382 1
CHRONOAMPEROMETRIC: CONSTANT FOR UNSHIELDED CIRCULAR ELECTRODES
Derivation of the Chronoamperometric Constant for Unshielded, Circular, Planar Electrodes
by Zoltan G. Soos and Peter James Lingane Diztision of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California (Received July 6. 1964)
We have calculated the chronoamperometric constant for a circular electrode of finite size and have shown that it can be expanded as a power series in -/PO. The coefficient of the term in m / p o is 2.26 and is in agreement with experiment. This coefficient is calculated by exploiting the equivalence betaween this problem and the corresponding heat conduction problem.
I. Introduction The faniilar Cottrell equation2
active species is essentially zero. Under these condit i o n ~the , ~ diffusion equation reduces to (1.3)
does not adequately describe the current-time behavior observed with electrodes of finite size at times longer than a few seconds.3 Therefore, we have calculated the current-time behavior for a circular electrode of radius po and have shown that the chronoainperoiiietric constant can be expanded as a power series in d a / p o ; the coefficient of the term in f i t t l p 0 is calculated. To do this, we have exploited the siiiiilarity between this problem and analogous heat conduction problems by adding current sourcei3 in direct analogy to the heat sources added in heat conduction problems. We discuss only the mixed boundary condition probleiii in which the concentration is fixed at the surface of the electrode; the solution to the Neuniann problem (chronopotentiometry) may be obtained analogously. If we neglect the effect of turbulence and of convective stirring, then the current passing through the area A is simply d?ji i = n5 ---.A = n3DVp.A (1.2) dt L
where i is the current in amperes, n is the number of equivalents per mole, D is the diffusion coefficient in ciii.2/sec., and ,E is the electrochemical potential. If the solution contains a large concentration of supporting electrolyte, the transference number of the electro-
where C, is the concentration in nioles/cc. of the ith electroactive species. In the presence of excess supporting electrolyte, the compact double layer5s5 can be approximated by a, parallel plate condenser whose plates are separated by one or two molecular diameters. Therefore, the charge on the double layer for a circular electrode of radius po will be uniform until p - po approaches the thickness of the compact double layer ( c j . Fig. 1). Since the metal-. lic electrode is an equipotential surface, the uniformity of the double-layer charge requires that the potential a t the outer Heliiiholtz plane also be uiiiforiii uiitili p - po approaches the thickness of the compact double layer. Very close to the edge, the potential a t the outer Heliiiholtz plane will decrease with respect to the solution potential. Therefore, it is very reasonable to assume that the concentration of the electroactive species will be uni(1) Contribution S o . 3141 from the Gates and Crellin Laboratories of Chemistr) . (2) F. G . Cottrell, 2. p h y s i k . Chem., 42, 385 (1902). ( 3 ) P. J. Lingane, A n d . Chem., 36, 1723 (1964). (4) H. L. Kies, J . Electrclanal. Chem., 4 , 156 (1962). (5) D. C. Gmhame, Chem. Rev., 41, 441 (1947). (6) R. Parsons, “Advances in Electrochemistry and Electrochemical Engineering,” Vol. 1, P. Delahay and C. W. Tobias, Ed., Interscience Publishers, Inc., S e w York, N. Y . , 1961, Chapter I.
Volume 68, Number 12 December, 1964;
ZOLTANG. Soos AND PETER JAMES LIKGANE
3822
]I 5-i Figure 1. Model for the circular disk electrode.
form at the outer Helmholtz plane of metallic electrodes of finite size and in the presence of excess supporting electrolyte if the concentration is governed by the Kernst equation. 11. Formal Solution We will first consider the oxidation of a single insoluble specics to give a soluble product, e.g., the generation of silver ion into a solution initially free of silver. Our electrode is a circular planar disk of radius po situated in the x = 0 plane (cj. Fig. 1). This electrode is potentiostated a t a fixed potentia1 E , and it is assumed that the Nernst equation is obeyed. This guarantees that diffusion and not the electron-transfer step will control the current. Observe that under these conditions, the concentration at the surface of the electrode will continuously increase with potential and no limiting current will be achieved. This approach is purely one of coiivenience and is taken to achieve greater generality as will become evident in the Discussion section. The cylindrical symmetry enables us to write the concentration as C ( p , x , t ) . The boundary conditions for C ( p , z , t ) are C(p,O,t) =
{cor)
P I P 0
tr0+
p>po
t=0+
lini C(p,x,t) = lim P-
m
2-
lirn t-
C(p,z,t) =
C’(p,O,O+)
=
{coy)
P P
I PO > PO
(2.4)
Since the diffusion equation contains only a first derivative with respect to time, it is convenient to inalie the partial separation CI(P,Z,t) = C0g(p,t)f(z,t) (2.5) where CO(E) is the concentration a t the outer Helmholtz plane and f(z,t) and g(p,t) are dimensionless functions to be determined. It is evident that under certain conditions, such as t -+ 0 or p + 0, the solution must reduce to the solution for the infinite planar electrode. Hence, f (x,t) corresponds to the well-known solution for the infinite planar electrode’
0
m
C(p,x,t) =
m
In general, the surface of an electrode can be represented by a continuuni of current sources whose distribution and strengths are tiine dependent. The source distribution for an infinite planar electrode is time independent and is uniform over the surface of the electrode. The time-dependent sources of C’( p,z,t) are chosen to be equivalent to such a uniform distribution for p 5 po a t t = O+. Therefore, C1(p,x,t) is equivalent, initially, to the concentration distribution at a circular section of an infinite electrode. At later times, radial diffusion causes the sources of C I ( p , z , t ) to decrease near the edge of the electrode and apparent sources to appear at p > po. The sources of C T 1 ( p , z , t ) could be chosen so that the boundary condition (2.1) is satisfied at all times. Alternatively, the sources of CI1(p,x,t) might be chosen so that the total flux from the finite electrode into the region p 5 po is the same as from the infinite clectrode. We shall adopt the second condition because we wish to approximate the sources for C I I ( p , z , t ) by a line source. Such an approximation gives a nonuniform concentration a t the surface of the electrode. Obviously, an exact solution must satisfy both Conditions siniult aneously . The analogous heat conduction problem is that of a circular hot plate whose sources are chosen so as to keep the heat flux from the plate into the region p I: PO the same as the flux from a circular section of an infinite hot plate whose temperature is uniform and time independent. The initial conditions for C I ( p , x , t ) are
Co(E)
and g(p,t) represents the effect of diffusion in the radial direction. Furthermore, the initial condition for g(p,t) is (7) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers, Inc., New York, K. Y . , 1954, Chapter 3.
The Journal of Physical Chemistry
CHRONOAMPERQNETRXC C O N S T A N T FOR UNSHIELDED CIRCULAR ELECTRODES
P P
5
Po
g(p,t) must satisfy the radial portion of the diffusion equation. g ( p , t ) may be obt,ained via a Hankel transform.* Only zeroth-order Bessel functions appear because of the cylindrical symmetry. The integrand is a solution to the diffusion equation for each value of k, and hence g(p,t) is a solution. g(p,t)
=
-
p'dp'g(p',O+)Jo(kp')
(2.7)
> Po
3823
= PO
7(3.1)
JdkPO)
The exponential in (2.10) is now expanded in a Hankel transform. Only xeroth-order Bessel functions appear because of the symmetry in the k-plane. The inverse transformation reduces to a Laplace transform. e-Dkzt =
lrn
(3.2)
qdpJo(kp)f(q)
lrn kdk f(k)e-Dk"Jo(kp)
As is shown in Fig. 2, g(p,t) is a step function at t O+; a t later tames, the time dependence of (2.8) gives different superpositions of the Bessel functions and the step-function decays. This decay expresses the fact that there is radial diffusion. Since C'( p&) satisfies the boundary conditions only at t = 0+, it is evident that a current source niust be added toward the edge of the electrode to maintain a uniform concentration distribution. This current source will produce the concentration C"(p,x,t). Once we have calculated C'(p,x,t), we will be able to determine the position and the strength of the current sources that produce C''(p,z,t). The current flowing through the electrode is given rigorously by the gradient with respect to z of the concentration evaluated a t z = 0, p _< PO. Therefore, the current density corresponding to C'(p,z,t) is
Combining (2.9), @.lo), (3.2), and (3.3), we obtain
=E
i
2
7r Po2
PO2
__ = n5D --
'bCI [( z)pdp
g(p,t)
=
Lrn
Got
s"
(p2
+;ip-"")
(p
> (P
+ q)2
+ p)2 > po2 > (p - p)2
pdpg(p,t)
Hence t
>0
(2.9)
0
kdke-Dk*tJo(kp)X
J :
jcos-1
PO2
=
z=o
-n ! !
The integral over the Bessel functions is a special case (of the formula of Sonine and Dougall."J
p'dp'g(p',O+)Jo(kp')
5 (cos-l
(2.10)
bq
zip-
(p"+
PO")
By reversing tthe orders of integration freely, it is possible to show that for an arbitrary g(p,O+) the current density is an expansion in powers of d E / p o .
111. Special Case : One-Component System A . Expressions for g(p,t). We now consider in detail the properties of C'(p,z,t). The initial conditions are fixed by (2.4); f(z,t) is given by (2.6); g(p,O+) is the step function defined by (2.8). We integrate (2.8) to obtain
(8) P. M. Morse and H. Feshbach, "Methods of Theoretical Physics," McGraw-Hill Book Co., New York, N. Y., 1953.
(9) W. Magnus and F. Oberhettinger, "Formulas and Theorems for the Functions of Mathematical Physics," Chelsea Publishing CO.,New York, N. Y . , 1949, p. 132. (10) W. Magnus and F. Oberhettinger, ibid., p. 37
Volume 68, Number 12 December, 1964
3824
ZOLTAN G, Soos AND PETERJAMES LINGANE
In order to evaluate the integrals in (3.7) and (3.8), we first evaluate bg/bp(p,t). We can easily verify (or observe from the boundary conditions) that ag/bp(p,O+) is a &function of unit strength.
0 plane. We replace g(p,t) in (2.9) by Ag(p,t); except for constants, we have
s,” pdpAg(p,t)
=
PO2
-
2
‘x
e-2
P02+P2
pdpe--.lot
C-
(g)
(:)’I,
u-0
(3.9) = -s(P
-
(3.15)
- PO)
(3.10) We integrate by parts repeat,edly to obtain The Laplace transform10 in (3.10) gives the modified Bessel function of the first order. We define Ag(p,t) to be the deviation of g(p,t) from the step function g(p,O+). A d P , t ) = g(p,t) =
-1
g(p,t)
P P
I PO
> PO
Subst’ituting(3.16) in (3.15), me have (3.11)
Po2
We now obtain an expression for Ag(p,t), p 5 po. It is evident from (3.7) that g(0,t) = 1 for all t > 0. Equation 3.10 then gives
ng(X,t)
=
-e-x@
(@2
-
X
2 D t e -4Dt 7 ”=O
vIy
(&)) (3.17)
Using the recursion relations for the modified Bessel functions, we find
(3.12)
If we set x = ppo/2Dt, xo = po2!2Dt, and p = Dt/po2 1/2x0,we find
pdpAg(p,t) = e-m
=
Substituting (3.18)in (3.17), we obt,aiii
s,” Il(x)e-Px*dz (3.13)
We use the properties of the modified Bessel functionsll and integrate (3.13) by parts repeatedly. Ag(x,t)
=
-e-xO’2
{ 2 ( 2 p ~ ) ~ I ~ ( ~ )1e}- ~ ~ * -
U = O
Hence
(3.14)
P
5
Po
where I , is the modified Bessel function of order v. B. Cowection to the Current Density. Expressions 3.13 and 3.14 will be used to evaluate g(p,t). Before we perform this calculation, which requires a variety of special conditions, we will evaluate the decay in the current density at the electrode due to the apparent motion of the current sources for C1(p,z,t) to p > po in the x = The Journal of Physical Chemistry
e + p ~ 2 / 4 D t } (3.20)
Equation 3.20 is an exact expression. The current sources for C1(p,x,t)have an initial uniform distribution over the surface of the electrode that is identical with the time-independent distribution of the current sources (11) H. B. Dwight, “Tables of Integrals and Other Mathematical Data,” The Macmillan Co., New York, N. Y . ,1961.
3825
CHRONOAMPEROMETRIC CONSTANT FOR UNSHIELDED CIRCULAR ELECTRODES
r-
I
I
I
I
Hence we observe that Ag(p,t) is negligible for p 5 po/4. Equation 3.21 is an upper bound for [ Ag(p,t)/, since, in it, we have replaced all of the modified Bessel functions by Io. For example, eq. 3.21 gives ‘Ag(0,t) 6 e-’O while Ag(0,t) = 0 (for all t) from eq. 3.14. For p0/4:. p 5 po, we may use the large argument expansion of the modified Bessel functions in eq. 3.12 throughout the entire range of the integration.
(3.23)
Since x0/4 > 5 , it is straightforward to integrate (3.23:l. Hence
p,cm
Figure 2. Plot of g( p , t ) as a function of redistribution of the sources of CI( p,z,t). 0.2 em., D = 1 X 10-6 cm.z/sec.
p
illustrating the
Ag(p,t)
=
pa =
for the infinite planar electrode. However, the current sources for C’(p,z,t>are redistributed in time because of radial diffusion; the sources near the edge of the electrode decay, and appaxent sources are found outside the electrode ( p *; Po). g(p,t) is identically unity for the infinite planar electrode since there is no radial diffusion in that case. C. Evaluation of g ( p , t ) for Experimental Conditions. In order to find g(p,t), p < po, and thus determine the position of the source8 that we shall use for C”(p,z,t), i t is convenient to specialize the experimental conditions, Le., p o 2 0.2 cm., D cm.2/sec., t 5 100 sec. Under these conditions, po2/4Dt = x0/2 > 10. From the definition of IV(x),it is evident that I , + l(x) < I,(x) for all v when x > 0. Hence, we have from (3.14)
-
I
(3.24)
I
0
P
PO
i --4
Expression 3.24, which has the expected form, may be obtained directly from (3.7) by taking p = po - A and letting A + 0 after the integration. If we set p = po in (3.7), the integrand is indeterminate at q = 0; when the integration is now performed, only the second term of (3.24),evaluated at p = pot is obtained. Since the derivative of g(p,t) is continuous for t > 0 , we may use an expression similar to (3.12) to evaluate g(p,t) for P > Po.
(P’) ____
e
+P,Z
4Dt
dp’
P
>
Po
(3.25) For p = p0/4, x0/4 2 5, and me may use the large argument expansion of Io.11 We then have, for 0 < P 5 p0/4 17011tme 68, Siinaber 12
December, 1.962:
ZOLTAN G. Soos AND PETER JAMES LINC-ANE
3826
The next term in eq. 3.24 or 3.25 is negligible for the times of interest. Since f(0,t) = 1, the deviation of C ' ( p , ~ , t from ) Co for p I po is CO times expression 3.24. D. Sources Jor and Calculation of CT1(p,z,t). g(p,l) is plotted as a function of p for various values of t in Fig. 2. It is evident that, for t 5 25 sec., deviations from the step-function initial condition are important only within 0.02 to 0.03 cin. of the edge of the electrode. Of course, this is precisely what is expected, especially if the analogous heat problem is considered. Therefore, we need a current source of radius of about 0.01 cm. at the perimeter of the electrode, placed so that the outer edge of the source is at p = po if the boundary condition (2.4) is to be satisfied for all times. Such a tiine-dependent source, which arises froin the decomposition of the concentration in (2.3),defines the diffusion problem for ~II(p,z,t). As a first approximation, we consider a time-independent source at p = po. This should he a good approximation for shorl times since, for short times, (3.24) shows that the source is essentially at the edge and, as is evident from the large argument expansion of the modified Bessel functions, (3.20) shows that Ai/npo2 is essentially time independent. .-
A
{
- n5D2Co ___ I + O Po2
(Et)) 7
ting a point source at the center of the distribution. We will approximate even further and place the source at the edge of the electrode instead of near the edge, The current flowing froin a straight wire is radial by symmetry. At a line source of strength a , the current per unit length is
i,
=
n5D2ar
b C"(r,t)lr-o br
-
I n the same spirit, we may, for short times, approxlinate the source by an infinitely long, straight wire and express the concentration at a distance r from the wire as
-n5DCoa
(3.28)
It is evident froiii Fig. 1 that the symmetry in the z = 0 plane requires that the flux flowing into the region where z > 0 is just half the total flux from a source whose intensity is twice the intensity of the actual source. Thus (3.28) will be correct if a is 1 aken to be the actual int'ensity per unit lengtJh of the line source. Furthermore, at least for short times, the flux into the region p < po is the same, by symmetry, as the flux into the region p > po. The total flux from a line source of length 2npo is 2 -i,- - - nSDCOa 7r Po2
(3.29)
Po
Half of this current will flow into the region z < po. Hence l/gi,
(3.26)
=
= -Ai
>
0,
p
(3.30)
where A i is given by (3.20). These two currents have opposite signs since the source for C"(p,z,t) is to correct for the effects of radial diffusion. Thus we obtain 2
a = -
(3.31)
T
The correction current density is therefore is
where Ei(x) is the exponential integral. (Y is the strength of the source and will be evaluated below. This solution is borrowed from the analogous heat conduction It is riot convenient to satisfy the boundary condition (2.1) by requiring that the sun1 of C1(p,z,l) and C I 1 ( p , z , t )be uniform and time independent at the face of the electrode, since, by lumping the corrections into a line source, we have introduced a singularity. We inay, however, as an alternative condition, match fluxes. Since the exact sources of C"(p,z,t) have "radii" of about 0.01 cin. for t 5 2.5 sec., the approximation of a line source should be excellent for r >, 0.01 cni. (cf. Fig. 2 ) . Once outside the source distribution, its dctailed coinposition becomes unimportant; as in a multipole expansion, the major effect is found by putThe Journal of Physical Chemistry
n- Po2
dE dad, po
n5DCo 4
(3.32)
E . Total Current Density and Concentration Profiles. The total current density is the sum of the current sources for C'(p,z,t) and eq. 3.32. The total current density due to the sources of C'(p,z,t) is (cf. eq. 2.9)
(3.33) Therefore, the total current is (12) L. R. Ingersoll, 0. J. Zobel, and A. C. Ingersol, "Heat Conduction," McGraw-Hill Book C o . , New York, N. Y., 1948, p. 146.
CHRONOAMPEROMETRlC CONSTANT FOR UNBHIELDED @IRCULAR
3827
7%b+ZT
_i_ - - --
"Po'
ELECTRODES
4
f i t
(3.34)
Although the boundary condition (2.1) is not satisfied in detail on the electrode, it is evident that, if the line charge is redistributed near the edge, the concentration could be calculated even at the face of the electrode. The current correction would, however, be very nearly the same since the flux from a source distribution niay be fourrd by calculating the flux from the equivalent point source. We have overestimated the correction in (3.30) since the center of the correction source is actually at p
