£1ectroanalytical Chemistry and lnter)dcial Electrochemistry Elsevier Sequoia S.A., Lausanne Printed in The Netherlands

447

L O G A R I T H M I C ANALYSIS OF TWO OVERLAPPING P OLAR OGRAPHIC WAVES

IV. QUASIREVERSIBLE ELECTRODE PROCESSES

D.C.

IVICA RUZI(~

Center for Marine Research, Institute "Rudjer Bogkovi(", Zagreb, Croatia (Yugoslavia)

(Received 18th June 1971 ; in revised form 1 l th November 1971)

INTRODUCTION

By logarithmic analysis of a d.c. polarogram composed of two overlapping waves, it is possible to obtain the electrochemical parameters (half-wave potential as well as the number of electrons involved in the electrode reaction) for each of the components. The methods for separation of two overlapping reversible or totally irreversible d.c. polarographic waves 1"2, and the d.c. polarogram of a multistep elec- trode process 3 have already been described. In these cases an exact mathematical analysis was performed. The logarithmic curve originally obtained has an inflection point and two linear parts, from which the limiting currents ratio m = idl/id2 of two overlapping waves can be determined by elimination of xl and x2 from the following equations :

Xl. 2 = (x• - x i ) x y ( x , - (1) and

x* = (rex, + x 2 ) / ( m + 1), x* = (m + l )x 1Xz/(,nx 2 + x , ) (2)

x* and x* correspond to the linear parts of the logarithmic curve, xi to the inflection point, and Xl and x 2 to the separated waves (see Fig. 1).

For Xl >>x2 eqns. (1) and (2) give:

m = x )xy(x - (3)

and for x* ~> xi ~> x* it follows that :

m = x i (4)

When the value of m is known, it is possible to calculate x 1 and x 2 from eqns. (2). If the condition Xl ~> x2 is fulfilled, eqns. (2) become:

x * = m X a / ( m + l ) and x * = ( m + 1)x z (5)

In the case when one of the two linear parts intersects the composite logarithmic curve inside the limits - 2 < log x < 2 (where x is the antilogarithm of the composite loga- rithmic curve, i.e. x = i/(i d - i)), it is necessary to apply a more complicated procedure 1. It is useful to see what kind of the procedure must be applied for the analysis of two overlapping d.c. polarographic waves where at lea;t one of them is quasireversible.

J. Electroanal. Chem., 36 (1972)

448 I. RUT.IC

tog (a2)' (11

log x i

x

Fig. 1. Logarithmic analysis of two (reversible or totally irreversible) overlapping d.c. polarographic waves. The inflection point is determined by ( ...... ) approximate, ( . . . . . ) iterative methods.

ANALYSIS OF TWO OVERLAPPING QUASIREVERSIBLE WAVES

For the case of a single quasireversible wave the following equat ion can be written :

1/x = (id-- i)/i = exp [ (nF/R T)(E- E~)] + 1.13/k~ (z/D) ~ • exp [(anF/RT)(E- E~)] (6)

If nF/RT and cmF/RT are denoted as a and b, and in the same way exp (a E~) and ks/1.13 (D/r)~'exp(bE~) as A and B, eqn. (6) becomes:

1Ix = exp (aE)/A + exp (bE)/B = 1Iv + 1/w (7)

Fo r the sum of two waves, x can be written as 1 :

[/'~/)1 W1/(U1 ~- W1)] [1 "-~/2 2 Wg/(U 2 ~- Wg)] Jr- [1)2 W2/(U2 "~ W2)] [1 "J-/]1 W1/(/]I -t- W1) ] (~)

llzl[1 Jr- 1)2W2/(U 2 ~- W2) ] "-~ 1 -[- U 1WI/(V 1 -[- W1)

At sufficiently positive potentials for which the condit ions v I ~ Wl ~ 1 and u 2 ~ w 2 ~ 1 are fulfilled, eqn. (8) becomes:

X -=-- X]' = (mY 1 +/92)/(m--[- 1) (9)

Analogously, at sufficiently negative potentials, supposing el ~> wl >> 1 and v2 ~> w2 ~> 1,

J. Electroanal. Chem., 36 (1972)

ANALYSIS OF OVERLAPPING D.C. WAVES. IV. 449

eqn. (8) becomes:

x = x * = (m+ 1)Wl w21(mw2 -t- w,) (10) In the special case when for all potentials Vl >> vz and wl >> w2, eqns. (9) and (10) yield :

x* = mvl/(m + 1) = [mA 1/(m + 1)] exp ( - a 1E) = A~ exp ( - a 1 E) (11) and

x*= (m+ 1)w2= [(m+l)B2] exp( -bzE)=B * exp ( - b2E) (12)

For the inflection point one can write:

, 2 2 2 w, w 2 v , + b # a 2 " z t v l v : + ( a : - Q f l a ] ' x t z * (13)

X i vlw2(bl- 1)2(1/x~ + i/z*)

It is impossible to eliminate z~, z, , rE, and w 1 from this equation, and using the procedure described earlier only vl and w 2 can be determined (Fig. 2).

ANALYSIS OF OVERLAPPING QUASIREVERSIBLE AND REVERSIBLE (OR TOTALLY IRREVER- SIBLE) WAVES

i. The case when the quasireversible wave Jbl lows the reversible (or total ly irreversible) wave

In this case i n s t ead of eqn. (10) the fo l l owing exp re s s ion can be wr i t t en :

m x , [1 + v 2 w2/(v 2 + w2)] --}- v2 w2/(v2 + w2)" (1 + x , ) (14)

x = m [1 + v2 w2/(v2 Jr- w2)] -~- 1 + x,

X

/ t ' / / / , / / /

/ / / / / / / / v l , , / , v 2 / /

,,~¢ ./ ,, / / , / / / / / ,'/ / / / / ,,,,/ j < / / / / /

/;'.J / / /.,/.7 , , / /

/ , f / / , . , d ' / /

¢ .4" ! _ _ . " / . " / ./.1

/ / / / . / I i ' ~ ¢ / / l fS ~ POTENTIAL / / / , ,"

. . ,,7,," , / / / / ,Y / / / .11" / / / / Wl/ / # J I w / z

/ / / / " /

/ / / I o o 5 o v I

Fig. 2. Logarithmic analysis of two overlapping quasireversible d.c. polarographic waves. The inflection point is determined by: (1) eqn. (17), (2) the method described earlier 1.

J. Electroanal. Chem., 36 (1972)

450 I. RUZIC

At more positive potentials using x 1 ~ 1 and 132 ~ w 2,~ 1 equation (18) becomes :

x = x* = (mx x + v2)/(m+ 1) (lS)

and at more negative potentials (Xx >> 1 and v2 >>w2 >> 1):

X = X~ = (m--t- 1)X 1 w 2 / ( g n w 2 -~- w1) (16)

Specially when the condition v2 ~ Xl >> w2 is fulfilled eqns. (15) and (16) become (11) and (12) respectively. The inflection point can be determined from the following expression :

(m+l)x,/32+ 2 2 (a2/b2) rex, w, + [(b2 - a2)2/b 2] rex* X 2 "~

2 2 2 (w2a2/b2)(a, - 1) 2 [1/x + 1/(m + 1)/323 (17)

It is impossible to eliminate v 2 from this equation. By analogy with the case of two quasireversible waves, only x 1 and w 2 can be determined (see Fig. 3).

When the limiting current of the first wave is very small with respect to the second (quasireversible) wave, then u 2 can also be determined (see Fig. 4). The following procedure can be used : The irreversible part of the second wave is back extrapolated (dashed straight line) and subtracted 4 from the original curve (©). The resulting curve (A) corresponds to the reversible part of the second wave which is separated from the first by using the above mentioned procedure 1 (Xl, t'2). The irreversible part of the

X

/ / /

/ /

! I

I i i

! I

I

11 I

/// 2 ~

f /

V2// /

/ /

/ /

/

POTENTIAL

0.050 V b ~4

Fig. 3. Logarithmic analysis of reversible (or totally irreversible) and quasireversible overlapping d.c. polarographic waves using (1) eqn. (23), (2) the method described earlier 1.

J. Electroanal. Chem., 36 (1972)

ANALYSIS OF OVERLAPPING D.C. WAVES. IV, 451

x

o~

+ I - -

0 - -

--I --

XI

150mY I

w2

POTENTIAL Fig. 4. Logarithmic analysis of two overlapping d.c. polarographic waves, when the first of them is very small. (O) Original logarithmic curve, (A) resulting curve after separation of the irreversible part of the second wave.

second wave is corrected (w2) similarly. The sum of l / v 2 and 1/W 2 gives the complete logarithmic curve of the separated second quasireversible wave. The method has been applied 5 for the analysis of the polarogram of the system U(VI) L iOH-H202 . For this system three waves are obtained ; the second of them is a catalytic wave which can never be obtained alone. In analysing the polarograms of various H z O 2 concentra- tions the reversible and irreversible parts of the second wave can be obtained separately, and a complete second quasireversible wave has been constructed.

ii. The case when the quasireversible wave precedes the reversible (or totally irreversible) wat, e

Instead of eqn. (8) we have"

.~ = [ D I U I W I / ( U I -}- WI) ] (1 -]-X2)-~-X 2 [ ] -~-UIWI/(U 1 -~- WI) ] (18) m(1 + x2) + 1 + v, w, / (v 1 + Wl)

At sufficiently positive potentials for which the conditions v a < w 1 < 1 and x2 < 1 are fulfilled, eqn. (18) can be written as '

X = X~ = (/nU 1 ~- X2)/(DI-~ 1) (19)

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452 I. RUZIC

At more negative potentials using v~ >>w~ >> 1 and x 2 >> 1 eqn. (18) results in:

X = X~ = ( In "1- 1) ~/91 X2/ (mX 2 Jr- W 1) (20 )

In this case the inflection point is given as:

(m+ 1)x 'w, 2 Jr- (b 1 / a ~ ) V l X 2 -~- [ ( a 2 - ba)2/a 2] x* x 2 ~ ' ' - (21)

v , { ( a , - 1)2/x * + (a2/a~) [ (a z - 1)2/(m + 1 ) w t ] }

It is impossible to eliminate w~ from eqn. (21). By an earlier procedure I it is possible to determine only vl and x 2 (see Fig. 5).

I / /

/ x ,

/ ,/U

/ / .// /

/ // / / / / / //~.//,

/ / / Z / / ~ //wl

/ / / /

/ / / / / /

' z/ / / /

/" f I

Yl/// //Xl~ / / / / / !

/ / .

&050 V b - i

POTENTIAL

Fig. 5. Logarithmic analysis of quasireversible and reversible (or totally irreversible) overlapping d.c. polaro- graphic waves using (1) eqn. (29), (2) method described earlier 1.

When the limiting current of the second wave is very small with respect to the first (quasireversible) wave, then wl can also be determined (see Fig. 6). The following procedure can be used: First, the theoretical slope of the reversible part is assumed and then the corresponding straight line (l/v1) is drawn so close to the original loga- rithmic curve that after subtraction of the reversible part from the composite curve, the best straight line results (in the potential range where the reversible part of the first wave still influences the composite logarithmic curve). The resulting straight line corresponds to the irreversible part of the first wave (1/w~), and by using the earlier procedure a it can be separated from the second wave. Finally the whole first quasi- reversible wave can be constructed as the sum of 1Iv 1 and 1/w~.

J. Electroanal, Chem., 36 (1972)

453

+1

0

X

-1

150 mV I I

Wl / /

¢

0

ANALYSIS OF OVERLAPPING D.C. WAVES. IV.

P O T E N T I A L

Fig. 6. Logarithmic analysis of two overlapping d.c. polarographic waves, when the second of them is very small. (©) Original logarithmic curve, (A) resulting curve after separation of the reversible part of the first wave.

SEPARATION OF CATHODIC ANODIC QUASIREVERSIBLE WAVE

The equation for the cathodic quasireversible single wave is given by relation (6). The anodic wave is defined by:

x = (id-- i)/i

= exp [ - (nF/RT)(E-Er)] + 1.13/k~(z/D)+'exp [ - ( f lnF/RT)(E-E~)] (22) from which

x = A exp ( - aE) + B a e x p ( - baE) = v + Wa (23)

For the sum of the cathodic and anodic waves it follows that :

x = m (v + %) [ 1 + VWc/(V + we) ] + [vwc/(v + wc)] (1 + v + %) (24) m [1 + VWc/(V+wo)] + 1 +v+w

At sufficiently positive potentials supposing 1 >> Wa >> Wc >> V eqn. (24) yields :

x=x*=mWa/(m+ l)= [mBa/(m+ l)] exp( -baE)=B* e x p ( - b a E ) (25)

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454 I. Ru~i6

Analogously, for sufficiently negative potentials (because 1 ~ w~ ~ w a ~ V ) :

x = x* = (m + 1)w~ = (m + 1)Be exp ( - b~E) = B~ exp ( - beE) (26)

IfE~ and E 2 are the potentials at which x* and x~' are equal to 1, then it follows that"

E~ = (1/ba)In B* = (1/ba)In B a + (1/ha)In m/(m + 1) (27) and

E2= (1/bE)In B~ = (1/b~)In Be+ (1/b~)In (m+ 1) (28)

from which

Er = baE 1 + b ~ E z - l n m = b'aE ~ + b ' c E z - l o g m (29) - baWb~ b'a+b'¢

and

or

bc In m ln[ks/l '13(D/r)~] - b~ bab~+ b~ ( E 2 - E x ) - I n ( m + 1) + ba+b ~ (30)

log [kJl .13 (D/T) ~'] -- b'ab'¢ (E 2 _ El) - l o g (m + 1) + b'c log m_m b',+b'c b',+b'e

where b'~ = b~ log e, and b'c = b E log e When the cathodic and anodic parts of the cathodic-anodic wave are close

to each other, from the logarithmic analysis it is possible to calculate the reversible half-wave potential (E~) and the rate constant of the electrode reaction at the reversible half-wave potential (ks) by means of relations (29) and (30). Figures 7 and 8 present such a cathodic-anodic wave and its analysis. Limitations of this method are similar to those previously given for the analysis of two reversible or totally irreversible waves 1.

The reversible part of the wave can be constructed as the straight line which in- tersects the abscissa at the reversible half-wave potential; it has a slope of(~ + fl) nF/R T

0.050 V Eo E~z2 I I

I I I 1 i idQ

I

3

i (%

i . . . . .

J , 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

El/2

Fig. 7. Anodic-cathodic quasireversible d.c. polarographic wave.

J. Electroanal. Chem., 36 (1972)

ANALYSIS OF OVERLAPPING D.C. WAVES. IV.

X

+I

0

Eo+IO0 Eo+50 Eo

Fig. 8. Logarithmic analysis of the polarogram from Fig. 7.

/ /~ /~( / / / / / j i/IV ~ ~ I I I I

s." l I I i i - !

Q

y'l i # i i l l~ ! j ) , . / / / / ~/z/ J / /:/: i

##1 i I i i / i

/ ..'_17 ,' > Y !

I i L

Eo-50 Eo-IO0

455

(for the sum o f , and fl equal to 1). The composite logarithmic curve of the anodic and the cathodic waves can be obtained as the sum of v and Wa for the anodic wave, and as the sum of 1/v and 1~we for the cathodic wave.

DISCUSSION

By using the logarithmic analysis of the overlapping quasireversible waves, it is not possible to perform any completely exact mathematical treatment. Application of the procedure described 1 for two reversible or totally irreversible waves offers the possibility of determining the reversible part of the first quasireversible wave (vl) as well as the irreversible part of the second quasireversible wave (w2). In Figs. 2, 3, and 5 such an analysis is presented. In order to test the suitability of the earlier procedure for the analysis of overlapping d.c. polarographic waves it is compared with the semi-exact relations for the inflection point, (13), (17) and (21). The procedure is not exact for quasireversible waves and its accuracy is extremely dependent on the separation be- tween the half-wave potentials, and on the slopes of the waves. This procedure can be applied with good accuracy only in the case when the conditions va >> v2 and w~ >> w2 are fulfilled. By using the logarithmic analysis of d.c. polarograms it is not always possible to see if one or both of the waves are quasireversible.

If one of the two overlapping d.c. polarographic waves is very small, the other quasireversible wave can be completely constructed (see Figs. 4 and 6). In the case when the first of the two overlapping d.c. waves is quasireversible, the slope of the second wave determined from the composite logarithmic curve very often differs from the exact value. This discrepancy is due to the influence of the irreversible part of the first wave on that slope. This effect is evident only when the irreversible part of the first wave intersects the composite logarithmic curve in the potential range +2 > log x > - 2. The earlier work a describes a method by means of which the inaccurate slope

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456 I. RU2Ad

can be corrected but it cannot be applied in the case of overlapping quasireversible waves. For illustration, the straight lines of the separated second wave (x2) are presented in Fig. 5 ; one of them is obtained by using the method proposed earlier 1 (dashed line) and the correct one (full line) calculated by using the expressions evaluated in the present paper.

The analysis of the anodic-cathodic quasireversible wave is also presented. Using this analysis both transfer coefficients (~ and fl), reversible half-wave potential (E~), and the rate constant of the electrode reaction at the reversible half-wave po- tential (ks) can be evaluated. The anodic and the cathodic waves can be completely constructed as a combination of the corresponding reversible and irreversible parts.

ACKNOWLEDGEMENT

The author is grateful to Dr. M. Branica for helpful discussions.

SUMMARY

The application of logarithmic analysis of two quasireversible overlapping d.c. polarographic waves is discussed. In this case, only the reversible part of the first quasireversible wave and the irreversible part of the second quasireversible wave can usually be separated from the original experimental logarithmic curve. One of the quasireversible waves can be completely constructed only in the case when the other is relatively small. The logarithmic analysis can also be applied to the anodic-cathodic quasireversible wave and a method for the determination of both transfer coefficients, reversible half-wave potential, and the corresponding rate constant of the electrode reaction is proposed.

REFERENCES

1 I. RU~I(AND M. BRANICA, J. Electroanal. Chem., 22 (1969) 243. 2 I. R~J£Id, J. Electroanal. Chem., 29 (1971) 440. 3 I. Ru~It~, J. Electroanal. Chem., 25 (1970) 144. 4 I. Ru£1d, A. BARIC AND M. BRANICA, J. Electroanal. Chem., 29 (1971 ) 411. 5 V. ZUTId, Ph .D. Thesis, University of Zagreb, 1970.

J. Electroanal. Chem.. 36 (1972)