Measurement of Half-Wave Potentials with Cylindrical Electrodes

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Measurement of Half-Wave Potentials with Cylindrical Electrodes EVAN MORGAN,' J. E. HARRARI2 and A. L. CRITTENDEN Department o f Chemistry, University o f Washington, Seattle 5, Wash.

b Cylindrical microelectrodes have been useful in voltammetry for the measurement of oxidation potentials where mass transfer is diffusion-controlled. Current-potential relationships are only slightly different from those obtained assuming linear diffusion a t small times of electrolysis,

T

use of statioilary, solid microelectrodes for voltammetry in quiet, unstirred solution offers the advantage that the mass transfer process is simple and may be attributed solely to diffusion, provided that certain experimental conditions are met. The concentration throughout the cell must be uniform a t the start of electrolysis. Electrolysis time must not be much greater than 10 seconds, if convective processes are to be avoided. Cylindrical electrodes, usually in the form of vertically mounted wires, are more convenient than planar electrodes, as planar electrodes must be shielded to avoid edge effects. Such shielding makes restoration of the original concentration a t the surface awkward. With cylindrical electrodes, a short translation of the electrode through the solution, as by mechanical tapping, adequately restores uniform concentration. This is easily done after each observation of current flow. Currents flowing a t cylindrical electrodes under conditions of diffusioncontrol decrease rapidly with time. It is more convenient and accurate to measure total charge transferred during a fixed increment of time after start of electrolysis than to measure instantaneous currents as functions of time using an oscillograph. Charge transferred is readily measured using an electronic current integrator. A similar method using disk electrodes has been used by Hush (5). Charges transferred at stationary wire microelectrodes in the diffusion current region of potential correspond closely to those calculated assuming that the mass transfer process is strictly diffusion-controlled (8). The assumption of linear diffusion often does

not permit satisfactory calculation of limiting currents and the cylindrical nature of the diffusion must be taken into account. The cylindrical diffusion pattern is of less importance in the measurement of half-wave potentials, the deviations from linear diffusion seldom causing errors larger than 1 or 2 mv.

HE

1 Present address, International Business Machines Corp., Poughkeepsie, N. Y. * Present address, University of California Radiation Laboratory, Livermore, Calif.

756

0

ANALYTICAL CHEMISTRY

MATHEMATICAL RELATIONSHIPS

For the case \There 1 mole of an oxidized species, 0, reacts reversibly a t the surface to form 1 mole of a reduced species, R, with transfer of n equivalents of charge, both species being soluble in the solution, the system may be described by Fick's second law of diffusion, written in cylindrical coordinates and considering concentrations to be functions of radius and time only. Using notations similar to that of Delahay (4),let the initial concentrations in bulk solution be C i and C i . At the surface in the region of the rising part of = e, where e the wave Co(ro.t)/C~(ro,t) is a constant a t constant potential. If diffusion is linear, Co(r0,t) and C+,t) are independent of time, but this is not the case with cylindrical diffusion. The conservation of mass a t the surface provides an additional boundary condition

(1)

where ro is the radius of the electrode. Let t 2 = Do/DR, v = C{ - Co(r,t), and w = C R ( r , t ) - CR. By substituting these variables into Fick's second law in cylindrical coordinates and following the method of Carslaw and Jaeger (S), except for the boundary conditions, one obtains via the Laplace transform, subsidiary equations having Bessel's functions as solutions. The particular Bessel's functions are chosen using Equation 1. Then they are expanded and the series is inverted and differentiated with respect to 7, term by term, to give as the flux of substance 0 at the surface for small values of $0:

where oo is the quantity Dotl r;. The first term in the series corresponds to linear diffusion. The second term may be obtained by assuming that Co(r0,t) and C ~ ( r ~ ,are t ) independent of time. The third term is not obtained correctly using this assumption. Use of this idea will be made below. If one lets io be the limiting cathodic current when e = 0, and i, be the limiting anodic current when 1'8 = 0, one obtains the ratio

+ ..

*

(3)

+

Because the potential E = E" RT/nF In e, the current-potent,ial relation is given by E =Eo

RT ie - - i + RT - In - + - l n nF z 2nF -to

DR Do

The first three terms are the same as those given by Delahay for linear diffusion. The last term contains time, but this term is usually rather small. For the measurement of charge, Equation 4 must be integrated over the time interval used. If charge is measured from a fixed time tl after start of electrolysis to a time tz, where the increment of time is small compared to tl, using the approximation A(t112) = 1/2 t l i 2 At, and using a power expansion of the logarithm of the last term, one obtains

+ R T G t (Dif2- DL") + . . .

(5)

where t is the average value of time, or better, the square of the average root. As is customary, one may plot In (qc q ) / ( q - q.) us. E , taking the potential where the logarithmic term is zero as the half-wave potential, El/*. The relation between the formal oxidation potential

E" and Ellz can be obtained from the remaining terms in Equation 5. The last term indicates a dependence of E I , ~ upon integration time, but the effect is usually small, as D E seldom differs widely from Do. In the more general case where x moles of an oxidized species 0 reacts a t the surface to produce y moles of a reduced species R with transfer of m equivalents of charge, the boundary conditions are more difficult. The problem is simplified if it is assumed that the concentrations a t the surface are constant a t any finite time. As far as diffusion control is involved, this is justified by the observation that in the case where x = y, no error is introduced by this assumption in the first two terms of the resultant series, and the second term is small. By use of suitable changes of variables, the solution given by Carslaw and Jaeger (3) becomes, for integrations over At where At < ti: 4 =

[Cz - Co(ro,t)]mFADo4t xro

-

+ + ... 1

(6)

(

+ + '/2

Fe(CN)B-4in L O M KC1 F e ( c x ) ~ in -~ 0.1M KC1 Fe(CN)6-g in 0.25M KC1 Fe( C?j)B-' in 1 . O M KC1

0.1

0.2

3.2

0.063 0.062

f O . 183

+ O . 183

3.0

0.80 0.80

0.7 12.0

0.8 12.8

0.062 0.060

0.201

0.201

0.80 0.80

0.1

0.2

3.2

0.060 0.061

0.231 0.230

0.2330 0,233"

0.80 0.80

0.7 12.0

0.8 12.8

0.061

0.058

0.178 0.181

0.183 0.183

0.80 0.80

0.1 3.0

0.2 3.2

0.067 0.060

0.190 0.195

0,201

0.80 0.80

0.7 12.0

0.8

0.060 0.058

0.229

12.8

0.231

0.233" 0.2330

0.7 12.0

0.8 12.8 0.2

3.2 0.8 12.8

0.033 0.030 0.035 0.033 0.034 0.032

0.378 0.380 0.374 0.381 0.382 0.379

0.381 0.381 0.381 0.381 0.381 0.381

0.035 0.032 0,030

0.836 0.844 0.844

0.845 0.845 0.845

0.030 0.030 0.031

0.853 0.850

0,845 0.845 0.845

I- in 0.05M HzSO, 0.25 0.80 0.50 0.80

3 .O

0.1 3 .O

1.oo 0.80

12.0

0.25

0.7

1.00 Br- in 0.05M 0.25 HzS0,

3.0

0.8 1.6 3.2

0.7 1.4 3.0

0.8 1.6 3.2

0.50 1 .oo

[Ci - C~(ro,t)]mFADdt Yro I(dR)-"'

F~(CN)P,-~in 0.25M KCl

0.80 0.80

0.50

{(T+O)-~/~

Y =

Reactant Fe(c?rT)~-~ in 0.1M KC1

Table I. Observed Formal Oxidation Potentials Concn., tl, E", E", t2r Lit. Calcd. Sec. Slope mM See.

, ,

.

1

(7)

From these may be obtained expressions for surface concentrations in terms of As E = E' the charges q, qc, and 4. RT/mF In ( C O ) ' / ( C R ) ~E, may be plotted us. In (so - q ) " / ( q - pq)y, but the potential a t which this quantity becomes zero depends strongly upon time of electrolysis and the radius of the electrode. It seems convenient to convert the term under the logarithm to a dimensionless quantity by multiplying by qc or q. raised to an appropriate power. Thus

+

0.7

1.4

0.200

0.849

0.183 0.201

0.201

Diffusion coefficients: ??e(Cx)~-'= 6.4 X 10-6 sq. cm. aec.-l(g); Fe(cx)~-~ = 7.6 x 10-4 sq. cm. see.-' (9); I- = 1.99 X 10-6 sq. cm. sec.-l(1); 1 2 = 1.11 x 10-6 sq. em. sec.-1 (1); Br- = 2.12 X 10-6 sq. em. sec-1, Br2 = 1.47 X 10-6 sq. em. see.-', both calculated from diffusion charges. Extrapolated value (6).

mation; the time dependence indicated by the last term in Equations 8 and 9 is usually small. However, such potentials offer little advantage and it seems preferable to report the formal oxidation potential, E", calculated directly from Equations 8 and 9. These equations are seen to reduce to Equation 5 whenx = y. EXPERIMENTAL

Charges transferred a t stationary platinum wires were measured using previously described apparatus (d), except that a student-type potentiometer was connected across the electrolysis cell to permit precise measurement of the applied potential. The electrode was 0.80 mm. in diameter and 5.4 mm. long. It was cleaned periodically in chromic acid solution. (9)

where t is the average time as before. Also, one may plot E us. the first logarithmic term in Equations 8 and 9, taking the potential a t which the first logarithmic term is zero as a "wave potential," not the same as the halfwave potential. Such a wave potential is independent of time to a first approxi-

0.182

RESULTS

The above equations were applied to several reasonably rapid electrode reactions. Residual charges measured in the absence of reactant studied were usually small and diffusion charges were corrected for these residual charges. In each case, the appropriate logarithmic

term involving charge transferred was plotted us. applied potential. I n all cases, straight lines were obtained, provided that the term under the logarithm was not less than 0.1 or greater than 10. Formal potentials were calculated from the point where the logarithmic term became zero. Calculated potentials referred to saturated calomel are given in Table I together with the slopes of the logarithmic plots. Formal potentials for the oxidation of ferrocyanide and fcr the reduction of ferricyanide in potassium chloride solutions agree well with those reported by Kolthoff and Tomsicek (6) and by Hush (6). The potential varies with salt concentration as previously reported. Some slight irreversibility is indicated by the difference between the potentials calculated from oxidation and from reduction processes, particularly a t low salt concentrations and a t shorter times of electrolysis where curente are larger. The oxidation of iodide to iodine in 0.05M sulfuric acid is complicated by the formation of triiodide ion, but the formation constant is not large and at concentrations on the order of 1 mM, very little change in current is VOL. 32, NO. 7, JUNE 1960

0

757

caused by triiodide formation (1). Potentials calculated assuming that iodine is the product of oxidation agree well with those calculated from the data of Latimer ( 7 ) . The oxidation of bromide or reduction of bromine in 0.05M sulfuric acid is similar to the iodide-iodine case except that, as found by Lingane and Anson (8), the reactions are much less reversible. The reactions a t stationary electrodes do give straight lines on plots corresponding to Equations 8 and 9 and the slopes are reasonable. However, a definite discrepancy exists betnTeen formal potentials calculated for oxidation and those calculated for reduction, again most noticeable a t short electrolysis times. I n the cases discussed here, the last

term in Equations 8 and 9 was never greater than 1 mv.; thus the convergent nature of the diffusion makes very little difference. If the logarithmic charge term is not made dimensionless as mentioned above, other terms appear in the charge-potential relationship in cases where 5 # y. These terms contain D, TO, and t to a more important extent and make it necessary to know these parameters more closely. Making the logarithmic charge term dimensionless has the disadvantage of introducing a concentration term, but this datum is . often readily available.

(2) Booman, G. L., Morgan, E., Crittenden, A. L., J . Am. Chem. SOC.78, 5533 (1956). (3) Carslaw, H. S., Jaeger, J. C., “Conduction of Heat in Solids,” p. 280, Oxford Press, London, 1948. (4) Delahay, P., [(New Instrumental

Methods in Electrochemistry,” Chap.

3, Interscience, New York, 1954. (5) Hush, N. S., 2. Elektrochem. 61, 738 (1957). \ - - - . I -

(6) Kolthoff, I. M., Tomsicek, W. J., J . Phys. Chem. 39,945 (1935). (7) Latimer, W. M., “Oxidation Poten-

tials,

2nd ed., Prentice-Hall, New

York, 1952. (8) Lingane, J. J., Anson, F. C., ANAL. CHEM.28,1871 (1956). (9) Stackelberg, M. von, Pilgram, M., Toome, V., 2. Elektrochem. 57, 342 (1953).

LITERATURE CITED

(1) Beilby, A. L., Crittenden, A. L., J . Phys. Chem. 64, 177 (1960).

RECEIVED for review November 9, 1959. Accepted March 7, 1960.

Si muIta neo us PoIa rogra phic Dete r mina ti o n of Lead a nd Azide Ions of Lead Azides in Aqueous Media JAMES 1. BRYANT and MARYLAND D.

KEMP

0.S. Army Engineer Research and Development laboratories, Fori Belvoir, Va. b A polarographic method has been developed for the simultaneous determination of lead and azide ions of lead azide in aqueous solutions. Its sensitivity allows safe low concentrations of lead and azide ions to b e effectively determined in a fraction of the time required by classical analytical methods. The polarographic method was standardized b y gravimetric determinations.

A

in the study of the properties of azides is the lack of an accurate and convenient method for the analysis of solutions of the more unstable salts and hydrazoic acid. Gravimetric and volumetric methods are time-consuming and generally dangerous, and often give low results (1). Gas volumetric methods do not give accurate results when very dilute solutions or very small samples, which are required for purposes of safety, are to be analyzed. Haul and Scholz (3) made polarographic studies of the azide ion of lead azide and reported the wave height to be concentrationto 10-3N. dependent from 4 X Later Vasiliev (7) determined the lead content of lead azide polarographically and calculated the azide concentration, assuming a stoichiometric relationship. The purpose of this study was to develop an effective and convenient MAJOR DIFFICULTY

758

ANALYTICAL CHEMISTRY

method for routine determination of lead and azide content of lead azide simultaneously with the polarograph. The purity of samples used was checked by gravimetric determinations. I n 1934 Revenda (6) showed that in the presence of depolarizers which have a great affinity for mercury ions, anodic depolarization at the dropping mercury electrode gives well defined polarographic waves. Later Kolthoff and Miller (4) extended this investigation in a more analytical manner and reported that within certain concentration ranges the polarographic step was concentration-dependent. These investigators worked with chloride, bromide, and iodide solutions among others, and found that upon the electro-oxidation of mercury in the presence of these anions the metal formed a difficultly soluble mercurous salt a t the surface of the drop. Because of their halogenoid character, similar electrode reactions should be expected for the azides. This expected similarity has been found to exist. Thus, if this property of an electro-oxidized dropping mercury electrode is utilized here, the mercury goes into solution as mercurous ion, making possible the formation of a precipitate of difficultly soluble mercurous azide adsorbed on the surface of the mercury drops. Because of the resultant depolarization, an anodic current develops and its magnitude is proportional to the

rate of diffusion of the azide ion to the electrode surface, which in turn is dependent upon the difference in the azide ion concentration in the bulk solution and a t the mercury drop solution interface. The limiting current is reached when the azide ion concentration a t the mercury drop surface becomes effectively zero. Then id =

KCNS-

(1)

where C N ~ is- the concentration of azide ion in the bulk solution. The equations for the electrode reaction for the azide ion are Hg+Hg+ Hg”

+ ?Js-

+

E

+ HgN;

(2)

(3)

and the net electrode reaction is Hg

+ N a - 4 HgNa +

(4)

The cathodic reduction of the lead ion is of general familiarity and is not discussed here. EXPERIMENTAL

A 10-3N stock solution of lead azide was prepared by dissolving 0.2914 gram of lead azide in O.1N potassium nitrate and diluting to 2 liters. The lead azide used was precipitated by mixing equal volumes of soIutions of 2N doubly recrystallized sodium azide and 1N certified lead nitrate. The precipitate was successively washed with distilled water, alcohol, and ether and