Equations of the polarographic waves of simple or complexed metal ions

ELECTROANALYTICAL CHEMISTRY AND INTERFACIAL ELECTROCHEMISTRY Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

457

EQUATIONS OF THE POLAROGRAPHIC WAVES OF SIMPLE OR COMPLEXED METAL IONS 1. THE METAL ION IS R E D U C E D W I T H A M A L G A M F O R M A T I O N A N D THE L I G A N D IS A NON-HYDROLYSABLE SUBSTANCE

M I H A I L E. MACOVSCHI

Institute of Physical Chemistry of Roumanian Academy of Sciences, Bucharest (Roumania) (Received May 4th, 1967; in revised form, July 21st, 1967)

LINGANE 1 has shown that during the polarography of a solution of the complex, MXq, with reversible reduction of the metal, the free ligand concentration at the dropping electrode surface must generally be different from that in the bulk solution,

i.e.: C° = Cx + ACx = Cx + (i/kx) q

(1)

LINGANE avoided the calculation of the concentration excess, ACx, by using the simplifying clause that there is a sufficiently high ligand excess; therefore, c ° ~ Cx

(2)

Using the same simplification, which we shall call "Lingane's approximation", DEFORD ANDHUMEZfurther developed the theory, solving the problem of the complex series, and later, SCHAAP AND MCMASTERS3 studied the mixed complex. When the paper of LAITINENet al.*, dealing with the effect of solution pH is also considered, it can be seen that for Lingane's approximation, the problem of the polarography of the complexes has been practically solved. But in the determination of stability constants there are many cases when only a slight excess of ligand concentration can be used; in such cases the theory developed on Lingane's approximation is insufficient. Among the methods for fitting the calculation to the cases of small ligand excess, the work of RINGBOMAND ERIKSON5 is of importance. The limits of application of Lingane's approximation have been indicated by BUTLER AND KAYE 6. They showed, theoretically and experimentally, that when only the complex exists in solution, the potential at the dropping electrode surface is a function of ln{i~+l/(id-- i)}, compared with ln{i/(i a - i)} for the usual case. Since an equation for the polarographic waves of the complexes which for the condition Cxtot = 0 leads to the classical equation of the polarographic wave of the metal ion does not yet exist, the theory in its present stage does not explain completely the polarographic behaviour of the complexed ions. J. Electroanal. Chem., 16 (1968) 457-47 o

458

~. r. MACOVSCrn

We have, therefore, tried to find a general equation of the reversible polarographic waves that will represent (as nearly as possible) the real polarogmphic behaviour of the complexed metal or simple (non-complexed) ions. The present work is the first stage in this investigation and deals with the case when only the complex MXa* exists in solution--the metal ion is reduced with amalgam formation and the ligand is a non-hydrolysable compound. For this case, the following special equations can be found in the literature: (a) The classical equation 7 for polarographic waves of the simple metal ions:

RT

L

-n~ In in- i

(3)

(b) LINGANE'S equation t for the case where the complexing agent is present in large excess: RTln E=e + ~

R T In t"t°t it" RT i q ~ff~'X JX -- ~ - In ia - i

flqfMXqkam

faro kMxq

(4)

where flq is the dissociation constant of the complex MX~ (see eqn. (8)). (c) BUTLER AND KAYE'S equation 6 for the case where there is no excess of complexing agent:

R T l n flqfMxqkam E=8 + nF famkMxq RT 1 q n kx

-q ~

RT afHX fx q-~ffln C n f H f x + a f n x

R T l n iq+l "7 nF tj-l

(5)

where a is the dissociation constant of the acid, HX, that forms the complex** and CH is the concentration of H + ions in solution; the other symbols have their usual significance. (d) BOCK'S equation a for a large excess of complexing agent where the complex is of low stability***:

RTln flqfuxq RT F~= ~ + ~ T ---Yam - q ~ In Cx fx

-

RTln nF

(1 +

flqf~x~ 7 - - q --

] + Rrlni"-i

Cx f¢, f~/

nF

z

(6)

* It must be emphasized that MXa is not the highest complex that could appear, but the simple

complex present in the solution under the given conditions; in other words, q is a parameter the value of which depends on the conditions imposed on the system (0 ~
459

EQUATIONS OF METAL ION WAVES. I

The constant, e, is related to the normal amalgam potential of the metal considered, E°m, by the equation:

= E°m + (RT/nF) In aug

(7)

The following symbols are used in the text (for the sake of simplicity, the charges are not included): C° - - concentration of metal ions at the electrode surface, C° - - concentration of complex-forming ion at the electrode surface, C MXa O - - concentration of the metal complex at the electrode surface, C~° - - concentration of amalgam at the drop surface; CM - - concentration of metal ions in solution, Cx ~ concentration of the ligand in solution, CMX~ - - concentration of the metal complex in solution, C~°t - - total concentration (analytical) of the ligand in solution, C~ t - - total concentration of metal in solution, fM, fx, fMX~,f~m - - the corresponding activity coefficients, kM, kx, kMxjk~m - - the corresponding Ilkovi~ constants. The dissociation constant of the complex, obviously the same in solution as at the electrode surface, is: /~q=(Cx

o o fx) q C~ fM/CMx ~fMxq = (Cxo fx) ~ CM fM/CMx, fMXq

(8)

I. EQUATIONOF THE POLAROGRAPHICWAVE

1. Derivation of equation The potential at the dropping electrode is given by the thermodynamic relation: E = e - ( R T / n F ) In (Cam o f~m/CM o fM)

(9)

To obtain the equation of the polarographic wave, that is E = E(i), the concentrations must be expressed in terms of current. The relationship between cathodic current and internal amalgam surface is:

i=k.~ C.°

(10)

or 0

Cam ~

I/kam

(:i)

Combining eqn. (11) with eqn. (9), gives:

E = e - (RT/nF) In (i fam[k~ fM C°)

(12)

The current is carried, on the solution side of the interface, by both the free and complexed metal ions; furthermore, the total current, i, is the sum of two partial currents: iM due to the metal ions and iMxq due to the complexed metal ions:

i=iM+iMX ~

(13)

j. Electroanal. Chem. 16 (1968) 457-47 °

400

M . E . MACOVSCHI

If migration does not exist and if the diffusion layer is sufficiently thin so that the concentration gradient is constant, then:

iM = kM(C M- C °) •

/MXq

----kMxq(CMxq - -

(14) 0

(15)

CMXq)

At the diffusion limit, eqns. (14) and (15) become: idM= kM CM

(16)

iMx,d = kMx, CMXq

(17)

resulting in: o id-- i = k MC ° + kMxqCMxq

(18)

If CMx° is expressed in terms of C ° and flq: 0

0

q

id-- i = k MC ° + kMx, CM(Cx fx) fM/flq fMX~

(19)

The concentration, Cx °, of the ligand at the electrode surface is equal to the concentration of the ligand in the bulk solution plus the variation of the concentration due to the release of the ligand out of the complex at the moment of metal ion reduction: C ° = Cx + ACx

(20)

Since the complex has the formula MXq, it is obvious that (see Fig. 1): ACx=q(iMxq/kx)

Mercury

Drop surface

Electrode surface zone

zone

CQrn:O

o Cam

c~

Solution

c~

c°MXq

CM

.

k× kMXq

CM×q

o

kM

+ o= Jr

Cx

#q

#q o ~

(21)

~ I I

Fig. 1. Distribution of concns, and diffusion of ions in the neighbourhood of the dropping electrode. (O), metal ion M; (O), ligand X; (QOO), complex MXq.

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EQUATIONS OF METAL ION WAVES. I

46X

From eqns. (20), (21), (13) and (14): C ° = Cx + q{i - kM(CM- C°)}/kx

(22)

The value of Cx ° may be introduced into eqn. (19), and suitable rearrangement gives: id - i = kM C ° + flq fMX~ku

(23)

an equation which relates CM° to the current. The concentration of metal ions and of complexing agent can be obtained from the total concentration by the following method: Cu = C ~ ' - C u x ~ Cx c,tot "~x - qCMx~

(24)

Cu c~, = ~ C,~xofuxq/ fMf~ If we note that: y = kM C °

(25)

Q=(kMxqfM/fMx kM) (qfxl kx) q

(26)

and

then relation (23) can be written in terms of y: Q---(kx C x - - k M C M + i + y

+y+i--io=O

(27)

and C ° will be given by: f, tot, fM, fx, fUX¢ kM, kx, k M x ) / k u C ° = q~(id,i, flq, q, C tuo t, "-'x

(28)

where ~0 is the solution to equation ~(y)= 0. The value of CM° obtained in this way may now be introduced into eqn. (12), and taking into account the half-wave potential relation of the simple metal ion: E~ = e - (RT/nF) In (faro kM/ fM kam),

(29)

the required equation of the polarographic reversible waves for simple and complexed metal ions is obtained: E = E ~ - ( R T / n F ) I n (i/q0

(30)

2. Function qb(r) has a single real solution To obtain results with physico-chemical meaning, the function #(r) must have only a single real solution. j . Electroanal. Chem., 16 (1968) 4-57-47 °

402

M.E. MACOVSCHI

Demonstration (a) 4)(r) has at the most only a single real solution. Function #(y) is defined in the domain y = 0, y = k MCM. O(y)'S derivative is:

dy

flq

(31)

" [~-Cx-kMCM+i+(q+I)y] + I Since i+y >/kMCM (i >~iM)and Cx, flq and Q cannot be negative, the derivative does not have real solutions, i.e., the function may have at most only one real solution. (b) The real solution, qL does exist. The values of the function O(y) at the end of the interval are: ~(o1 = i - ida< 0

(32)

~)(kMCM)= kM CM ~q (~-- Cx + i)q-}-kM CM-I-i-- id

(33)

Introducing into eqn. (33) the expressions of flq and Q from eqns. (8) and (20, and writing the parenthesis in the form ((kx/q)Cx} q+ iPq_1 results in:

Q ~(kMCM)= kMX~CMX~+ kM CM ~ iPq_ 1 + kM CM+ = kM CM ~ iPq_ 1 + i >10

i

--

id

(34)

The sign of the function Cry) at the ends of the interval varies with the parameter, i: i=0 0< i < ki= id

id

#(o)< 0 ~(0) < 0 q3(o)=O

(~(kMCM)= 0 ¢(kMCM)> 0 (~(kMCM)> 0

(35)

It is obvious that within the definition domain, ~(y) changes its sign, therefore the real solution cp exists.

3. A possibility of simplification in solving equation ~(y~=O with numerical coefficients If we use the notation:

3 = (kx/q) C x - kM CM

(36)

~(y) can be written:

4~(y)=y (O/flq) (/~+ i +y)q + y + i - id = 0

(37)

The change of variable:

z=~÷i+y j. Electroanal.Chem., 16 (1968) 457-47°

(38)

EQUATIONS OF METAL ION WAVES. I

463

and rearrangement of terms gives: -(~+i)z q + ~-z

fl~(6+id) = 0 Q

(39)

For q >2, the function hv(z) is simplified; it has only four terms instead of q + 2 for function ~(y). C ° is related to the solution ~ of function 7s(z) by the expression: (40)

C ° = (~, - t5 - i)/kM

Introducing C ° from eqn. (40) into eqn. (12) and writing: *d = ~-- ~

(41)

we obtain: E = E ~ - ( R T / n F ) In {i/(i* - i)}

(42)

Thus, the variation of the potential of the dropping electrode in terms of current has the same form as the case of the simple metal ion, with the difference that instead of id there appears id*. This can formally be considered as a corrected diffusion current. II. SPECIAL FORMS OBTAINED FROM THE GENERAL EQUATION WITH DIFFERENT LIMITING CONDITIONS

For simplification, in the following section, the notation: (43)

0 = k___~xC x _ k M ( C M _ C O ) + i

q

will be used. 1. The case o f the simple m e t a l ion

The limiting conditions: there is no ligand in solution and thus no complex is formed, i.e., ctot = 0 X

(44)

Taking into account eqns. (24), (25), (13-15), gives: ( k x / q ) C x - kM Cu + i + k u C ° = 0

(45)

q~= id - i

(46)

and subsequently:

E=E~-(RT/nF) In {i/(id--i)}

(47)

which is the equation of the polarographic wave for simple metal ions (eqn (3)). J. Electroanal. Chem., r6 (z968) 457-47 °

4o4

M . E . MACOVSCHI

2. Lingane case The limiting conditions are: the ligand is in large excess (Cx = Cxt°t) and the current transport to the dropping electrode is due to the metal ions from the complex (flq-low*), i.e.,

{

O=(kx/q)C~ t

y,~(Q/flq)Oqy

(48)

tp will be ~o = ( i a - i)flq fMx~ kM kMx~fM(fX ct~t) q

(49)

and RT flqfMXqJCM E=E~ + ~ffln

RT RT, i q-n-if-In c~t fx - ~ - m i d - - i

fMkMx~

(50)

and taking in account the expression for E~ (eqn. (29)) R T l n flqfMx, kam E = e + nF famkMx~

RT RTln i q nff -ln C~t fx -- nF ia--i

(51)

which is the Lingane equation (eqn. (4)). 3. Butler and Kaye's case The limiting conditions are: there is no ligand excess in solution (Cx=0) and the current transport to the dropping electrode is accomplished by the metal ions from the complex (to obtain these conditions it is also necessary that CM= 0 and flq should be low), i.e., O=i Y ~ (Q/fl~)Oqy

(52)

The resulting value for ~0 is given by: ¢p= (id-- i)flq f MXqkM k~

(53)

iqqqkMx,fMf~ and for E:

RTln ~qf~x. kM

E=Et~ + nF

kMX~fM

RTIn fxq q nF kx

R~I n iq+l ia - i

(54)

or:

RT flq fMX~ kam E = e + -ffff-In YamkMx~

-- q

RT q ~ff- In kx

RT q ~-_~In fx

(55)

R T l n i q+l nF id - i

* fla is considered low if y can be ignored in comparison with (Q/flq)Oay; otherwise flq is considered high. It must be emphasized that a general numerical value which would separate flq-low from fl~-high cannot be given, because QOq is not a constant.

j . Electroanal. Chem., 16 (1968) 457-47 °

EQUATIONS OF METAL ION WAVES. I

465

The difference between eqns. (55) and (5), is that the term eqn. (55) instead of the term

RT q -ffff In fx

q(RT/nF) In fx appears in

o~fi-ix Cnfnfx + ~fnx

in eqn. (5), owing to the fact that BUTLER AND KAYE considered the ligand a weak acid dissociated according to the equilibrium: HX # H + X

(56)

with the constant c~, and only the ionized form X active as complexing agent. The present work does not consider such an equilibrium.

4. Buck's case The limiting conditions are: the ligand is present in large excess (Cx = C~°t) and the complex has low stability (flq is large), i.e.,

( O=(kx/q)C~t

(57)

y cannot be neglected in comparison with

(Q/flq)Oqy

The value for ~p will be: tp =

(id -- i)flqfMX~kM flqfMX,lkM+ fMkMxqf•( Cx,or),~

(58)

and for E:

E=E~ -- RT

-n-ff in

[ kMx, fMf~ fMXq kM

(c~t) q ] RT i .. flq + 1 - ~ In id-- i

(59)

Equation (6) given by BUCK is valid when all the diffusion coefficients are equal. For various diffusion coefficients, BUCK obtained the relation:

RT

E~ lxq = e + - ~

fMXqflq

RT

In faro f ~ -- q ~

In C x (60)

RT ~--~ln kix" { 1 + fam ~

k i f i x , flq kMX, fMC~f~ ]

which by taking into account eqn. (29), can be rearranged and rewritten EMX,_E M

- ½ -~

RTln / kMXqfMf~ C~¢ ) \ kMfMxq -- fl---~+ 1

(61)

i.e., relation (59) in which i = ½id. When Cx is very large or fl~ decreases, term 1 from the parenthesis may be neglected and eqn. (59) becomes Lingane's equation. Equation (59) can be transformed into the polarographic equation of the

j. Electroanal.Chem., 16 (1968)

457-47 °

466

M.E. MACOVSCHI

simple metal ion by neglecting the term kMxqfMfxq Cxq/kMfMxqflq in the presence of term 1. As there are two parameters, Cx and flq, two possibilities will occur: (a) flq~ oo. The supplementary condition flq~oo is not in contradiction with the fundamental condition,/?q-high, for which eqn. (59) was inferred. Therefore, this method of obtaining the simple metal ion equation has a real physico-chemical significance. (b) Cx decreases to O. For Cx decreasing, i - kM(C M -- CM 0) from eqn. (43) cannot be neglected in comparison with (kx/q)C x, and thus eqn. (59) inferred for the case when (kx/q)Cx ~>i-kM(C M-- CM°) no longer has a real meaning. Consequently, BucK's affirmation that eqn. (59) is a general equation is not valid; as it has been shown, eqn. (59) is a limiting case.

5. The case of a slight ligand excess, the complex being stable (not yet described in the literature) The limiting conditions are: the excess of the ligand is small (Cx is low) and the transport of the current to the dropping electrode is due to the metal ions from the complex (flq-low), i.e.,

0 = (kx/q)Cx+ i Y ~ (Q/flq)Oqy

(62)

This leads to: ¢P =

(id-- i)flqfMX, kM

(63)

kMxJM ( q k ~ f (-~--Cx+i) ~ and

RT

flqfMx~ki

RTln fxq kx

E = E ~ t +-ffff-ln kMx~fM -- q nF

RTln i ( ~ - C x + i f nF id - i

(64)

or:

RTln flqfMx~kM E=Er~ + nF kMx.f~ R T ln nF

Cx

id - i

RTln qfx q nF kx +

1 l(q - 1) !

(65)

Cx

id- i

q-2 i3 iq+l] q! {kxc x + + + 2 !(q - 2) l \ q id- i "'" i d - iJ According to whether the values of Cx are very high or equal to zero, the parenthesis will reduce to the first or the last term, and the cases of LINGANE, and BUTLER AND KAYE, respectively, will result. For Cx-low, the terms have values comparable with each other and must all be taken into account. J. Electroanal. Chem., i6 (1968) 457-47 °

EQUATIONS OF METAL ION WAVES. I

467

6. There is no other limiting case

If the following notation is used: A = (kx/q)C x B = - k~(CM - c °)

(66)

C=i

then: (67)

O=A+B+C

The special forms of the derived equations depend on the flq-value and the form of the expression for 0, by neglecting different terms. All the forms that 0 can take and their meaning when fl~ is high or low are as follows: 1.0 = A. flq-Iow--LINGANE'S case--II.2, flq-high--BucK's case--II.4. 2. 0 = B. Impossible, independent of flq-value: C cannot be neglected in the presence of B because always C/> [BI (i 1>iM). 3. O=C. flq-Iow--BUTLER AND KAYE'S case--II.3, flq-high--impossible: 0 = C would mean that in the solution there exists only a non-dissociated complex; but as flq is high, this situation is impossible. 4. 0 = A + B. Impossible, independent of flq-value: see point 2. 5. O= A + C. flq-low--case II.5. flq-high--impossible: O= A + C means i >/iu, i.e., the whole of the metal ion is complexed but in the given conditions (flq-high and Cx-low so that C cannot be neglected in comparison with A) this situation is impossible. = 0 - - t hecase of simple metal i o n - - I I . l . - - C ~ ° ~ q C i tot--flqlow--the general case: although Cx ~-,0, still 0 = 0 and then equation in y is of q + 1 grade.--flq-high--impossible: if flq is high and ,,,tot •--x # 0, then A cannot [be neglected the presence of B + C. 7. 0 = A + B + C.--the general case. Therefore, the five limiting cases presented (four from the literature and one inferred in this paper) are the only limiting equations that exist, Figure 2 shows schematically, the limiting equations field of applicability. 6. O = B + C . - - C x

tot

III. DISCUSSION

The inferred equation (of the polarographic waves for simple and complexed metal ions when the metal ion is reduced with amalgam formation and the ligand is a non-hydrolysable substance) involves all the limiting equations reported in the literature. They are: the classical equation for the simple metal ion; the equation deduced by LINGANE;BUTLER AND KAYE'S equation and that deduced by BUCK. Another limiting equation not so far presented in the literature has been inferred from the same general equation. It refers to the case of a slight ligand excess. This last limiting equation is of special interest because it is available from the domain of low ligand concentrations. The inferred equation includes all ligand concentrations and allows the disj . Eleetroanal. Chem., 16 (1968) 4 5 7 - 4 7 °

468

M.E. MACOVSCHI The field of single

-Emet°l.1.-ion e q u a t i o r J

~q high

The field of equation

ffq low

-If .4.-

ffq ~ o----~ Y

c~°t = cx

c~,°t

cl°'=q.c~ °,

cI=°o the field

Butler end

of single

equation's

metal ion -If .1.-

Koye field

-if.3.-

Fig. 2. Schematic representation of the field of applicability of the limiting equations. C ~ t, satd. ligand concn. For the non-hatched area, only the general equation may be applied, i.e., the general equation has no limiting form and must be applied as such.

/'/-//-S

a

b

c

d

e

Fig. 3. The theoretical shape of the polaro~'aphJc wave, vs. the ligand concm in soln. (a), C~t=0; (b), 0 < C~tot < qCM, tot. (C), C~tot= q C M tot. tot
cussion of the shape of the polarographic waves for C~°t varying from zero to C~"t (Fig. 3). When CJ~°t =0, the potential at the dropping electrode is dependent upon i/(id-i) and the polarographic wave is classical in shape; the slope of the wave depends on the metal ion charge. For C~°t qC~°lt (BUTLERAND KAYE) the potential at the dropping electrode depends on iq+l/(id - i) and the slope of the polarographic wave decreases with the increasing value of q. For a large ligand excess, i.e., CJc°t~Cx, the initial relation E = E{i/(i d- i)} is found (LINGANE). Consequently, in the concentration range O
J. Electroanal. Chem., 16 (1968) 457-47 °

EQUATIONS

OF METALION WAVES. I

469

electrode will be a function of a sum of terms with/-powers from 1 to q + 1,

q

i.e.,

iJ+l

E=E j~=oAJid_i Therefore, with increasing ~xr~t°t,the polarographic waves slope continuously, reach a maximum slope for C~°t=qC~lt and then flatten out. The relation

q

iJ+l

E=E j~=oAJia_i for the domain of concentration qC~lt< C~(°t<(C~°t~CX)is clearly shown by eqn. (65). For the domain 0 < C~°t< qC~.lt this relation is expressed implicitly in eqns. (27) and (30). It is evident from the above discussion that small quantities of substances present as impurities and able to complex metal ions can strongly interfere with the shape of the polarographic wave, especially at small metal ion concentrations*. Therefore, the estimation of the number of electrons changed at the electrode or the test of reversibility can lead to erroneous results if the reagents used as supporting electrolyte are not of the highest purity. SUMMARY

In the polarographic determination of a solution containing the complex, MX~, with dissociation constant flq, the total current, i, is a sum of two currents: iM due to the simple metal ions and iMxq due to the complexed metal ions. Since, on reducing the metal ion the complex is broken down, the free ligand concentration at the dropping electrode is: C ° = C x + qiMxq/kx, Cx being the concentration of free ligand in the bulk solution. These considerations lead to the following equation of the polarographic waves:

E= E ~ - (RT/NF)

In

(i/q~)

being the solution of the equation:

1 kMX. f M ( ~ x X ) q ( ~ C x _ k M C M + i + y ) q + y + i _ i d = ~ " = Y flq fMX~kM

0

where CM is the concentration of the metallic ion in the bulk solution, f the activity coefficients and k the respective Ilkovi~ constants. It has been demonstrated that the equation has only one real solution. * The method of DETRICK AND FREDERICK9 gives polarographic waves for metal ion concentrations of 107-109 M.

J. Electroanal. Chem., 16

(1968) 457-47o

470

M.E. MACOVSCHI

All the known special equations appear as limiting conditions; i.e., the equation of (i) the simple metal ion; (ii) LINGANE; (iii) BUCK; and (iv) BUTLER AND KAYE, as well as another equation for the case of slight ligand excess, not dealt with in the literature. It has been demonstrated also that no other limiting equation of any physico-chemical meaning can be deduced. The relation between the polarographic wave shape and the ligand concentration is also discussed. REFERENCES 1 2 3 4 5 6 7 8 9

J. J. LINGANE, Chem. Rev., 29 (1941) I. D. D. DEFORD AND N. D. HUME, J. Am. Chem. Soc., 73 (1951) 5321. W. B. SCHAAP AND n . L. McMASTERS, J. Am. Chem. Sot., 83 (1961) 4699. H. A. LAITINEN, E. I. ONSTOTT, J. C. BAILAR, JR. AND S. SWANN JR., J. Am. Chem. Soe., 71 (1949) 1550. P. KIVALO AND J. RASTAS, Suomen Kemistilehti, B 30 (1957) 128. C. G. BUTLER AND R. C. KAV~, J. EleetroanaL Chem., 8 (1964) 463. J. HEYROVSK'~AND J. K6~rA, Tratat de Polarografie, Academy R.P.R., 1959, p. 118. R. P. BucK, J. Eleetroanal. Chem., 5 (1963) 295. F. DETgICK AND M. FREDERICK, Anal. Chem., 36 (1964) 1143.

j . Electroanal. Chem., 16 (1968) 457-47 °