Application of Time-dependent Rate Constant Theory to Reactions of Solvated Electrons Reaction Distances, Rate Constants and Diffusion Coefficients in Concentrated Aqueous Solutions BY GEORGE V. BUXTON," FRANK C. R. CATTELL? AND FREDERICK S. DAINTON~ Cookridge Hospital, Leeds LS16 6QB University of Leeds, Cookridge Radiation Research Centre, Received 20th May, 1974 The rates of reaction of e; with a number of solutes have been measured in the temperature range 190 - 260 K in concentrated aqueous solutions of LiCl and NaOH + KOH. The rate coefficients are observed to be time-dependent and fit the equations developed by Noyes. Using these equations we have obtained estimates of diffusion controlled rate constants, reaction distances and diffusion coefficients for each reaction. These suggest that e; reacts, and may also diffuse, by a tunnelling mechanism. The temperature dependences of the rate constants and diffusion coefficients are similar to those of other transport processes in glass forming solutions. For rapid bimolecular reactions in solution diffusive transport can be so slow that the average distance between potentially reactive molecules soon exceeds that pre- dicted for a random equilibrium distribution.The measured rate coefficient will decrease, therefore, from an initial value k , characteristic of the random distribution, to a steady state value k' given by eqn (1),l 47cnaD' 1 + 4naD'/k k' = where a is the reaction distance and D' is the sum of the diffusion coefficients of the two molecules. This eqn reduces to the familiar Smoluchowski eqn (2)2 k' = 47cuD' (2) when k 9 47caD'.Noyes efficient from its initial value k to its final value k' is given by eqn (3) has shown that the change with time of the rate co- 1 k k, = k' 1 +-, ex' erfc x [ 4xaD where k, is the rate coefficient at time t, and x = k( JD't)/k'a. For very short times when t 4 (a2/D')(k'/k)2 the observed rate coefficient approaches k. Typically, for diffusion controlled reactions of the hydrated electron at room temperature, Q - 0.5 nm and D' - 7 x lo-' cm2 s-l, implying that a time resolution better than 10-l' s is needed to explore this region. When t 9 (a2/D')(k'/k)2 eqn (3) reduces to (4) t present address : C.S.I.R.O., Division of Cloud Physics, P.O. Box 134, Epping, New South Wales $ present address : University Grants Committee, 14 Park Crescent, London W1N 4DH.Australia. 115116 TIME-DEPENDENT RATE CONSTANTS For typical diffusion controlled reactions of the hydrated electron * with values of CT and D’ given above and k + 4naD’, the observed rate coefficient is expected to be greater than k‘ by 1 % at s, 3 % at lo-’ s, 11 % at s and 34 % at 10-l’~. Through the pulse radiolysis technique it is possible to form essentially instan- taneously, a strongly absorbing reactive species (e;) in a solution containing randomly distributed solute molecules, and to measure its rate of reaction. For time-dependent changes in the rate coefficient to be conveniently observed on a timescale of tens of nanoseconds or longer, D’ must be reduced by several orders of magnitude.This is most satisfactorily achieved by the use of liquids which on cooling progressively become more viscous and eventually vitrefy, and several aqueous solutions fall into this category. Rate coefficients which have already been reported over a very wide temperature range for one such aqueous system are consistent with a variation of D’ with temperature, over at least eight orders of magnitude, given by where To = 135 K and B w 800 K. D’ = Dd exp[ - B/(T- To)] (5) The disappearance of solvated electrons, e;, by reaction with a solute, S, is given by eqn (6) and if k, is given by eqn (4) and [S] % [e;], the decrease in optical density, OD, with time due to e; is given by eqn (7) where ODo is the optical destity at t = 0. The variation of OD with time allows the evaluation of k’ and D’, and provided k % 4noD’, a can be calculated from In this paper we have applied the time dependent rate constant theory of Noyes to the reactions of e; with a number of solutes in 9.5 mol dm-3 LiCl and in 10 mol dm-3 OH- (5 mol dm-3 NaOH + 5 mol dm-3 KOH) over a wide temperature range and have obtained estimates of o and D‘.eqn (2). EXPERIMENTAL APPARATUS The irradiation cell and assembly has been de~cribed.~ Solutions were irradiated with pulses of 2.9 MeV electrons from a Van de Graaff accelerator. Pulse durations of 5 ns, 0.2 p s and 0.6 ps were used, and the dose per pulse (1 to 2 krad) was monitored by a secondary emission chamber which was calibrated with the iodide and ferrocyanide dosimeters6 Corrections were made for the different electron densities of the solutions. Details of the pulse radiolysis apparatus have been reported el~ewhere.~ MATERIALS All solutions were prepared from triply distilled H20 and were purged with argon or LiCl was Eastman certified reagent grade or Merck analytical grade.All other reagents 10 mol dm-3 OH- solutions con- With this composition the solutions exp [U(r)/kT]r-’dr)-’ nitrogen. were AnalaR grade (B.D.H. or Hopkin and Williams). sisted of 5 mol dm-3 NaOH+ 5 rnol dm-3 KOH. could be cooled slowly without crystallisation occurring. * When eiq reacts with an ion, B in eqn (1) -(3) is replaced by3 of= 1 where U(Y) is the potential energy of the reacting ions at separation Y.G . V. BUXTON, F. C . R . CATTELL AND F . S . DAINTON 117 RESULTS Conditions were chosen such that (i) the pulse length was very short compared with the lifetime of e;, and (ii) the decay of e; in the absence of reactive solutes was negligible on the timescale of its decay in the presence of such solutes.This latter condition was generally satisfied for solute concentrations 3 Kinetic measurements were generally made in the temperature range 190 to 260 K. The decay of e; was independent of wavelength in this temperature range, although the spectrum of e; does change with time at lower temperature^.^^ Fig. 1 illustrates typical decays of e; in the presence and absence of reactive solutes. Because of possible distortion of the oscilloscope trace depicting these decays, due to the risetime of the detection system (-4 ns) and to the possible contribution of spur reactions to the decay of e; at very short times, eqn (7) was modified to (8) to permit kinetic analysis to be made a finite time after the pulse.In eqn (8) ODi is the optical density at time ti after the end of the pulse, which is taken as time zero, and t2 > t l . mol dm-3. k''[S] (t$-tb>-' In ___ OD1 = k'[S](tf+t$)+--- OD2 2(nD)+' In fig. 2 we show typical data plotted according to eqn (8). From the slopes and intercepts of such plots values of k' and D' have been obtained between 190 and 260 K. Also shown in fig. 2 is a first order plot of the same data. At first sight such plots may appear to be acceptably linear, but close inspection reveals a systematic decrease in slope with time and the mean slope is appreciably larger than that predicted by eqn (8)..I .I CI) 4 5 ps per division mol dm-3 Cr02,. FIG. 1.-Decay of e; in 10 mol dm-3 OH- solution at 200 K containing (a) no solute, (b) 2 x The temperature dependences of k' and D' are illustrated in fig. 3 for the LiCl system, and in fig. 4 for the OH- system. Also shown in fig. 3 is the temperature dependence of T/r, where y is the shear viscosity derived from data in ref. (8). For both solutions the temperature dependence of D' is given by eqn (3, and of k' by eqn (9) k' = A exp[ - B/(T- To)]. (9) In these equations A. B, D6 and T are constants and are listed in table 1.( t t + 4 4 5 . 5 3 x 10-~~3) FIG. 2.-Kinetics of e, decay in 9.5 mol dm-3 LiCl solution at 211 K containing mol dm-3 Cr02,. ( 0 ) Plot of kinetic data according to eqn (S), (0) first order plot, (- - -) first order line having the same slope as plot (0) i.e.k' [S]. I I I I - *. - ...- FIG. 3.-Temperature dependence of k'(0) and D'(0) for the reaction of e; with acetone (1.09 x lo-' mol dm-3) in 9.5 mol dm-3 LiCl solution. Also shown is the self diffusion coefficient of H20 (H) from ref. (9), and log,, [8RT/3m)/dm3 mol-' s-'1 (..--) taking values of from ref. (8).G . V. BUXTON, F . C. R . CATTELL AND F . S . DAINTON 119 1 0 3 ~ / ( ~ - 135 K) FIG. 4.-Temperature dependence of k' (open points) and D' (solid points) for the reaction of e, with NO; in 10 mol dm-3 OH- solution, [NO;] = 4.7 x mol dm-3 (0) and lo-' rnol dm-3 (0). Also shown are values of loglo [(8RT/30007)/dm3 mol-' s-'1 w) taking values of 7 measured in this laboratory.rnol dm-3 (A), 5 x TABLE VA VALUES OF THE PARAMETERS IN EQN (2), (5) AND (9) FOR THE REACTIONS OF e ; WITH SOLUTES IN 9.5 mol dm3 LiCl AND 10 mol d ~ n - ~ OH- SOLUTIONS eqn (9) eqn (5) eqn (2) log 10 system solute (A/cim3 mol-1 s-1) B/K p$Z:s-1) B/ K of Inm 9.5 mol dm-3 LiCl (CH&CO 10.68 f 0.05 613 f 7 - 3.65 f 0.15 656k 20 0.60+ 0.27 (f = 1) To = 129K Hf 10.50+0.12 557+16 -4-02f0.29 578f39 0.6850.17 NO; 11.03+0.18 666&41 -3.61 f0.54 688k106 0.4220.07 NO; 11.35t0.44 716f88 -3.77f0.19 631 A37 0.81 k0.22 CrO2- 11.30f0.22 655f49 -3.91kO.30 633f71 1.73k0.33 10mol dm-3 OH- NO; 11430f0.12 905f21 -2.2050.42 1020k74 0.77f0.24 CrOi- 11.93f0.10 824f17 -2.78k0.18 902f31 2.01f0.54 To = 135K NO, 12.15k0.12 956f21 -2.5720.31 986f51 1.05f0.37 Quoted errors are standard deviations from least squares fits.Values of To are best values common to all the data. DISCUSSION Since the kinetic data fit eqn (8) (see fig. 2) it follows that eqn (4) is a good repre- sentation of the time-dependence of the rate coefficient kt on the timescale of our experiments. Thus the data are consistent with the time-dependence theory developed by Noyes for diffusion controlled reactions, implying that the reactions we have studied are diffusion controlled. This might have been anticipated from the fact that the majority of the reactions of solvated electrons in dilute aqueous solution are accepted as being diffusion controlled, but the following points provide firm evidence that this is the case in the present work. (i) Eqn (5) and (9) are examples of the general phenomenon that mass transport processes, P, in glass forming liquids are accurately described by equations of the form P = AP exp[ - B/(T- To)] (10)120 TIME-DEPENDENT RATE CONSTANTS where the parameters B and To are scarcely dependent on the particular transport process, e.g.fluidity, conductance, diffusion, mobility, dielectric relaxation time etc. (ii) Within the limits of our measurements B and To have the same values in eqn ( 5 ) and (9) for a given reaction (see table) and similar values for all the reactions studied in each system. has a similar temperature dependence (fig. 3) as does the self diffusion of H20 (fig. 3), although this was measured in a higher temperature range (280-370 K), where slightly different parameters may be expected.O Eqn (1) may be rewritten as For the LiCl system the shear viscosity 1 1 1 - = -+- k’ k 4naD” and the fact that k’ and D’ have the same temperature dependence indicates that k % 4noD’, unless k also has the same temperature dependence, in which case no conclusion can be drawn about the relative magnitude of k and 4noD’. However, there is evidence from other work l 1 on the effect of solutes on the yield of solvated electrons that e; reacts effectively instantaneously with the solute when formed at the encounter distance, i.e. when no diffusion is necessary for reaction to occur. We conclude, therefore, that eqn (2) is appropriate. Combining eqn (2) and (5) gives eqn ( 1 3 , and comparison with eqn (9) shows that A = 4ncrfDb. Values of afare listed in the table, and because of (ii) above are independent of temperature in the range 190- 260 K.The value o f f depends on the magnitude of U(r)/kT which is equal to Zie$j/kT, the ratio of the electrical and thermal energy of an ion i of charge Z i e in the field $ j of an ionj. Unfortunately one cannot calculate 1+9~ with any confidence except in dilute solutions l2 so we are unable to compute the magnitude off. How- ever, it is likely that there will be a large degree of association of cations and anions in 10 mol dm-3 salt solutions so that ionic solutes may behave as neutral species, in which casefwill be close to unity. The similar values of of for the reactions of e; with H+, acetone and NO: provide some support for this conclusion and we tenta- tively equate of with the actual reaction distance.The striking feature of the reaction distances listed in the table is the very large value for the reaction of e; with CrOi- in both systems. Such distances are only compatible with a mechanism in which the electron transfers from its solvent trap to the solute by tunnelling. Hart and Anbar l 3 have discussed possible mechanisms of the transfer of hydrated electrons to solutes and favour a tunnelling mechanism. It is interesting that o = 0.98 nm l 3 for the reaction of e; with CrOi- in dilute aqueous solution. This is also an abnormally large value which was estimated from a comparison of the measured rate constant with that predicted by the Debye- Smoluchowski equation. The similar values of Db for all the reactant pairs in the LiCl system imply that the mechanism of the diffusion of H+ is predominantly the hydrodynamic type rather than the proton transfer type in these concentrated solutions.This is not unexpected since a similar conclusion has been drawn by Lown and Thirsk l4 from the effect of pressure and concentration on the electrical conductivity of solutions of alkali metal hydroxides and orthophosphoric acid. They suggest that the proton transfer mechanism is suppressed at high solute concentrations because of the very small fraction of water molecules which are free to rotate in a manner which allows this mechanism to operate. k’ = 4nofDb exp[ - B/(T- To)]. (12)G . V. BUXTON, F. C. R. CATTELL AND F . S . DAINTON 121 It is interesting that the values of Db (see table) for the hydroxide solutions are about an order of magnitude larger than those for the LiCl solutions.This is most likely to be due to a difference in the values of Do for e, rather than Do for the solutes in the two systems. If the solvated electron diffuses in these media by tunnelling from trap to trap, as has been suggested for e; in H,0,13 then Do for e; will reflect the density of trapping sites in the medium, There is evidence from the depression of the yield of solvated electrons in these systems by reactive solutes that the OH- system is more efficient than the LiCl system at trapping electrons, particularly as the temperature is lowered, implying that there is indeed a higher density of electron trapping configurations in the OH- system. In this respect it is known that LiCl and KOH have profoundly different effects on the water structure.15 Thus Li+ breaks down the tetrahedral structure whereas Kf, being of similar size to HzO, can substitute into the water lattice.In addition C1-, because of its large size, contributes more to structure breakdown that does OH- which is also close in size to H20. If all these effects are causally related then they point to the importance of the tetrahedral water structure to the trapping of electrons in aqueous media. Several theoretical models have been proposed to account for the properties of glass forming liquids and to interpret the physical significance of the parameters in eqn (10). Among these are the free volume theory of Cohen and Turnbull," the configurational entropy theory of Adam and Gibbs,'7 and more recently the bond- lattice model of Angel1 and Rao l8 which can be grouped with the entropy theory.In each case To represents the temperature at which mass transport ceases. In the free volume model this is associated with the free volume falling to zero,I6 in the entropy case it is the temperature at which the configurational entropy falls to zero,17 corresponding to no broken bonds in the bond-lattice model. The parameter B in the free volume model is proportional to v*/av,, where v* is the minimum volume of the hole required to permit molecular displacement, and a and v, are the mean values of the coefficient of thermal expansion and molecular volume respectively. In the configurational entropy model lo* l8 B relates inversely the number of broken lattice bonds, or configurational excitations, to the temperature T when T > To and is shown to be proportional to To for a given system in agreement with experimental findings. lo In our experiments B for LiCl solutions is about 30 % lower than the hydroxide value, although a is larger for the hydroxide solution,* which seems contradictory to the requirements of the free volume theory.This theory has also been criticised for other reasons.19 In terms of the entropy theory a smaller value of B indicates a larger configurational entropy change with temperature, and more configurational entropy at a given temperature. This is in keeping with less disruption of the solvent network in the hydroxide solutions. We are grateful to the S.R.C.and General Electric Research for financial assistance. R. M. Noyes, Progr. Reaction Kinetics, 1961, 1, 129. M. von Smoluchowski, 2. ghys. Chem., 1917, 92,129. P. Debye, Trans. Electrochem. Soc., 1942, 82,265. G. V. Buxton, F. C. R. Cattell and F. S. Dainton, Truns. Furuduy Soc., 1971, 67, 687. G. V. Buxton, Proc. Roy. SOC. A , 1972, 328, 9. G. E. Adams, J. W. Boag and B. D. Michael, Trans. Furaday SOC., 1965, 61,492. ' G. V. Buxton, F. C. R. Cattell and F. S. Dainton, to be published. * C. T. Moynihan, N. Balitactac, L. Boone and T. A. Litovitz, J. Chem. Phys., 1971, 55, 3013. A. Weiss and K. H. Nothnagel, Ber. Bunsenges. phys. Chem., 1971, 75, 216. * On cooling from room temperature to 190 K 10 mol dm-3 OH- contracts by 3.3% and 9.5 mol dm-3 LiCl by 1.5 %.122 TIME-DEPENDENT RATE CONSTANTS lo C.A. Angell and R. D. Bressell, J. Phys. Chem., 1972, 76,3244. G. V. Buxton and K. G. Kemsley, J.C.S. FaraCiay I, in press. l2 R. A. Robinson and R. H. Stokes, Electrolyte Solutions (Butterworth, London, 2nd edn., 1959). l3 E. J. Hart and M. Anbar, The Hydrated Efectron (Wiley-Interscience, New York, 1970). l4 D. A. Lown and H. R. Thirsk, Trans. Furaday SOC., 1971,67, 132, 149. l5 G. W. Brady, J. Chem. Phys., 1958,28,464. l6 M. H. Cohen and D. Turnbull, J. Chem. Phys., 1959,31,1164. l7 G. Adam and J. H. Gibbs, J. Chem. Phys., 1965,43, 139. l8 C. A. Angell and K. J. Rao, J. Chem. Phys., 1972,57,470. l9 M. Goldstein, J. Chem. Phys., 1969, 51, 3728. Application of Time-dependent Rate Constant Theory to Reactions of Solvated Electrons Reaction Distances, Rate Constants and Diffusion Coefficients in Concentrated Aqueous Solutions BY GEORGE V.BUXTON," FRANK C. R. CATTELL? AND FREDERICK S. DAINTON~ Cookridge Hospital, Leeds LS16 6QB University of Leeds, Cookridge Radiation Research Centre, Received 20th May, 1974 The rates of reaction of e; with a number of solutes have been measured in the temperature range 190 - 260 K in concentrated aqueous solutions of LiCl and NaOH + KOH. The rate coefficients are observed to be time-dependent and fit the equations developed by Noyes. Using these equations we have obtained estimates of diffusion controlled rate constants, reaction distances and diffusion coefficients for each reaction. These suggest that e; reacts, and may also diffuse, by a tunnelling mechanism.The temperature dependences of the rate constants and diffusion coefficients are similar to those of other transport processes in glass forming solutions. For rapid bimolecular reactions in solution diffusive transport can be so slow that the average distance between potentially reactive molecules soon exceeds that pre- dicted for a random equilibrium distribution. The measured rate coefficient will decrease, therefore, from an initial value k , characteristic of the random distribution, to a steady state value k' given by eqn (1),l 47cnaD' 1 + 4naD'/k k' = where a is the reaction distance and D' is the sum of the diffusion coefficients of the two molecules. This eqn reduces to the familiar Smoluchowski eqn (2)2 k' = 47cuD' (2) when k 9 47caD'. Noyes efficient from its initial value k to its final value k' is given by eqn (3) has shown that the change with time of the rate co- 1 k k, = k' 1 +-, ex' erfc x [ 4xaD where k, is the rate coefficient at time t, and x = k( JD't)/k'a. For very short times when t 4 (a2/D')(k'/k)2 the observed rate coefficient approaches k.Typically, for diffusion controlled reactions of the hydrated electron at room temperature, Q - 0.5 nm and D' - 7 x lo-' cm2 s-l, implying that a time resolution better than 10-l' s is needed to explore this region. When t 9 (a2/D')(k'/k)2 eqn (3) reduces to (4) t present address : C.S.I.R.O., Division of Cloud Physics, P.O. Box 134, Epping, New South Wales $ present address : University Grants Committee, 14 Park Crescent, London W1N 4DH.Australia. 115116 TIME-DEPENDENT RATE CONSTANTS For typical diffusion controlled reactions of the hydrated electron * with values of CT and D’ given above and k + 4naD’, the observed rate coefficient is expected to be greater than k‘ by 1 % at s, 3 % at lo-’ s, 11 % at s and 34 % at 10-l’~. Through the pulse radiolysis technique it is possible to form essentially instan- taneously, a strongly absorbing reactive species (e;) in a solution containing randomly distributed solute molecules, and to measure its rate of reaction. For time-dependent changes in the rate coefficient to be conveniently observed on a timescale of tens of nanoseconds or longer, D’ must be reduced by several orders of magnitude. This is most satisfactorily achieved by the use of liquids which on cooling progressively become more viscous and eventually vitrefy, and several aqueous solutions fall into this category.Rate coefficients which have already been reported over a very wide temperature range for one such aqueous system are consistent with a variation of D’ with temperature, over at least eight orders of magnitude, given by where To = 135 K and B w 800 K. D’ = Dd exp[ - B/(T- To)] (5) The disappearance of solvated electrons, e;, by reaction with a solute, S, is given by eqn (6) and if k, is given by eqn (4) and [S] % [e;], the decrease in optical density, OD, with time due to e; is given by eqn (7) where ODo is the optical destity at t = 0. The variation of OD with time allows the evaluation of k’ and D’, and provided k % 4noD’, a can be calculated from In this paper we have applied the time dependent rate constant theory of Noyes to the reactions of e; with a number of solutes in 9.5 mol dm-3 LiCl and in 10 mol dm-3 OH- (5 mol dm-3 NaOH + 5 mol dm-3 KOH) over a wide temperature range and have obtained estimates of o and D‘.eqn (2). EXPERIMENTAL APPARATUS The irradiation cell and assembly has been de~cribed.~ Solutions were irradiated with pulses of 2.9 MeV electrons from a Van de Graaff accelerator. Pulse durations of 5 ns, 0.2 p s and 0.6 ps were used, and the dose per pulse (1 to 2 krad) was monitored by a secondary emission chamber which was calibrated with the iodide and ferrocyanide dosimeters6 Corrections were made for the different electron densities of the solutions. Details of the pulse radiolysis apparatus have been reported el~ewhere.~ MATERIALS All solutions were prepared from triply distilled H20 and were purged with argon or LiCl was Eastman certified reagent grade or Merck analytical grade.All other reagents 10 mol dm-3 OH- solutions con- With this composition the solutions exp [U(r)/kT]r-’dr)-’ nitrogen. were AnalaR grade (B.D.H. or Hopkin and Williams). sisted of 5 mol dm-3 NaOH+ 5 rnol dm-3 KOH. could be cooled slowly without crystallisation occurring. * When eiq reacts with an ion, B in eqn (1) -(3) is replaced by3 of= 1 where U(Y) is the potential energy of the reacting ions at separation Y.G . V. BUXTON, F. C . R . CATTELL AND F . S . DAINTON 117 RESULTS Conditions were chosen such that (i) the pulse length was very short compared with the lifetime of e;, and (ii) the decay of e; in the absence of reactive solutes was negligible on the timescale of its decay in the presence of such solutes.This latter condition was generally satisfied for solute concentrations 3 Kinetic measurements were generally made in the temperature range 190 to 260 K. The decay of e; was independent of wavelength in this temperature range, although the spectrum of e; does change with time at lower temperature^.^^ Fig. 1 illustrates typical decays of e; in the presence and absence of reactive solutes. Because of possible distortion of the oscilloscope trace depicting these decays, due to the risetime of the detection system (-4 ns) and to the possible contribution of spur reactions to the decay of e; at very short times, eqn (7) was modified to (8) to permit kinetic analysis to be made a finite time after the pulse.In eqn (8) ODi is the optical density at time ti after the end of the pulse, which is taken as time zero, and t2 > t l . mol dm-3. k''[S] (t$-tb>-' In ___ OD1 = k'[S](tf+t$)+--- OD2 2(nD)+' In fig. 2 we show typical data plotted according to eqn (8). From the slopes and intercepts of such plots values of k' and D' have been obtained between 190 and 260 K. Also shown in fig. 2 is a first order plot of the same data. At first sight such plots may appear to be acceptably linear, but close inspection reveals a systematic decrease in slope with time and the mean slope is appreciably larger than that predicted by eqn (8). .I .I CI) 4 5 ps per division mol dm-3 Cr02,.FIG. 1.-Decay of e; in 10 mol dm-3 OH- solution at 200 K containing (a) no solute, (b) 2 x The temperature dependences of k' and D' are illustrated in fig. 3 for the LiCl system, and in fig. 4 for the OH- system. Also shown in fig. 3 is the temperature dependence of T/r, where y is the shear viscosity derived from data in ref. (8). For both solutions the temperature dependence of D' is given by eqn (3, and of k' by eqn (9) k' = A exp[ - B/(T- To)]. (9) In these equations A. B, D6 and T are constants and are listed in table 1.( t t + 4 4 5 . 5 3 x 10-~~3) FIG. 2.-Kinetics of e, decay in 9.5 mol dm-3 LiCl solution at 211 K containing mol dm-3 Cr02,. ( 0 ) Plot of kinetic data according to eqn (S), (0) first order plot, (- - -) first order line having the same slope as plot (0) i.e.k' [S]. I I I I - *. - ...- FIG. 3.-Temperature dependence of k'(0) and D'(0) for the reaction of e; with acetone (1.09 x lo-' mol dm-3) in 9.5 mol dm-3 LiCl solution. Also shown is the self diffusion coefficient of H20 (H) from ref. (9), and log,, [8RT/3m)/dm3 mol-' s-'1 (..--) taking values of from ref. (8).G . V. BUXTON, F . C. R . CATTELL AND F . S . DAINTON 119 1 0 3 ~ / ( ~ - 135 K) FIG. 4.-Temperature dependence of k' (open points) and D' (solid points) for the reaction of e, with NO; in 10 mol dm-3 OH- solution, [NO;] = 4.7 x mol dm-3 (0) and lo-' rnol dm-3 (0). Also shown are values of loglo [(8RT/30007)/dm3 mol-' s-'1 w) taking values of 7 measured in this laboratory. rnol dm-3 (A), 5 x TABLE VA VALUES OF THE PARAMETERS IN EQN (2), (5) AND (9) FOR THE REACTIONS OF e ; WITH SOLUTES IN 9.5 mol dm3 LiCl AND 10 mol d ~ n - ~ OH- SOLUTIONS eqn (9) eqn (5) eqn (2) log 10 system solute (A/cim3 mol-1 s-1) B/K p$Z:s-1) B/ K of Inm 9.5 mol dm-3 LiCl (CH&CO 10.68 f 0.05 613 f 7 - 3.65 f 0.15 656k 20 0.60+ 0.27 (f = 1) To = 129K Hf 10.50+0.12 557+16 -4-02f0.29 578f39 0.6850.17 NO; 11.03+0.18 666&41 -3.61 f0.54 688k106 0.4220.07 NO; 11.35t0.44 716f88 -3.77f0.19 631 A37 0.81 k0.22 CrO2- 11.30f0.22 655f49 -3.91kO.30 633f71 1.73k0.33 10mol dm-3 OH- NO; 11430f0.12 905f21 -2.2050.42 1020k74 0.77f0.24 CrOi- 11.93f0.10 824f17 -2.78k0.18 902f31 2.01f0.54 To = 135K NO, 12.15k0.12 956f21 -2.5720.31 986f51 1.05f0.37 Quoted errors are standard deviations from least squares fits. Values of To are best values common to all the data.DISCUSSION Since the kinetic data fit eqn (8) (see fig. 2) it follows that eqn (4) is a good repre- sentation of the time-dependence of the rate coefficient kt on the timescale of our experiments. Thus the data are consistent with the time-dependence theory developed by Noyes for diffusion controlled reactions, implying that the reactions we have studied are diffusion controlled. This might have been anticipated from the fact that the majority of the reactions of solvated electrons in dilute aqueous solution are accepted as being diffusion controlled, but the following points provide firm evidence that this is the case in the present work. (i) Eqn (5) and (9) are examples of the general phenomenon that mass transport processes, P, in glass forming liquids are accurately described by equations of the form P = AP exp[ - B/(T- To)] (10)120 TIME-DEPENDENT RATE CONSTANTS where the parameters B and To are scarcely dependent on the particular transport process, e.g.fluidity, conductance, diffusion, mobility, dielectric relaxation time etc. (ii) Within the limits of our measurements B and To have the same values in eqn ( 5 ) and (9) for a given reaction (see table) and similar values for all the reactions studied in each system. has a similar temperature dependence (fig. 3) as does the self diffusion of H20 (fig. 3), although this was measured in a higher temperature range (280-370 K), where slightly different parameters may be expected. O Eqn (1) may be rewritten as For the LiCl system the shear viscosity 1 1 1 - = -+- k’ k 4naD” and the fact that k’ and D’ have the same temperature dependence indicates that k % 4noD’, unless k also has the same temperature dependence, in which case no conclusion can be drawn about the relative magnitude of k and 4noD’.However, there is evidence from other work l 1 on the effect of solutes on the yield of solvated electrons that e; reacts effectively instantaneously with the solute when formed at the encounter distance, i.e. when no diffusion is necessary for reaction to occur. We conclude, therefore, that eqn (2) is appropriate. Combining eqn (2) and (5) gives eqn ( 1 3 , and comparison with eqn (9) shows that A = 4ncrfDb. Values of afare listed in the table, and because of (ii) above are independent of temperature in the range 190- 260 K.The value o f f depends on the magnitude of U(r)/kT which is equal to Zie$j/kT, the ratio of the electrical and thermal energy of an ion i of charge Z i e in the field $ j of an ionj. Unfortunately one cannot calculate 1+9~ with any confidence except in dilute solutions l2 so we are unable to compute the magnitude off. How- ever, it is likely that there will be a large degree of association of cations and anions in 10 mol dm-3 salt solutions so that ionic solutes may behave as neutral species, in which casefwill be close to unity. The similar values of of for the reactions of e; with H+, acetone and NO: provide some support for this conclusion and we tenta- tively equate of with the actual reaction distance.The striking feature of the reaction distances listed in the table is the very large value for the reaction of e; with CrOi- in both systems. Such distances are only compatible with a mechanism in which the electron transfers from its solvent trap to the solute by tunnelling. Hart and Anbar l 3 have discussed possible mechanisms of the transfer of hydrated electrons to solutes and favour a tunnelling mechanism. It is interesting that o = 0.98 nm l 3 for the reaction of e; with CrOi- in dilute aqueous solution. This is also an abnormally large value which was estimated from a comparison of the measured rate constant with that predicted by the Debye- Smoluchowski equation. The similar values of Db for all the reactant pairs in the LiCl system imply that the mechanism of the diffusion of H+ is predominantly the hydrodynamic type rather than the proton transfer type in these concentrated solutions.This is not unexpected since a similar conclusion has been drawn by Lown and Thirsk l4 from the effect of pressure and concentration on the electrical conductivity of solutions of alkali metal hydroxides and orthophosphoric acid. They suggest that the proton transfer mechanism is suppressed at high solute concentrations because of the very small fraction of water molecules which are free to rotate in a manner which allows this mechanism to operate. k’ = 4nofDb exp[ - B/(T- To)]. (12)G . V. BUXTON, F. C. R. CATTELL AND F . S . DAINTON 121 It is interesting that the values of Db (see table) for the hydroxide solutions are about an order of magnitude larger than those for the LiCl solutions.This is most likely to be due to a difference in the values of Do for e, rather than Do for the solutes in the two systems. If the solvated electron diffuses in these media by tunnelling from trap to trap, as has been suggested for e; in H,0,13 then Do for e; will reflect the density of trapping sites in the medium, There is evidence from the depression of the yield of solvated electrons in these systems by reactive solutes that the OH- system is more efficient than the LiCl system at trapping electrons, particularly as the temperature is lowered, implying that there is indeed a higher density of electron trapping configurations in the OH- system. In this respect it is known that LiCl and KOH have profoundly different effects on the water structure.15 Thus Li+ breaks down the tetrahedral structure whereas Kf, being of similar size to HzO, can substitute into the water lattice.In addition C1-, because of its large size, contributes more to structure breakdown that does OH- which is also close in size to H20. If all these effects are causally related then they point to the importance of the tetrahedral water structure to the trapping of electrons in aqueous media. Several theoretical models have been proposed to account for the properties of glass forming liquids and to interpret the physical significance of the parameters in eqn (10). Among these are the free volume theory of Cohen and Turnbull," the configurational entropy theory of Adam and Gibbs,'7 and more recently the bond- lattice model of Angel1 and Rao l8 which can be grouped with the entropy theory.In each case To represents the temperature at which mass transport ceases. In the free volume model this is associated with the free volume falling to zero,I6 in the entropy case it is the temperature at which the configurational entropy falls to zero,17 corresponding to no broken bonds in the bond-lattice model. The parameter B in the free volume model is proportional to v*/av,, where v* is the minimum volume of the hole required to permit molecular displacement, and a and v, are the mean values of the coefficient of thermal expansion and molecular volume respectively. In the configurational entropy model lo* l8 B relates inversely the number of broken lattice bonds, or configurational excitations, to the temperature T when T > To and is shown to be proportional to To for a given system in agreement with experimental findings.lo In our experiments B for LiCl solutions is about 30 % lower than the hydroxide value, although a is larger for the hydroxide solution,* which seems contradictory to the requirements of the free volume theory. This theory has also been criticised for other reasons.19 In terms of the entropy theory a smaller value of B indicates a larger configurational entropy change with temperature, and more configurational entropy at a given temperature. This is in keeping with less disruption of the solvent network in the hydroxide solutions. We are grateful to the S.R.C. and General Electric Research for financial assistance. R. M. Noyes, Progr. Reaction Kinetics, 1961, 1, 129. M. von Smoluchowski, 2. ghys. Chem., 1917, 92,129. P. Debye, Trans. Electrochem. Soc., 1942, 82,265. G. V. Buxton, F. C. R. Cattell and F. S. Dainton, Truns. Furuduy Soc., 1971, 67, 687. G. V. Buxton, Proc. Roy. SOC. A , 1972, 328, 9. G. E. Adams, J. W. Boag and B. D. Michael, Trans. Furaday SOC., 1965, 61,492. ' G. V. Buxton, F. C. R. Cattell and F. S. Dainton, to be published. * C. T. Moynihan, N. Balitactac, L. Boone and T. A. Litovitz, J. Chem. Phys., 1971, 55, 3013. A. Weiss and K. H. Nothnagel, Ber. Bunsenges. phys. Chem., 1971, 75, 216. * On cooling from room temperature to 190 K 10 mol dm-3 OH- contracts by 3.3% and 9.5 mol dm-3 LiCl by 1.5 %.122 TIME-DEPENDENT RATE CONSTANTS lo C. A. Angell and R. D. Bressell, J. Phys. Chem., 1972, 76,3244. G. V. Buxton and K. G. Kemsley, J.C.S. FaraCiay I, in press. l2 R. A. Robinson and R. H. Stokes, Electrolyte Solutions (Butterworth, London, 2nd edn., 1959). l3 E. J. Hart and M. Anbar, The Hydrated Efectron (Wiley-Interscience, New York, 1970). l4 D. A. Lown and H. R. Thirsk, Trans. Furaday SOC., 1971,67, 132, 149. l5 G. W. Brady, J. Chem. Phys., 1958,28,464. l6 M. H. Cohen and D. Turnbull, J. Chem. Phys., 1959,31,1164. l7 G. Adam and J. H. Gibbs, J. Chem. Phys., 1965,43, 139. l8 C. A. Angell and K. J. Rao, J. Chem. Phys., 1972,57,470. l9 M. Goldstein, J. Chem. Phys., 1969, 51, 3728.