GENERAL DISCUSSIONDr. W. J. Albery (Oxford University) said: I would agree with Dr. Niirnbergabout the importance of the effect of the electric field in the double layer on rateconstants determined by electrochemical techniques but I would like to make threecomments. First, I do not think that a plot of kHet against [A-l-t as shown in fig. 2should in all cases give a straight line, especially when the diffuse double and reactionlayer thicknesses are not very different. One may show that when [A14 tendsto zero kH& must also tend to zero and not to a positive intercept as in fig. 2.Writing down the diffusion equation for Hf in the vicinity of an electrode,we may describe the dissociation field effect by a series expression of the form :kD/kg = 1 -k 0, eXp (- i l Z / p ) ,n = lwherep is the thickness of the diffuse double layer. We then substitute this expressioiiin the diffusion equation and by means of Laplace transformations we arrive at acorrecting factor for the dissociation field effect which has the formDr.Nurnberg's correction obtained by integrating the series expression over thereaction layer isa , 1+C- kHet KPk;; k; ! I = 1 nplp'-=-=For small values of p/p, i.e., small values of [A-1, the corrections are approximatelyequal, but for higher values of [A-1, on the left of fig. 2, p/p rises to about 0.3 aiidhence the correcting terms may differ by 20 % or so. Calculations carried out bycomputer since show that this difference is about 10 % for p = 4p and 20 % forp = 2p. When [A-]-%+O, p+O, and whereas Dr.Niirnberg's expression for kHettends to a positive intercept of k&Z,cc,/n, in fact kHet must decrease to zero despitethe dissociation field effect. The fact that this is not predicted by the formuladerived from consideration of electrical analogues and transmission lines must Ithink throw some doubt on this approach especially when p is not much greaterthan p.My second point is that the static and dynamic $ effects cannot be neglectedeven though oppositely charged ions are being produced from a neutral species.The field close to the electrode repels the A- and its concentration there may bedepleted by a factor of 50 or so. Because the H+ is being reduced at the electrodethere is no compensating rise in [H+] ; hence the recombination term is reduced anddissociation is enhanced.We have worked out a treatment 1 for both the staticand dynamic $ effects using hypergeometric functions. They give the followingcorrecting function to kHet for the high field case of a Hg electrode at -2 V :PIP 0 0-1 0.2 0.3f(PlPt) 1.00 1-19 1.35 1 -50At Dr. Niirnberg's highest value of [A-1, p/p is 0-2-0-3 : thus, these correctionsare not negligible and, if included, would pull the left-hand points in fig. 2 down,1 Albery, Trans. Faraday SOC., 1965, 61, 2063.15160 GENERAL DISCUSSIONleading to a smaller intercept and therefore implying that the dissociation fieldeffect was less important.This is quite sensible since the Onsager formula is only applicable as Onsager 1himself stated when " the concentration of free ions is sufficiently small so that theDebye-Huckel radius of the ionic atmosphere is much greater than the effectiverange q of the ions ".In aqueous solution, q is 3.5 A, and in 1 M salt solutions theDebye length is much the same and thus the necessary condition is not fulfilled.Hence the ions are to a certain extent shielded by the electrostatic interactions inthe solution and the dissociation field effect will be less than that predicted by theOnsager formula.To help sort out the interplay of these various field and kinetic effects it seemsto me important to carry out kinetic investigations in which not only [A-] is variedbut also the total ionic strength. Furthermore, the use of electrodes of differentmaterials may be useful since the kinetic effects should remain unchanged butowing to the different overvoltages for the discharge of H+ there will be large variationsin the field effects.Dr.H. W. Niirnberg (Kernforschungsanlage Jiilich) said : Concerning the com-ments of Dr. Albery I would give the following answers. Certainly the relationshipbetween khet/ J D H A and [A-]A will deviate from linearity if the reaction layer isreduced to the region of the diffuse double layer as has been stated already in thepaper. In this connection the equivalent thickness of the diffuse double layer pdrefers only to the distance where the @-potential has decayed to a value given byeqn. (15) while the whole range of the diffuse double layer is somewhat larger andmay be expected to equal say 4pd.However, the present paper is mainly concernedwith the case p>4pd. The lowest experimental point in fig. 2 for acetic acid at[A-]-t = 2-55 corresponds to ,Y = 5.7pd. For all reaction layer thickness adjustedto ,u > 4 pd the amount of all additional influences on khet exerted by effects restrictedto the diffuse double layer region will remain constant and independent of ,u or[A-]+. Therefore they can be accounted for by a constant additional term B/ JD=*,and the experimentally obtained khet/ JDHA will then depend linearly on [A-]+according to eqn. (la) :Consequently the true value of ka free from any side effects caused by the diffusedouble layer of the electrode may be deduced either from the slope of the linearpart of the relationship between khet/JDHA and [A-]+, or the intercept obtainedby extrapolation to the ordinate.This correction is essentially independent ofpossible defects in the interpretation of the nature and relative magnitude of theeffects contributing to B. The situation changes for p<4pd. B now becomes avariable and it is to be expected that it will decrease with decreasing p. The de-crease of B will be for the greater part of 4pd relatively small and would reachappreciable amounts only if p corresponds to very small distances x from the outerHeImholtz plane. However, in practice, p never goes to zero but to a constantminimum value which may equal in the limit the distance Xlim. Therefore B, kLetand consequently khet, will have a finite minimum value which will possibly be smallerthan the intercept obtained by the extrapolation of the linear part of the khet against[A-]+ relationship.As Dr.Albery has stated, the equivalent circuit given in the paper does not account1 Onsager, J. Chem. Physics, 1934,2, 599GENERAL DISCUSSION 161for this decrease of B in the double layer region, because it was designed for thecase p>4pd, i.e., the linear part of the plot khet against [A-]-+, in which we wereexperimentally primarily interested and for which the present equivalent circuitis correct and sufficient. Therefore it is not justifiable to raise fundamental doubtsabout the capability of the equivalent circuit approach. The more complicatedcase of accounting for the situation of p<4p& can be achieved by a more sophisticatedequivalent circuit where RFD, Rh and TLH+ become functions of x for the diffusedouble layer region 4pd.In my paper, uncharged acids B were completely attributed to the dissociation fieldefTect.I agree with Dr. Albery that this is a somewhat too crude approxim-ation. Though the d.f.-effect certainly remains dominant, especially the contribution toB of the static $-effect on the anions should not be neglected, while the dynamic $-effecton H+ seems negligible at least for p > +&as a re-examinationof the problem has shown.1For instance, for an ionic strength of 1.0 we obtain, after allowing only for thestatic $-effect on the anions, a contribution of 20 % to B for p>4pd.Practicallythe same result is obtained from equations derived by Matsuda,;! which accountin principle for both the static $-effect on A- and the dynamic $-effect on H+, whilea similar general correction function tabulated by Albery 3 * yields a $-contributionto B which is 7-3 % larger. This seems to be a satisfactory agreement betweenthe various approaches to the complicated problem of sorting out the differentcontributions of the various double-layer effects for the given conditions. However,any errors in calculating the relative contribution of the various double-layer effectsto B will not affect the accuracy of the correction of khet.After the preparation of my paper the problem has been further examined allow-ing also for the decrease in the dielectric constant E due to the electrical field in thediffuse double layer.2 While details of this treatment will be published elsewheresome essential results are communicated here.For 1 = 1.0, Pd reduces from 2.7to 2.12 A due to the decrease of E . Also for distances x<4 A the ki/kd, againstx-relationship becomes considerably more steep than in fig. 4 due to the increasingdielectric saturation when approaching the outer Helmholtz plane (x = 0). Thedissociation field effect in the portion of the diffuse double layer close to x = 0will therefore be of dominating weight in the d.f.-contribution to B. The decreaseof B from its constant value for p>4pd to its constant minimum value correspondingto the minimum value of p might be therefore slight and difficult to detect.Themain result of the refined treatment allowing for the dielectric saturation is thatthe value for the distance of closest approach XI- of the carboxylic group of theacid molecule to the outer Helmholtz plane is altered for the acetic as well as forthe benzoic acid type to the more reasonable figure of xlimzl*6A. However,none of the general conclusions drawn in my paper is altered by this. Furthermorewith respect to Dr. Albery’s remarks on the limitations of the applicability ofOnsager’s equation due to shielding effects in solutions of high ionic strength, itcan be shown3 that over the greater part of the diffuse double layer due to thedecrease of E the effective external field is strong enough to peel off the shieldingionic atmosphere.Therefore, the demands of the Onsager theory seem to befulfilled in this region.* The supply of these data by Dr. Albery prior to publication is gratefully acknowledged.1 Nurnberg and Wolff, unpublished results.2 Matsuda, J. Amer. Chem. Sac., 1960, 64, 336. Senda and Delahay, J. Amer. Chem. SOC.,3 Albery, Trans. Faraduy SOC., 1965, in press.F1961, 65, 15871 62 GENERAL DISCUSSIONDr. W. J. Albery (Oxford University) said: Taking Dr. Nurnberg’s equation,B is not constant but is a function of the ratio p / p . Hence it is not necessarilytrue that akHet/ap must equal k ~ . For p&p, aB/ap will be unimportant butthis condition is not fulfilled for all of Dr. Nurnberg’s experimental points.Dr. H. W. Nhberg (Kernforschungsanlage Jiilich) said: I agree with Dr.Alberythat in the equation above for khet, kfd and klL are not strictly constants for thewhole reaction layer p. However, for all experimental points given in my papera(kfd+k+)/dp = aB/ap may still be regarded as negligible (see fig. 2, 3 and 4 ofmy paper). The deviation from linearity at the lowest experimental point in fig. 2(acetic acid, [A-]-3 = 2-55) due to the small finite value of aB/ap would fall withinthe stated experimental error of +4 %. This is in excellent agreement with newcalculations employing a computer by Albery 1 * (see table 2 in ref. (1)).Dr. M. Fleischmann (Newcastle upon T’ne) said: There is a general difficultyin the interpretation of electrochemical measurements of the rates of fast homo-geneous reactions which may be illustrated by the evaluation of the effect of highfields on the velocity of dissociation.The mass transfer of the species involved inthe equilibriumkH,t = pkD +BYk lk2HA+A- +H+will be governed by the differential equations,taking into account diffusion and migration. These one-dimensional equationsare solved (usually with simplifying assumptions) with the appropriate boundaryconditions so as to derive the reaction-limited rate. For high field dissociation theelectrical field gradient is additionally determined by the solution of Poisson’sequation for the inert electrolyte. It is instructive to consider the particular con-dition of a steady state (differential with respect to time zero) in the absence ofcurrent. Excluding discontinuous solutions the left-hand side of all the equationsis then zero and the only possible self-consistent solution of the first two is CH+CA- =constant.2 With CHA also a constant, it is impossible to make kl a function of thedistance from the electrode unless k2 exactly compensates the variation of kl.Under these conditions we therefore conclude that there can be no high-fielddissociation in the field of the double layer.While there is difficulty in considering high-field dissociation in the absence ofcurrent flow (or indeed the dynamic 1G/ effect separately from high-field dissociation),this conclusion appears to be unlikely.The cause would seem to lie in the dualuse of the concepts of diffusion, first to derive the variation of kl with field from aformulation in three-dimensions in the bulk of the solution (as well as the evaluation* The supply of these data by Dr.Albery prior to publication is gratefully acknowledged.1 Albery, Trans. Faraday Suc., submitted for publication.2 Bass, Trans. Faraday Soc., 1964, 60, 1656GENERAL DISCUSSION 163of k2) followed by the use of these quantities in the one-dimensional differentialequation in the surface region. A completely self-consistent treatment wouldhave to consider the three-dimensional case in the region of the surface of theelectrode, the kinetically-limited current being derived directly.Dr. W. J. AIbery (Oxford University) said: I think that there is an obviousfallacy in Dr. Fleischmann’s paradox.He states that, because there is no currentflowing, terms of the form D3[X]/W are equal to zero. But this is not true. Theincreased tendency of the acid to dissociate in the region of high field near the elec-trode leads to a fall in [HA] and a rise in both [H+] and [A-] and thereby the estab-lishment of steady-state concentration gradients. Thus 82[HA]/az2 is positivewhile 8[H+]/dz2 and d2[A-]/az2 are both negative; this is quite compatible in allthree equations with an increase in k D , caused by the dissociation field effect, leadingto kD[m] being slightly greater than ~R[H+][A-].Prof. M. Eigen (Giittingen) said : After receiving the preprints, Mr. Ilgenfritzat our laboratory has tried to remeasure the data reported by Bewick, Fleischmannand Wynne-Jones.The error limits are quite high since these measurements haveto be made around pH 7 in unbuffered solutions at small concentrations of the acidsor bases. Furthermore, the measured relaxation times are very short and in manycases close to the heating time. In general, our results agree with those reportedby the authors. The evaluation of the data, however, has to take into considerationthe protolytic (H+ + base) as well as the hydrolytic (OH- + acid) reactions, sincethe pK of the system and the pH of the solution are close to 7. By using the correctexpressions 1 for z, one obtains values whose orders of magnitude agree with themeasured data throughout the whole range of concentration. The agreement isquite good at low concentrations of HIn (phenol red) and OH- (< 10-5 M).How-ever, there is a systematic deviation at higher concentrations : the experimentalvalues of l/z become almost constant whereas they should increase with increasingconcentration of (HIn + OH-). A possible explanation is that at high dye concentra-tions association occurs. Peculiar rate phenomena of this type have been observedby us for a number of dyes (at concentrations around 10-5-10-4 M), and it seemsthat such irregularities exist in phenol red also.The conclusion that these deviations disprove the Debye-theory of diffusion-controlled reactions does not seem to us appropriate. We have studied more thana hundred, more or less simple, acid-base systems, covering about 6 orders ofmagnitude in concentration, and have always found excellent agreement with thetheory.These studies include quite different methods such as sound absorption,high-field pulse and temperature and pressure jump techniques. The results agreealso excellently with those from n.m.r., fluorescence transformation and-wherethe rates are slow enough-flow studies. Some systems (e.g., NH3) have beenstudied in the concentration range from 10-5 to 1 M (employing different tech-niques). At low concentrations we have never observed any concentration de-pendence of the rate constant ; at higher concentrations ionic strength effects, etc.,of course, are present. Deviations of the above-mentioned kind have been observedin some systems, but they could always be quantitatively related to secondaryreactions (such as dimerization or keto-enol tautomerism).We suggest that ainore detailed study of these dye systems be made in order to clarify the nature ofthese deviations. Such T-jump measurements could be complemented also byhigh-field relaxation studies, which can resolve the time range below 1 psec.Dr. A. Bewick (Newcastle upon T’ne) said: We are indebted to Dr. Eigen forrepeating some of our T-jump measurements and eliminating any remaining doubts1 cf. Angew. Chem. int. ed., 1964, 3, 1164 GENERAL DISCUSSIONconcerning conclusion (a) in the discussion section of our paper. We would makethe following reply to his comments concerning the results for phenol red. Al-though we use very dilute solutions which are only self-buffered, we have developeda technique for circulating the solution through the cell from a large capacity reservoirin which the pH is monitored.This enables a good reproducibility to be maintainedwith these solutions.We have calculated relaxation times for the complete protolysis-hydrolysiskinetic scheme represented bykzkik5kikSk iH' + In2 - +HIn-H'+OH-+H,OHIn-+OH-+In2- +H20The literature values were used for kz and k2, and several values for k; wereinvestigated. Table 3 shows the calculated and experimental values for the longerrelaxation time 22, using for k; the value which gives the closest correspondencebetween the calculated and the experimental results. Whereas for the simpleTABLE 3 .-RELAXATION TIMES CALCULATED USING A PROTOLYSIS-HYDROLYSIS KINETICSCHEMEP H 10-5 qn2- mole L-1 ' 2 , psec dc.72, psec expt.7.1 0.23 60 577.1 0.46 38 407.1 0.88 25 257.5 0.44 14 -7.5 0.88 7.7 557.5 1-72 4 287.75 0.28 31 20.67.75 0.56 19 17.97.75 1.10 10 13.87.75 2.2 5 11.47.75 4.1 2.8 10.48-0 0.34 30 18.58-0 0.68 21 15.58.0 1.34 12.5 12.78-0 5.04 3.6 12.28.0 9.3 2.1 10.2k2 = 3 x 1010 1. mole-1 sec-1, k i = 1.4 x 1011 1. mole-1 sec-1, kz = 1010 1. mole-1 sec-1.8-0 2-61 7.3 10.9protolysis scheme, table 1, the calculated z2 values show about the correct percentagevariation with phenol red concentration at fixed pH but are several orders of mag-nitude too long, the protolysis-hydrolysis scheme gives results having the correctorder of magnitude but their concentration dependence is now incorrect, e.g., atpH 8.0 the calculated and experimental values for 22 vary by factors of 15 and 1-8respectively. Therefore our conclusions concerning the inadequacy of the presenttreatment remain as stated.No other case has been reported in which the kinetics of two coupled diffusion-controlled reactions has been investigated over a wide range of concentration, anGENERAL DISCUSSION 165indeed, the data for a single diffusion-controlled reaction are inadequate for veri-fying unequivocally the current theoretical treatment.The accepted theory mustbe regarded as unproved except insofar as the present paper casts considerabledoubt on its validity.The electrochemical perturbation method 5 has also been analyzed using theprotolysis-hydrolysis scheme in order to try to explain the pH dependence of theresults for phosphoric acid. This treatment leads to the following expression forthe currentIf the terms due to the water are small, this gives as before the observed dependenceof the current on the total buffer concentration. Depending upon the relativemagnitudes of the various terms, the pH dependence is either as before or differentin such a sense as to increase the variation of the rate constant with pH.It was suggested also that at higher concentrations phenol red showed deviationsfrom the Beer-Lambert law due probably to association. We have tested this anddo find some small deviations. Over a twenty-fold concentration range extendingfrom the region of the lowest concentration to greater than the highest concentra-tion used in our T-jump experiments, there is only a 20 % discrepancy between thetotal phenol red concentration and that indicated by the optical density. It wouldrequire more than an order of magnitude change over this range in order to accountfor the relaxation times in terms of the theoretical model