SIDNEY BARNARTT AND CHARLES A. JOHNSON
4430
Kinetics of Higher Order Electrode Reactions at Constant Potential
by Sidney Barnartt and Charles A. Johnson Edgar C. Bain Laboratory for Fundamental Research, United States Steel Corpmation, Research Center, Monroeville, P e n m y h a n k (Received May 2.4, 1967)
The effects of charge-transfer parameters and reactant concentrations on the kinetics of higher order electrode reactions are determined. The approximate closed-form solution of the diffusion problem for constant potential yields current-time curves which are compared with exact curves generated numerically. It is shown that the range of validity of the approximate solution is unaffected by the exchange current density or the transfer coefficient, but is sensitive to reactant concentrations and to the sign and magnitude of the overpotential. A method of obtaining accurate values for the ultimate concentrations of the reactants a t the electrode surface is given. From these values, which depend only on the initial concentrations and the applied-potential step, a good assessment of the accuracy of the approximate solution can be obtained. Even in unfavorable cases, the range of validity of the approximate solution encompasses the range of experimental data most useful for evaluating charge-transfer parameters.
Introduction I n a recent paper, we have generalized the theory of electrode kinetics at constant potential to include charge-transfer mechanisms of higher order, coupled with linear diffusion. The closed-form solution of the diffusion problem given therein is an approximate one, valid for small concentration changes. Its utility is determined primarily by the length of time and the corresponding current range, over which it reproduces the true current-time curve within a stated maximum deviation. It was shown that this range of validity decreases as the reaction "pseudo-order" with respect to one of the diffusing reactants increases. The effects of charge-transfer parameters or reactant concentrations on the range of validity, however, were not established. A study of these effects is presented in the present communication. Consider the general electrode reaction
+
~ B Z+' z. + ne- = yy[(bz-n)/vl+ww
(1)
with the restrictions that only substances B and Y exhibit significant concentration changes at the electrode and that all of the electrical work involved in the reaction occurs during the rate-determining step. The net anodic current density a t overpotential 7 may be written2 The Journul of Physical Chemistry
i
=
io exp( -p:q){
(s)"~ exp( CY0
-
(")7 CBo
(2)
where io is the exchange current density, p is the transfer coefficient, CY and CB are the time-dependent concentrations a t the electrode surface, cyo and cBO are the bulk concentrations, v is the stoichiometric number, and c = F/RT. Reaction 1 may be described' as being of pseudo-order y/v with respect to Y and b/v with respect to B. The approximate solution for the current-time curve a t constant potential is1
i/i,=o = exp(i2t) erfc(x4i) with
where DY and D B are the diffusivities of substances Y and B in the solution. This has the same general form as the rigorous solution for simple first-order reac(1) S. Barnartt and C. A. Johnson, Trans. Faraday Soc., 63, 431 (1967). (2) S. Barnartt, Electrochim. Acta, 11, 1531 (1966).
KINETICS OF HIGHER ORDERELECTRODE REACTIONS AT CONSTANT POTENTIAL
UPPER CURVES
443 1
UPPER CURVES
-
-?
0.6-
.
.--
.-
0.4
c
-4 0.2
-
*.
t t-
-1
LOWER CURVES _ .
Figure 1. Effect of the concentration of the second-pseudo-order reactant Y on anodic current-time curves: solid curves, exact numerical solution; dotted curves, approximate solution.
In order to establish how closely eq 3 approximates the true current in specific cases, we have developed exact numerical solutions using the Schmidt meth~d.~ The cbxact current-time curves are compared with the corresponding approximate curves calculated from eq 3, for a reaction of pseudo-order 2 with respect to one of the diffusing species and of pseudoorder 1 with respect to the other. ti on^.^,^
1
fi
(sec)”2
Figure 2. Effect of overpotential on anodic current-time curves: solid curves, exact numerical solution; dotted curves, approximate solution.
UPPER CURVES
Validity of the Approximate Solution The curves in Figure 1 illustrate the effect of varying the concentration of the second-pseudo-order reactant, other parameters remaining constant. The small bar \ on each exact curve gives the time to.osa t which the I *.\ 0.4 . ‘ * corresponding pair of curves differ by 5%; we desig1 J nate the range of current ratio from unity to il=to.o,/ it=o as the range of validity of eq 3. Figure 1 shows that an increase in concentration of the pseudo-secondo.2 order reactant extends the range of validity. The two pairs of curves plotted in Figure 1 are the 0 2 4 6 8 10 12 first two cases listed in Table I. From cases 2, 3, and 4 LOWER CURVES of this table it is seen (last column) that a change in Figure 3. Anodic current-time curves for reaction of transfer coefficient from 0.25 to 0.75, for 7 = 0.01 v and pseudo-order 2 with respect to anodic reactant Y (lower io= 5 X low3A/cm2,has no effect on the range of validcurves) or cathodic reactant B (upper curves): solid curves, ity of the approximate equation. Cases 7, 8, and 9, exact numerical solution; dotted curves, approximate solution. which refer to a higher overpotential ( X 10) and lower io(XO.OOZ), also show no effect of p ; note that the range of validity is appreciably shorter here. (3) H. Gerischer and W. Vielstich, 2. Physik. Chem. (Frankfurt), 3 , The effect of changing iowhile keeping the other pa16 (1955). rameters constant, is evident from cases 2, 5 , and 6. A (4) C. A. Johnson and S. Barnartt, J. Phys. Chem., 71, 1637 (1967). 500-fold increase in ioproduced no appreciable effect ( 5 ) J. C. Jaeger, Proc. Cambridge Phil. SOC.,46, 634 (1950). 8-
t
I
t
1
Volume 71. Number 13 December 1967
SIDNEY BARNARTT AND CHARLES A. JOHNSON
4432
Table I : Range of Validity of Eq 3 for Reaction Second Pseudo-Order in Y" Case no.
1 2 3 4 5 6 7 9 10
Dy
io,
V
A em-4
0.01 0.01 0.01 0.01 0.01 0.01 0.1 0.1 0.1 0.1
8
a
?1
=
5 5 5 5
B
x x x x
10-6 and DB = 2
10-3 10-3 10-3 10-3 10-4 10 -6 10-6 10-6 10 -6 10-6
x
0.5 0.5 0.25 0.75 0.5 0.5 0.5 0.25 0.75 0.5
CY0,
CBo9
mole cm-8
mole
x,
cm-8
sec-1'2
10 -4 10 -6 10-6 10-5 10-5 10-5 10 -6 10-6 10-6 10 -*
5 5 5 5 5 5 5 5 5 5
10-6 cmz sec-1; y = n = 2 and b =
on the range of validity, although it caused a decrease of more than 5 orders of magnitude. in The effect of overpotential is illustrated by cases 6 and 7, the results for which are plotted in Figure 2. An increase in 9 , other parameters being constant, causes the range of validity to contract appreciably. In all of the cases listed in Table I the deviations of eq 3 are negative; i.e., the approximate solution always yields too small a current. In Table I the second-pseudo-order component is the main anodic reactant, the first-pseudo-order component being an anodic product. The effect of reversing this situation, so that b = n = 2 and y = v = 1, is shown in Figure 3. The concentrations, overpotentials, and charge-transfer parameters v, and io) are exactly the same in the two cases. It is seen that the approximate solution exhibits negligible deviations, even at i/itz0 as low as 0.3, when the anodic reactant is the first-pseudo-order component. These effects may all be explained with the help of the general i/?relation as formulated in eq 2. Consider first the condition y / v = 2, b / v = 1. (a) Anodic reactant Y will contribute little to the mass-transfer effect (decrease of current) as long as cy/cy0 remains close to unity. I n this case the system approaches the behavior of a first-pseudo-order reaction, and eq 3 approaches the true current-time behavior. A decrease in cyo at constant CBO is associated with a more rapid decrease in cy/cy0 with time and thus contracts the range of validity of eq 3, as in Figure 1. (b) When cy/cy0 is appreciably less than unity, any increase in the first term in braces in eq 2 relative to the second term, i.e., any increase in the anodic: cathodic current ratio, enhances the higher pseudoorder effect and contracts the range of validity. This current ratio is not affected by @ or io; hence neither P
(a,
The Journal of Physical Chemistry
x x x x x x x x x x
10-6 10-6 10-6 10-6 10-6 10-5 10-5 10-5 10-6 10-5
Y
= 1.
0.5622 4.916 5.972 4.046 0.09831 0.009831 0.3218 2.255 0.04593 321.9
tQ.06,
88C
4.9 0.042 0.026 0.059 97 9 . 6 x 103 3.1 0.063 153 3.0 x
it-ro.ar/ it-0
0.390 0.45 0.460 0.455 0.460 0.460 0.615 0.615 0.610 0.615
nor i o should affect the range of validity, as observed (Table I). (c) The anodic:cathodic current ratio is quite sensitive to 7, so that the range of validity should contract markedly as ? is increased. This is in accord with Figure 2. Here the curve at q = 0.1 v is completely dominated by the second-pseudo-order substance (anodic:cathodic current ratio = 2411 at t = 0) and hence exhibits a minimum range of validity. Under these conditions, any changes which greatly reduce the mass-transfer effect of the first-pseudo-order component should cause no further contraction in the range of validity. That this is so may be seen in Table I, cases 7 and 10, where a decrease in the concentration ratio CYO/CBO from 0.2 to 2 X lo-* caused no change in the range of validity while decreasing t 0 . 0 6 by a factor of lo6. Next consider the inverse condition: y / v = 1, b / v = 2. At high q the first-pseudo-order component controls the current; hence eq 3 should exhibit little deviation from the true current-time curve, as was found (Figure 3). In like manner, eq 3 could be made accurate for y / v = 2 and b / v = 1 by working at large negative values of 1, where again the first-pseudoorder component controls the current. As pointed out by Gerischer and Vielstich3 and emphasized by the work of Oldham and Osteryoung16 first-order reactions become predominantly diffusion controlled below i/it=o rr-' 0.5. Data a t current ratios above 0.5 should be utilized to extract the chargetransfer kinetic parameters. This rule may be applied as a rough guide for reactions of higher order as well. The examples discussed above show that the range of (6) K. B. Oldham and R.A. Osteryoung, J. Electroanal. Chem., 11, 397 (1966).
KINETICS OF HIGHER ORDERELECTRODE REACTIONS AT CONSTANT POTENTIAL
validity of t,he approximate solution includes almost all of the current range of interest.
Analysis of Experimental Current-Time Curves The approximate solution permits us to analyze an experimental current-time curve to obtain the charge, well as A. At short times, transfer current i t = O as > 0.85, eq 3 reduces to3 such that i/its0
i/i,=o= 1 - (2/dX)Xdi (4) Thus, a plot of i us. d$ is linear with intercept it=o, and the slope yields A. In this current region any inaccuracies of the approximate solution are expected to be within the uncertainty of the experimental curve. When current ratios >0.85 are inaccessible, the experimental current-time curve at longer times must be analyzed. A convenient analytical procedure for this range, reported previously, utilizes the ratio of experimental currents at times t and 4t. From a master curve4 of i(t)/i(4t) against X d t we obtain a value for X; then In applying from eq 3, a corresponding value for it this method to higher order reactions, we note that the larger the selected value of t, the larger the deviations of' eq 3 from the true curve and of the apparent vstlups of X and it,o from their true values. We have found, however, that the i ( t ) / i ( 4 t ) method will yield good results if the apparent values of itE0 or X are plotted against z/i and a linear extrapolation is made to t = 0. As an example, case 7 of Table I was analyzed in this way over the range 0.86 > i / i t = o > 0.61, the lower limit of current being that at time The "raw" current-time data were taken from the exact numerical solution, which corresponds to the A and X = 0.3218 == 4.909 X true values it=o - '/? . Figure 4 shows the plots of apparent i t = o and apparent X against dt and illustrate that linear extrapolation back to t = 0 yields the true values (indicated by horizontal bars) to a good approximation; i t 5 0 so obtained is 1% high and X is 2.5% low. Procedures for evaluating the charge-transfer parameters io, p, and Y from the quantities it,o and X have been described previously.
U1timate Concentration Changes We examine next the problem of calculating the maximum concentration changes which can occur potentiostatically (as t + rn) at the electrode surface. The approximate analytical solutions for the concentration changes may be written' uy
=
cy -
cy0
CY0
---
UB =
4433
- CBo Cgo CB
-{ 1 - exp(X2t)erfc(Xdi)]
Zt-0 (n/b)FcBOXdDB
(6)
As t + , the term in braces approaches unity, so that the maximum fractional concentration change U B ,or~ UY,,,,is predicted to be just the term in front of the braces. This can be grossly in error, however, because eq 5 and 6, as well as (3), are not accurate for large changes in the concentration of reactants of pseudo-order greater than unity; in fact, maximum concentration changes determined in this manner were found to be 50% too low for cases 7-9 of Table I.
.'. .
5 4.90'
2 4.75 2
' L
L
0.341 i
I
8 0.32+-
0.30 c
0.28 0.26 0
0.4
0.2
0.6
0.8
JT, Figure 4. Apparent values of
and A, case 7.
Accurate values of the maximum concentration change are calculable by use of the following two relations. The first is a general theorem, derived in a previous paper:' for all t
u B = - hod%UY yCB0d& -
(7)
--a
The second is based on the fact that as t + the current goes to zero for the conditions considered here (planar electrode, mass transfer by semiinfinite linear diffusion, and higher order reactions with electrical work confined to the rate-determining step). In the case of first-order reactions, proof that i + 0 as t -+ 03 has been given.3 A general proof that i 0 as t -+ for reactions of any pseudo-order and that there is
-
Q)
Volume 71, Number 19 December 1967
SIDNEY BARNARTT AND CHARLES A. JOHNSON
4434
precisely one pair of realizable values U Y , m and UB,m, which correspond to i = 0, is given in the Appendixes. The condition of zero current at very long times signifies that, the usual thermodynamic relation exists between the electrode potential and the actual surface activities of reactants and products -2/
?I = R T ln[*l”[*]
(8)
nF
We may assume that the activity coefficient ratios involved are unity under the conditions for which eq 2 is valid (large excess of supporting electrolyte) and may rewrite this equation as (1 4-
UB,m)b(l
4-
aUy,m)b(l
+
uY,m)-’
= exp(nFq/RT)
This result permits direct evaluation of UB,m
=
U Y , ~as
well as
The approximate solution given by eq 3 is based on the assumption that the fractional concentration change of the higher pseudo-order reactant is small. Thus U Y , (the ~ nux xi mum change) should be a good measure of the accuracy, or range of validity, of eq 3. According to eq 9 u Y , ~is a function only of overpotential and initial concentration ratio. In Figure 5 we have plotted contours of constant u Y , ~as a function of q and a = bcyo~%/ycBo& for the case n = y = 2, b = 1. The qualitative behavior of these contours of constant U Y , is~ the same for any set of pseudo-orders. As appears from eq 9 for a > 1 bRT In a 7 -- A + =
(-Uy,m)
- y In (1 + UY,,)]
(12)
and the contour is an ascending straight line on the q us. log a plot. For values of q and a which lie anywhere on, say, the U Y , ~= 0.5 contour, the range of validity of eq 3 is expected to be roughly the same. As an example of the reliability of this prediction, we have computed values of q and a for which the range of validity (as determined by comparison of approximate and numerical solutions) extends precisely to a current ratio of 0.5. These values are plotted in Figure 5 as the dashed curve which, particularly a t high and a t low a,does tend The Journal of Physical Chemistry
100
1000
to be parallel to the contours of constant U Y , ~ . Thus Figure 5 (or corresponding plots for other values of g and b) provides a reasonably good a priori choice of overpotential and concentrations which will make eq 3 represent the true current-time behavior within any desired accuracy. It has previously been mentioned that, when the overpotential is sufficiently high, the current is almost completely controlled by the higher pseudo-order component, and the range of validity of eq 3 is reduced to a minimum, namely, 1 > i/it,o > 0.62. The shaded region of Figure 5 shows the values of ?I and a for which the range of validity is within -0.2% of this minimum. Throughout the shaded region eq 3 is essentially as poor an approximation to the true currenttime behavior as it can be. The specific cases 1-9 of Table I are shown as points on Figure 5 . The ranges of validity found for these cases are clearly reflected in the positions of the corresponding points relative to the contours of constant UY,m.
where A is a constant a t constant U y , m
A = (RT/nF)[b In
IO
(9)
-QUY,m.
-UY,m
I
Figure 5. Contours of constant UY,, for net anodic reaction, y = 2 and b = 1.
Combination of the eq 7 and 8A yields
(1 -
0. I
(W
= exp(nh/RT)
UY,m)-’
0.01
Conclusions This study reveals some general rules concerning the validity of the approximate current-time relation for higher order electrode reactions a t constant potential. (1) The range of validity depends only on the pseudo-orders, the initial concentrations, and the overpotential. It is independent of the exchange current density and the transfer coefficient. (2) Accurate values of the ultimate concentration changes at the electrode surface are obtainable (eq 9 and Figure 5 ) . These values provide a good indication of the accuracy of the approximate solution: the smaller the ultimate fractional changes in concentra-
KINETICS OF HIGHERORDERELECTRODE REACTIONS AT CONSTANT POTENTIAL
tion of higher pseudo-order components, the better the accuracy of the approximate solution. (3) The range of validity of the approximate solution is extended greatly by ensuring that the current is controlled predominantly by the component (reactant or product) of lower pseudo-order. Thus if the higher pseudo-order component is a cathodic reactant, it is desirable to apply positive overpotentials (net anodic reaction), and the range of validity increases with increasing overpotential. (4) If, on the other hand, the current is controlled predominantly by the higher pseudo-order component (as in most of the cases examined in this paper): (a) the range of validity generally increases with increasing concentration of the higher pseudo-order component but is insensitive to concentration at low concentrations; (b) the rmge of validity decreases with increasing overpotential and reaches a minimum value at sufficiently high overpotentials. For reactions second pseudo-order with respect to one of the diffusing reactants and first pseudo-order with respect to the other, this minimal range spans 1 > i / i t 3 0 > 0.62 and covers most of the current range over which the reaction is controlled to an appreciable extent by the chargetransfer mechanism.
4435
To facilitate the application of the uniqueness theorem we make a change of variable. Let U Y , , be the solution of i ( t ) = 0 which lies in the interval -1 2 u y 5 0; that there is always precisely one such solution is shown in Appendix 11. We now proceed to show that lim (uY) = UY,, and, therefore, that t-
m
lirn (i(t))= 0. We define a new variable v ( x , t ) by 1-
m
UY(X,
t)
= uy,,[v(x,
t ) ] , and the problem becomes b2v
Dy-
=
ax2
with v(x, 0)
=
0 , v ( x -+
00,
t)
bv at
---c
0, and
This last condition can be written (note that negative) as
-(E)
=
z=o
[ks(l
io
G(v) =
+
( n / v ) F c y 0 D y ( -U Y ~y,m~)’/”
- kc(1 -
UY,,
is
X
,d
C Y U Y , ~ U ) ~ / (-45) ”]
G(v) has the three properties required for the applicability of the uniqueness theorem:’ (1) G(v) is analytic in the interval 0 2 v 5 1; (2) G(l) = 0, since U y , m is a root of i(t) = 0; (3) G(v) is a monotonically decreasing function of v in 0 5 v 5 1 , since
Appendix I Properties of the Exact Solution for Higher Order Reactions We show here that the diffusion problem’ for reactions of any positive pseudo-orders y/v and b / v has a unique solution and that lirn ( i ( t ) ) = 0. The proof is t-
m
by appeal to a powerful uniqueness theorem established by lllanri and Wolf? The diffusion problem, written in terms of component Y only, is1
With these conditions established, we now use theorem 11 of the Alann-Wolf paper and conclude that: (1) v ( x , t) is unique; (2) v(0,t ) is nondecreasing for all t > 0; (3) v(0,t) < 1 for all t > 0; (4) lim (v(0,t ) ) = 1. 1-8
with U Y ( X , 0) = 0, U Y ( X -t
00
, t>
+
0, and
m
In terms of the original problem, this means that there is a unique solution for u Y ( x , t ) , that lim (uY(0,t ) ) = t-m
uY,,, and therefore that lirn ( i ( t ) ) t-
io [ka(l (nlv)F Here
UY
= (CY
-4-
bCyo
a = y cnO
+
- kc(l -
~y)”’”
a~y)~’” ] (A2)
- cy0)>cyoand
Dy. DB’
k, = exp[(l - P)(n/v)qJ; k, = exp[-@(n/V)Eq]
=
0.
m
Appendix II Uniqueness of the Solution of Eq 9 The existence of precisely one value of UY in the interval -1 5 UY 5 0, corresponding to 0 5 CY 5 cyo, (7) W. R. Mann and F. Wolf, Quart. Appl. Math., 9 , 163 (1951).
Volume 71, Number 13 December 1967
MARIOGOFFREDI AND THEODORE SHEDLOVSKY
4436
which satisfies eq 9 will be proved by showing that the function f(Uy)
= (1
-
auyy
[exp(nFrl/RT) 10
+
UY)'
throughout this interval. Differentiation of eq A7 yields
647)
where a 2 0, 7 1 0, b 1 1, and y 2 1, has precisely one root in -1 I UY I 0. There is at least one root of eq A7 in this interval, sincef(-1) = (1 > 0, andf(0) = 1 - exp(nPq/ RT) 2 0. To show that there is a t most one root in -1 2 uy 2 0, we show that df/duy is negative
+
y[exp(nFrllRT) 1(1
+
UY)"'
(AB)
+
Now (1 is positive and (1 uY)#-' 2 0 in -1 I UY 5 0 which, together with the conditions a 2 0, b 1 1, y 1 1, and 7 2 0, yields df/duy < 0 in -1 I UY I 0. Hence eq 9 has precisely one solution, designated UY,,, in -1 5 UY I 0.
Studies of Electrolytic Conductance in Alcohol-Water Mixtures.1 V. The Ionization Constant of Acetic Acid in 1-Propanol-Water Mixtures at 15, 25, and 35"
by Mario Goffredi and Theodore Shedlovsky The RockejeEler UnCersity, New York, New York 10081 (Received M a y 8.4, 1967)
Measurements are reported on the conductance of dilute solutions of acetic acid and of sodium acetate a t 15, 25, and 35' in 1-propanol-water mixtures over the entire range of solvent composition. From these data and those on hydrochloric acid and sodium chloride solutions in the same solvent system previously reported by us, values for A. and the ionization constants, K, for acetic acid have been computed for these temperatures over the entire solvent composition range. The results are discussed briefly from theoretical considerations.
Introduction I n this paper, we present data on the electrolytic conductance of dilute solutions of acetic acid and also of sodium acetate a t 15, 25, and 35" in 1-propanol-water mixtures over the entire range of solvent composition. The acetic acid ionization constants in the mixed solvents that contained 0,20,40,60,80,90, or 100 wt % 1propanol were obtained from these data supplemented by those from our work on sodium chloride2 and on The Journal of Physical Chemistry
hydrochloric acida solutions in a manner previously de~cribed.~
Experimental Section The preparation of the 1-propanol-water mixtures and their physical properties have been detailed in a (1) This research was supported by the National Science Foundation through Grant No. GB-3062. (2) M. Goffredi and T. Shedlovsky, J. Phys. Chem., 71, 2176 (1967).
