Flow coulometry on a porous electrode under condition of limiting current

Electroanalytical Chemistry and lnterfacial Electrochemistry Elsevier Sequoia S.A., Lausanne

51

Printed in The Netherlands

FLOW COULOMETRY ON A POROUS ELECTRODE UNDER CONDITION OF L I M I T I N G C U R R E N T

R. E. SIODA* AND T. KAMBARA

Department of Chem&try, Faculty of Science, Hokkaido University, Sapporo (Japan) (Received 1st December 1971)

The application of flow electrolysis on porous electrodes for coulometry has been proposed by several authors I -4. This new coulometric techniqUe seems to offer advantages over older coulometric methods as the time of the measurement can be significantly shortened, and interference of slow, secondary electrochemical reactions can be eliminated. The importance of the speed of a coulometric determination was first considered by Eckfeldt s, Bard 6 and Johansson 7, who devised fast coulometric cells. The role of secondary reactions in coulometric determinations has been theoretically considered by Geske and Bard s, and Bard and Mayell 9. In the present paper we describe a theoretical model which allows the performance of a flow coulometric cell employing a porous electrode working under condition of a limiting current to be compared with that of a traditional, non-flow cell. THEORY

In a batch-type electrolytic cell, the electrolytic current of a single electrode reaction at a potential corresponding to a mass transfer control of the overall reaction rate, decays exponentially with time: I = I o exp ( - p t )

(1)

where I is the electrolytic current at time t, I o the "initial current" extrapolated to t = 0, and p a "cell factor" dependent on : the electrode area, the volume of the cell, the rate of mixing of the solution, and the diffusion coefficient of the electroactive species2,6,~ o. The decay of the current with time corresponds to an exponential decay of the concentration of the electrolyzed species. The higher the cell factor, the faster is the electrolysis. Specially built fast electrolytic cells 6'7 had p of the order of 0.1 s-1. A cell employing a porous electrode had p even higher, namely of the order of 1 s- ~, as measured from the decay of the electrolytic current with time 2'3. The application of eqn. (1) and the calculation of p for a flow electrolytic cell employing a porous electrode is not straightforward. In a typical flow electrolytic experiment, the current of electrolysis does not decay with time, as a fresh solution of the electroactive species is constantly supplied. Hence eqn. (1) cannot be generally * On leave from Institute of Physical Chemistry, Polish Academy of Sciences, Warszawa, ul. Kasprzaka 44, Poland,

J. Electroanal. Chem., 38 (1972)

52

R . E . SIODA, T. K A M B A R A

applied to the total current of electrolysis. It can, however, be applied to the distribution of the current density along the porous electrode. Let us imagine a porous electrode along the length of which, parallel to the x-axis, the solution flows with a uniform specific flow rate: (2)

q=v/a

where q is the specific flow rate, v the volume flow rate, and a the cross-sectional area of the porous electrode. The average interstitial flow velocity is equal to:

(3)

w=q/

where e is the porosity of the porous electrode, corresponding to interconnected, open to flow pores. We assume that the flow of the solution through the porous electrode is described by the "plug flow" approximation, meaning that mixing of the solution in the porous electrode is not present, and consequently the elements of volume of the solution travel with a uniform flow velocity on parallel paths. The solution contains, besides the electroactive species, a supporting electrolyte, so that the migration current can be neglected. We apply to the porous electrode such a potential that it works at the limiting current, reducing (dr oxidising) the electroactive species. The conditions which are necessary for obtaining a limiting current have been described before 11 14 Les us imagine a small element of volume of the solution, which entered the porous electrode at time t = 0. The element moves along the electrode, and at time t it will be at a distance: (4)

x = wt = qt/e

if x = 0 corresponds to the beginning of the porous electrode. During the movement of the element of volume of the solution, the concentration of the electroactive species in it will change, due to the electrolysis. It has already been calculated la that at a distance x from the beginning of the porous electrode the concentration will be equal to :

c = Co exp (--jsq ~

1 x)

(5)

where Co is the initial concentration of the solution before entering the porous electrode, j a mass transfer constant, s the specific internal surface of the porous electrode, and c~a constant having a value between 0 and 1. On substitution of x from eqn. (4) we obtain : c = c o exp ( - j s q ~ e-1 t)

(6)

Equation (6) describes the concentration at time t of the element of volume of the solution which entered the electrode at t = 0, and travels along it. As has been postulated before 11, the local limiting current density in the porous electrode can be represented by the following equation: iI=jsnFcq

~

(7)

where i I is the local limiting current density, n the number of electrons transferred pe~ molecule of the electroactive species, and F the Faraday. On substitution of c from eqn. (6) to eqn. (7) we obtain: J. Electroanal. Chem., 38 (1972)

F L O W COULOMETRY ON POROUS ELECTRODE i 1~- i 0 e x p

( - - j s q % - I t)

53 (8)

where (9)

i°=jsnFcoq ~

corresponds to the limiting current density at x = 0. Equation (8) describes the limiting current density at an element of the internal surface of the porous electrode facing the element of volume of the solution which entered the porous electrode at t - 0 . By equating eqns. (1) and (8) we obtain an expression for a "cell factor" of a flow electrolytic cell with a porous electrode working under condition of a limiting current: p =jsq~ e - 1

(1 O)

The cell factor p is higher, the higher is the specific internal surface of the porous electrode and the specific flow rate, and the smaller the porosity. The derivation of eqn. (10) has been done on the assumption that the role of the tortuosity of the porous electrode can be neglected. RESULTS A N D DISCUSSION

As an example of the application of eqn. (10), values ofp have been calculated using the experimental data of one of the authors14. The flow electrolytic cell employed had a porous electrode composed of 12 parallel 80-mesh platinum grids of crosssectional area 0.90 cm 2. The value of the product of the constantsjs has been determined experimentally as 0.85, and e was found experimentally to be 0.373. The porous electrode had a porosity of 0.76. For this electrode and for the range of specific flow rates employed, 0.0086-0.97 cm s -a, the cell factor p calculated according to eqn. (10) ranges between 0.19 and 1.11 s -1. In the application of flow electrolysis for coulometry it is of importance to predict the length of the porous electrode needed for obtaining the given degree of conversion of the flowing electroactive species. Let R denote the limiting---obtained under condition of a limiting c u r r e n t ~ e g r e e of conversion in a flow electrolytic cell: (1 l)

R = 1 - %/%

where cL is the outlet concentration of the electroactive species at the end of the porous electrode of length L. The length of the porous electrode needed for achieving this degree of conversion can be calculated using eqns. (5) and (11) as: L = --In (1--R)q'-~/js

(12)

The average time spent by an element of volume of the solution in the porous electrode (transit time) we shall denote by z. It can be calculated using eqns. (4), (10) and (12) as: z=Le/q=p-l

ln(1 - R )

(13)

As an example of the application of eqns. (12) and (13), L and z have been calculated for R = 0.99 and 0.95 based on the values o f j s and ~ given above. The results are presented in Table 1. From eqns. (12) and (13) and from Table 1 it can be seen that to achieve the given value of the limiting degree of conversion, elg. R = 0.99, although the necessary J. Electroanal. Chem., 38

(1972)

54 TABLE

R.E.

S I O D A , T. K A M B A R A

1

CALCULATED LENGTHS OF A TYPICAL POROUS ELECTRODE AND TRANSIT TIMES OF SOLUTION CORRESPONDING TO LIMITING DEGREES OF CONVERSION R = 0.99 AND 0.95*

Specific flow rate q/cm s x

Length/cm

Transit time/s

/99

/95

1799

"~95

0.01 0.1 1.0

0.24 1.28 5.42

0.16 0.83 3.53

23.0 9.7 4.1

14.9 6,3 2.7

* T h e r e s p e c t i v e v a l u e s of t h e degree of c o n v e r s i o n a r e d e n o t e d as s u b s c r i p t s 99 a n d 95 to L a n d z

length of the electrode increases with increasing specific flow rate, the time spent by an element of volume of the solution in the porous electrode decreases with increasing specific flow rate. Hence, if during the coulometric measurement some secondary electrochemical reactions (e.g. of the e.c.e, mechanism ) occur, influencing the value of n determined, it is advisable to work with higher specific flow rates, as the time of contact of the solution with the electrode will be shorter, and the role of the secondary electrochemical reactions can be diminished. We have tried below to evaluate approximately the maximal "permissible" value of the chemical rate constant in an e.c.e, mechanism 15, so that only the first electron transfer can be observed by the flow coulometric method with a limiting current. The evaluation is made on the assumption that only the chemical reaction occurring in the bulk of the solution is of significance, and that the role of the chemical reaction taking place in the diffusion layer can be neglected. Let us suppose that the chemical reaction is a first-order one with respect to the product of the first electron transfer:

dcs/dt=k(c o - c )

(14)

where cs is the local concentration of the chemical reaction product. Substitution for c from eqn. (6), use of eqn. (10), and integration in the limits from 0 to t leads to:

c~= kc o { t - p - 1 [1 - exp ( - p t ) ] }

(15)

The concentration cs given in eqn. (15) is an "accumulated" concentration, which means that the consumption of the product of the chemical reaction by the second electron transfer is neglected. Substituting for t in eqn. (15) by z from eqn. (13) leads to :

c~=kco P-~ [ - R - l n

(l-R)]

(16)

It seems reasonable to assume that only the first electron transfer will be observed if the "accumulated" concentration of the product of the chemical reaction at the end of the porous electrode does not exceed 1% of the initial concentration c 0. Thus from eqn. (16) we obtain the following relation for the upper limit of the rate constant of the chemical reaction:

k< O . O l p / [ - R - l n (1 - R ) ]

(17)

The dependence of the value of the denominator in relation (17) on R is given in Fig. 1. J. Electroanal. Chem., 38 (1972)

55

FLOW COULOMETRY ON POROUS ELECTRODE

-R-In(I-R)

30

2,0

1.0

l

|

i

0.0

0.2

0.4

I

0.6

|

I

0.8

O0 1.0

R

Fig. 1. Dependence of the expression [ - R - I n ( 1 -

R)] in eqn. (16) and relation (17) on R.

Using relation (17) one can evaluate roughly the range (starting with k = 0) of the rate constant of the chemical reaction in an e.c.e, mechanism that should not influence the flow coulometric determination corresponding to the initial electron transfer. As can be seen from relation (17) and Fig. 1, the upper limit of this range is higher, the higher is p, i.e. the higher the specific flow rate, and the smaller the limiting degree of conversion. Further it is expected that the method of flow electrolysis on a porous electrode can be of advantage if applied to the external generation of reagents in coulometry. ACKNOWLEDGEMENT

One of us (R.E.S.) is grateful to the Japanese Society for the Promotion of Science for a scholarship which led to this work. SUMMARY

A theoretical model is described for coulometry employing a flow electrolytic cell with a porous electrode working under condition of a limiting current. The model allows the "cell factor" of a flow electrolytic cell to be calculated. The possibility of eliminating secondary electrochemical reactions in the flow coulometric determination is discussed. A relation is derived for the upper limit of the chemical rate constant in an e.c.e, mechanism, so that the second electron transfer will not be observed in the flow coulometric determination. J. Electroanal. Chem., 38 (1972)

56

R . E . SIODA, T. KAMBARA

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J . A . SrmOPSHIRE,J. Electroanal. Chon., 9 (1965) 90. T. FUJINAGA,Bunseki Kagaku, 17 (1968) 651 ; Pure Appl. Chem., 25 (1971) 709. S. OKAZAKI, Rev. Polarogr. (Kvoto), 15 (1968) 155. R. E. S~ODA,Eleetroehim. Acta, 13 (1968) 375. E. L. ECKFELDT, Anal. Chem., 31 (1959) 1453. A. J. BARD, Anal. Chem., 35 (1963) 1125. G. JOrIANSSON,Talanta, 12 (1965) 163. D. H. GESKEAND A. J. BARD, J. Phys. Chem., 63 (1959) 1057. A. J. BARD AND J. S. MAVELL,J. Phys. Chem., 66 (1962) 2173. J. J. LINGANE,Electroanalytical Chemistry, Interscience, London, 2nd ed., 1958, p. 224. R. E. S~ODA,Electrochim. Acta, 15 (1970) 783. R. E. StODA, Electrochim. Acta, 16 (1971) 1569. R. E. SIODA, J. ElectroanaL Chem., 34 (1972) 399. R. E. SIODA,J. Electroanal. Chem., 34 (1972) 411. R. N. ADAMS, Electrochemistry at SolMElectrodes, Marcel Dekker, New York, 1969, p. 247.

J. Electroanal. Chem., 38 (1972)