A Unified Treatment of Electrolysis at an Expanding Mercury Electrode Keith B. Oldharn Science Center, North American Rockwell Corporation, Thousand Oaks, Calif. 91360 This article describes a general treatment of voltammetry at an electrode expanding in accord with any power law: the stationary electrode and the dropping mercury electrode are special cases. Reversible, irreversible, and quasireversible electrode processes are em braced by the treatment. An equation of very wide generality is derived, the solution of which under appropriate conditions gives the equations associated with the names of Ilkovic, Cottrell, Heyrovsky-llkovic, Matsuda-Ayabe, Delahay-Strassner, Koutecky, Smutek Karaoglanoff, Sand, Delahay-Berzins, and Reinmuth. Many new voltammetric relationships are derived, several of which have attractive possibilities in electroanalysis.
THOUGH the dropping mercury electrode is the prime example of an electrode whose area increases with time, other types of electrode which share this property are conceivable. Thus, mercury could be allowed to flow into the base of a funnel (a conical funnel will yield a t213area dependence; a paraboloidal funnel will give an electrode area proportional to t 1 / 2 ) or a vessel holding a mercury pool electrode could be progressively inclined. These devices do not necessarily share the dropping electrode’s t2/3 area law. Moreover, recent work (1) in which a piston driven by a synchronous motor was used to replace the usual gravity feed, opens up the possibility of introducing other than t2i3dependence of area, even for the capillary mercury electrode. Here, therefore, the electrode area is considered to obey the general law A(t) = atp
where a and p are positive constants. [For a conventional dropping electrodap = 2/3 and a = (36n)”3 m213p-2/3 where h and p are, respectively, the mass rate of flow of mercury and its density.] The constant area case is also covered by including thep = 0 possibility. Consider the n-electron electroreduction Ox(so1n)
+ ne-(Hg)
=
Rd(so1n)
(2)
occurring at an expanding electrode, where Ox is present in solution at an initially uniform concentration C; Rd is soluble either in the solution or by amalgamation in the electrode phase, but is initially absent. It will be assumed that the expanding electrode is either planar, or can be approximated by an expanding plane. [Several investigations (2, 3) suggest that, for a dropping mercury electrode, the effect of sphericity is minor]. Moreover, linear semi-infinite diffusion is regarded as the sole mode of transport of Ox and Rd relative to their solvents. Yet another assumption is that the expansion of the electrode is uniform in the sense that in a given time interval all subregions of the electrode surface increase in (1) S. Roffia and E. Vianello, J. Electroanal. Chem. & Interfacial Electrochem.,17,13(1968). ( 2 ) I. M. Kolthoff and J. J. Lingane, “Polarography,” Interscience, New York, 1952, p 44. (3) J . Koutecky and J . Cizek, C d . Czech. Chem. Comrnun., 21, 836 (1956). 936
ANALYTICAL CHEMISTRY
area by the same factor. Unlike stretched flat membranes, planar or spherical liquid-liquid interfaces may expand uniformly, though nonuniform expansion is also possible. Inasmuch as uniform expansion has been prescribed, the concentration of Ox at any point in the solution depends only on the distance r separating that point from the electrode and the time t since the birth of the electrode: C(r,t) will be the symbol used to denote this concentration. In this terminology C(0,t) and aC(O,t)/ar then denote the surface concentration and the surface concentration gradient. The latter is related to the cathodic faradaic current (which may be time-dependent) by D
ac ; (0,t) =
40 ~
nFA(t)
(3)
which follows from the requirements of Fick’s and Faraday’s laws. All symbols are fully defined in Appendix A. To derive the partial differential equation appropriate to the concentration of Ox in the vicinity of a uniformly expanding plane, let us first consider the situation in the absence of diffusion. Since the concentration is a function solely of distance and time we can write (4)
Now, if no diffusion occurs, the concentration of Ox will remain constant in a moving plane defined such that a constant volume is enclosed between that plane and the electrode surface. We therefore insert the conditions dC = 0 and d[rA(t)] = 0 into equation (4),whence, by use of relation (l), (aC/at), = (pr/t)(aC/ar)lis derived. In the presence of diffusion but without expansion, the usual Fick‘s second law relationship (aC/at), = D(a2C/ar2)zapplies. In the present situation the effects of expansion and diffusion are additive and hence pr aC a2c ac - (r,O = - - (r,t) + D - (r,t)
at
t ar
ar
We seek to solve this equation subject to the initial condition C(r,O) = C
(6)
and to a variety of boundary conditions. There are two boundaries to consider. One is a plane at an infinite distance from the electrode: within a finite time the Ox concentration here will not be perturbed by the electrode reaction and hence C( m , r ) = C
(7)
The other boundary is the electrode itself and the appropriate condition here is determined by the electrolysis regime (potentiostatic, galvanostatic, etc.). We choose not to specify this second boundary condition yet. In Appendix B it is demonstrated that the surface Ox concentration is related to the electrode potential and to the
c
cathodic faradaic current (either or both of which may be time-dependent) by the equation
C(QJ,S)= S result, where S-*and C(X,S) are the transforms of Y and C(X,Y). Subject to the requirement of equation (18), the general solution to differential equation (17) is
where all new symbols are fully defined in Appendix A. Certain assumptions concerning the kinetics of reaction (2) are implicit in equation (8), but this equation is general in the sense that it applies equally to reversible, irreversible and quasireversible ( 4 ) electrode reactions. However, should the reaction be reversible the first right-hand term in equation (8) may be ignored, whereas if the reaction is totally irreversible the first term is dominant and the second may be omitted. To aid in the solution of equation (9,it is convenient to define new independent variables
X S
(2p
rtPkh2Pf1 Dpfl
+
(9)
where Q(S) is a function of the dummy variable determined by the as-yet-unspecified boundary condition. This unknown function may be eliminated by combining equation (19) with the result of its own X-differentiation; this yields
C
C(X,S) = - =
s
1 4;axG!
(X,S)
Inserting the X = 0 condition, and adopting the symbol L1 ) to indicate the operation of Laplace transformation with respect to Y , equation (20) becomes
and
to replace r and t, respectively. The choice of these definitions is motivated by the simplification to equation (6) which thereby results, and by a wish to undimensionalize the variables. This is also a convenient juncture at which to replace the current and potential variables by the following dimensionless analogues L(
Y )=
I(r)khZp-l nuFCDP(2p
+
and
Equation (21) is a very general result interrelating the surface Ox concentration and its surface concentration gradient, as functions of time, at an expanding electrode under any electrolysis regime. Usually, however, these two variables are not accessible to measurement or to direct control. Accordingly, a more useful relationship is obtained by solving equation (8) for C(0,t) and inserting this expression together with equation (3) into (21), thereby producing an equation interrelating potential and current. In terms of the dimensionless quantities defined in identities (10) through (1 3) the interrelationship is found to be
(12) Yet a further convenience is the introduction of a parameter w,
to play the role hitherto assumed by p . Note that w kalues in the range 0 < w 6 l/2 only are permitted. Particularly interesting values of w are 3/14 (corresponding to a conventional dropping mercury electrode), l / , , and r / 2 (corresponding to a stationary electrode). In terms of the new independent variables, equations (5) through (7) become
aC -
l3Y
(X,Y)
- a'c ( X , Y ) = 0
and
c=
C(o0,Y)
(15, 16)
When equations (14) and (16) are Laplace transformed with respect to Y , SC(X,S)
- ax2 (X,S)= C(X,O) = c
+
QJ,
a x 2
C(X,O) =
The generality of this equation is extremely wide and the remainder of this article is devoted to its solution under specific electrolysis regimes. Though all degrees of reversibility are embraced by equation (22) as it stands, the lefthand side may be simplified to L(e(Y)fll 4Y)I) or to L ( Y"- "( Y)e('(Y)) , respectively, under conditions of total reversibility or irreversibility. We shall consider four electrolysis regimes: very negative potential, constant potential, constant current, and linearly increasing current. These regimes correspond to electrolysis orders ( 5 ) of 0 , 2 , 3 , and 5 . Very Negative Potentials. As E approaches c b e comes vanishingly small and the left hand side of equation (22) becomes zero, leading to the simple transform equation
Hence c ( Y ) = Y t - 2 W / l / K which on insertion of the original variables becomes
(17)
I(t) = naFCtP-'/S
(the latter equality following from (15)) and (4) P.Delahay, 1.Amer. Chem. SOC.,75, 1430 (1953).
(5)
(24)
K.B. Oldham, ANAL. CHEM., 40,1912 (1968), VOL. 41, NO. 7, JUNE 1969
937
/
I,(O
I
I
I
I
I
I
1
/
microamps
b
IO
0
I
/ 2
3
4
Figure 2. Q(w,z) plotted semilogarithmically as a function of z for four values of W : reading downward w = l/2, l/,, l/e, and 0 These graphs also depict the irreversible behavior of I(i)/&(r) for p values of 0, 1/2, 1and ; the abscissa in this case is linear in potential. The plot for the dropping mercury electrode case (o = a/l4, p = '/a) is not included but it lies between the second and third curves
t secs
Figure 1. Current-time relationships for very negative potentials and variousp values
The following constant values were assumed: n = 2 equivmole-l,C= lO*molecm-*,a = 10-2cm*sec-~, and D = 10-Kcm2sec-l
For the case p = 2/a, this is the equation derived 34 years ago by Ilkovic (6). For the p = 0 case, equation (24) correctly reduces to the Cottrell expression (7). For the intriguing p = '12 case, equation (24) predicts a time-independent current of magnitude naFCd2Dl?r, Expression (24) is independent both of potential and of the kinetics of reaction (2). Diffusion alone limits the extent of reaction and such a current is termed a "diffusion limited current," I&). Figure 1 shows graphically how Zdt) depends onp, for p values of 0, l / ~ 1/3, , ' 1 2 , 2/3, 6 / 6 , and l. Reversible Potentiostatic Regime. At constant potential, t does not depend on Y , which leads to a marked simplification of equation (22). If, additionally, the reversibility simplification is introduced, there remains
Ilkovic equation (8) of polarography, with potential, equal to En. A time-independent current
Ella,
the half-wave
is again the case if p = '12. It should be appreciated that this time-independence does not result from a steady state having been established. Rather, the concentration profile is timedependent in such a way that the fall-off in current density is exactly compensated by the area increase. Irreversible Potentiostatic Regime. The situation here is less simple. The prescription of constant e reduces the irreversible version of equation (22) to
Clearly no simple power expression r ( Y ) = Ya wiu satisfy this equation. However, it is easy to show that either of the (25)
+
TherebyLi Y Z w - lr ( Y ) } = l/dg(l e), which gives r ( Y ) = Yfi-2"/(1 e)& on inversion. Returning to the original variables :
+
Z(t)
-- n a ~ ~ t ~ - * d+( 21)D/?r p 1 + exp -[E - Eh]
{I;
1
(26)
m
BjYb'jw do satisfy equation
two power series r ( Y ) = 0
(28) provided that, in each case, the index b and the coefficients Bj are suitably chosen. The procedure for finding b and each Bj is given in Appendix C, wherein is also defined a function Q(w,z) in terms of which the solution to equation (28) is i(Y) = ? r ~ ' ~ 2 Y 1 ~ 2 ~ z 2 0 ~ ( w In , Y oterms t ~ a )of. the original variables the solution is
The numerator of this equation is identical with the expression given in equation (24) for the diffusion-limited current Zd(t). Rearrangement therefore gives E = Eh [RT/nFl ln([l&) - I(r)]/Z(t)}, which is the well-known Heyrovsky-
+
(6) D. Ilkovic, CON.Czech. Chem. Commun., 6,498 (1934). (7l F. G. Cottrell, 2.Physik. Chem., 42,385 (1902). 938
ANALYTICAL CHEMISTRY
(8) J. Heyrovsky and D. Ilkovic, COIL Czech. Chem. Commun., 7, 198 (1935).
The pre-3 term on the right-hand side of this equation is zd(t). For p = '/a, equation (29) proves to be identical with that derived by Matsuda and Ayaba (9) for a totally irreversible reduction at a dropping mercury electrode [see Appendix C, also see Smith, et al. (IO).] There is some simplification to the Q function when p = 'Iz, but no pseudo-steady-state is developed as in the reversible case. As demonstrated in Appendix C, equation (29) reduces drastically when p = 0, yielding
p*'
I (bt 1 microomps
t
/p=5/6
//
(304 where
This result is identical with that first derived by Delahay and Strassner (11) for an irreversible reduction at a stationary electrode. Figure 2 shows the shapes predicted by equation (29) for irreversible current-voltage curves. Notice that the asymmetry of the curve is marked when p is small, but that symmetry improves as p increases, becoming perfect in the p + limit (as is apparent in equation (C12) of the appendix). little asymmetry exists, a fact that has Already for p = been made use of in an empirical method (12, 10) for the analysis of irreversible polarographic waves. The term p appears in both arguments of the Q function [equation (2911 and this leads to a complex dependence of the half wave potential [the potential, &z, at which Z(t) = la(t)/2] on p . As p increases, becomes more negative, in accordance with the equation
-
Figure 3. Current-time relationships for an irreversible reduction, at various values o f p In annotating the axes the same parameter valw were used as in Figure 1, together with khD-'/* exp(-anF [E-Eh]/RT) = 1.0 see.-'/* The constant potential, E, used in this example, lies somewhat negative of El/,
s"L(y"-f&(Y)) - - + L(Y2"-"(Y)) 3 :
s
E In {.sa*
-
.201
anF which is obtained by combining equations (29), (C13) and the definition of El/z. Equation (31) suggests the possibility of determining kinetic parameters from a study of the variation of Elizwith t and p : this suggestion was made previously (13). Because t occurs in equation (29) both as an argument of the s2 function and as a pre-Q term, the dependence of current on time is complex. The behavior is most striking when 0 < p < l / ~ for , then the pre-Q term decreases with time while s2 increases. Figure 3 shows examples of chronoamperometric relationships for a constant potential somewhat more negative than El,z. General Potentiostatic Regime. For a reduction under quasireversible conditions, neither of the foregoing simplifications apply and it is necessary to solve the complete equation (22) subject to constancy of 4 Y). The relationship (9) H. Matsuda and Y. Ayabe, Bull. Chem. Soc. Japan, 28, 422 (1955'1. ,- - - - ,. (10) D. E. Smith, T. G. McCord, and H. L. Hung, ANAL.CWM., 39, 1149 (1967). (1 1) P. Delahay and J. E. Strassner, J. Amer. Chem. Soc., 75, 5219 (1951). (12) L. Meites and Y. Israel, ibid., 83,4903 (1961). 40,65 (1968). (13) K. B. Oldham and E. P. Parry, ANAL.CHEM.,
(32)
43
is easily derived and will be Seen to differ from equation (28) only by the presence of the constant factor (1 6). The solutions of equation (32) therefore parallel closely those given in Appendix C and lead to L( Y) = [ Yt -2"/(1 e) 4&2(0,[1 4 Y " e - 3 in dimensionless variables, or
+
+
+
in experimental variables. When p = equation (33) becomes identical to the expression derived by Koutecky (14) for the general case of reduction at a dropping mercury electrode. (See Appendix C for the relation between Q and the Koutecky function F). When p = 0, equation (33) reduces to
where
(14) J. Koutecky, CON.Czech. Chem. Commun., 18,597 (1953).
VOL. 41, NO. 7, JUNE 1969
939
Let us first consider 0 6 p < '/2. In this case, the final term in equation (36) is initially infinite, so that E(0) = +a. As time passes, however, this term will diminish in magnitude, leading to a negatively-shifting potential. At the time t = T , where T is termed the transition time (17) and is given by
t
the logarithmic argument in equation (36) becomes equal to zero and hence E(T) = - a . For t > T , equation (36) cannot be satisfied-Le., it is impossible for reaction ( 2 ) to continue to deliver a faradaic current of magnitude I. Combination of equation (36) with definition (37) gives, E(r) = Eh
0
I
I
I
I
2
3
4.
t sacs
A temperature of 25 "C has been assumed and other parameters take the valws given in the legend of Figure 1. The dashed lines show the transition time asymptotes for thep = 0, and cases. No tramition time exists for p = '/z and fwp > '/z COMht m t chr0n0potentiometry is Impossible
This equation is equivalent to that originally derived by Smutek (15)and others (16). Reversible Galvanostatic Regime. A constant current implies a constant i and if this condition is inserted into the reversible version of equation (22))there results
+
whence 4Y) = [Y*-zwY(2w 1/z)/rr(2u)] - 1. Written in terms of experimental variables, this becomes
(38)
and shows that (provided Z < n a F C d Z D / r ) the potential should remain constant indefinitely. This condition corresponds to the pseudo-steady-state discussed earlier in this article. Equations (39) and (27) are, in fact, equivalent. equation (36) has no short-time solution, For p > 'Iz, corresponding to the inability of a rapidly expanding electrode to provide a constant faradaic current. This is the state of affairs which pertains for a dropping mercury electrode ( p = */a) and "constant current" electrolysis at such an electrode is feasible only if precautions are taken to avoid high polarization early in the drop-life (20). Irreversible Galvanostatic Regime. When i is a constant, the irreversible version of equation (22) becomes
whence ea( Y) = [ Y*-"/i] - [ Y"r(2w)/r(2w ing to the original variables, we find
E((1)=
(36) This is the equation which expresses the potential-time relation at an expanding electrode at which a reversible reaction is occurring at a constant current 1. Examples of the E us. t relationships are shown as Figure 4. (IS) M. Smutek, Chem. Lisry, 45, 241 (1951); Coll. Czech. Chem. Commun., 18, 171 (1953). (16) P.Delahay, "New Instrumental Methods in Electrochemistry," Interscience,New York, 1954,Chap 4.
ANALYTICAL CHEMISTRY
11
For p = 0, equation (38) is identical with the chronopotentiometric equation for a stationary electrode, first derived by Karaoglanoff (18), and for this case equation (37) gives T = lm2aZF3C2D/41z, the well-known result due to Sand (19). With p = 1/2, equation (36) gives
Figure 4. Chronopotentiogramsfor reversible reductions at a constant current I of 4.00 microamperes
940
+ RT nF In {(:)'-'-
Eh
+ '/z)]. Revert-
-k
As in the previous section, it is convenient to examine equation (41) separately for different p values: here we shall consider successivelyp = 0,O < p < '12, p = 7'2, and p > '12. (17) J. A. V. Butler and G. Armstrong, Proc. Roy. Soc. A , 139, 406 (1933). (18) Z.Karaoglanoff, 2.Elekrrochem., 12, 5 (1906). (19) H.J. S.Sand, Phil. Mug,1,45 (1901). (20) H.B. Mark, E. M. Smith, and C. N. Reilley, 1. Electrounul. Chem., 3,98 (1962).
When p is zero, the argument of the logarithmic term in equation (41) is initially positive, but becomes progressively less positive until it vanishes at a transition time r given by the Sand expression mZa2FzC2D/4Z2.Accordingly, equation (41) may be rewritten as
where
a result originally due to Delahay and Berzins (21). For p lying between 0 and lI2, the initial potential E(0) is - a,but E rapidly becomes more positive, passes through a maximum and then drifts negative again, reattaining - a at a transition time r given by equation (37). The chronopotentiometric relation is
an equation of which (42) is seen to be a special case. Figure 5 shows plots of this chronopotentiometric relationship for p values of 0, ' 1 6 , and l / a , as well as the uniquep = ' 1 2 case. With p = l / 2 and provided that Z < n a F C d F r the initial potential, E(0) = - a, but no transition time is developed. Instead the potential drifts to ever more positive values in accord with the relationship
As for the reversible case, p > ' 1 2 leads to no meaningful solution of equation (41). General Galvanostatic Regime. A solution of the unsimplified equation (22) subject to the constancy of 1 is
1 + rr (( 224~+Y~~- * [1 + 4 Y ) I = 1
(45)
and this may be converted to experimental variables by employing equations (10) through (13). The result, representing a situation intermediate between those of the two preceding sections, cannot be written to give the potential as an explicit function of time. Moreover, the expressing of time as an explicit function of potential is possible only for the special cases p = 0 or ' 1 2 . These special expressions are, respectively
kh2
[+ 1
"' expt;
[E(t) -
(46)
'%I}]
3
4
Figure 5. Chronopotentiograms for totally irreversible electroreductions at constant current In addition to the parameter values used in constructing Figure 4, the values a = 0.5 and kh = 8.63 X 10- cm sec-l were assumed. Notice that the transition times are identical with those in Figure 4, but that there is a more leisurely approach to the asymptotes. Note also that all E(0) = - m ,except for the p = 0 case wherein E(0) is finite (and equal to En 200 mV for the values used here)
the first of which has been reported earlier (22). However, it is evident from equation (45) that a transition time is developed generally for 0 p < ' 1 2 and that its magnitude is given by equation (37). Reversible Linear Current-Scan Regime. In this section and the next two, the situation is treated in which the current through the expanding electrode increases linearly with timeLe.,
