Chronopotentiometric Transition Times and Their Interpretation W. H. REINMUTH Department of Chemistry, Columbia University, New York 27, N. Y.
F A discussion is given of chronopotentiometric processes in which charge transfer is preceded b y adsorption or chemical kinetic complications. Exact equations are derived for a number of cases of interest. Diagnostic criteria are proposed by which various types of mechanisms may b e distinguished from one another. Implications in the development of analytical procedures are considered.
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work ( 8 )involved examining the effect of complications other than simple diffusion on chronopotentiometric processes for cases in which the complication follows the charge transfer reaction. It seemed of interest to extend the investigation to cases in which the complication precedes the electrode reaction. I n cases of the former type the transition time is unaffected by the complication so that detection rests on close scrutiny of the potential-time curves. I n the latter circumstance, however, variation of transition time with experimental parameters is often of value in elucidating mechanism. When feasible, investigation of variation of transition times seems preferable to analysis of potential-time curves because of the greater precision and accuracy attainable [compare, for example, (IO) and (11) for investigations of both types on the same system]. The present discussion will therefore consider only transition time measurements. The work is divided in four sections dealing with chemical kinetics of two types, with adsorption, and finally with comparison of the characteristics of these and other cases. ECEST
MECHANISMS OF THE TYPE p Y S 0
R
Gierst and Juliard (4)treated the particular case of first order chemical kinetics ( p = 1) by the approximate Nernst diffusion layer method. Delahay and Berzins (2) revived the more rigorous treatment of Rosebrugh and Miller (9) and applied i t experimentally to a number of cases of chemical interest. Qualitatively, no matter what the value of p , i t is readily apparent that at low current densities (or large values of
322
ANALYTICAL CHEMISTRY
the chemical rate constant) i+12 approaches a limiting value identical to that i t would have if no complication were present and all of the reactant were in the 0 form. At very high current densities (corresponding to short transition times) the amount of Y converted to 0 during electrolysis becomes negligible and i+ approaches a limiting value identical to that it would have if no complication were present and the total concentration of the oxidized form were the initial concentration of 0. At intermediate current densities i ~ 1 1 2 varies between these limiting values. From these facts i t is apparent that if the chemical equilibrium favors species 0 the chemical complication has little effect on the experimental results since the upper and lower limiting values of iT1” approach one another under these conditions. Moreover, by restricting consideration to cases in which the equilibrium strongly favors Y , major simplification of the theoretical treatment can be achieved. The theoretical derivation for the general case of any p with the afore-mentioned restriction is given in Appendix I. Only the results will be discussed in the present section. For the limiting case in which the rate constants for the chemical process are infinitely rapid and the chemical equilibrium favors the Y form, the surface concentration of 0 is given by the equation
x
=
0 ; Co = ppKe(C7 - 2 i 0 t “ ~ / rl%FD1/2)p (1)
and the transition time (time a t which Co = 0) behaves in the same way as i t would if no complication were present. The function iW is independent of current density and directly proportional to concentration. When the rate of attainment of chemical equilibrium is so slow that no interconversion of 0 and Y can occur during the time of electrolysis, then
x = 0; Co
=
( p C ~ ) p K e2iot’i ’/ rl/%F D
(2)
The transition time is shortened considerably from its value for rapid equilibrium (Equation 1). The function i + / 2 is still independent of cur-
rent density, but is proportional to concentration to the p power. When the rate of attainment of chemical equilibrium is intermediate between the two extremes, the expression for transition time is
which for the particular case p = 1 simplifies to the result given by Delahay and Berzins ( 2 ) under the same conditions. It also simplifies to the limiting cases of Equation 1 or 2 as k, approaches infinity or zero, respectively. When k-7 2, Equation 3 simplifies to
>
This equation says effectively that at constant concentration a plot of W2us. i l / p gives a limiting slope at low current densities which is linear and equal to slope
=
7rl
- (nFD”2)l-l’p(k,’/2K,)--lip ( 5 ) 2P
The diffusion coefficient can be determined from the extrapolation of this plot to zero current density so that if the stoichiometry and equilibrium constant are known, the rate constant can be determined. An alternative experimental method of accomplishing the same end is to hold the current density constant and plot +2as a function of concentration. Since ~ , 1 / 2 is directly proportional to concentration under these circumstances, extrapolation to T = 0 gives the second term on the right hand side of Equation 4 directly. The slope of this plot yields the diffusion coefficient and again k, can be deduced if K , and p are known. This procedure was followed, apparently unwittingly, by Moorhead and Furman ( 7 ) in the case of gallium in thiocyanate medium. For that case, the extrapolation a t two current densities yielded the same intercept a t T = 0. It follows therefrom that
p = l . Presumably the kinetic complication in this case is partial predissociation of the thiocyanato complex. I n the more general case, can be determined from the variation of intercept with current density. It is worthy of emphasis that when kinetic complications precede the charge transfer reaction i + 2 is not directly proportional to concentration (though, as in the gallium thiocyanate case, they may be linearly related if p = 1). This is in contrast to the situation in polarography where limiting current (at least for first order kinetics) is proportional to concentration. It is therefore imperative that the constancy of i+!Z with current density be established before proportionality t o concentration is assumed. Fortunately. when unwanted kinetic complications appear in the development of an analytical method a simple procedure suffices to negate the effect. It is only necessary that i + / 2 for each sample be determined at several current densities and extrapolated to zero current density. For first order kinetics. extrapolation is linear for i + 2 us. i. For higher order kinetics, irl!? us. i 1 ' p gives linear extrapolation. The appropriate plot can be deduced readily by inspection of the data. Davis and Garichoff ( 1 ) recommend another approach in the same circumstances. They suggest adjusting the current densities in such fashion that the transition times are approximately equal for all samples studied. The practicality of this approach depends on the equipment employed. If currents are conveniently variable only in fixed increments the approach may not be feasible. Although ostensibly only one cshronopotentiogram need be run 011 a sample with this procedure as compared with several for extrapolation, the necessity of finding the current density required t o produce the desired transition time makes the advantage illusory in many cases. The estrapolation procedure might be expected to yield better reproducibility between samples because the result obtained by extrapolation is independent of the characteristics of the kinetic process while that from the constant transition time method is not. Results in the latter case are thus inherently more susceptible to changes in temperature and solution characteristics. I t should be mentioned that, although Equations 3 and 4 formally represent the scheme p Y 0 R, they more generally describe the behaT-ior of any system of the type
-
*
ZX
+ p Y e 0 + 22;
0
-
R
(6)
provided that the concentrations of all substances X and 2 can be held constant during the electrolysis or are in
large excess of the concentrations of 0 and Y . MECHANISMS OF THE TYPE Y S p 0
+
R
Mechanisms of this type are similar in concept to those of the preceding section. Distinction is made here only on the basis of the mathematical approach in treating the problems. For the cases of infinitely rapid and infinitesimally slow reactions, the treatment is entirely analogous to that of the preceding section. For kinetics of intermediate rate, the mathematical approach of the previous section is unfruitful except in the special case p = 1. Koutecky and Cizik (5) have treated the case p = 2 by assuming conditions analogous to Equation A13 and making further approximations. Dracka (3) has restated and elaborated on their results. The behavior of i+I2is the same qualitatively as in the cases of the preceding section. However, distinction can be made from the fact that the rate ~ io is higher of decrease of i ~ 1 'with than first power in io rather than lower. ADSORPTION WAVES
Electrode processes in which adsorption is a factor can yield variations of i & 2 with i which are similar to those obtained with kinetic complications. Therefore, it is instructive to consider these cases. Unfortunately, the number of possibilities makes consideration of all of the alternatives unfeasible. Among the complications are the variety of adsorption isotherms which can be obeyed by the adsorbant, the possibility of slow kinetics of adsorption, preferential adsorption of either the oxidized or reduced forms, the greater or lesser facility of reduction of the adsorbed species as compared with its soluble analog. On the other hand, because the electrode is stationary some of the complications of polarography are avoided-e.g., adsorption equilibrium is generally established prior t o electrolysis. We shall consider first the two cases which would be expected to be the most frequent on the basis of polarographic results, those in which a complete monolayer is established prior to electrolysis, and that monolayer is reduced either before or after the solvated analog to a monolayer of reduced form, the cases of polarographic pre- and postwaves, respectively. Secondly, we shall consider as representative of cases in which adsorption is less than monolayer coverage and adsorption equilibrium is maintained during electrolysis, the case of a linear isotherm. The mathematical details of the treatments are given in Appendix I1 and only the results will be discussed here. Assuming both oxidized and reduced
forms to be adsorbed and the adsorbed form to be more easily reduced, two potential breaks are observed-the first for reduction of the adsorbed form, the second for the soluble species. Because the amount of adsorbrd species is limited by the surface area rather than by the concentration of the corresponding form in the solution, the total charge needed to reduce the adsorbed layer is a constant. That is, by Faraday's law iorl =
nFr*
(7)
When reduction of that layer is complete, reduction of the solvated species begins and obeys the conventional chronopotentiometric relations for diffusion limited waves with the restriction that observed times must be measured from 71. If the adsorbed oxidized form is less easily reduced than the solvated species, then the conventional equations are obeyed for the first transition time, due to the soluble species. However, the second wave. due to the adsorbed form, is complicated by the fact that a large fraction of the current is still devoted t o reduction of the soluble spec7ies even after its surface concentration is reduced to zero. The expression for transition time takes the form
2
dZ2
(8)
which, of course, reduces to Equation 7 for the special case T~ = 0. The behavior of the second transition time is qualitatively similar to that of the second transition time for reduction of two soluble species. It is disproportionately prolonged on increase of the concentration of the solvated form-that is, by increase of 71. The case of a linear isotherm is of particular interest not only because it may describe the adsorption of species commonly thought of as adsorbable, but also because it gives a t lcast qualitative insight into the behavior of ions which may be specifically adsorbed in the electrical double layer. The expression for transition time, given originally by Lorenz (6),is
When the argument of the last term is large, this simplifies to 7112
=
rrn112
+ 7r112Ka/2D1'2
(10)
For very small 7 , the last term of Equation 9 can be expanded in a McLaurin series t o yield 7 = 2K,,r,l/2/~l/~D1!2 = K,C*/aFio (11)
which says, as might be expected intuitively, that a t sufficiently high curVOL. 33, NO. 3, MARCH 1961
* 323
rent densities only the species originally adsorbed contributes to the observed transition time. The feature of interest in all of the adsorption cases above is the fact that the contribution of adsorption to the observed result increases and that of diffusion decreases as the current density increases. This means that a plot of i+12 us. current density has a positive slope rather than a negative one. A plot of us. C extrapolates to a negative intersection with the abscissa rather than a positive one. I n addition, the temperature coefficient of 7 tends to be negative in the case of adsorption, rather than positive as in the case of a kinetic effect. DISCUSSION
The problem facing the experimentalist is to determine for the particular system in which he is interested vvhether complications other than diffusion are involved in the electrode process. Two restrictions are placed on this aim by the technique itself. First, by examining only transition times the investigator limits himself to complications preceding charge transfer because transition times are directly dependent on surface concentrations of the unreacted species. This generalization breaks down nhen the reaction products can themselves undergo further electrochemical reaction or can react with substrate or initial reactant t o produce new negative species. These more complicated cases are neglected in the present discussion. Second, in reaction schemes involving chemical kinetic steps, the effect of these steps on the scheme is observable only when the rate constant for the chemical reaction and the transition time are of comparable magnitudes. K h e n the reaction rate is large, the behavior is the same as if no complication were present and both I’ and 0 were reducible. For those familiar with polarographic practice it may be convenient to consider the similarity of the two techniques and the analogy of correspond~~
~~~
Table I. Diagnostic Criteria for Distinguishing Chronopotentiometrically between Reaction Mechanisms hIech- ( b i T 1 / 2 / ( b z i ~ 1 / 2 / bi) ai2) C r - ~ a ( d ~ / b t ) anism
0-R 0 0 0 + Y*0R 0 pY*O -tR Y ep o ; O+R 0 (adsorbed) +R Variable a Obtained by extrapolation of plot of
+
+ ++ + ++ + ++
+
708.
CtO
7
= 0.
ing variables between them. I n polarography for diffusion limited currents the product i l H - l l 2 is independent of H for a given system, where H is the height of the mercury head above the dropping electrode, and il is the limiting current. For adsorption control, the product &H-l is independent of H. For first order kinetic control, i~ is independent of H. Variation of H yields inversely proportional variation in drop time of the electrode (when H is appropriately corrected for back pressure) so that i t might be expected that the analogous variable in chronopotentiometry would be the transition time. This expectation is borne out by the fact that i+ is constant for a diffusion controlled process and i7 is constant for an adsorption process. For kinetic processes the correspondence breaks don-n, not through the failure of the analogy, however, but rather because of the inversion of the variable of interest. It is still true that i tends to become independent of 7. Paradoxically, this does not mean that 7 becomes independent of i but quite the contrary. Viewing the two methods in this light clearly indicates one of the major advantages of chronopotentiometry as a diagnostic technique. Currents can be varied over several orders of magnitude, while variation of H is restricted to three- or fourfold a t best. Distinction between various reaction schemes herein discussed can be made simply on the basis of two experimental plots: i+ vs. i a t constant C and z’s. C a t constant i. The criteria to be applied to these plots and the expected results are given in Table I. Inspection of the table shows that unambiguous assignment can be made of any of the cases mentioned. It is true that many more complicated kinetic schemes are possible. All of them, however, obey the same qualitative criteria as the ones in the table.
ANALYTICAL CHEMISTRY
-
> 0, x
> 0, x
=
CT --c C$
m;
(AS)
0 ; D ( ~ C T / ~=X io/nF )
(AT)
The solutio11 of this problem is ne11 k n o m and the concentration at the surface is x = 0; CT-= C*, - 2iot1/2/.rr1/ZnFD1/2 (A81
If the 0 - Y equilibrium is infinitely rapid in attainment C,/(Cy)P = K , = k!’& (A9) By combination of Equations 8 3 , A8, and A9 the concent,rations of 0 and Y at any time can be evaluated. For the case of slow attainment of equilibrium it is convenient to define
a,
=
Co, - c, so that C y = C y ,
+ p&
(-410)
v-here C,, and C y , are the concentrations n-hich n-odd be observed if equilibrium betn-een 0 and Y n’ere at,tained at infinite rate. Substituting Equation A10 into Eqiiation A 2 leads to (bAo/bt) = D ( d 2 A o / d ~ 2 )- ,kT& ki(CYrn
-
+ PA,)’ + k / ( C ~ m )(411) ~
If the equilibrium strongly favors Y , then CY,
>> co, >, ao;
0, 5
1
t > 0, x
=
03;
A,
0
(A15)
0 ; D(dA,/dx) = D ( d c o m / h ) & / n F G -io/nF
where the approximation of the last boundary condition is valid n-hen Equation A12 is satisfied. The problem can be solved readily by Laplace transform methods and the result is z = 0 ; A. =
n F k 7i0 1 , z D 1 ,erf!k,t)1/2 2
(A16)
APPENDIXES
I. Theory of P U S 0 +- R. For the reaction scheme in question, the concentrations of 0 and Y are governed by Fick’s IaTv equations of the form
Under condition A12 Combination of
A16, and -417 gives
(acy/at)= D ( P c ~ / ~ + x ~p )k , ~ , p k i ( C y I P (-41)
+
c,
z = 0;
=
Equations AS, -110,
K,pp(C*, -
(dC,/dt) = D(d2C,/dx2) - k,Co k f ( C y I P (-42)
2iotl/2/Tl/2nFDl’2)P-
vhere the diffusion coefficients of 0 and Y are assumed equal. It is convenient to define
11. Adsorption Processes. For an adsorption process, Fick’s 1aTY still applies to the species in solution
CT =
co
+ cY/p
(A3)
Combination of Equations A l , A2, and A3 yields ( d C T / b t ) = D(d2CT/bX2)
(-44)
vith the boundary conditions t = 0;
324
t t
CT
=
C;
(-45)
jbC,/dt)
=
> 0;
-*
t
x
1=0;
io erf ( / ~ , t ) ” ~ nFk,‘ 2D112 (A181
D ( d 2 C o / d ~ * ) (A19) m;
C + C*
c=c*
(.AZO)
(-421)
but the boundary condition describing the flux at the electrode surface must be rrritten in the form
1:
=
0; zo/nF
=
D(dC/dx)
-
(dr/dt) (-422)
Kith the aid of conventional Laplace transform methods the condition a t the electrode surface can be restated in the form of the integral equation C*Dll2
~zotl12/nFnl/2
- CDll2 -
or alternatively
&t/nF
=
r* -
After T ~ ,the time at which C reaches zero, Equation A6 can be written in the form
(A301 The integral assumes zero value after T~ because its argument vanishes at that point. The argument of the integral can be given explicit form by substitution of Equation 410. Making that substitution and performing the indicated integration leads to iot/nF
To solve either of these equations for
the transition time, it 16 necessary to assunie some explicit relation betm-een C at the electrode and I’. CASE I. Assume r and C not to be related b j an equilibrium expression and 1’ to react electrochemically before C. ZIathematically this means that Then r > 0, then C = C* (A25) IYhile r has a finite value, Equatioii A24 siniphfies to znt/nF
=
I*
-r
Ixcause the argument of the integral is Aero. hfter T ~ ,the time at which r hcconies zero, Equation A25 can be put in the form 2C(t
- n)/nF
=
D’/2
L1-r‘ x
( C * - C ) / + l 2 ( t - 71 - B ) W B (A27) nhich is cxactly the same form as the expression n-ould have if there nere no adsorption and elt.ctrolysis began at time 71rather than time zero. CASE 11. Assume l7 and C not to be related by an equilibrium expression rind C to be reduced to zero before reaction of r begins. IhIathematically, when C > 0, then ( d r / d t ) = 0 (A28) \Yhilc tlierc is still a finite concentration, C, a t the electrode, Equation A23 can be bimplified to 2 ~ o t ” ~ / r ~ F ( r r D= ) ” C* ~ - C (A29) lmausc the argument of the integral is zero.
%n
=
1’*
-
I?
+
[arcsin(2+t) 41 -
(? -
1)’
+ 71/21
(A431)
The expression for 7 2 can be obtained from Equation A31 by noting that a t t = T p J r = 0.
+
CASE 111. Assume r and C to be related by an equilibrium of the form
r = K,C at z = 0 (A32) The integral equation resulting from the substitution of Equation A32 into Equation A23 or A25 can be solved readily by Laplace transform methods and the result has been given by Lorenz (6). NOMENCLATURE
C*
=
C, C*,
= =
D
= = =
erf erfc exp E
= =
E”’
=
F H
= =
20
= = =
ti
kj
concentration of species 0 prior to electrolysis instantaneous concentration of 0 hypothetical Concentration of 0 prior to electrolysis assuming complete conversion of Y to 0 diffusion coefficient error function error function complement natural exponential function instantaneous cathode potential us. unpolarized reference formal standard potential of the 0-R couple Faraday’s constant height of mercury head in polarographic experiment currentdensity polarographic limiting current formal rate constant for chemical reaction in forn-ard direction
formal rate constant for chemical reaction in reverse direction R, = equilibrium constant for chemical reaction ( k , / k f ) K O = equilibrium constant for adsorption obeying linear isotherm n = number of electrons per molecule in reduction of 0 to R 0 = oxidized form of electrochemically reactive couple = stoichiometric constant for chemp ical reaction R = reduced form of electrochemically reactive couple t = time from start of electrolysis z = linear distance from electrode surface X, Y = electrochemically inactive participants in chemical reaction A0 = defined by Equation A10 r = instantaneous amount of adsorbed species per unit area r* = initial amount of adsorbed species per unit area 7 = transition time (numerical subscript indicates chronological order) T~ = hypothetical transition time n-hich mould be observed if all chemical kinetic steps were infinitely rapid k,
=
LITERATURE CITED
(1) Davis, D. G., Ganchoff, J., J . Electroa m l . Chem. 1, 248 (1960). ( 2 ) Delahay, P., Berzins, T., J. Am. Chem. SOC.75, 2486 (1953).
(3) Fischer, O., Dracka, O., Fischerova, E., Collection Czechoslav. Chem. Communs. 25, 323 (1960). (4) Gierst, L., Juliard, A,, J . Phys. Cliem. 57, 701 (1953). (5) Koutecky, J., Cizik, J., Collectzon Czechoslov. Chem. Communs. 22, 914 (1959). (6) Lorenz, W.,2. Elektrochern. 59, 730 (1955). ( 7 ) Moorhead, E. D., Furman, N. H., ASAL. CHEW32, 1506 (1960). (8) Reinmuth, W. H., Ibid., 32, 1514 (1960).
(9)’ Rosebrugh, T. R., Miller, W. L.. J . Phys. Chem. 14, 816 (1910). (10) Testa, A. C., Reinmuth, SIr. H., A N A L . CHEM. 32, 1512 (1960). (11) Ibid., p. 1518. RECEIVEDfor review October 21, 1960. lccepted December 22, 1960.
Application of Stripping Analysis to the Determination of Iodide with Silver Microelectrodes IRVING SHAlN and S. P, PERONE Deparfment o f Chemistry, University of Wisconsin, Madison, Wis. The extension of stripping analysis to the determination of halides with a silver microelectrode has been investigated. During the pre-electrolysis step, a portion of the halide was deposited b y a controlled potential oxidation of the silver electrode. Two methods of stripping the silver halide deposit from the electrode
were investigated: electrolysis with constant potential, and electrolysis with linearly varying potential. The quantity of electricity measured in the stripping step was a direct function of the pre-electrolysis time and the bulk concentration of halide. The method was applied to iodide solutions as dilute a s 4 X 10-8M.
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RESEARCH on stripping analysis has shoum that the method is very sensitive for the determination of electroactive materials. The technique consists of a pre-electrolysis step, during which the sample is concentrated by electrodeposition on a n electrode. The actual analysis takes place during a subsequent elecECEST
VOL. 33,
NO. 3, MARCH 1961
325
