria associated with the reactions (7), (8) and (9) may be considered, where cr’lcr amounts to +, 3, *, respectively. Single values of (T may be obtained * Mean lifetime measurements have been carried out at the University of Giessen, Germany, by Dr.A. Schmillen, whose valuable co-operation is thankfully acknowledged.32 REACTIONS OF EXCITED MOLECULES approximately from the y-values of the first four reactions for which p is very near unity, due to the high value of the equilibrium constant ; thus y = U. Since no steric requirements are involved with H3O+ and OH- as reactants, u = 3 can be assigned to the acetate ion and to the naphtholate ion, and u = 3 to the acridine molecule. If u = 1 is assumed for the ammonium ion, u = + for ammonia follows from the statistical factor U'/U for the equilibrium. By similar arguments the other U-values of table 2 have been obtained. The molar rate constants which can now be calculated by means of are listed in table 2. The agreement with the experimental values is remarkably good, except with the reactions (9) and (13), in which the proton is transferred between nitrogen atoms.Quite analogous results are obtained with acid- base reactions of 3-acetylaminopyrene-5 : 8 : 10-trisulphonate (DH3-), in water 17 (table 3). For reactions (l), (6) and (7) the agreement is good, con- trary to reactions (2)-(5), in which, again, the proton is transferred between nitrogen atoms. no. 1 2 3 4 5 6 7 AH+B * DH3- + OH- Dh3- + (CH&NH Dk3- + CH3NH2 Dk3- + (CH3)3N P a DH3- + NH3 CH3COOH + 64- Dk3- + CH3COO- TABLE 3 K 1.3 x 107 8 x 104 2 x 104 800 210 140 0.0072 kexpt. 1. mole-1 sec-1 14.8 i 0.9 X 108 4-5 f 0.3 x 108 4.6 I0.3 x 108 2.2 f 0.15 x 108 2.34 f 0-15 X 108 11 1 5 x 108 0.077 & 0.008 X 108 kcalc. 1. mole-1 sec-1 14.5 X 108 6.5 X 108 7.8 X 108 5.8 X 108 10.4 x 108 12 x 108 0.087 x 108 These results suggest that the " inner " proton transfer equilibrium is estab- lished in a time much less than the mean lifetime of the excited molecules (so that eqn.(9) is valid), except in the proton transfer between nitrogen atoms. These remarkable exceptions are certainly connected with the absence of abnormal conductivity in liquid ammonia and amines. Finally, a few remarks are necessary about the influence of the environment on the inner mechanism. For acid-base reactions in water at room temperature no such influence seems to be involved. This is probably due to the Grotthuss- type migration of the proton through the hydration spheres and to the short dielectric relaxation time of water.It is, however, very likely that changes of the solvent configuration which are connected with the transfer of charges (proton or electron), become rate-determining, when the dielectric relaxation time of the solvent is longer than the mean lifetime of the excited molecules. In fact, it hasA . WELLER 33 been observed 18 with excited aromatic amines Ar*NH2 in different akohols, that the rate constant k2 of the reaction : k2 Ar*NH3+ * HOR + Ar*NH2 H2OR-t k-2 depends on the dielectric relaxation time of the solvent. The excited ammonium compounds are extremely strong acids with pK-values between - 2 and - 6, therefore k2 > k-2. Acidic alcoholic solutions of the ammonium compound exhibit at room temperature the fluorescence of the amine. This indicates that reaction (IV) goes practically to completion.When, however, the temperature is lowered, the fluorescence of the ammonium compound appears and the fluor- escence of the amine decreases. In different alcohols this fluorescence trans- formation occurs at different temperatures, depending on the temperature de- pendence of the dielectric relaxation time of the respective alcohol. This effect can be explained by the assumption that the configurational changes of the solvent, which are connected with the proton transfer become rate-determining at these temperatures. 1 Forster, Fluoreszenz org. Verb. (Vandenhoeck and Ruprecht, Gottingen, 1951), 2 Forster, 2. Elektrochem., 1950,54,42 and 531. 3 Weller, 2. Elektrochem., 1952, 56, 662 ; 1956, 60, 1044. 4 Weller, 2. Elektrochem., 1957, 61, 956. 5 Forster, 2. Naturforsch., 1949, 4a, 321. 6 Weller, unpublished results. 7 Weller, 2. physik. Chem., 1957, 13, 335. 8 Debye, Trans. Electrochem. Soc., 1942, 82, 265. 9 Umberger and La Mer, J. Amer. Chem. Soc., 1945, 67, 1099. 10 Sveshnikov, Acta physicochim., 1935, 3, 257. 11 Weller, 2. physik. Chem., 1958, 15 (Bonhoeffer-Gedenkband), 438. 12 Eigen and de Maeyer, 2. Elektrochem., 1955,59,986. 13 Eigen and Schoen, 2. Elektrochem., 1955,59,483. 14 Othmer and Thakar, Ind. Eng. Chem., 1953, 45, 589. 15 Weller, 2. physik. Chem., 1958, 17, 224. 16 Gurr, Diplomarbeit (Stuttgart, 1959). 17 Weller, 2. physik. Chem., 1958, 18, 163. 18 Urban and Weller, unpublished results. p. 199.