Coulostatic Titrations Using a Dropping Mercury Electrode Richard W. Sorensenl and R. F. Sympson Department of Chemistry, Ohio Uniaersity, Athens, Ohio End points of titrations have been located using coulostatic relaxation curves measured during the course of titrations of zinc, cadmium, lead, and manganese with EDTA. A DME was used as the working electrode, and an equation was derived for the relaxation process at a DME. Coulostatlc relaxatton curves produted By reduction of the metal ion and relaxation curves produced by anodic depolarization of the DME by EDTA were both used in locating end points of titrations.
THISINVESTIGATION was begyn to determine whether end points of titrations could be,hcated by measuring soulostatic relaxation curves during the c&se of a titration. It was hoped that such titrations would produce a reasonably accurate method of analysis in the concentration range 10-4 to 10F6M. Although it was Barker ( I ) who first suggested the coulostatic or charge-step refaxation method, it was developed into a workable means of studying the kinetics of rapid electrode processes by Delahay (2) and Reinmuth and Wilson (3). Delahay (4) first demonstrated the potential of the method as a n analytical tool, and Delahay and Ide (5) reported the first analytical applications of the method and suggested the possibility of using the technique to indicate the end point of a titration. Anson (6) demonstrated the usefulpess of the technique in obtaining chronocoulometrk, chafge-time curves indirectly from potmtial-time relaxation curves. By this method he eliminated the error arising from uncompensated cell resistance that is encountered in conventional potential-step chronocoulometry. H e showed the amount of absorption of Cd+* from thiocyanate medium as determined by this method agrees well with that obtained by convebtional chronocoulometry. The present authors believed that using the technique to follow a titration would yield more precise and accurate analytical results than could be obtained by direct measurement just as the accutacy and precision of amperometric titrations are generally better than that of direct polarographic analysis. At the present state of devel6bment in this laboratory the treatment of the titration data is too cumbersome for the method to be used routinely, but it is believed the results are worth reporting. Incorporation of a digital readout system into the apparatus shbuld be possible, and this would simplify data t r e a t m n t immensely. Delahay’s article ( 4 ) should be consulted fc# a thorough description of the principles of the method. A brief summary of the essential features of the method will be made here. The coulostatic technique involves the application of a pulse of charge to an electrode initially at a potential a t which no faradaic current can flow. The pulse of charge shifts the Present address, Texas Instruments, Inc., Attleboro, Mask
(1) G. C. Barker, “Symposium on Electrode Processes,” E. Yeager, Ed., Philadelphia, 1959, Wiley, New York, 1961, p. 325. (2) P. Delahay, ANAL.CHEM., 34, 1161 (y962). (3) W. H. Reinmuth and C. E. Wilson, Ibid., p. 1159. (4) P. Delahay, Ibid., p. 1261. ( 5 ) P. Delahay and Y. Ide, Ibid., p. 1580. ( 6 ) F. C. Anson, Ibid., 38, 1924 (1966).
1238
ANALYTICAL CHEMISTRY
potential of the electrode t o a value a t which oxidation or reduction of some species in the solution can occur. Following the pulse the change in potential of the electrode as a function of time is observed under virtually open-circuit conditions. The potential decays in the direction of its original value because of consumption of charge by a faradaic process. When the concern is with chemical analysis, as opposed t o electrochemical kinetics o r absorption phenomena, the charge pulse should be of sufficient magnitude that the surface concentration of electroactive species becomes zero instantaneously after application of the pulse. The relaxation process is thus under diffusion control, and the rate of potential decay is related to the concentration of electroactive species. It was found in exploratory experiments that a hanging drop mercury electrode was too tedious to use during titrations, so a dropping mercury electrode (DME) was used as the working electrode. The use of a D M E complicated the equation for the voltage-time curve because of the changing area of the DME. An equation was derived relating potential and time for the DME during the relaxation process.
THEORY Equation 1 was given by Delahay ( 4 ) and it is the fundamental equation used in deriving voltage-time equations for the relaxation process for various types of electrodes. It is based on the conservation of charge. PI
I n this equation Cdl is the integral double layer capacitance per unit area, A is the electrode area, E(r) is the electrode potential a t any time t after application of the pulse of charge, E, is the e1ectrQde potential a t the point of zero charge, and E(0) is the potential a t zero time. Time, t , is measured from the instapt the pulse of charge reaches the electrode. The derivatiun of different voltage-time equations is simply a matter of supplying the proper expression for current as a function of time in Equation 1 and performing the integration. Fquation l and all equations derived from it neglect the small variation of C ~with E potential. The same assumptions will be made as were made in deriving the Ilkovic equation (7). The limiting current under conditions of semi-infinite linear diffusion is given by Equation 2 (8). ( M ) refers to the bulk concentration i(t) = i.
nFAD1/2(M) (,t)“*
of electroactive species, and the other symbols have their conventional meanings. The area, A , will be expressed as a function of the mass rate of flow of mercury and the age of the drop. It should be noted that in the derivation of the Ilkovic equation A is expressed as a function of m and t’ where m is the mass rate of flow of mercury and t’ is time measured (7) P. Delahay, “New Instrumental Methods in Electrochemistry,” Int science, New York, 1954, p. 63. (8)
&.,
p. 51.
from the birth of the drop. I n the present derivation, t is time measured from the application of the pulse of charge which occurs late in the life of the drop. If the age of the drop is a seconds when the pulse is applied, the drop volume, V , will be given by
v
=
(4/3) 7rr3
=
(3)
P
where r is the drop radius and p the density of mercury. Solving this equation for r and substituting the result into the equation for the area of a sphere gives
+ t)2/3
A = @(a
(4)
where,
p
=
47r
($)
2/ 3
Substituting Equation 4 into Equation 2 and multiplying by (7/3)lI2to correct for the growth of the drop in the direction of undepleted solution yields
+
(7/3)1/2nFD1/2(M)P(at)*/3 (7rt)”2
i =
(6)
Equations 6 and 4 can be substituted into Equation 1 yielding Cdl(a
should yield a straight line of slope proportional t o the concentration of electroactive species. The value of the zerocharge potential, necessary for computation of the value of Y, can be determined from electrocapillary curves obtained for the particular supporting electrolyte-buffer medium. I n applying the method t o following the course of a titration, potential decay curves were measured a t various stages of the titration. Slopes of Y us. X plots were determined for each potential decay curve and these slopes were plotted against volume of titrant. Two straight lines were obtained that intersect at the end point. Most of the calculations and determination of slopes were performed by a computer. For a D M E the intercept of the X-Y plot, E(O), is a function of the area of the D M E a t t = 0. The charge residing on the electrode before application of the charge pulse is given by CdlA(Ei - E,) where Et is the starting potentialLe., the potential of the electrode before the charge is applied. The charge to be supplied to the cell initially resides on a capacitor, C,, that is small compared to the double layer capacitance, Cdl.A. If the capacitor is charged to voltage Vc,the charge on it is C,V,. After closure of the relay which connects the capacitor across the experimental cell, the voltage drop across C , is E(0) - Et, and the drop across the cell equals E(0). The total charge the instant before appli-ation of the charge t o the electrode is equal t o the charge the instant after application-Le., at t = 0; therefore
+ t ) 2 / 3 [ E ( t-) Ezl - C ~ Z ~ ~ ’ ~ -[ EELI ( O=)
I n order t o evaluate the integral on the right side of Equation 7 , the numerator of the integrand may be expanded as a Taylor series with a > t. The integral can then be evaluated by integrating the resulting series term by term. The result of this evaluation can be written as
where S(t)
=
1
+ 2t/9a - t2/45a2+ 4t3/567a3- .
,
. .
(9)
Substituting Equation 8 into 7 and rearranging gives
It is assumed that the area is the same the instant before application of the charge pulse and the instant after its application. Solving for E(0) gives
To be consistent with usual sign conventions for the current and potential, V, must be greater than zero for anodic processes and less than zero for cathodic processes. Since c,
