JULY, 1973 THE AhTALYST Vol. 98, No. 1168 Mass and Charge Transfer Kinetics and Coulometric Current Efficiencies Part V.* Comparison of Pattern Theory, Tafel, Allen and Hickling and Lewartowicz Methods, and Apparatus and Procedures for Ramping Voltammetry BY E. BISHOP AND P. H. H1TCHCOCK-J. (Chemistry Department, University of Exeter, Stocker Road, Exeter, EX4 4QD) A comparison is made of the provenance and application of pattern theory, Tafel, Allen and Hickling and Lewartowicz methods for the deter- mination of charge-transfer kinetic parameters. A general-purpose electro- chemical cell is described together with a ramping potentiostat - galvanostat for slow single-sweep voltammetry. The mass-transfer characteristics, which are governed by the geometrical placing of the various appurtenances that dip into the solution and by the stirring speed, were examined by determining the apparent diffusion layer thickness at various stirring speeds for a system of known mass-transfer properties, copper(I1) - copper(1).It was further established that the limiting current of reduction of hexacyanoferrate(II1) is reproducible and linearly related to concentration. IN the derivation of pattern theory,1,2 the thermal diffusion coefficient, Dx, of the species X and the diffusion-layer thickness, ax, were retained as discrete entities. The diffusion-layer thickness in a turbulently stirred solution is a pure fiction, Mass transport is nevertheless a definite and, if conditions such as the stirring speed, solution volume and viscosity and the geometrical placing of the various appurtenances in the cell are held constant, reproducible phenomenon.Although no hydrodynamic treatment of turbulently stirred solutions is possible, L e ~ i c h ~ - ~ has shown that mass transport is proportional to the concentration, [XI,, of the active species in the bulk of the solution. It is therefore proper to use an over-all conditional mass-transfer rate constant, J2,,,,,7 which can be determined in situ from a voltammetric scan to the limiting current, ILx, and absorbs all indeterminate quantities, such as the roughness factor, r, of the electrode and the transport number, tx, of the species X, so that, for a projected area, A , of the electrode This relationship can be directly substituted in the behaviour equations and solutions for the charge-transfer rate constant, k , and the charge-transfer coefficients, a, for cathodic processes, and (1 - a), or /3, for anodic processes.A realistic treatment of mass transfer in electrode processes is thus secured in the practical context. There are many methods in use for the determination of charge-transfer rate parameters. A small selection of the simpler and more popular of these methods will be reviewed together with pattern theory in the context of stirred solutions and coulometric current efficiencies. PATTERN THEORY Pattern theory is entirely rigorous when the conditions are satisfied,1,2 viz., that the charge-transfer overpotential, ri)a, is substantial so that the backward reaction is without effect, and [ nqn I 2 0.059 log,, (lOO/.r), where .r is the percentage experimental error in the measurement of the current in a voltammetric scan.The method for the determination of k and a (or /3) is extremely simple and rapid.2 A single voltammetric scan is made in the appropriate direction, cathodic or anodic, and the substitution of two points on each wave, including the background for the calculation of current efficiencies, into the appropriate * For details of Parts I11 and IV of this series, see reference list on p. 474. For Part VI, see p. 475. t Present address: Ever Ready Co. (G.B.) Ltd., Central Research Laboratory, St. Ann’s Road, @ SAC and the authors. London, N.15. 465466 BISHOP AND HITCHCOCK: MASS AND CHARGE TRANSFER KINETICS [Analyst, VOl. 98 will give a quick check. In a careful study, as many points can be taken as desired and subjected to statistical appraisal.The calculation is almost as quickly performed manually as by computer when the time taken to punch the data cards is taken into account. The scan speed should be slow enough to avoid maxima and to allow a reasonable approach to quasi-equilibrium, but not so slow as to produce any significant change in [XI,. Scan speeds between 0-2 and 2.0mVs-l are suitable. When the process is applied during the course or at the start of a coulometric determination, the quantity of electricity passed and performing desired reactions is easily integrated from the corrected (for unwanted reactions) area under the voltammogram. The method has been thoroughly tested experimentally, as will be shown in later papers, and gives results in concordance with other methods when they have been duly corrected for their deficiencies.One application not mentioned earlier1s2 is the determination of n', the number of electrons involved up to and including the rate- determining step.7 This determination simply involves back-computation of the experimental voltammogram with different integral values of n, excluding those which give values of a greater than unity. Non-integral values of n can be diagnostically useful in working out reaction mechanisms that involve chemical steps. Pattern theory takes full cognisance of mass transfer, but applies only to slow reactions. It applies also to background reactions2 and therefore to current efficiency calculation^.^ It is not restricted to cases for which the limiting currents and conditional potentials are known.TAFEL PLOT METHOD In this well known method,* the logarithm of the total current is plotted against the working electrode potential. All too frequently it is applied ignorantly, or with the bland assumption that mass transfer is insignificant; the current is often taken beyond the limiting current of the reaction under examination. Extrapolations of the linear parts, if there are any, of the cathodic and anodic wings of the plot cross at the equilibrium zero-current potential and the logarithm of the exchange current, i0,7 from which k can be calculated. The slopes of the cathodic and anodic wings of the plot give a and p. The fallacy lies in the dismissal of mass-transport and mass-transfer overpotential.For extremely fast reactions in solutions that contain a high concentration of the active species, it is possible to have mass-transfer overpotential without detectable charge-transfer overpotential. However, unless a net current passes, i, + i a # 0, there can be no charge-transfer overpotential, and if a net current, however small, flows, there must always be mass transport to carry the current, and therefore mass-transfer overpotential. A charge-transfer overpotential can never arise without an accompanying mass-transfer overpotential, which cannot be ignored. If the potentials in the Tafel plot are corrected for mass transfer, which can be easily done but involves more calculation, then the plot and the parameters derived therefrom will be valid; the plot will have straight parts of the wings unless there are other complications.Without this correc- tion, the wings of the plot can never be linear, and the extrapolations will be of dubious validity. Fundamentally, the Taf el relationship depends on the same assumptions as pattern theory, that I nva I 2 log,, (lOO/n), so that one of the exponential terms in equations (20) and (22) in reference 7, and equation (1) in reference 1, can be neglected with respect to the other. The Tafel method is therefore restricted to the pattern region. Furthermore, the part outside the exponentials is falsely identified as the exchange current, io, which is where the subscript B refers to conditions in the bulk of the solution, and equation (3) is true only when Ta = 0.To avoid confusion, the relevant section in equation (2) is identified in pattern theory as 10s and the subscript S identifies concentrations at the plane of closest approach to the electrode surface : Only when this correction is made do Tafel plots become valid, and it should be noted that equation (4) contains the unknown parameters that the plot is intended to reveal. This i, = nFAk [Ox]g-") [Redlg . . .. .. ' * (3) IOS = nFAK [Oxlg-.) [Red]: .. . . * (4)July, 19731 AND COULOMETRIC CURRENT EFFICIENCIES. PART V 467 difficulty is resolved by using equation (5) in reference 7 in order to separate qa and to correct the potential for mass transport : where the concentration term, qc, refers to the plane of closest approach’: .. - * (5) y/a = Ewe - Eo’ - .... .. . . (6) I I n FAhnass ox [ox]B - I + [RedIB -- n FAk mass red 2.303 RT q c = nF log,, This procedure is possible only if values for the two limiting currents IL,, and I L r e d can be obtained and the conditional mass-transfer rate constants calculated. Then q a becomes a logarithmic function of the current or current density [equations (16) and (41) in reference 13. The corrected Tafel relationships become equation (7) for cathodic and equation (8) for anodic reactions : a n F .. . (7) In I = (In 10s) -- RT ya m e . . The Tafel method is therefore restricted to slow reactions, and is not helpful with background reactions. It requires knowledge of limiting currents and conditional potentials. ALLEN AND HICKLING METHOD Allen and Hicklingg expanded the second exponential in equation (Z), extracted the common term in cc and, like Tafel, identified the non-exponential terms as the exchange current, io, writing then rearranging and taking logarithms : They then plotted the left-hand function in equation (10) against qa, the slope of the line giving cc and the intercept giving io, from which k could be calculated.This method does not, therefore, neglect one of the exponential terms, and is therefore applicable to faster reactions, in the categories fast and m0derate.l However, it makes the assumption that the non-exponential terms are constant and equal to i, (which is true only when I = q a = 0), whereas the second term on the right-hand side of equation (10) is properly I O S [equation (4)], which is not constant.In evaluating the functions in equation (lo), it is again necessary to know qa, which can be calculated from equation (5) and the mass-transfer equation (6), but again only if the limiting currents can be measured for the calculation of the mass-transfer rate constants. The further assumption is made that a + p = 1, although a change in slope as the plot passes from positive (cathodic) to negative (anodic) currents will reveal any change in the charge-transfer coefficient. Even at extremely low currents, the “constant,” In io, in equation (10) is not constant unless the reaction is very slow, which vitiates the retention of both exponential terms, and [ox]B and red]^ are large and equal. Otherwise, the “constant,” In (nFA k OX]^ [RedIE), requires evaluation and therefore pre-knowledge of the unknowns, k , cc and p.LEWARTOWICZ METHOD Reviewing earlier work,l0 Lewartowicz took as his starting point the work of Audu- bert,llJ2 and developed m e t h ~ d s l * J ~ - ~ ~ for linearising the Tafel plots of the logarithm of the current versus potential with or without making a correction for diffusion, and with or468 BISHOP AND HITCHCOCK: MASS AND CHARGE TRANSFER KINETICS [Analyst, VOl. 98 without making the concentrations of Ox and Red equal. He referenced his potentials to the equilibrium, zero-current potential of the solution of Ox and Red, so that 7 = Ewe - Eeq, the total overpotential arising from the passage of current, and then corrected for diffusion by subtracting yes, for the evaluation of which anodic and cathodic limiting currents must be known: The current in equation (2) is then split into partial currents, cathodic I , and anodic I , (which is negative), and their values are calculated by using the value of 7 a calculated from equation (11) : ... . (12) I and I , = 1-exp [ _____ --;g%] e x p [ $ F ] - - l I I c - Anodic currents being negative, the logarithm of the modulus of the partial current is plotted against Ewe. The slope of the cathodic wing gives cc, that of the anodic wing ,B, and the intercept of the two curves gives the corrected exchange current, io, from which k can be calculated and the equilibrium potential of the particular solution, Eeq. Lewartowicz took the process a step further by calculating the “ideal” partial currents, which sum to the total “ideal” current Ii; using his notation of a single prime for cathodic and a double prime for anodic reactions, Ii = I’ + I” .. . . .. .. . . (13) Plots of the logarithm of the “ideal” partial currents against Ewe give more effective linearisa- tion than the use of the “real,’ currents, I , and I,. Both current and potential are therefore “corrected” for mass transport, and there is no neglect of either exponential term in equa- tion (2), so the treatment is applicable to moderate and fast reactions. Knowledge of the limiting currents of both anodic and cathodic directions and of the equilibrium, zero-current potential of the system is required, but if the electrode process is fast it is usually possible to measure these quantities experimentally.Lewartowicz’s approach deserves more attention : it has tended to be neglected in view of similar approaches by other workers a t about the same time. CURVE-FITTING METHODS Of the foregoing methods, only pattern theory can be used on background reactions,2 which come into the pattern region at moderate charge-transfer overpotentials because of the larger number of electrons. Before pattern theory had been developed for background reactions, and in default of better methods, curve-fitting methods involving the use of curves computed from the full rigorous theory’ were perforce used. A family of curves for a series of values of k and a first guess for a, or (1 - a), was plotted by computer using the program VOLTAMMETRY 9 G/P.’ These curves were then matched with the experimental curves, and the best value of k was chosen.The process was repeated for a series of values of cc, the computed curves were offered to the experimental voltammograms and the best value of cc was chosen. The process could be repeated as often as necessary in order to refine the charge- transfer parameter values further, but in practice no more than two such essays were required. After development of pattern theory for background reactions, curve fitting was no longer required, because background waves usually came well into the pattern region at solid electrodes. SUMMARY It cannot be said that the problem of measuring the rate parameters for all the reactions , including the solvent molecule and ion reactions, in vigorously stirred solutions under the conditions of coulometry has been solved for fast reactions.However, the method of patternJuly, 19731 AND COULOMETRIC CURRENT EFFICIENCIES. PART V 469 theory is the simplest of all, involves the least calculation, can be applied at any time before, during and after a coulometric determination thus to detect any change in rate parameters and therefore current efficiencies, and is completely rigorous at adequate charge-transfer overpotentials. It does deal with background reactions,2 and can be applied without know- ledge of limiting currents or equilibrium or conditional potentials. The implementation of the method involves merely a single-sweep voltammetric scan in the appropriate direction using the working electrode and the electrolyte and species under the conditions pertaining to an ensuing coulometric determination, which additionally allows selection of the working potential for potentiostatic determinations and the working current for amperostatic deter- minations productive of the maximum current efficiency.Equipment has therefore been designed and tested for the examination of coulometrically useful reactions. The design of a ramping potentiostat - galvanostat, together with certain tests, will be described in this paper, and the applications in later papers. EXPERIMENTAL The thermostatically controlled cell for voltammetry is shown in Fig. 1. Water main- tained thermostatically at 25 & 0.05 "C is pumped through the jacket. The lid is machined from Perspex and B19 cones are cemented with Perspex dissolved in chloroform into holes drilled in the lid. The stirrer paddle is a magnetic follower covered with PTFE and fitted A€ I R S WE I, J' r - Fig.1. Thermostatically controlled voltam- metric cell: AE, auxiliary electrode in filter stick ; RE, reference electrode (in thermostat tank); WE, working electrode; C, Luggin capillary; D, Perspex stirrer disc; L, machined Perspex lid ; M, PTFE-coated magnetic follower; N,, nitrogen inlet to disperser; S, salt bridge between reference electrode and Luggin capillary ; and T, jacket of cell thermostat470 BISHOP AND HITCHCOCK: MASS AND CHARGE TRANSFER KINETICS [Analyst, Vol. 98 into a Perspex disc machined so as to be a close fit in the bottom of the cell : this device gave much smoother stirring and therefore lower signal noise than the follower alone.Platinum working electrodes are made of bright platinum sheet, one side and all edges and corners of which are covered with lead-glass, so that one face alone is exposed to the solution. Electrical connection is made by platinum wire spot-welded to the back of the plate, welded in turn to tinned copper connecting wire, and sheathed with lead-glass. The dimensions were measured by means of a travelling microscope. Platinum-wire electrodes consist of 25-mm lengths of 22 s.w.g. wire sealed into soft glass. Gold working electrodes consist of lengths of 22 s.w.g. wire, 1000 fine, on to which a bead of cobalt glass (Plowden and Thomson) is sealed by winding a thread of glass round the heated wire. The hot bead is then sealed to a stem of soft glass and the whole annealed.The exposed wire is trimmed to a length of 25 mm. The auxiliary electrode is a spiral of platinum wire that dips into supporting electrolyte in a porosity 4 filter stick. The reference electrode is either a saturated calomel or a saturated mercury(1) sulphate electrode immersed in a beaker filled with salt-bridge electrolyte and placed in the thermostat, and connected to the Luggin capillary by means of a polythene tube. The Luggin capillary is placed close to the working electrode. In normal operation, the reference electrode is fully protected from polarisation. The cell contents are de-aerated with white- spot nitrogen, scrubbed first with chromium(I1) chloride solution and then with distilled water, passing through a porosity 2 disperser.The gas escapes via the rim of the lid. RAMPING POTENTIOSTAT- The essentials of a simple potentiostat are shown in Fig. 2, the heart of which is the control amplifier, CA, which operates in the manner of all operational amplifiers so as to maintain the potential of the inverting input or summing junction, S, at the same potential as its non-inverting input, which is shown as connected to earth. The summing junction is therefore a virtual earth. The negative feedback loop includes the auxiliary and reference electrodes and the cell electrolyte, and the amplifier passes a current through auxiliary and working electrodes so as to bring the potential of the working electrode with respect to the reference electrode to equality with a command potential, which is pre-set by a signal genera- tor, R,E,.The amplifier C h is therefore in the configuration of a linear combiner. In order -=_L - Y Y Fig. 2. A simplified practical potentiostat based on a linear combiner: CA, potentiostat control amplifier; S, summing junction ; VF,, voltage follower on the reference electrode; CF, current follower; X,X, output to X-axis of recorder; Y,Y, output to Y-axis of recorder; Rj (Ri’), current-measuring standard resistor ; Rf, feedback resistor of CA; R, and R,, input resistors for signal generators El and E,, respec- tively; RD, dummy load resistors ; AE, auxiliary electrode : RE, reference electrode; WE, working electrode; and CA, VF, and CF, Philbrick P65AU differential operational amplifiersJuly, 19731 AND COULOMETRIC CURRENT EFFICIENCIES.PART V 471 to prevent the reference electrode from being polarised, a second operational amplifier, VF,, is connected between the reference electrode and the feedback resistor, Rf, of CA. VFI pro- vides a low impedance output to supply current to Rf while offering a very high input impedance and therefore drawing a minimum current through the reference electrode. VF, is a voltage follower in the non-inverting mode a t unity gain. The potential of the working electrode can be measured against the reference electrode either directly with a high-impedance meter with a recorder output to the X amplifier of an X - Y recorder, or at the points XX, which can be taken direct to the recorder, or via a pH meter. The measurement of the cell current poses certain problcms, as do the earthing and placement of the signal generators.These topics have been reviewed by Schwarz and Shain.22 The current is measured in terms of a voltage drop across a standard resistor, Ri, which could be placed between the working electrode and earth. This arrangement, however, would form an extra component in the command voltage provided by the signal generator, and an unwanted voltage in the output of the control amplifier. Such a resistor could be placed in the line to the auxiliary electrode as represented by Ri’ in Fig. 2 . Neither end of the resistor is then a t earth potential and, although the input to the Y-axis amplifier of the recorder can be floated, this position of the resistor was found to be productive of excessive noise.Instead, therefore, an operational amplifier in the current follower mode, CF, is placed in the working electrode to earth path, thus holding the working electrode at virtual earth, and the calibrated resistor, Ki, forms the feedback loop of the current follower, the output of which with respect to earth is taken to the Y-axis of the recorder. Unless the cell with its electrodes and electrolyte solution are in circuit, the control amplifier will slam into saturation when trying to obey impossible commands, and so dummy load resistors, R,, are switched in in place of the cell when the latter is not in use. A second signal generator, R,E,, is included, which can be used to set the starting voltage of a scan, and forms the third input to the linear combiner, CA.In the complete seven-amplifier circuit shown in Fig. 3, a second voltage follower, VF,, is added between the output from the current follower and the X - Y recorder, because the latter is a low-impedance instrument (lo4 to lo8 Q). The amplifiers used are capable of producing currents of only 5 2 . 2 mA at kll V, and so booster amplifiers, B, are included in the feedback loops of the current-driving amplifiers, the control amplifier, CA, and the current follower, CF, so as to increase the available current to 100 mA. The pre-set starting potential signal generator is retained in the lorm of a Mallory cell and a 15-turn Helipot potentiometer. The potential command signal generator shown as R2E2 in Fig. 2 is replaced by a ramp generator, RG, which takes the simple form of an integrator, for which the output potential is given by t n .. (15) By integrating a constant, stable input voltage, the output voltage increases linearly with time. The ramp speed is controlled by R, and C ; in practice, it was found convenient to use a fixed capacitance of 10pF and a four-decade resistance box, such as was formerly used in d.c. differential electrolytic p~tentiometry.~~ The integrator can be arranged so as to hold a given command potential by open-circuiting the input, and then the instrument becomes an ordinary potentiostat. Ramping in either direction is secured by reversal of the polarity of the input voltage to the integrator; re-setting to zero is unnecessary and the ramp can be reversed at any potential. The integrator is re-set to 0 V by discharging the capacitor, C, through the shorting resistor and re-set switch.The loose circuitry, batteries, resistors, capacitors, switches, etc., are mounted on eircuit boards inside an aluminium box, and connections to the cell electrodes and the recorders are co-axial. The box is provided with a number of Cinch sockets, into which the canned operational amplifiers and boosters and the power supplies can be plugged as required. The whole assembly is very flexible and circuit alterations can be made very rapidly. OTHER EQUIPMENT- E.I.L. Vibron 39A pH meters were used for the measurement of potential at high impedance, or as impedance transducers. Honeywell-Brown 10-mV strip-chart recorders with chart speeds of 1 to 360 in h-l, and a Hewlett-Packard Moseley 7035A X - Y recorder472 BISHOP AND HITCHCOCK: MASS AND CHARGE TRANSFER KINETICS [Analyst, Vol.98 b b Y Y Fig. 3. The seven-amplifier ramping potentiostat : RG, Philbrick P65AU integrating operational amplifier ; B, Philbrick P66A current booster amplifiers; VF,, voltage follower on the working electrode; XX, output to X-axis amplifier of recorder: YY, output to Y-axis ampli- fier of recorder; C, integrating capacitor, 10 x 1 p F rf 10 per cent. 160-V d.c. polyester capa- citors, Mullard Series C296 AA/A ; El, starting potential signal generator, 5.4-V Mallory cell or 2.0-V accumulator; E, supply battery for ramp generator, as El; Rf, feedback resistor of CA = R, = 1 to 100 ksZ in five steps; R,, 100 to 1 M a resistors in seven steps; R,, integrating resistor, 100 ksZ to 100 MsZ in decades; R4, calibrated current-measuring resistor, 10, 100 or 1000 Q; other components as in Fig.2 of sensitivity 1 mV to 10 V in-l, input impedance lo4 to lo8 IR, depending on range, and a writing area of 10 by 7 inches were used. The operational amplifiers in the present phase of the work were as shown in Table I. TABLE I OPERATIONAL AMPLIFIERS Type D.c. open loop gain Philbrick P65AU 2 2 x 104 differential chopper-stabilised chopper-stabilised current booster Philbrick SP456 2 108 Solartron AA1023 2 108 Philbrick P66A - output Drift &ll V a t 50 pV d-l 12.2 mA 30 pV 'C-l f l O V a t 1 pV week-' &lo0 V at &12mA &12 V a t - A100 mA f 2 0 mA 0.1 p v "C-1 20 pV d-l Power supplies included either a Philbrick PU3, 3 1 2 to 15 V at 1 A per line, or lead - acid accumulators, k 1 4 V and 80 A h, and transformer-supplied 6.3 and 115 V, 50 Hz for the SP456 amplifiers.A Solartron AS 141 1 constant-current source2* was used for integrator calibration. The high-precision p~tentiometer~~ was a Cropico P3, and Cropico RS 1 resistance standards were used. The stroboscope was an EM1 type 6, and the thermoregulator a B.T.L. C' ircon. VOLTAMMETRIC PROCEDURE- The voltammetric cell was initially thoroughly leached so as to condition it, cleaned with chromic acid, very thoroughly washed and again conditioned with pure electrolyteJuly, 19731 AND COULOMETRIC CURRENT EFFICIENCIES. PART V 473 solution. The supporting electrolyte was placed in the cell and de-aerated for a t least 20 minutes.During this period, the working electrode was given the appropriate pre- treatment so as to “clean” or “activate” it. The Luggin capillary end of the salt bridge was filled with cell electrolyte by application of suction to the Y-piece shown in Fig. 1. The ramping potentiostat was run in to thermal equilibrium with the dummy load resistors switched in, and the integrator re-set switch closed. The balance of the differential amplifiers was checked and adjusted if necessary by means of the built-in pre-set. At the end of the de- aeration, the working and auxiliary electrodes were positioned with care so as to ensure constant geometry. The electrodes were then connected to the potentiostat, the starting potential was pre-set and the cell switched into circuit.After holding the starting potential for 15 s, the shorting switch on the integrator was opened and the Mallory cell switched into its input. The required portion of the current - voltage curve was recorded on the X - Y recorder. The scan was then repeated after addition of the de-aerated sample. If required, the working electrode was pre-treated before each scan. RESULTS OF PERFORMANCE TESTS- The characteristics of the cell and the method of stirring were investigated by scanning a M solution of copper(I1) chloride in 1.0 M potassium chloride adjusted to pH 2.0 with hydrochloric acid, because this sample gives well defined voltammetric waves at platinum electrodes, and the thermal diffusion coefficient of copper(I1) ion in this medium has been determined.25 At first, the PTFE-covered follower alone was used for stirring, but gave an uneven effect and fluctuations on the current axis of the voltammogram.When fitted into the Perspex disc, much smoother stirring was achieved. This disc was used thereafter unless the electrolyte was such as to attack Perspex. In 7 and 10 M sulphuric acid, the follower alone proved satisfactory because the high viscosity of such solutions smoothed out the stirring to acceptable noise levels. The limiting current at constant stirring speed varied with the positioning of the working electrode, and it was therefore necessary to ensure constancy of geometrical placing of the various appurtenances that dipped into the solution. The orienta- tion of the working electrode with respect to the flow of the solution was adjusted to that giving the maximum limiting current, which proved to be the condition with the exposed face of the electrode at right-angles to the direction of flow of the solution.The speed of rotation of the stirrer disc was measured with a stroboscope. The effect of stirring speed on the limiting current for the reduction of copper(I1) was investigated. Assuming that r and tx in equation (1) are constant, a value of 6x was calculated with the aid of the reported value of Dx.25 The relationship between 6% and stirring speed is shown in Fig. 4. Whatever interpretation may be placed on ax, its value approaches a minimum of about low3 cm at high stirring speeds, and is nearly constant at speeds in excess of about 6 Hz. This is fortunate in that it permits an ordinary stirrer motor to be 0.008 - 0.006 - E 1 cg % 0.004 - 0.002 - 0 1 2 3 4 5 6 7 8 9 Speed of rotation of stirrer/Hz Fig.4. Effect of stirring speed on the apparent thickness of the diffusion layer. Reduction of M copper(I1) in 1.0 M potassium chloride solution a t pH 2.0474 BISHOP AND HITCHCOCK run on an unstabilised mains supply at a sufficiently high speed without small fluctuations in speed having any material effect on the mass-transfer process. The dependence of the limiting current on the concentration of the electroactive species was examined for potassium hexacyanoferrate(II1) in 1.0 M hydrochloric acid. I t was found that the relationship was obeyed over the concentration range of indicates that a stable and reproducible mass-transfer condition is readily attainable. funds to provide a 3-year maintenance grant for this work. I L [Fe(CN)i-]1*00 i O * O o ~ to 6 x M hexacyanoferrate(III), which We are pleased to record our gratitude to Imperial Chemical Industries Limited for 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. REFERENCES Bishop, E., Analyst, 1972, 97, 761. -, Ibid., 1972, 97, 772. Levich, B., Russ. J . Phys. Chem., 1947, 21, 689. -, Ibid., 1948, 22, 575. -, Ibid., 1948, 22, 711. Bishop, E., Chemia Analit., 1972, 17, 511. Tafel, J., 2. phys. Chem., 1905, 50, 641. Allen, P. L., and Hickling, A,, Trans. Favaday SOL, 1957, 53, 1626 Lewartowicz, E., J. Chivn. Phys., 1952, 49, 557. Audubert, R., Ibid., 1924, 21, 351. -, Ibid., 1952, 49, 106. Lewartowicz, E., Ibid., 1952, 49, 551. -, Ibid., 1952, 49, 573. -, Ibid., 1954, 51, 267. -, C . R. Hebd. Acad. Sci., Pavis, 1954, 238, 1580. -, Ibid., 1948, 22, 721. -, Ibid., 1952, 49, 565. -, Ibid., 1954, 238, 1812. -, Ibid., 1959, 248, 2996. -, CITCE Colzf., Lindau, 1955, 144. -, CITCE Conf., Paris, 1957, 267. Schwarz, M. W., and Shain, I., A.Pzalyt. Chem., 1963, 35, 1770. Bishop, E., and Short, G. D., Analyst, 1962, 87, 467. Bishop, E., and Riley, M., Ibid., 1973, 98, 305. Peters, D. G., and Cruser, S. A., J . Electroanalyt. Chem., 1965, 9, 27. NOTE-References 1, 2 and 7 are to Parts 111, IV and I of this series, respectively. Received December 29tlt, 1972 Accepted Mavch 20th, 1973