and ranged from 10 to 15%, with the Gd determination the least precise. The accuracy of the method could only be assessed by the method of known additions, the data for which are listed in Table V.
G d analytical curve has been corrected for a residual Gd content of 0.1 ppm, as determined by the graphical extrapolation method (27). Precision and Accuracy. The coefficient of variation for the line pairs listed in Table I1 was calculated for the repetitive preparation of a standard sample on five different days,
RECEIVED for review September 22, 1969. Accepted December 11, 1969. Work performed in the Ames Laboratory of the u. s. Atomic Energy Commission, Iowa State University, Ames, Iowa.
(27) T. E. Beukelman and S. S. Lord, Jr., Appl. Spectres., 14, 12 (1960).
Bipolar Pulse Technique for Fast Conductance Measurements D. E. Johnson and C . G . Enke Chemistry Department, Michigan State University, East Lansing, Mich. 48823
The applications and limitations of several ac bridge techniques and models are analyzed, especially with respect to polarization, the parallel (C,) and series (c,) cell capacitances, and the frequency required. I n the case of using a phase-angle voltmeter as a null detector, the ideal frequency is shown to be proporA new technique, which is fast tional to (cpcs)-1'2. (40 psec), accurate (O.Ol%), wide ranged (100 Q-IMQ), and independent of C, and C,, has been developed to overcome some of the limitations of ac methods. This technique consists of applying consecutive constant voltage pulses of equal magnitude but opposite polarity to a standard conductance cell. The current/voltage ratio is measured at the end of the second pulse. Theoretical and actual errors are discussed. The rate of ethanolysis of acetyl chloride is studied and compared to results by others. An EDTA titration of Zn2+ in a highly buffered medium is followed conductometrically and an end-point change of approximately 40 pmhos out of a total conductance of 40,000 Nmhos is recorded.
Figure 1. AC conductance bridge Cell consists of C,,R,, and C,
RECENTLY, there has been wide interest in systems where conventional conductance techniques cannot be used, or can be used only with great difficulty. These systems include those with very high resistance (such as much non-aqueous work), with very low resistance (such as work in molten salts), or where platinized electrodes cannot be used (if surface adsorption is a problem). Furthermore, traditional bridge techniques are not readily applicable to systems which require continuous or instantaneous conductance measurements (such as in following reactions, titrations, flow systems, or ion exchange). The conductance techniques which are used today are quite limited in applications since each was developed for a particular model of the conductance cell. The conductance measurement is valid only as long as the model truly represents the cell. The most appropriate model, in turn, depends on the experimental conditions used, especially the cell design, resistance range, and frequency. The purpose of this paper is to show the limits of applicability of some of the models and methods which are commonly used for the measurement of conductance, and then to demonstrate a completely new technique which is not only less restricted in application than the traditional methods, but is also extremely fast (40 psec/ measurement) and accurate (0.01 %).
was not considered, and C, was used only to obtain a null (not as a correction factor). This method is still the method most commonly used, as it works quite well on aqueous solutions whose resistance is between 200 $2 and 10 kn when platinized electrodes are used. Jones and Josephs (2) extended this technique by using a modified Wagner ground to make meaningful measurements up to 60 k$2. They demonstrated that for bridge balance, the reactance in any arm must be balanced in another arm, and that resistive balance occurs only if the phase angle between the current and voltage is the same in two adjacent pairs of arms of the bridge. They concluded that the Kohlrausch method was the best approximation to resistive balance. They discounted the method of Taylor and Acree (3) who used a n inductor in series with the cell to compensate for C,. Adjusting C, and R, in Figure 1 until a null is obtained ideally causes an error of (RzCzw)-2,where w is the angular frequency ( 2 ~ f ) . The equations for balance also require:
or where
AC MODELS AND METHODS
In 1893, Kohlrausch ( I ) developed the ac bridge technique shown in Figure 1. In his model, Cp (parallel cell capacitance) (1) F. Kohlrausch, Wied. Ann., 69,249 (1893).
CsRs/Rz"Cp Kz
=
+ Cz/Kz2
(2)
RzCfl.
Thus, knowing C, and w , it is possible to calculate the error term and compensate for it to obtain more accurate results. (2) G . Jones and R. Josephs, J . Amer. Chem. SOC.,50,1049 (1928)~ (3) W. Taylor and S. Acree, ibid., 38,2403 (1916). ANALYTICAL CHEMISTRY, VOL. 42, NO. 3, MARCH 1970
0
329
Figure 2. Complex equivalent circuit of conductance cell Definition of symbols
R, solution resistance Cd double layer capacitance ZF Faradaic impedance R, contact and lead resistance C, contact and lead capacitance Ci inter-electrode capacitance C t filling tube capacitance Rt filling tube resistance
Simplified circuit (Figure 1) remains the same forms C, >> impedance of C d reduced to zero
I
combine to form C ,
low or non-existent
It is obvious from Equation 2 that two frequencies must be used in order to calculate both C, and C,, which would be necessary for highly accurate results. Usually C, is ignored, which can lead to substantial errors for high R , and/or high frequency. In any case, the frequency used should be high enough so that the correction term is very small, but not so high that C, carries significant current. Whenever C, carries substantial current, the balancing of its reactance becomes very tedious (if not totally impractical) since C, must then be adjusted precisely in the pF rather than pF capacitance range. Jones and Bollinger (4) were first to realize that a conductance cell has a parallel capacitance associated with it. Their model consisted of a series resistance ( R , ) and capacitance (C,) across a cell of pure resistance (R,). They were working with a capillary cell which had large filling tubes (the source of R t and C c ) ,and by its nature, a large K,,so that C, can be ignored. They used R t and C t to explain the Parker effect which other authors (5-7) reported as a changing cell constant with increasing resistance. Their solution to this problem was to design the cell in such a way that close proximity of parts of opposite polarity is avoided. When working with solutions of low resistance, one must be concerned not only with C, and R,, but also with the lead and contact resistance, R,, and the Faradaic impedance, ZF, if there is any possibility for a reaction taking place. A complete conductance cell can be represented by the equivalent circuit shown in Figure 2. ZFconsists of several terms, including a frequency dependent Warburg impedance (8) which varies inversely as the square root of the frequency. Winterhager and Werner (9) have used a high frequency (50 kHz) bridge to eliminate the importance of ZF since at high frequency, the admittance of Cd is infinite with respect to that of Z F . No technique has been used to compensate for ZF: the experimental conditions are adjusted to minimize its effects (high frequency, low e,, etc.). Feates, Ives, and Pryor (10) have devised a method to eliminate the effects of lead and contact resistance by using a special switching arrangement in their bridge. Probably the best method to date for the (4) G.Jones and G . Bollinger, J. Amer. Chem. Soc., 53,411 (1931). (5) H.Parker, ibid., 45, 1366,2017(1923). (6) M.Randall and G. Scott, ibid., 49,636(1927). (7) F.Smith, ibid., p 2167. (8) H.Warburg, Wied.Ann. Physik., 67,493 (1899). (9) H. Winterhager and L. Werner, Forschungsber. Wirtschafts, No.341 (1956). (IO) F. Feates, D. Ives, and J. Pryor, J. Electrochem. Soc., 103, 580 (1956). 330
ANALYTICAL CHEMISTRY, VOL. 42,
NO. 3, MARCH 1970
measurement on low resistance solutions which have low c d is that recently developed by Robbins and Braunstein (11). They use a specially designed bridge with a series (instead of parallel) R , and C,, and have obtained an accuracy of better than 0.5 % ( R , = 100 Q). All of the methods so far have ignored either C, or Cz or both. Wershaw and Goldberg (12) used the equivalent circuit shown in Figure 1 for the cell model and devised a very ingenious technique using an inductive divider bridge to obtain very precise measurements of conductance changes. They have a variable capacitor in each of two bridge arms which is varied to balance the reactance of C, or C,. The frequency must be varied to ensure that each cell capacitance is balanced precisely. The main disadvantage of this method is the length of time required for a measurement ( R , and two capacitors must be balanced and the frequency varied). Strong support for the simple model is indicated as they were able to achieve resistive balance sensitive to 0.2 part per million. One of the main problems of the ac bridge techniques, the tedious capacitance balancing, has been greatly reduced with the introduction of the phase-angle voltmeter to respond only to that portion of the difference signal which is in phase with the signal source. Klaus Schmidt (13) has shown that this technique can be used with a fair amount of success without C, in the bridge circuit, leaving only R , to be varied until the “in phase” signal is zero (a phase difference of 90” between e d and e,). The equations for the bridge can be solved, and for a 90” phase difference,
+ (C,/C,)2 +2(C,/CZ) + ( 1 + Q)Rz/RiI) - ( 1 + Q ) 2 R z / R ~ 0
KZ4(C,/C,>2
K,2[
QU
=
(3)
where Q = R,R1/R,R2 - 1 = relative deviation from resistive balance. At high frequencies, C, acts as a nearly perfect shunt, so that the only error is the positive conductance deviation caused by the shunting capability of C,. At low frequencies, C, carries insignificant current and the only error is the negative deviation due to the impedance of C,. Thus, to be within a desired error limit, the frequency should be adjusted so that:
*em,,,
[ Q m A
+ R ~ / R Z ) I - ~ ~ ~
