The application of linear sweep voltammetry to a rotating disk electrode for a first-order irreversible reaction

J. Electroanal. Chem., 124 ( 1 9 8 1 ) 9 5 - - 1 0 1

95

Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in T h e N e t h e r l a n d s

T H E APPLICATION O F L I N E A R S W E E P V O L T A M M E T R Y TO A R O T A T I N G DISK E L E C T R O D E F O R A F I R S T - O R D E R I R R E V E R S I B L E REACTION

P.C. A N D R I C A C O S a n d H.Y. C H E H

Department of Chemical Engineering and Applied Chemistry, Columbia University, New York, N Y 10027 (U.S.A.) (Received 6 t h J u l y 1 9 8 0 ; in revised f o r m 2 n d D e c e m b e r 1 9 8 0 )

ABSTRACT T h e r e s p o n s e of a first-order irreversible r e a c t i o n o c c u r r i n g o n a r o t a t i n g disk e l e c t r o d e t o a linear sweeping o f t h e p o t e n t i a l is a n a l y z e d w i t h t h e aid o f a N e r n s t d i f f u s i o n m o d e l . T h e peak c u r r e n t d e n s i t y n o r m a l i z e d w i t h r e s p e c t to t h e d i f f u s i o n - l i m i t i n g c u r r e n t d e n s i t y is s h o w n t o be a linear f u n c t i o n o f a d i m e n s i o n l e s s sweep rate, w h e n t h e l a t t e r assumes h i g h values. T h e slope d e p e n d s o n t h e t r a n s f e r c o e f f i c i e n t o f t h e r e a c t i o n . T h e p o s i t i o n of t h e p o t e n t i a l at w h i c h t h e c u r r e n t p e a k a p p e a r s is c a l c u l a t e d a n d s h o w n t o d e p e n d o n t h e s w e e p rate. A m o d e l is d e v e l o p e d for t h e d e t e r m i n a t i o n o f t h e t r a n s f e r c o e f f i c i e n t a n d t h e d i m e n sionless rate c o n s t a n t at t h e initial state.

INTRODUCTION

The study of transient techniques on a rotating disk electrode (RDE) has not been developed similarly to that existing for systems without convection. A probable cause is the complexity of the mathematical models which describe these techniques. In addition, resulting correlations to be actually e m p l o y e d for the analysis of experimental data are usually more complicated than those derived from stagnant system theory. However, certain properties of the RDE render it an attractive alternative for transient studies. These are primarily the suppression of natural convection effects, even at low values of the Reynolds number, and the attainment of a periodic state response to a periodic input. Previous analyses [ 1 ] as well as more recent developments [ 2,3] of the application of linear sweep voltammetry (LSV) to a RDE for simple reversible reactions have indicated the existence of relations between the dimensionless current ratio, ip/il, and a dimensionless sweep rate, a. Here, ip is the peak current density, il is the diffusion-limiting current density and o is defined by (1)

o = nFv82/RTD

where v is the sweep rate, D the diffusion coefficient of the reacting species and 8 the thickness of the diffusion layer; n, F, R, and T have their usual meaning. The values taken by o can be divided into three regions [2,3]. When o is less than approximately 3, the current density increases monotonically to il without peaks appearing at intermediate times. When o is high, ip/i~ is a linear function 0022-0728/81/0000--0000/$02.50,

© 1 9 8 1 , Elsevier S e q u o i a S.A.

96

of o i n . The slope depends only on the reaction mechanism and is independent of any physical property of the system. In the transition region, ip/il is a complicated function of o. At high sweep rates, the ip/il vs. o 1/2 correlations are identical to those predicted by stagnant system theory when expressed with dimensional quantities. In other words, the LSV--RDE correlations can be obtained from those predicted by the theory in stagnant systems upon normalization of the relevant variables with the appropriate characteristic quantities associated with the RDE. It should be noted that when a takes values lower than those for which the linear dependence of i,/il on a 1/2 holds, extraction of mechanistic information is difficult. This is so because either ip/il = 1 when o ~ 3, or ip/il does n o t exhibit a simple dependence on o when o is within the transition region. In general, the higher the slope of the ip/il vs. o 1/2 correlation, the smaller the transition region. Thus, for reversible depositions, the slope is 0.611 [2] and the transition region occurs at 2.7 ~ o < 4 [2]. For the Sevcik--Randles problem, the slope is 0.446 [1,3] and the transition region is at 3 < a < 10 [3]. In this paper, the response of a first-order irreversible reaction occurring on a RDE under LSV conditions is analyzed. Results conform with the description of the LSV--RDE technique presented above. The corresponding theory in stagnant systems has been developed by Delahay [4], Matsuda and Ayabe [ 5], Reinmuth [6,7], Gokhstein [8] and Nicholson and Shain [9]. Furthermore, the results of this work can also be applied to the analysis of thin-layer electrochemical cells [ 10], and anodic stripping voltammetry [ 1 1]. THEORETICAL

It is assumed that the kinetic parameter, k, of the irreversible reaction

0 + ne -* R

(2)

exhibits a Tafel dependence on the overpotential, 77:

k = ke e x p ( - - ~ n F ~ / R T )

(3)

where a is the cathodic transfer coefficient and ke is a characteristic of the equilibrium. Since the electrode potential, E(t), is varied linearly with time away from the initial value, El, the dependence of k on time for the cathodic reaction is given by

k(t) = ki e x p ( a n F v t / R T )

(4)

where k i = k e exp (--anF(E i -- Eo)/R T}

(5)

where E0 is the equilibrium potential. If E~ = E0, it follows that ki = k~. The boundary condition on the electrode surface dictates t h a t kinetic and diffusion currents must be equal. If the kinetics of reaction (2) is first order with respect to the surface concentration of species O, it follows that

i(t) = n F k ( t ) c ( O , t) = n F n ~ - (0, t) ox

(6)

97 where i(t) is the current density, c(x, t) the concentration of O as a function of the distance x away from the RDE surface and t h e time. The characteristic quantities of the system are the transfer coefficient, a, the initial rate constant, ki, of the reaction, the sweep rate, v, the bulk concentration, c b, of O, and the thickness of the diffusion layer, 5, derived by Levich [12] to be 6 = 1.612D 1/3 v 1/6 co-1/2

(7)

where v is the kinematic viscosity of the electrolyte and co the rotation speed of the RDE. The following dimensionless variables can then be introduced:

(8)

C(}, T) = c(}, T)/c b

(9)

= x/5

(10)

=Dr~62 Oir = a n F v S Z / R T D = ~ o

(11)

Ki = k i S / D

(12)

where C(~, T), ~ and T are dimensionless concentration, distance and time, respectively, Oir is a new dimensionless sweep rate incorporating the value of the transfer coefficient and simply related to o, and ~i is a dimensionless initial rate constant characteristic of first-order irreversible reactions. Under the assumptions of the Nernst diffusion model [2,3] and in terms of quantities introduced in eqns. (8)--(12), the boundary value problem describing LSV on a RDE for a first-order irreversible reaction can be formulated as follows: OC

02C

aT

0} z

(13)

C(}, O) = 1

(14)

C(1, T) = 1

(15)

OC -0} - ( 0 , ~) = f(T) = ~i exp(airT)C(0, r)

(16)

The RHS of eqn. (16) involves the product of two functions of time, the u n k n o w n being one of them. The difficulty of its analytical treatment was recognized by Delahay [4], who converted the corresponding problem for a stagnant system to an integral equation. A similar treatment is followed here. Using the Laplace transform m e t h o d , it can be shown that T

C(0, T) = 1 -- f f(},) 02(0, ~i(r -- h)) dh 0

(17)

where 02 is one of the theta functions defined by [13] j=oo

02(0, 7ri(r -- h)) = 2 ~

exp (--(2j -- 1) 2 7r2(T -- }Q/4}

j=l

and f(h) is defined by eqn. (16).

(18)

98 Equation (17) is a Volterra integral equation of the second kind [14]. By letting = oi~X

(19)

and = oi:

(20)

and using a transfbrmation property for 02 [ 15], we can rewrite eqn. (17) as follows: C(~) = I -- ~ir-'/2a~/2 f

exp(~')C(~')(@ -- ~.)-,/2

o

X 1 +2 ~

(--1)Jexp(--j2oir/(~--f))

d~"

(21)

.i=1

In terms of dimensional quantities, we see that (22)

Ki ~ - 1 / 2 Oi-rI/2 = k i ( T r f l D ) -1:2

where (23)

fl =- a n F v / R T

i.e. this quantity is independent of the rotation speed and is equal to the corresponding quantity in a stagnant system. Letting KiTr -1/2 a51/2 = exp(--a)

(24)

and ¢(~) = exp(~

-

-

(25)

a)C(~)

we can rewrite eqn. (21) as follows: 0 ( ~ ) = e x p ( ~ -- a) -- exp(~ -- a) f

{-

X 1 +2 ~

o

0(~)(ff -- ~.)-in

}

(--1) i e x p ( - - j 2 o i r / ( ~ -- ~')) d~"

/'=1

(26)

By comparison one sees that if the series is eliminated from the expression inside the integral, the resulting kernel is identical to that describing the stagnant system. Furthermore, 0 is only a function of ~ once a is fixed. On the other hand, if the series is retained, ¢ is still parametrized with respect to oir. Finally, all dimensionless quantities outside the series are independent of the value of the diffusion layer thickness. The current density and the solution of eqn. (26) can be easily related by using eqn. (16): i( ~ )/i 1 = ~rl/2a'/2 •. ~ 0 ( 4 )

(27)

where il --- n F D c b / 5

(28)

99

A simple technique was used for the numerical solution of eqn. (26) (see Appendix) for a particular value of a = 7. As explained by Delahay [4], this choice of a describes the fact that ki is usually much smaller than (TROD)1/2 . The behavior of q~(~) for various values of oi~ is shown in Fig. 1. For oi~ < 3, ¢ increases monotonically from exp(--a) at ~ = 0 to ¢ __> ~.-1/2 0.51/2

as ~ -> ~

(29)

Combining eqns. (27) and (29) we obtain:

i/i~ -* 1

as ~ -+ oo

(30)

which is expected. For air > 3, a maximum appears. With increasing values of oir, the m a x i m u m decreases in magnitude and moves to smaller values of 4. For oir > 10: ~max ----0.280

when ~max = 7.20

(31)

while eqn. (30) still describes the asymptotic behavior of ¢. Combining eqns. (27) and (31) we obtain:

ip/il = 0.496oi~ 2 = 0.496~ ~2 o in ,

Oir > 10

(32)

In the region 7 < oir < 10, deviations from eqn. (32) are around 1%. Equation (32) indicates that the normalized peak current density is a linear function of Oir-1/2. The slope is a constant independent of the physical properties of the system. If ip/il is plotted vs. o ~n, the slope of the resulting straight line depends only on the u n k n o w n transfer coefficient, a. However, the e x t e n t of the transition region also depends on a. It can be shown after Delahay [4], that for values of a > 7, the relation: ~max = a + 0.20

(33)

valid for stagnant systems, is also valid for the RDE. Equation (33) can be used

oi, =1

0.~

4

10

0

5

10

15

Fig. 1. R D E c u r r e n t f u n c t i o n for a f i r s t - o r d e r irreversible r e a c t i o n u n d e r L S V c o n d i t i o n s . N u m b e r s o n lines i n d i c a t e values o f t h e d i m e n s i o n l e s s sweep r a t e i n c l u d i n g t h e t r a n s f e r coefficient.

I00

to evaluate the peak potential, Ep, at which ip occurs:

Ep -- E i = - - ( R T / n F a ) {0.77 + ln(a 1/2 01/2 K~I)}

(34)

Contrary to the behavior of reversible reactions, Ep depends on the sweep rate. Combining eqns. (32) and (34):

ip/il = 0.23•i e x p { - - a n F ( E p - - E i ) / R T }

(35)

Equation (35) indicates that for a given rotation speed, a plot of ln(ip/il) vs. Ep -- E i will result in a straight line. Values of a and Ki can be extracted from its slope and intercept respectively. If the experiment is conducted at a different rotation speed, a line parallel to the first will be obtained but with a different intercept. This behavior is due to the dependence of i~ and 5 on the rotation speed. However, ip and Ep are independent of the rotation speed. CONCLUSIONS The theoretical treatment of the application of LSV to a RDE for the simple reaction mechanisms has indicated some advantages and disadvantages of this method. Among the advantages one can cite: (1) The suppression of natural convection even at low values of the Reynolds number. (2) The weak dependence of the current ratio, ip fib on the physical properties of the electrolyte. (3) The simultaneous determination of both ip and/1 in a single LSV--RDE experiment. (4) The relative simplicity of performing a LSV experiment with a RDE than with an absolutely stagnant system. The disadvantage stems from the fact that the information provided by a LSV--RDE experiment become accessible to easy quantitative analysis only when convection contributes insignificantly. This is judged by the relative value of the sweep rate with respect to the characteristic voltage, R T / n F , and the characteristic time, 62/D. When convection predominates over diffusion, the LSV--RDE experiment yieldsvalues only for i~, since no current peaks appear. After the appearance of peaks, there is a transition region in which the current ratio is a complicated function of the dimensionless sweep rate, o. When convection is overcome, ip/il depends linearly on 01/2, and clear diagnostic criteria for reaction mechanisms can be developed. Despite this disadvantage -- namely, the exclusion of a range of values of the dimensionless sweep rate -- the wide availability of RDEs, as well as the reliability and reproducibility of RDE measurements, make the LSV--RDE technique an attractive alternative to stagnant system experimentation. APPENDIX Equation (26) was solved numerically by Huber's method [16]. The kernel can be more conveniently written as j=oo

g(@, ~) = 2~ 'n air'-/2 ~ j=l

exp(--(2] -- 1)2~2(~ -- ~')/4oir }

(A1)

101

Then, the integrals: h

Io(n, ~) = f K(nh, vh + z) dz

(A2)

0

and h

Is(n, v) = / (z/h)K(nh, vh + z) dz

(A3)

0

which are necessary for the computations, can be evaluated analytically. This property makes computer calculations much faster. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

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