REDUCTIONOF Co(II1) COMPLEXES
1091
which is a fairly strong base, hydrogen bonds to a On the other hand, the value of 4 ~ ~ 0 4is2 -unreasonable both in magnitude and its relation to + M ~ P O ~ - . number of solvent molecules more strongly than they hydrogen bond to each other. This would be likely to Acids as weak as HP042- generally have 4 values well above l.O.’a The values of K H ~ P O ~ - / K Dwith ~PO an~ 1- , lead to a positive AFO of transfer from H 2 0 to DzO,l9 and could account for the observed effect. Direct, of 0.69, and the assumption that AFO of transfer is zero, experimental determination of KHPO ,~-/KDPO p 2 - , and requires that ~ M P O(2 - be substantially smaller than also + a f p o 4 2 - might help to clarify this situation. 4 ~ ~ regardless ~ 0 ~ -of the value assigned to the latter. The assumption of a zero AFo of transfer is the likely culprit. It has long been known and recently re(18) D. M. Goodall and F. A. Long, J . Amer. Chem. SOC.,90, 238 emphasized that this is an inexact approximation.18 (1968). Structurally an attractive hypothesis is that H P O P , (19) M.M . Kreevoy, J . Chem. Educ., 41, 636 (1964).
Reduction of Cobalt(II1) Complexes by Monovalent Zinc, Cadmium, and Nickel Ions in Aqueous Solutions1
by D. Meyerstein2 and W. A. Mulac Chemistry Division, Argonne National Laboratory, Argonne, Illinois, and the Nuclear Research Centre, Negev, Israel (Received June 26, 1 9 6 8 )
The specific rates of reaction of Zn+, Cd+) Ni+, and e,, with a series of Co(II1) complexes have been determined. The mechanisms of reduction and their dependence on the electronic structure of the monovalent cations are discussed.
The specific rates of reduction of a series of cobalt (111) complexes by different cations3-* and by hydrogen atoms,9-11 have been recently measured. The results have been used as guides for the elucidation of the reaction mechanism. Several criteria based on the kinetic evidence have been suggested in order to differentiate between “outer-sphere” and “innersphere” reducing agent^.^-^,'^ Recent studies have shown that monovalent zinc, cadmium, and nickel ions, formed by the reduction of the corresponding bivalent ions by the hydrated electrons, are powerful reducing agents.13J4 Many reactions of these cations have been measured.lk17 It has been of interest therefore to measure their rates of reaction with a series of cobalt (111) complexes in order to obtain a better understanding of the mechanism of reduction by these cations. The specific rates of reaction of Zn+, Cdf, and Ni+ with a series of Co (111)(“3) &Xand Co (111)(En) 2XY complexes have been determined. The results suggest that Zn+ is mainly an outer-sphere reducing agent, Ni+ is mainly an inner-sphere reducing agent, and Cd+ reacts via both mechanisms.
Experimental Section iWateriuls. The water has been triply distilled. ZnS04,CdSOr, NiSOp, and CH30Hwere Baker Analyzed (1) Based on work performed under the auspices of the U. S. Atomic Energy Commission. ( 2 ) Reprint requests to be sent to D. Meyerstein, Nuclear Research Centre, Negev, Israel. (3) J. F. Endicott and H. Taube, J . Amer. Chem. Soc., 86, 1686 (1964). (4) A. Zwickel and H. Taube, ibid., 83, 793 (1961). (5) J. P. Candlin, J. Halpern, and D. L. Trimm. ibid., 86, 1019 (1964). (6) J. P.Candlin and J. Halpern, Inofg. Chem., 4, 766 (1965). (7) J. H.Espenson. ibid., 4, 121 (1965). (8) J. P. Candlin, J. Halpern, and S. Nakamura, J . Amer. Chem Soc., 8 5 , 2517 (1965). (9) G. Navon and G. Stein, J . Phys. Chem., 69, 1391 (1065). (10) M.Anbar and D. Meyerstein, Nature, 206, 818 (1965). (11) J. Halpern and J. Rabani, J . Amer. Chem. Soc., 88, 699 (1966). (12) H. Diebler, P. H. Dodel, and H. Taube, Inorg. Chem., 5 , 1688 (1966). (13) J. H.Baxendale and R . 8 . Dixon, 2. Physik. Chem. (Frankfurt am Main), 43, 161 (1964). (14) D.iMeyerstein and W. A. Mulac, J.Phys. Chem., 72, 784 (1968). (15) J. H.Baxendale, J. P. Keene, and D. A. Stott in “Pulse Radiolysis,” Academic Press, London, 1965. p 107. (16) J. H. Baxendale, J. P. Keene, and D. A. Stott, Chem. Commun., 715 (1966). (17) G. V. Buxton, F. 9. Dainton, and G. Thielens, (bid., 201 (1967). Volume 7S,Number 4 April 1969
D. MEYERSFEIN
1092 Reagents. The complexes [Co ("3) ~HzO] (C104)3, [Co ( " 3 ) sBr] (Clod 2, [Co * [Co ("3) 5Cll (C104)2, ("3) s N ~(Clod) ] 2, [Co ("3) &N] (C104)2, and [Co (En) zCOa]Cl have been obtained from Professor H. Taube. The complexes [Co ("3) 6 1 (C104)3, [Co ("3) sNCSIS04, [Co (En) 3]C13, cis-[Co (En) 2Cl,]ClO4, and trans-[Co (En) aCl2lCl have been obtained from Professor F. Basolo. The complexes [Co ("3) 5 acetate] ( ClO,) z and [Co ( NH3) fumarate]. (ClO,) 2 have been obtained from Professor J. Halpern. The complex [Co ( NH3)5F] ( C104) has been obtained from Dr. G. Navon. The purity of the complexes has been checked by measuring the absorption spectra of their solutions and comparing them with the spectra reported in the literature. Proeedure. For the determination of the specific rate constants of the complexes with the hydrated electron, solutions containing 5 X to 2 X 10-5 M of the complexes and 1 X 10-3 M methanol have been used. For the determination of the rate constants of M+ ( M = Zn, Cd, or Ni) with the complexes, solutions M MS04, 1 X loF3M methanol, containing 2 X and the Co(II1) complexes in the concentration range of 5 X to 3 X 10-41M have been used. The solutions were deaerated by shaking with argon gas in a syringe and then expelling the gas. This procedure was repeated four times leaving a residual concentration of oxygen less than 3 X 10-7 M . In order to minimize the extent of hydrolysis of the complexes, all solutions were prepared immediately before the irradiation. (At most 3 hr passed between dissolving the complex and the irradiation, the extent of hydrolysis, determined spectrophotometrically, being smaller than 1O%.) The pulse-radiolytic experiments were carried out using a O.4-psec, 15-MeV electron pulse from a linear accelerator yielding a dose of 0.6-6 X 1018 eV 1.-' per pulse (producing 0.3-3 pi%! reducing radicals). The decay of eaq- was followed spectrophotometrically at 5750k using a 4-cm double-path cell giving an 8-em total light path. A 450-W xenon lamp was used as an analyzing light source, and a Bausch and Lomb monochromator followed by a IP28 photomultiplier and a Type 5.55 Tetronix oscilloscope fitted with a Polaroid camera was used for recording the absorption changes in the solution. In order to minimize photochemical decomposition of the solutes which might be induced by the analyzing light, a Corning 7-54 glass filter was introduced between the xenon lamp and the cell when an absorption was followed a t 5750A. This filter was replaced by the filters 0-63 and 3-68 when the absorption was followed at 3100k. Furthermore, a mechanical shutter was kept closed until shortly before the pulse and the solutions were kept in the dark. The calculations were carried out using Chloe, an automatic photoelectronic scanner,'S for reading the Polaroid pictures and transferring the data to a magThe Journal of Physical Chemistry
AND
w. A.
MULAC
netic tape which was then used in a computer program for the analysis of the kinetics.lg All traces were analyzed for first- and second-order decay and pseudofirst-order rates were calculated when a good firsborder decay plot was obtained for at least three half-lives. A correction for the decay of M+ in the absence of oxidants was included when needed; this correction was always smaller than 25%. For every rate constant a t least ten traces were analyzed, which were obtained by pulse radiolyzing ten sample solutions, prepared by using at least two different stock solutions to dilute to several different concentrations. Some of the complexes have a relatively high absorption coefficient at 3100 A, and their reactions with M+ were followed therefore at 3 3 0 0 k . In some cases the light transmittance at the end of the reaction was slightly higher than before the pulse, due to the decomposition of t,he complex. In these cases one also has to correct partially for the reaction of hydrogen atoms and CHzOH radicals with the complex. As the latter correction is not very accurate though always smaller than lo%, a larger error limit is given for these results.
Results and Discussion The radiolysis of water may be described by H2O
-+
ea,-, H, OH, Hz, H2Ozj H30aq+
(1)
the yields of the products being GeaQ-= 2.6, GOH = 2.65, GH = 0.6, G H = ~ 0.45, and GH,o, = 0.75. In solutions containing MS04 salts, Co (111) complexes, and methanol the following reactions are expected e aq- + MZ+-+M+
+ Co(II1) -+ Co(I1) H + CH30H -+ CHzOH + Hz H + Co(II1) Co(I1) + H+ OH + CHaOH -+CHzOH + HzO eaq-
-+
CHzOH
+ CO(111) -+CHzO + CO(11)
+ CHzOH HOCHzCHzOH M+ + Co(II1) Co(I1) + Mz+ M+ + M++M2+ + M M+ + HzOz M2+ + OH + OH-
CHzOH
-+
-+
-+
(2) (3) (4) (5)
(6)
(7) (8) (9)
(10)
(11)
When the Co(II1) complexes are added in concentrations low enough to avoid a major contribution of reaction 3 as compared to reaction 2, one can study (18) The Ohloe system was developed a t the Argonne National Laboratory Applied Mathematics Division, by an Engineering group led by Donald Hodges. A detailed description of this machine is given in Technical Memorandum No. 61. by Donald Hodges, Applied Mathematics Division, Argonne National Laboratory, Nov 1963. (19) M. 0.Sauer, ANL-7146 Report (1966).
REDUCTION OF Co(1II) COMPLEXES
1093
Table I: Specific Rates of Reduction of Co(II1) Complexes" ZIl+
CO(NHa) eat Co(NHa)6HzO3+ Co(NHa)aOH2* Co(NHa)6F2+ CO(NH~)~C~~' Co(NH3)6Br2+ CO("3) .5Na2+ CO("a) 6CNa+ CO(NH~)~NCS'+ CO(NH~)~OOCCH~~+ Co ("2) 6fumarate+ Co(En)a3+ cis-Go(En)2NH3C12t ci~-Co(En)~NH~N02~+ Co(En)2FH2O2+ Co(En)zCOa+ cis-Co(En)zFz+ cis-Co(En)zClz+ trans-Co(En)zG12+
8.4 X 108 1.56 x 109b 1.i o x 109d 8.2 X 108 109 2.2 2.6 x 109b 1.49 X 10gi 1.30 x 109 1.65 109i 5.0 X 108 1.21 x 109' 2.5 X 108 1.47 x 109 2.7 x 109j!k 4.7 x 108 4.7 x 108 5.4 x 10s 1.91 x 109i 2.3 x 109i
x
x
Ni+
Cd'
5 5 x 108 5 5 x lO8b 1.3 x 107, 5 5 x 108 6.5 X 108 1.05 X 10% 5.8 x l o s i t f 3.3 x 1081 4.1 X l O S i * i 5 5 x 108
1.72 X108 6.2 X lo& 9.0 1086 5.4 x 108 2.2 x 109 2.5 x 109b 1.41 109d 9 . 1 x 108 1.32 x 109i 9.0 x 107 8.3 X 1081 1.60 x 107 1.75 x 109 2 . 8 X lO9i9k 4.1 X 108 6.7 X 108 6.0 X 108 2.3 x 109; 2.6 X l O V i
x
x
...
x
5 5 106 4.7 x 108 3.3 x lO8j*k 5 5 x 108 5 5 x 106 5 5 x 106 5.9 x 10s; 8.3 X 108i~j
H
%a-
8 . 5 X 1010 8.0 x 1010" 6.0 X 10'0' 6.6 X 10'0 7.8 X 10'0 8.0 X 1Olo 7 * 2 x 1010 7.4 x 1010 7.3 x 1010 7 * 3 1010 6.5 X 1O1O 8 . 5 X 10'0 6.6 X 1O1o 6.6 X lO1o 6.3 X 10'0 4.8 X 1O1O 4.9 1010 7.3 x 1010 7.7 x 1010
x
x
5 1 . 6 X 106m~"~o 51 x 106mfO 2.5 107% 5 1 . 2 106m,n'o 0.73%-16 X 1080 0.46m-14 X 1090 3.9%-11 X 108" 3.7m-6.1 X 1070 6.3 x 1090 1 . 3 x 1060 6.1 x 1090
x x
... ... 2
x 106"
...
... ...
M CHaOH, plus the complexes in different concentration and were 811 solutions contained 0.02 M MSOa, 1 X a In units of M-l sec-l. at, pH 5-6. The reactions of M+ were followed a t 310 mp and of esa- at 575 mp without MSOa. The maximal standard deviation is &1501,. From ref 23. Measured Measured at pH 4.0. The rate constants for the reaction of hydrogen atoms are from the literature (ref 9-11). i Measured at 350 mfi. 0 Measured at pH 10.0. at pH 6.6. e Measured at pH 7.1. f Measured a t pH 7.6, standard deviation &%yo. i Standard deviation 125%. k Measured at 350 mp. 1 The products of the reaction had a relatively large absorption, which decayed slowly and could not be identified, decreasing the accuracy; the results have therefore an accuracy of &25%. Due to the large residual absorption Reference 9. n Reference 10. 0 Reference 11. the rate of the reaction Ni" f Co(NHa)h fumarate+ could not be determined.
reaction 9 by following the disappearance of M+. Reactions 10 and 11, in the absence of Co(II1) complexes, have been studied earlier.14 It was found that one can determine the specific rate constant of reaction 9, kg, by introducing a small correction for reactions 10 and 1l.14 In the case where the final light transmission was slightly larger after the reaction than before the pulse, due to the decomposition of the Co(II1) complex, an additional correction due to reactions 5 and 7 must be added. The specific rates of reaction of Zn+, Cd+, Ni+, and em- with a series of Co(II1) complexes are summarized in Table I. The known corresponding rates of reaction of hydrogen atoms with the same complexes are included for comparison. It should be mentioned that the rates of reaction of Zn+, Cd+, and Ni+ were measured in solutions containing 2 X 10-2M of M2+S042-, having therefore an ionic strength, = 0.08. The rates of reaction of the hydrated electron with the Co(II1) complexes seem to be all diffusion controlled and therefore one can learn very little from the small differences in rates between the different complexes. Most of these rates are in fair agreement with other results cited in the literature.20-23 The diffusion-controlled rate of reaction between M+ and a divalent cation has been calculated by means of the Debye equation for two extreme cases: (1) the radius of M+ is 3.0 A, assuming that it is fully hydrated, and (2) the radius of M+ is 0.8 A, assuming that it is
not hydrated when reacting. After correcting for the salt effect it was found that the diffusion-controlled rate for such a reaction will range between 4 and 1 X log M-I sec-l, respectively. From the results in Table I it can be seen that some of the reactions have a rate of about 2 X logM-' sec-I and it can be assumed that these reactions are diffusion controlled, or at least approach this limit. From the results it is evident that for the series of the Co (111)(NH,) 6X complexes the order of reactivities is Zn+ 2 Cd+ > Ni+, the same order found for the reactions of these cations with other oxidants studied earlier.14 However, for the series Co (111)(En) 2XYthe order of reactivities is Cdf 4 Zn+ > Ni+ with the exception of Co (En) 2+ and Co (En) zFH202+. This change in the relative reactivities of Zn+ and Cd+ toward these two groups of very similar oxidants seems to suggest a difference in the mechanism of reduction by these cations. It is generally accepted that the reduction of CO("3) #+ and Co (En) as+ proceeds via the outer(20) J. H. Baxendale. E . M. Fielden, 0.Oapellos, J. M . Francis. J. V. Davies, M. Ebert, C . W. Gilbert, J. P. Keene, E. 5. Land, A. J. Swallow, and J. Nosworthy, Nature, 201, 468 (1964). (21) A. Szutka, J. K. Thomas, S. Gordon, and E. J. Hart, J. P h y s . Chem., 69, 289 (1966). (22)
J. H. Baxendale, E. M . Fielden, and J. P. Keene, Proc. Rou.
SOC.,
A286, 320 (1965).
Fielden, and E. J. Hart, unpublished results cited in M . Anbar and P. Neta. Int. J . A p p l . Radiation Isotopes, 18, (23) M . Anbar. E. M.
493 (1967). Volume 75,Number 4 April 1969
1094
sphere r n e c h a n i ~ m . ~It , ~ is evident from these results that Zn+ and Cd+ are very efficient reducing agents through this mechanism. The results also suggest that Ni+ is a much weaker reducing agent via this mechanism. The fact that the rates of reduction of Co (NH3)63+ and Co (En)83+ by Zn+ are within one order of magnitude of the diffusion-controlled limit naturally diminishes the effects of changing the ligands, and therefore decrease, even more than usual, the possibility to deduce from them the mechanism of reduction. The same is true to a lesser degree also for Cd+. Still it seems that some suggestions can be made based on the effect of ligands on the reaction rates. It is suggested that Zn+ reacts mainly via the outersphere mechanism. This is reasonable in view of the very high rates of reaction with C O ( N H ~ ) ~and ~+ Co (En) 33+ and the behavior of the aquo relative t o the hydroxo c ~ m p l e x . ~The fluoro and acetate complexes are less reactive than the C O ( N H ~ ) complex, ~~+ also suggesting an outer-sphere mechanism. Still one cannot exclude the possibility that a contribution of the inner-sphere mechanism exists in some of the Zn+ reductions. For Cd+ it is suggested that most of the reductions are via the inner-sphere mechanism or at least have a large contribution of it. This is in accordance with the fact that the hydroxo complex is reduced faster than the aquo c ~ m p l e x ,and ~ the relatively large effect of fluoride as a ligand. Furthermore, this explains the fact that Cd+ and Zn+ change their relative reactivities when passing from the C O ( N H ~ ) ~group X to the Co (En)2XY group. If Zn+ is really mainly an outersphere reducing agent, the effect of changing the ligands on the oxidant is expected to be relatively small, whereas for Cd+ as an inner-sphere reductant the effect should be larger.6 This effect is naturally larger for the Co(En)2XY group as the rate of reduction of C O ( E ~ ) is~ ~lower + than that of C O ( N H ~ ) ~ ~It+ is . therefore reasonable that the rate of reduction of Co(En)2XY complexes, where X and/or Y are good bridging groups, would be higher by Cd+ than by Znf. It is further suggested that as Cd+ reduces Pb2+faster than Zn+,16that Cd+ reacts in this case mainly via the inner-sphere mechanism. The relative rates of reaction of Xi+ with the different complexes suggest that it is an inner-sphere reducing agent. The effect of the ligands in this case i s large, changing the rates of reaction by several orders of magnitude, though the absolute rates are high. Furthermore, the rate of reduction of the hydroxo complex is higher than that of the aquo ~ o m p l e x . ~When comparing the specific rates of reaction of Cd+ and Ni+ with those of hydrogen atoms, which has been suggested to be an inner-sphere reducing agent,”“ the similarity in the effects of ligands is evident, especially for Ni+. It might be added that the rate of reaction of Cd+ with the fumarato complex is signifiThe Journal of Physical Chemistry
D. MEYERSTEIN AND W. A. MULAC cantly lower than would be expected from the comparison with the hydrogen atoms; this may be due t o the latter adding to the double bond on the ligand. The suggestion that Zn+ and Cd+ are good outersphere reductants is reasonqble as they have one electron in an S orbital,14and it is also expected that Zn+ would be a better reducing agent by this mechanism, as it is expected that in the Zn, Cd, and Hg group the stability of the lower oxidation state will increase with the atomic weight. On the other hand, the larger cation Cd+ is more polarizable and therefore probably a better reducing agent via the inner-sphere mechanism in accordance with the experiment. The added electron in Ni+, having a 3dgc~nfiguration,’~ is located in an inner orbital and it is reasonable therefore that Ni+ is less reactive than Zn+ and Cd+. If the suggestion that Cd+ and Ni+ do react with some of the complexes via the inner-sphere mechanism is true, it has t o be concluded that the rate of exchange of the water molecules in the inner hydration shell with the bulk is at least of the order of 5 X lo6 sec-’, as reactions with half-lives of the order of 1 x 10-e sec have been observed. Finally, it is of interest to note that the effects of ligands on the rate of reduction are similar for these very fast reactions with a large - AF to those found for other reducing agents which act by the same mecha(e.g., compare nisms but at much lower reaction the relative rates of reduction by Cd+ and Co (CN)b3which both react via the outer- and inner-sphere mechanism). As expected, though the order of reactivities is kept, the relative effects of the ligands decrease as the reaction rates approach the diffusion-controlled limit. Still, it is found that the ratio
equals 3.4 and 10.7 for Red = Zn+ and Cd+, respectively, very similar values to those found for the relative rates of reduction of these two complexes by Cr2+, V2+, Eu2+, and Cr(Bipy)32+,6where R = 4.5, 19, 4.0, and 3.8, respectively, though the specific reaction rates of the latter reductions are five to thirteen orders of magnitude lower than those for Zn+ and Cd+. The fact that R is not changed over such a wide range of reactivities can be used as an argument in favor of the Marcus theory, relating the rates of reaction via the outer-sphere mechanism to the free-energy gain in the reacti0ns.2~ This agreement with the Marcus theory might be fortuitous as most other Co(II1) reductions do not fit the theory, presumably due to the change in the spin multiplicity when reducing Co(II1) to c o (11). 3 3 (24) R. A. Marcus, J. Phys. Chem., 67, 863 (1963). (26) I?. Basolo and R . G . Pearson, “Mechanism in Inorganic Reactions,” John Wiley and Sons, Inc., New York. N. Y . , 1967, p 507.
TRAKSPORT IN CONCENTRATED SOLUTIONS OF 1 :1 ELECTROLYTES Acknowledgments. We are indebted to Dr. M. S. Matheson for his encouragement and discussions throughout this work, to Professor H. Taube, Professor J. Halpern, Professor F. Basolo, and Dr. G. Navon for providing the complexes thus enabling this study, to
1095
Professor M. Anbar for helpful discussions, and to Dr. M. C. Sauer for his assistance in the use of his computer program. We wish also to thank the Linac group for their careful operation and Mr. S. Petrek for maintaining the electronic equipment.
Transport in Concentrated Solutions of 1 :1 Electrolytes
by P. C. Carman National Chemical Research Laboratory, South African Council for Pretoria, South Africa (Received J u l y I , 1 9 6 8 )
:ientific an I n
strial Research,
By neglecting the relaxation effect in concentrated solutions, it is shown that ionic mobilities determined from ionic self-diffusion coefficients give a consistent interpretation of conduction and diffusion measurements. Ionic hydration is assumed and is found to lead to hydration numbers which are reasonably constant over wide ranges of concentration and are consistent with the known behavior of the ions concerned. Where ionic hydration occurs, it should affect transport numbers, and the corresponding relationship is derived and applied. Empirically determined electrophoretic factors agree with those calculated by the Onsager-Fuoss model up to the highest concentrations for neutral salts and only fail for the highest concentrations of HCl. The values of a which must be assumed can be used for accurate calculation of equivalent conductivities in dilute solutions.
Introduction In the classical paper of Onsager and Fuoss’ on diffusion and conduction in electrolytes, it was recognized that, except in very dilute solutions, solution viscosity and solvation of ions had to be taken into account as well as interionic effects, Much later, Robinson and Stokes2 presented a theory of diffusion in concentrated electrolyte solutions which recognized that, apart from solvation effects, diffusion of free solvent must be allowed for. Consequently, if each diffusing species possesses an intrinsic mobility, the appropriate reference frame is that proposed independently by Darken8 and by Hartley and Crank.4 -4severe limitation to application of this theory is that the intrinsic mobility qi of ion i is calculated by dividing the limiting mobility pio by the relative viscosity of the solution, vr. Basically the same theory was applied by Tarn& and l&ziszy,6 but they used self-diffusion coefficients Di* to calculate intrinsic mobilities and thereby found that interdiffusion ~ such electrolyte solutions as could be coefficients D I for tested could be correlated excellently up to concentrations of 3-4 gfw/l. This diffusion theory neglected interionic effects, though self-diffusion coefficients are subject to a relaxation correction and interdiffusion coefficients to an electrophoretic correction. Unfortunately, these corrections can be calculated accurately only in very dilute solutions, so that a rigorous approach
is not possible. However, it is well known that the electrophoretic correction is small if the ions involved do not differ widely in mobility, so that if relaxation corrections in self-diffusion are also small in concentrated solutions, the theory is well based. Evidence for this assumption is discussed in the following section. The present paper is based on the same assumptions, applied to the correlation of equivalent conductivities and transport numbers as well as of diffusion coefficients.
Relaxation Corrections and Intrinsic Ionic Mobilities I n conduction, the time of relaxation effect gives rise to a field AE, opposed to the applied field E. The limiting value of AE,/E derived by Onsager is given by
where e is the electronic charge, x i is valence of ion i, D is the dielectric constant of the solvent, and IC is the Boltzmann gas constant. K is the reciprocal mean (1) L. Onsager and R. M. Fuoss, J . Phys. Chem., 36, 2689 (1932). (2) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed, Butterworth and 00. Ltd., London, 1959, p 325. (3) L. 9. Darken, Trans. Amer. Inst. Min. Met. Eng., 175, 184 (1948). (4) G. 8 . Hartley and J. Crank, Trans. Faraday SOC.,45, 801 (1949). (5) J. Tames and K. fJjss6szy, Act. Chim. Hung., 49, 377 (1966).
Volume 75,Number 4 April 1969
