J . Phys. Chem. 1987, 91, 3 108-3 1 11
3108
0
100
to( m in)
200
300
Figure 5. The plot of In h vs. to for the system methane/hexadecane at 324.7 K with V, = 5 cm3. The symbols 0 are experimental points, while - is the curve calculated from the data obtained with V, = 10.4 cm3. even at the lowest possible attenuation of the amplifier. It has been observed experimentally that better resolution of each curve into two straight lines is achieved with a big volume of liquid. For instance, in the system methane/hexadecane the two slopes have a ratio 4.6 when VL = 10.4 cm3, but with 5 cm3 of liquid the ratio is much smaller and a plot of only slight curvature is obtained after the maximum (cf. Figure 5 ) . The slope of this plot, considered as a straight line, is -(1.79 f 0.02) X lo4 s-’ but no value of KL,KG,or K can be calculated from this single slope. To see whether this slope is consistent with those obtained when VL = 10.4 cm3 was used, the curve with V , = 5 cm3 was calculated by using eq 34 and the values of X , Y, and Z computed from the data with 10.4 cm’ of Table I. The necessary value of N2 was obtained as the mean of the two relative values of N2 found from the intercepts of the straight lines with 10.4 cm3 of liquid,
being 6.1 19 X lo4 and 5.808 X lo4 (in arbitrary units). The calculated plot, given in Figure 5, is not far from the experimental points. Of course, it is drawn for times after that of the maximum, since eq 34 was derived and is applied only at such times. In the other two gases,. butane and propene, the values of KL, KG, and K obtained with 10.4 and 5 cm3 of hexadecane are of the same order of magnitude and their difference can be attributed to small variations in aL,i.e. the extent of the gas-liquid boundary. A comparison of the Henry’s law constants @ found in the present work with those calculated from H (dimensionless) as given in a nomogram2* (see last column of Table I) shows that the difference between the two is not big and can be due to composition variations. Coming now to the results obtained with water as the liquid phase, one can see from Table I that the slope of the single straight line after the maximum differs little from that obtained with no liquid in the diffusion column. Therefore, mass transfer phenomena across the boundary hydrocarbon/water are negligible, as expected from the very low solubility of these gases in water at temperatures 52-54 OC. Conclusion By using a very simple experimental arrangement and a simple mathematical analysis, the reversed-flow gas chromatographic technique leads to the determination of mass transfer coefficients across a gas-liquid boundary, together with the partition coefficient (or the Henry’s law constant) for the distribution of a solute between the gas and the liquid phases. Practical applications of these coefficients can be found in gas chromatography, gas absorption, evaporation, etc.
Acknowledgment. The generous help of Mrs. Anna SinouKarahaliou is gratefully acknowledged by the authors. Registry No. Butane, 106-97-8;propene, 115-07-1;methane, 74-82-8; hexadecane, 544-76-3. (28) Reference 26, p 145.
Theory of the Current at Microelectrodes: Application to Ring Electrodes Attila Szabo Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892 (Received: October 13, 1986)
An approximate but analytically tractable approach to the calculation of the general (quasi-reversible) i-E characteristics of microelectrodes is developed. The flux at the electrode is assumed to be spatially uniform and the usual boundary condition at this surface is required to hold only on the average. With this approach, a simple expression for the long-time behavior of the current at a microring electrode is derived. The steady-state diffusion-limited chromoamperometric current at thin rings obtained when using the uniform flux boundary condition is within a couple of percent of the exact result. On the basis of a variety of considerations, it is conjectured that the long-time behavior of the diffusion-limited current at any microelectrode with a closed surface lying on an effectively infinite planar support is described by i(r)/nFDCo* = lo(1 + lo/(4r3Dt)1/2),where lo has the dimensions of length and depends on the geometry of the electrode (e.g., lo = 4b for a disk with radius b; lo = 2nR for a hemisphere of radius R; lo = r Z ( a+ b)/ln [32a/(b - a) + exp(r2/4)] for a ring with inner radius a and outer radius b).
Introduction The derivation of analytical expressiom for the t h e dependence of the current to is a challenging problem because the diffusion equation must be subject to boundary conditions (Le., the flux is specified on one part of a surface and the concentration on another). Numerical finite-difference approaches can handle such boundary conditions; however, since one is often interested in the long-time behavior of the current, one
has to use algorithms that are more powerful than the familiar explicit one, such as the explicit hopscotch’s2 and the implicit alternating direction’ methods. Although it is always Wssible to simulate the current numerically, the availability of simple ana(1) Shoup, D.; Szabo, A. J . Electroanal. Chem. 1982, 140, 237. (2) Shoup, D.; Szabo, A. J . Electroanal. Chem. 1985, 160, 1. (3) Heinze, J. J . Electroanal. Chem. 1981, 124, 73.
This article not subject to U S . Copyright. Published 1987 by the American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 3109
Theory of the Current at Microelectrodes lytical expressions greatly facilitates the analysis of experiment data. The general (quasi-reversible) i-E characteristic is known analytically only for a hemispherical microelectrode lying on an infinite planar support (by symmetry, it is just one-half of the result for a spherical electrode4). The short-time behavior of the diffusion-limited current at planar inlaid electrodes (i.e., ones that are flush with the inert support) has been obtained analytically by Oldham,5 but his method does not appear to be generalizable to longer times. It has recently been shown6 that, at long times, the current a t a band electrode of width w is equal to that at a hemicylindrical electrode with radius w/4. Simple analytical expressions for the diffusion-limited current at these electrodes, which are accurate to within 1.3% over the entire time range, have been devised.6 For microdisk electrodes, several terms in the asymptotic (long-time) expansion have been obtained’ via a sophisticated application of the Wiener-Hopf method.’,7 It appears extremely difficult to adapt the mathematical techniques used to derive the rigorous results mentioned above to obtain the general i-E characteristics of microelectrodes or to treat more complicated geometries (e.g., microrings). In this paper we describe an approximate approach to the problem that has been successfully used to describe diffusion-influenced reactions between species with nonuniform surface r e a c t i ~ i t i e s . To ~ ~ ~illustrate the basic ideas of this approach, let us consider the calculation of the diffusion-limited current at an electrode with surface SElying on an inert support with surface SI. In the usual approach, one must solve the time-dependent diffusion equation subject to the mixed boundary conditions that
C = 0 on S E
(1)
&VC = 0 o n S I
(2)
where C i s the concentration of the electroactive substance and i? is a unit vector perpendicular to the surface. The current to the electrode is then obtained by using i = n F D j & V C dSE
(3)
Since it is very difficult to solve partial differential equations subject to mixed boundary conditions 1 and 2, let us modify the boundary conditions. If condition 1 is replaced by the requirement that the flux is spatially uniform over the surface of the electrode
&VC = J(t) on S E
(1’)
then solution of the differential equation becomes considerably easier. Once this is done, the unknown function J(t) is determined by forcing condition 1 to be valid on the average over the surface of the electrode
jcdsE=o
(4)
Finally, the current is given by (see eq 3 and 1’)
i = n F D l J ( t ) dSE = nFD J(t) A
uniform (it behaves as (b2 - r2)-1/2so it is infinite at the edge) and it would appear that it is a poor approximation to assume that the flux is a constant independent of r. In the next section, we shall see that this approximation works even better for thin microring electrodes.
Theory W e now present a detailed application of the ideas presented above to the problem of describing the general i-E characteristic of a ring electrode. W e consider a ring electrode with inner and outer radii a and b, respectively, lying on an inert surface in the x-y plane. The reaction at the electrode is described by
where, in the notation of Bard and Faulkner,lo the rate constants are kf = kOe-unF(E-EO‘)/RT (7) kb = koe(l -u)nF(E-@)/RT
(8)
We assume that Do = DR = D. In this case, the problem can be formulated solely in terms of the concentration of species 0. This concentration, denoted simply by C(r,z,t), obeys the diffusion equation (in cylindrical coordinates)
ac
at = D[!
$r $+$]C
(9)
with mixed boundary conditions ..
.
(kf
+ kb)C(r,O,t) - kb(CO* + CR*) =0
a
otherwise
< r < b (10) (1 1)
where C,* is the bulk concentration of species I. The current is given by i/nFD = 2 . 1 ‘
(E)
a
r=O
r dr
(12)
+
Since for any A, exp(-(h2 s/D)’/*z)Jo(hr) is a solution of the homogeneous part of the Laplace-transformed diffusion equation, the general solution is of the form CO*
Qr,z,s) = -- ~ q f ( h ) e ~ ~ ” i f s ~ D ~ 1dh ’ 2 z ~(13) o(hr) S
where J,(x) is a Bessel function of order m andflh) is unknown. When eq 13 is substituted into eq 10 and 11, one obtains a set of dual integral equations for f ( h ) that appear analytically intractable. As discussed in the Introduction, we can circumvent this difficulty by replacing conditions 10 and 11 by
(5)
where A is the area of the electrode. The current calculated by this approach agrees well with available exact results. For example, for a microdisk electrode of radius 6, one finds* that the steady-state diffusion-limited current is i(-)/nFDCo* = 3u2b/8 r 3.7b as compared with the exact result of 4b. This is somewhat surprising, because the exact steady-state flux over the surface of the electrode is highly non(4) Shain, I.; Martin, K. J.; Ross, J. W. J . Phys. Chem. 1961, 65, 259. (5) Oldham, K. B. J . Electroanal. Chem. 1981, 122, 1 . (6) Szabo, A.; Cope, D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, R. M. J . Electruunal. Chem., in press. (7) Aoki, K.;Osteryoung, J. J . Electroanal. Chem. 1981, 122, 19. (8) Shoup, D.;Lipari, G.; Szabo, A. Biophys. J . 1981, 36, 697. (9) Szabo, A.; Shoup, D.; Northrup, S . H.; McCammon, J. A. J . Chem. Phys. 1982, 77,4484.
=0
otherwise
(15)
where .?(s) is independent of r. As we shall se,e below,f(h) and henceC can readily be expressed in terms of J(s). Once this is done, J(s) is found by requiring eq 10 to be satisfied on the average over the electrode surface
(10) Bard, A. J.; Faulkner, L. R.Electrochemical Methods; Wiley: New York, 1980.
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987
3110
The current is then calculated by using (see eq 12 and 14) ?(s)/nFD = a ( b 2 - a2).?(s)
(17)
Szabo TABLE I: The Long-Time Limit of the Chronoamperometric Current at a Microring Electrode a < r < b i(m)/nFDCo*b = l o / b
Substituting eq 13 into eq 14 and 15, multiplying both sides by rJo(h’r), integrating over r from 0 to 03, and using the identities JmJo(Ar) Jo(A’r) r d r = X-I 6(X - A’)
(18)
where 6 ( x ) is the Dirac 6 function and l b J o ( X r ) r d r = X-] [bJ,(Ab)- aJl(Xa)]
(19)
we find f(X)(X2
+ s / D ) ’ / ’ = .?(s)[bJl(Xb)- u J ~ ( A u ) ]
bla
exact”
2 1.5 1.25 1.20 1.125 1.0909 1.0213
4 3.924 3.798 3.590 3.510 3.330 3.204 2.661
m
ea 35 4 3.911 3.801 3.596 3.516 3.335 3.208 2.668
ea 34
ea 31b
3.515 3.500 3.326 3.202 2.667
3.495 3.424 3.259 3.141 2.626
From ref 14.
(20)
Solving eq 20 for f(X), substituting eq 13 into eq 16, solving for J(s), and using eq 17, we finally have
integrals. For thin rings ( ( b - a ) small) eq 29 simplifies to (see Appendix)
a ( b 2 - a2)(kfCo*- k b C ~ * ) ?(s)/nFD = s ( D + 2 ( k f + k b ) F ( s ) / ( b 2- a 2 ) )
(21)
Finally, using eq 28 in eq 21 shows that the long-time behavior of the current to a ring electrode with inner and outer radii a and b, respectively, is given by eq 25 and 26 with
where we have defined
10
To get a feeling for this result, let us examine the special case of a microdisk with radius b. Setting a = 0 in eq 21, we have ab(kfCo* - kbCR*) i(s)/nFD = (23) s ( D / b + 2(k, + kb) G ( S b 2 / D ) )
=
37r2(b2- u ~ ) ~
+
8(b3 + a’) - 4 ( + ~ b ) [ ( a 2 b 2 ) E ( k )- ( b - ~ ) ~ K ( k ) l (31a) a 2 ( a + 6)
In
where9
[
4e3/2(
s)]
( b - a ) / b
