A Signal-Independent Electroanalytical Method SIR: There exist several electroanalytical techniques which share the following features. (i) The supply of the electroactive species to the electrode is controlled by semiinfinite diffusion. Though the electrode is rarely planar, its curvature is usually so much larger than DT where D is the diffusion coefficient of the electroactive species and T is the duration of the analysis period, that it is legitimate to approximate the diffusion conditions by those of planar diffusion to an electrode of constant area A . (ii) The analysis is preceded by a n equilibrium [or steadystate] situation characterized by a constant potential E,,. No electrochemical reaction of the pertinent electroactive species is possible at EO and therefore there preexists a uniform concentration C of this species throughout the electrolytic medium. (iii) During the analysis period, 0 < t 5 7, a signal f ( r ) is applied to the cell. The controlled variable f ’ is usually either the electrical potential E or the current i, and the imposed time-dependence is often a very simple function such as a step or a linear sweep. (iv) During the analysis period, some electrical property g is monitored as a function of time. This property must, of course, be distinct from the controlled propertyf. it is usually potential E, current i, or charge q. The time dependence g(t) is normally more complex than that of the applied signal
departure from a perfect step function in the chronopotentiometric or chronocoulometric techniques will have repercussions on g(t) and hence introduce error into the determined concentration. Likewise any nonlinearity of potential us. time in stationary electrode polarography will affect i, and hence the calculated C value. Inasmuch as modern instrumentation can generate extremely accurate f ( t ) signals, the burden imposed by this exactness requirement may appear inconsequential. However, it must be appreciated that the formulas used in the calculation of C presuppose that the signal applied to the cell is the signal applied to the faradaic element of the electrode interface. The inevitable presence of a capacitative element in parallel and a resistive element in series with the faradaic element implies that, notwithstanding the perfection of the signal applied to the cell, impeifections are necessarily present in the effective f(t). The consequence of these imperfections on g(t) [and thence o n the calculated concentration] may be trivial in some circumstances, but they become increasingly disturbing as C is decreased and as the conductivity of the electrolysis medium is decreased. Against the foregoing background, this letter reports a new electroanalytical method which has the merit of being completely independent of the signal f ( t ) , provided that the above six criteria are met. The method is summarized as the final tabular entry and the theory is presented below, together with an experimental verification. Theory of the Method. The new method, for which the name “semiintegral electroanalysis” is suggested as most appropriate, uses the semiintegral of the current as the observed, g(t), function. Because the concept of semiintegration will be new to most analytical chemists, a few words concerning this mathematical operation will be given to preface the description of the new method. The operation of semiintegration with respect to time [denoted by the operator symbol d-1’2/dt-1/2 because it is equivalent to differentiation to order minus one half] generates a function which possesses an intermediacy between the original function and its time-integral. Thus the semiintegral of a time-dependent current,
4,
f(0. (v) At the conclusion of the analysis period, the potential has a value E, corresponding to virtually complete concentration polarization with respect to the electroactive species in question. Because of this, the final concentration of the species a t the electrode surface is effectively zero. (vi) The concentration is found by examination of a plot of g(t) us. t. Often there is some characteristic feature of the graph [a maximum, asymptotic limit, or point of inflection] from one or the other coordinate of which C may be determined by application of a known formula. I n other cases any point on the g us. t graph may be employed to determine C, or the data may need to be replotted in some different fashion in order to calculate C with optimum precision. Table I lists examples of electroanalytical methods which satisfy these six criteria. It is crucial to the success of all such existing methods that the applied signal f ( t ) has the requisite form. Thus any
Table I. Electroanalytical Methods
Stationary electrode polarography or Linear potential sweep chronoamperometry
E(t) = EO
Chronopotentiometry
i(t) =
Potential step chronocoulometry
At)
.f(t)
Name of method
E(t)
=
t + ( E , - EO);
0,
t O
i
0
Eo, t E,,
i(t)
dt)
196
Arbitrary provided E(r)
=
E,
di -Go dt (defines ip)
J, i(t)dt
m(t)
ANALYTICAL CHEMISTRY, VOL. 44, NO. 1, J A N U A R Y 1972
C calculated from
‘2
RT r d n F D ( E , - EO)
reversible processes only
d-E + m dt (defines r ) none (7 = any t )
2nAF
m(t) + const, m (defines m)
n
ft
= New method: semiintegral electroanalysis
Characteristic feature
+-
Or
m i
is intermediate between i(r)itself and the electric charge
For example, if i(t) is a power function of time, i(t) a t*, tP-1 and m(t) a rP-l/*. With equal validity, we may regard m(r) as the semiderivative of the charge q(t), semidifferentiation being an operator intermediate between the identity operator and differentiation:
q(t)
i(t) nz(t)
dq - (t) dt d 2q dt”2
__ ( t )
Another way of regarding a semiintegral is as a weighted average. As is well known, an integral over the interval 0 t o t may be closely approximated by taking a sum of values of the function at equally spaced abscissa values, thus for example
+ i(t - A) + i(t - 2A) + . . . . + i(2A) + i(A)l,
q(r) = A[i(t)
Figure 1. Current 6s. time curve recorded during cappedramp study of cadmium ion electroreduction time T the potential has acquired a value E, at which concentration polarization is complete, Le. CO(T)= 0, then
wherein all ordinate values are seen to be equally weighted. A corresponding representation,
L
I
>
is possible for the semiintegral of i(t), but in this case the weighting is unequal, favoring later i values over earlier ones. Just as there are numerous algorithms for integration [the trapezoidal formula, the above equation, Simpson’s rule, etc.], so there are alternative ways of approximating a semiintegral, including m(t) = (t/A) -1
a- l i 2
Measurement of m ( ~thus ) permits a ready calculation of the bulk concentration, t o which m(T) is proportional. Though it is not a n essential feature of the method, semiintegral electroanalysis is particularly successful if the potential is constant, or almost constant, a t the end of the analysis period. There are two reasons for this. First, m ( f T ) is then a constant, m, obviating the need for any timing procedure t o identity the time range over which the potential equals E,. Second, the nonfaradaic current going to charge the doublelayer is then zero at time T so that nonfaradaic contribution to the semiintegral is minimal and decreasing. Two of the many ways in which a signal may be applied to give a desirable variation of potential with time are: (a) Capped-ramp. Here the potential applied to the cell is constant both before t = 0 and after t = r but changes linearly in between
2
j=O
[i(iA
+ A) + i(jA)l [dr- j A -
Eo, t
6
0
Em,t
>
T
d;- j A - A], the so-called RLO-algorithm. For a more exact mathematical introduction to semiintegration, the interested reader is referred to Reference I and to the literature cited therein. Provided that conditions (i) and (ii) are met, it may be shown ( I ) that the surface concentration of the electroactive species is given by
(b) Resistor-limited-step. Here a resistor of magnitude p is inserted in series with the cell and the potential applied to the series combination is stepped from EOto E,. Thereby the potential experienced by the cell is
i
Eo,
Em -
where n is the number of electrons involved in the reduction [oxidation] of the species and i(t) is the cathodic [anodic] faradaic current. The proof of this equation involves n o assumptions about the kinetics of the electrode process, nor does it involve any stipulation whatsoever [apart from those necessary to ensure conditions (i) and ($1 about the signal applied t o the cell. It follows directly, therefore, that if by (1) Keith B. Oldham and Jerome Spanier, J. Electroanal. Chem. Interfacia[ Electrochem., 26, 331 (1970).
pi(&
t
0
The value of p is not critical; but if it is too small, inconveniently large currents will be passed; whereas if it is too large, the value E, will be approached only after too long a n interval. It must be emphasized that the above are cell potential differences: whether or not the potential across the double layer of the working electrode closely duplicates the cell potential is irrelevant to the success of semiintegral electroanalysis. Experimental. A thorough evaluation of semiintegral elec-
ANALYTICAL CHEMISTRY, VOL. 44, NO. 1, JANUARY 1972
197
troanalysis is in progress in these laboratories and the results will be published. The purpose of the present brief report is to show that the theory is validated by a t least one system. The full curve in the accompanying Figure 1 shows the current us. time curve recorded during a capped-ramp study of the cadmium ion electroreduction from 1.00mM C d Z + in 0.10M K N 0 3 solution at a mercury drop electrode of approximately 0.042 cm2 area. The ramp rate was 100 mV per second from a potential of -490 to -770 mV cs. SCE. The cell was unthermostated at a 21.5 “ C ambient. The dashed curve shows the result of semiintegrating the full curve by the RLO-algorithm. Observe that the semiintegral is finally a constant, as theory predicts since -670 millivolts corresponds to virtually complete concentration polarization. The value, m,of this constant is seen to be approximately 23 microamplombs. [The amplomb is the name given to the unit intermediate between the ampere and
the coulomb; amplornb = ampere second1i2 = coulomb s e c ~ n d - ~ / ~Use ] . of this value in the formula m = n A F C 4 5 of calculate the diffusion coefficient of cadmium ion leads to cmp sec-I, close to the an approximate value of 8 X accepted figure. ACKNOWLEDGMENT It is a pleasure to acknowledge the experimental assistance of Dr. Morten Grenness and the financial assistance of the National Research Council of Canada. KEITHB. OLDHAM Trent University Peterborough Ontario, Canada
RECEIVED for review December 28, 1970. Accepted March 22, 1971.
AIDS FOR ANALYTICAL CHEMISTS ~~~~
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The Circulation Cell-A New Device for Electroanalysis, Spectrotitrimetry, and the Study of Fast Reaction Kinetics Anne-Marie Gary, Etienne Piemont, Michelle Roynette, and Jean-Paul Schwing’ Institut de Chimie, I , rue Blaise Pascal, 67-Strasbourg, France
A NEW CELL and its applications to voltammetry, coulometric titrations, fast reaction studies, and spectrotitrimetry is described. The versatility of this “circulation cell” is due t o the fact that the investigated solution is forced, by the action of an integrated centrifugal pump, into an adequate hydrodynamic circuit whose special design allows the establishment of reproducible limiting currents on immersed solid electrodes and the contiruous feeding of an external device as, for instance, a spectrophotometric circulation cell. The described circulation cell is of modular design to allow the practice of the majority of the electroanalytical technique. as listed, for instance, by Kolthoff and Elving ( I ) . Several cells were constructed having volumes between 30 and 100 ml. EXPERIMENTAL Description of the Circulation Cell. Figure 1 shows the realization of this device: the centrifugal pump P is introduced into the cell through the ground joint J6 and gives the solution a rapid movement in the direction of the arrows a. The ground joint J j allows continuous injection of a reactant when the capillary Ca is introduced into Jj. Electrolytic generation of a reactant can be realized when the generating electrode E is introduced into Js, the auxiliary electrode, AE, being placed into J4. The fritted glass disk F separates the auxiliary electrode compartment from the solution. The joints Jl and JZ are intended t o receive the electrodes for endpoint detection or for the preliminary recording of currentvoltage curves. The grounded cylinder C allows the expanl
sion of the solution when a reactant is injected as a solution through Js. If deaeration of the solution is necessary, this can be achieved in the following way: the solution at rest should not be higher than level Lp; the lower part of the cylinder C is raised to level L4 and an inert gas such as nitrogenis admitted through tubing Tq and escapes through T3. The centrifugal pump is then switched on and thus leads to deaeration of the strongly stirred solution. When deaeration is completed, the gas stream through T4 is stopped, and the cylinder C is lowered until the level of the solution rises in the capillary tubing T3 which is then closed. Tubing Tg is used to empty the circulation cell, and tubing T j can be used to connect the cell to a mercury reservoir in order to set up, for certain studies, a mercury pool electrode at the bottom B of the cell. If thermoregulation is necessary, this can be achieved by water circulation inside of the cylinder, C, the water entering through TI and leaving through T2. Thermoregulation can also be realized by a water jacket, J, built around the main body of the cell. When the absorption spectrum of the reacting solution is of interest (spectrotitrimetry, spectrophotometric study of electrochemically reduced or oxidized species), inlet and outlet tubes, shown on Figure 2, can be placed into J1 and Js. The special design of these tubes allows the deviation of a part of the solution, with a small and reproducible delay, through a spectrophotometric observation cell. RESULTS AND DISCUSSION
Correspondence to be addressed to this author.
(1) I. M. Kolthoff and P. J. Elving, “Treatise on Analytical Chemistry,’’ Interscience, New York, N.Y., 1965, Vol. 4, p 2225. 198
Figure 3 shows the current-voltage curve obtained for the rapid system 12/1-, in acidic medium (0.05M H2SO4) when a three-electrode system is used. In this case, the
ANALYTICAL CHEMISTRY, VOL. 44, NO. 1, JANUARY 1972
