High Resistance and Derivative Polarography Derivation aind Verification of Derivative Equations Useful in Evaluating lhcompensated Resistance in Polarographic Cells WARD 8. SCHAAP and PETER S. McKINNEYl Department o f Chemistry, Indiana University, Bloomingfon, Ind.
b The theory and techniques of derivative polarography are investigated as a means of studying the effect of electrode plcicement on resistance compensation iri three-electrode polarographic cells. Equations a r e derived describing the: maximum height and the width a t half-peak height of the derivative of (1 polarographic wave under conditioiis of uncompensated solution resistarice and appreciable residual current, The equations a r e verified experimentally using a resistance-compensatiiig and derivative-taking polarograph of the Kelley, Jones, and Fisher design. The results show that uncompensated solution resistance can b e easily and accurately evaluated from the shapes of the recorded first derivatives of polarographic waves. The fechnique is useful in studies involving solutions of high specific resistance -e.g., in nonaqueous studies-and in studies of potential fields in polarographk cells.
I
TEARS, one of the most significant developments in the instrumentation for conventional (d.c.) polarography has b e t a the design of circuitry capable of compensating instantaneously for iR losses incurred in the polarographic circuit. Though a number of resistance-compensating circuits involve the use’ of two electrodes (7, 1 4 ) ) the most sutcessful are those which employ three-electrode potentiostats. Of the latter type, the circuits designed by -4rthur and T’anderkam (2) and the control1:d potential and derivative polarograph developed by Kelley, tJones, and Fisher (9-11) are the most familiar and are both commercially available. The development of controlled potential, three-electrode circuitry makes it possible to consider precise polarography in nonaqueous, high resistance solutions or in aqueous solutions containing minimal imounts of supN RECEST
Present address, Department of Chemistry, Harvard University, Cambridge, Mass.
porting electrolyte. Also. highresistance electrode or cell designs can be used without difficulty. The application of three-electrode potentiostats to polarography in truly high-resistance solutions has not yet been thoroughly evaluated. In particular, the importance of electrode placement has not been carefully investigated. It is possible to use a resistance-compensating instrument which is entirely satisfactory from the point of view of electronic design, but to overlook in its application the importance of electrode placement within the electrochemical cell itself. It has been recognized that the potential developed by the cell (reference electrode-D.M.E.) is not independent of the reference electrode position but will include any ohmic potential existing in the solution between the controlled electrode and the orifice of the reference electrode (1, 9). The polarographic cells developed for use with three-electrode circuits have accordingly been designed so that the D.M.E. is located directly between the reference and the auxiliary counter electrodes, thereby keeping the reference electrode out of the principal cell-current path. I n aqueous solutions, or in nonaqueous solutions of moderate specific resistance, no significant iR distortion is introduced at normal cell currents with this electrode configuration and satisfactory resistance compensation is attained. When solutions of very high specific resistance are employed, however, complete resistance compensation may not be obtained. I n the case of very high resistances (megohm range), potentiostat performance has generally been evaluated in one of tm-o n-ays (1,9) : either external resistors nere placed in series with the cell, or long electrolyte paths were provided between the cell electrodes. I n both cases i t is possible to obtain high total cell resistances in one or both cell arms-i.e , in series with the reference and counter electrodes-even though solutions of low or moderate specific resistance are used. Though these experimental arrangements constitute
electrical equivalents of high-resistance, electrochemical systems and do indeed test potentiostat performance, they do not truly represent the situation encountered in a solution of high specific resistance. In such a solution the potential gradient, due to iR drop in the vicinity of the D.M.E. and normal to it, is significant, whereas i t would not be in the case where high resistance is introduced “synthetically” at locations relatively far from the D.1I.E. Because the D.M.E. is a very small diameter electrode, all points on its surface are essentially equidistant from the counter electrode, causing the cell current to flow radially from its entire equipotential surface for some distance out into the solution before turning in the direction of the counter electrode. Thus, in the region of appreciable current density-Le., near the D.M.E. surface-the ohmic potential gradient may be considered to be spherically symmetrical. This model for representing current flow around the D.1I.E. has been verified experimentally in this study and elsewhere (9, 5, 12) and is significant because it allows the effective solution resistance a t a given distance from the D.M.E. to be calculated using the concentric sphere treatment of Ilkovic (6) or of Kaspar (8). The experimental measurement of this resistance a t a giren distance by the usual a x . bridge method is meaningless because the value obtained is proportional to the area of the electrode (other than the D h1.E.) used I n view of these considerations, when the specific resistance of the medium is high, an appreciable ohmic potential gradient exists near the surface of the mercury drop even on the side opposite the counter electrode. In this situation, though the tip of the reference electrode salt bridge is located out of the major cell-current path. some ohmic potential would still be included in the total potential difference measured betrTeen the D A L E . and the salt bridge tip and this would lead to a n iR dirtortion of the recorded polarogram. .Ilthough the magnitude of such a residual. uncompensated, ohmic potential loss may be inferred from the shape of the polaroVOL. 36, NO. 1 , JANUARY 1964
b
29
graphic wave, as discussed below, the complete elimination of this solution resistance effect b y any direct method appears to be impossible at the dropping mercury electrode. (Further consideration of the current distribution around the mercury drop as well as theoretical calculations of the ohmic potential and a proposed indirect procedure for its elimination will be presented in subsequent papers.) A convenient experimental approach to the detection and evaluation of uncompensated resistance is possible through the use of derivative polarography. The slope of the conventional polarographic current-voltage curve, and hence the derivative, is very sensitive to the presence of uncompensated resistance. I n this paper general equations are derived, for the case of reversible polarographic reactions of known n a n d i d , which describe the effect of uncompensated solution resistance on the shape of the first derivative of the polarographic mave-i.e., on derivative peak height and half-peak width. These theoretical equations are then verified by comparison with experimental results obtained with cells having known, measured values of uncompensated resistance. Application of the verified derivative equations to studies in nonaqueous systems of high specific resistance-Le., to the measurement of uncompensated resistance as a function of reference electrode placement-is made in the second paper of this series.
EQUATIONS OF DERIVATIVE POLAROGRAPHY INCLUDING EFFECTS OF UNCOMPENSATED RESISTANCE A N D
RESIDUAL CURRENT
Peak Height. T h e HeyrovskyIlkovic equation of t h e polarographic wave relates the true electrode potential-i.e., t h e potential a t the electrodesolution interface, E,,-to the faradaic current flowing a t tlie electrode. I n the absence of uncompensated resistance, the applied potential IS equal to the true electrode potential. E , = E,,,, and undistorted polarograms can be recorded. I n the presence of uncompensated resistance, the true electrode voltage differs from the applied potential by the value of the ohmic potential developed across the uncompensated resistance ( I S ) . For a cathodic m v e , the true electrode potential can be expressed as EdL =
E. -t R z i
(1)
where Zi is the total current flowing at the electrode-Le., the sum of the faradaic current, i, and the residual current, i,. R is the total uncompensated resistance through which the total current flon-s. It should be noted that although the Heyrovsky-Ilkovic equation relates only i (faradaic) and Edr, the ohmic potential correction depends on the total current. The half-wave potential must also 30
ANALYTICAL CHEMISTRY
be corrected for the effect of ohmic losses and is given by
(E1iz)ds =
(El/z)ob
+ (id12 f irl/Z)R
(2)
where (El,2)deis the true half-wave potential, (El,z)oa is the half-wave potential observed from the recorded polarogram, id is the average faradaic diffusion current and irl,2 is the average residual current at the observed halfwave potential. Substitution of Equations 1 and 2 into the Heyrovsky-Ilkovic equation, applied to the reduction of a metal ion in solution to a metal soluble in mercury, yields the equation of the polarographic wave corrected for iR losses Eo f R z i = (El/r)ob ( i d / 2 ir112) R f
+
+
Differentiation of the expression with respect t o E , yields an expression for the derivative of the polarographic wave which includes both resistance and residual current
where di,/dEa is the observed slope of the residual current line, and K = R T I F . The second derivative of Equation 3 is
e =1( + di)
dE',
dE,
di dE,
If there is no appreciable lag or resistance, then dEac = dE,, or dEd,/dE, = 1, and the magnitude of the derivative is thus proportional to the rate at which the potential is scanned. I n the presence of uncompensated resistance this is no longer true. Since Equation 8 is derived from the Sernst equation, it is apparent that dEd,/dt must be represented in terms of the applied scan rate and an ohmic potential correction in order to account for uncompensated resistance. From a consideration of Equation 1, it is clear that Equation 8 could be rewritten
The presence of uncompensated resistance decreases the effective scan rate, which in turn diminishes the value of the derivative. Since d i / d t di,/dt = d 2 i / d t , Equation 9 can be written
+
and solution for (di/dt),,, yields Equation 7 . It is of interest to examine the significance of each term of Equation 7 . Multiplication by 4K PZRid and rearrangement give
+
x
and it becomes zero when i = id(2. Thus the first derivative attains Its maximum value at the observed halfwave potential even in the presence of uncompensated resistance. Substitution of id/2 for i gives an expression for the maximum value of the derivative
in which i t is now understood that (di,/dE,)E1/2 is the slope of the residual current curve at the half-wave potential. Introduction of the scan rate, dE,/dt, yields a n expression for dildt, the experimentally measurable quantity
+
showing that the difference between the true scan rate seen a t the electrode surface and the scan rate applied by the potentiostat is equal to the rate of change of the total ohmic potential drop, R(dZi/dt). This latter quantity, dzildt, reaches a maximum value at the half-wave potential, but is approximately constant and minimum before and after the nave. The true scan rate is therefore not constant during the rising portion of a polarographic wave and is a minimum at the half-wave potential. Because of the relation E d c= E , RZi, it can be seen that the difference or lag between the absolute values of the applied and true scan potentials changes sharply during each polarographic wave and in proportion to i, so that a plot of this difference due to iR loss, as a function of time, would have the same shape as a polarographic wave. The usefulness of Equation 7 is limited by the experimental difficulty in measaring (di/dt',,, and (di, l d t ) B I ~ Z independently. I n the absence of uncompensated resistance, the slope of the residual current is reasonably constant throughout the wave. K h e n resistance
+
Equation 7 can also be derived from an intuitive consideration of the resistance effect. It is useful to consider the derivation from this standpoint also since i t gives a physical picture of the situation which is better than that of the strictly mathematical approach. The maximum value of the derivative in the absence of uncompensated resistance can readily be shown to be
+
Remembering that Xi = i i,, then = di di,. Thus, the left hand From term is equal to R(dzi/dt),,,. Equation 8, it is seen that (-4Klnid) (di/dt),,, = dE,,/dt, so that Equation 11 can be written
dzi
is a factor however, tk.e value of di,/dt, which is itself dependent on the scan rate, changes during the course of the wave because the effective scan rate changes with d i l d t . [t is not strictly correct in this case to make a linear interpolation of the residual current to serve as a base line from which (dildt),, can be evaluated independently. (Actually, the derivative plot of the residual current curve, considered independently, should be bowl shaped-i.e., concave upward-because of the changing scan rate during a polarographic wave recorded in the presen1:e of uncompensated resistance.) It kiecomes necessary to solve Equation 7 for d z i / d t , which can be experimentally evaluated and compared with theory. By definition d-i = d-.2 i dt dt
- di, _. dt
and substitution for ( d i l d t ) in Equation 7 gives dEa dt
+ 4K +
nad ZRid
stant slope over the potential interval during the rising portion of the wave, but it is more difficult to evaluate di,/ d E d , if there is an appreciable change in slope, such as can occur if the Ell2 is near to the electrocapillary maximum in a given solution. Fortunately it is generally possible to change the supporting electrolyte and to shift the E112 value and the potential of the electrocapillary maximum relative to each other so that the residual current slope can be more easily evaluated. The importance of expressing the residual current slope in this way-i.e., in terms of the true electrode potential-is that the slope is then essentially constant and independent of resistance lowes which may be present and that it can be experimentally evaluated in the present study. The evaluation procedures are discussed below. Under other conditions, Equation 19 can be simplified as follows: If R >> 0 and i,
