A BIPOLAR PULSE TECHNIQUE FOR - FAST CONDUCTANCE MEASUREMENTS Thesis for the Degree of .Ph’. D. MiCHiGAN STATE UN‘S‘IERSITY DONALD E._JOHNS_0’N I 9 79 LI B R A R Y Michigan Sum University ,HESIQ This is to certify that the thesis entitled A BIPOLAR PULSE TECHNIQUE FOR FAST CONDUCTANCE MEASUREMENTS presented by has been accepted towards fulfillment of the requirements for Ph.D. degree in Chemistry 4.4 \_/ r7 Malcor professor Date February 27, 1970 0-169 mesme- ° ABSTRACT A BIPOLAR PULSE TECHNIQUE FOR FAST CONDUCTANCE MEASUREMENTS By Donald E. Johnson The applications and limitations of several a-c bridge techniques and models are analyzed, especially with reSpect to polarization, the parallel (Cp) and series (Cx) cell capacitances. and the frequency required. A complex model of the conductance cell is developed, and the necessary ex- perimental parameters needed in order to use a simplified model are given. It is shown that a-c techniques are accurate only over a relatively narrow frequency range. This range is especially dependent on Cx and C for all of the a-c bridge P techniques. In the case of using a phase-angle voltmeter as a null detector, the ideal frequency is shown to be propor- tional to (CPCX)7§. Recently. there has been considerable interest in ex- tending conductance measurements to new areas. Several attempts to measure rapid conductance changes are cited. These techniques use unbalancedbridges and assume that any change in conductance is due purely to the resistive com- Ponent. At best, this is only a very rough approximation. Other areas 0 high conducta very low conc‘ heasurements pecialiy with The bi; overcome many niques. This secutive cons OPPOSite pola current/yo 1m Pulse to give Combines the 3-0 techniQue Nsec. the eff Same as When Slightly p01; is Constant C913 the sa there is no on Cx’ Where A Pro1 aPplicetions make aCCUra. lMOhm). T “Lance on C titration g n-w _ ”Wt '13 Q Donald E. Johnson Other areas of recent interest include solutions with very high conductance (such as molten salts) and solutions of very low conductance (such as many non-aqueous solvents). Measurements on these systems take special precautions (es- pecially with respect to C and Cx)° P The bipolar pulse technique has been developed to overcome many of the limitations of the traditional tech- niques. This technique consists of applying two con- secutive 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 to give the conductance directly. This technique combines the best features of both high and low frequency a-c techniques. Since the pulses can be as short as ten usec, the effect on Cx and the Faradaic impedance is the same as when using very high frequency, since Cx is only slightly polarized. On the other hand, since the potential is constant by the end of the second pulse, the effect on Cp is the same as when using very low frequency. In theory, there is no dependence on C and a (t/Rxcx)2 dependence P on Cx' where-t=pulse width, and Rx=solution resistance. A prototype instrument was built to test the theory and applications of this technique. This instrument is able to make accurate (0.01%) measurements over a wide range (100 ohm- 1 Mohm). It was found to have excellent linearity. no depen- dence on Oh. and only slight dependence on Cx' An acid-base titration gives a typical titration curve (continuous analog output is obtained through a sample and hold amplifier). The rate of etl different ' results by buffered I! point chat me of 1+1 Ano prototype has a one Dr ..0ve Donald E. Johnson rate of ethanolysis of acetyl chloride is studied at three different temperatures and is found to be consistant with results by others. An EDTA titration of Zn2+ in a highly buffered medium is followed conductometrically and an end- point change of approximately #0 umhos out of a total conduct- ance of #0.000 umhos is recorded. Another instrument, which has many advantages over the prototype instrument. was then built. This new instrument has a one MHz crystal oscillator to control the pulse width and the pulse repetition frequency. In addition. it has a high absolute accuracy as 0.005% resistors are used. Also, it has provisions for automatic temperature compensation over a narrow (lo) temperature range. Many tests have been run on dummy and real systems. It has an absolute accuracy of better than 0.05% over the range from 10 ohm to l Mohm. falling to 0.1‘ at 10 Mohm. It has a maximum sensitivity of 1 ppm at 1 kohm. This instrument is nearly independent of Cp and Cx’ It also can be completely independent of the effects of lead and contact resistance (using a four-lead system). Measure- ments of solution conductances with this instrument and with a Wayne - Kerr Bridge agree within the accuracy of the bridge (0.1%). Several applications of this technique are mentioned. Since this technique permits fast, accurate conductance measurements which are independent of ex and Cp, it should prove extremely useful under a wide variety of experimental conditions. A BIPOLAR PULSE TECHNIQUE FOR FAST CONDUCTANCE MEASUREMENTS By . \ Donald E: Johnson A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1970 Olav ; Av Arl/ II I o R l .yfldrt‘ Unisex" a on", \vc uh x u ‘ - l y .s u. A YA .m A... e. 0v 5 n. V“' ~ 4.. 2‘ “U. . t r” a; a; a}. n. 6627?? 74-70 ACKNOWLEDGMENTS The author would like to express his appreciation to Professor C. G. Enkc for his help and encouragement throughout the course of this study. He also gratefully ackrowledges a National Science Foundation Traineeship, which supported him during his tenure at Michigan State University. The author wishes to rive Special thanks to his wife, Sharon, for all the help and encouragement she provided, and especially for typing and proof-reading the drafts of this work. ii II. TII. IV. TABLE OF CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . . . HISTORICAL DEVELOPMENT OF CONDUCTANCE TECHNIQUES A. 0-0 Techniques .,. . . . . . . . . . Bo A-C TeChnlqueS o o o o o o o o o o o 1. Traditional Bridge Techniques. . a. Typical Solutions. . . . . . b. Solutions of High Conductivity 2. Unique Bridge Techniques . . . . . 3. Phase-Sensitive Bridge Techniques. 4. Techniques Used to Follow Conductan Changes. . . . . . . . . . . . . . a. Slow Changes . . . . . . b. Rapid Changes. . . . . c. Conductivity Detectors . 6‘ 00000000000. THEORY OF A-C BRIDGE TECHNIQUES. . . . . . . . . A. General Bridge Equation. . . . . . . . . B. Kohlrausch Bridge Balance. . . . . . . . C. Phase-Sensitive Detector Balance . . . . D. Applicability of Traditional Techniques. THE BIPOLAR PULSE TECHNIQUE. . . . . . . . . . o A. Theory . . . . o . . . . . B. Theoretical Error Analysis C. The Instrument . . . . . . D. Instrumental Tests . . . . l. Linearity Test . . . . 2. Dependence on Parallel Cell Capacita 3. Dependence on Series Cell Capacitanc (4. A‘C Bridge Comparison. 0 o o o o o E. Instrument Tests on Chemical Systems . . 1. Acid - Base Titration. . . . . . . . 2. Reaction Rate Studies. . . . . . . 3. Conductometric EDTA Titration of Zin A NEW INSTRUMENT . . . . . . . . . . o o o o . C0 . . . o o o C A. Improvements . . . . B. The New Circuit. . . 1. Analog Circuit . 2. Digital Circuit. iii 70 71 71 75 . . FWD 9. i’. C : 'O L. V. C. Temperature Compensation . . . . . . . . . . 78 D. Obtaining Absolute Conductance . . . . . . . 82 E. Instrument Tests . . . . . . . . . . . . . . 83 l.‘ AbSOlUte Accuracy. 0 o o c o o o o o o o 83 2. SenSitiVity. o o o o o o o o o o o o o o 85 3. Dependence on the Parallel Cell CapaCitance. o o o o o o o o o o o o o o 86 U. Dependence on Series Cell Capacitance. . 86 5. Use of Four Cell Leads . . . . . . . . . 86 6. A-C Bridge Comparison. 0 o o o o o o o o 89 7. Temperature Compensation . . . . . . . . 90 VI. CONCLUSION O O O O O O O O O O O O O O O O O O O 92 REFERENCES 0 O O O O O O O O O O O O O O O O O I O O O 95 APPENDICES. O 0 O O O O O I O O O O O 0 O O O O O O O 99 A. PROGRAM FOR CALCULATING PSEUDO FIRSTEORDER RATE CONSTANTS . . . o . o . . . . o o o o o 99 B. CIRCUIT DESCRIPTION OF IMPROVED BIPOLAR PULSE CONDUCTANCE INSTRUMENT . . . . . . . o 105 iv 7. C O .2. Table l. 2. 3. u. 5. 6. 7. 8. 9. 10. ll. 12. 13. 14. A-l. LIST OF TABLES Complex Conductance Cell Symbols . . . . . . . Phase-Angle Voltmeter Dimensionless Parameters Examples For Using Table 2 . . . . . . . . . o Constants for Calculating the Ideal Frequency. Phase-Angle Voltmeter Frequency Dependence . . Dependence on Cx . . . . o . . . . . . . . . . Reaction Rate Studies -- Acetyl Chloride and EthanOIoo00000000000000... Absolute Accuracy Test . . . . . . . . . . . . Sensitivity gs. Resistance . . . . . . . . . . Dependence on Cp . . . . . . . . . . . . . . . Dependence on Series Capacitance . . . . . . . Elimination of Contact and Lead Resistance . . A-C Bridge Comparison. . . . . . . . . . . . . Temperature Compensation in KCl Solutions. . c Three-Parameter, Least-Squares, Curve-Fit Programoooooooooooooooooo. Input and Output of Least—Squares Program. 0 o Page ll 26 28 31 52 57 84 87 88 LIST OF FIGURES A-C Conductance Bridge. . . . . . . . . . Complex Equivalent Circuit of Conductance A-C Bridge Analysis . . . . . . . . . . . A Bipolar Pulse Conductance Device. . . . The Prototype Bipolar Pulse Instrument. . Timing Sequence During Measurement. . . . Constant Potential Generator. . . . . . . Linearity Test. . . . . . . . . . . . . . Acid - Base Titration . . . . . . . . . . Reaction Rate Studies . . . . . . . . . . Temperature Dependence of Reaction Rate . Temperature Profile During Reaction . . . Conductometric EDTA Titration of Zinc . . The Improved Bipolar Pulse Conductance Instrument Digital Logic Circuit for Improved Instrument Waveforms Of Logic CirCUit. . o o o o o 0 Circuit Diagram of the Improved Conductance InStrumentocoo-000000000. The Bipolar Pulse Instrument Front Panel. vi Page 106 110 a‘ I e and the :4 VA‘ .1. I. INTRODUCTION Conductance measurement is one of the oldest of the accurate electrochemical techniques. The basic a—c bridge technique was developed nearly a hundred years ago, and hasn't changed significantly since that time. Throughout the history of conductance techniques, minor modifications have been made in the methods and the models in order to more accurately determine and characterize the conductance and the equivalent circuit of the conductance cell. The conductance techniques which are used today are quite limited in application 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 cello The most appropriate model, in turn, depends on the experi- mental conditions used, especially the cell design, resis- tance range, and frequency. In discussing the history of conductance techniques in Chapter II. a complex model for the conductance cell will be developed. Several examples of what has been done to simplify the cell model will be cited. and the best model for normal conductance work will be critically analyzed in Chapter III. It will be shown that the traditional Kohl- rausch method and the new phase-sensitive techniques have . c'v c. S xyfiv‘i.’ VSa' ' a “3’0““ M-‘ CI' 3' v. e e w... t C t t i :u a» I AM“ w...“ .r» A» V!‘ s y .1 cl 9,.“ .‘ st. : 2 very distinct frequency ranges in which the results fall within reasonable error limits. The frequency range is especially dependent on the parallel and series cell capa- citances. Recently. there has been wide interest in systems where the traditional 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 conductance changes in reactions. titrations. flow systems. or ion exchange). Several attempts to measure conductance under these condi- tions will be cited. A new technique, which is less restricted, will then be introduced in Chapter IV. This technique uses the best characteristics of both high and low frequency measurements. It is demonstrated on dummy and real systems and is found to be very fast. accurate, and wide-ranged. In addition. it is much simpler to use than traditional techniques as it requires no balancing (either capacitive or resistive). Finally, a new instrument that is extremely accurate will be described. This model includes a crystal oscillator timing device. provisions to completely eliminate contact and lead resistance, and automatic temperature compensation over a narrow temperature range. *7 .5. ductanc a ConS' {if H ‘0 L‘ l n (/l 1.) H .4 r‘ II. HISTORICAL DEVELOPMENT OF CONDUCTANCE TECHNIQUES A. D-C Techniques Thus far. there has been only one technique which can compete with the a-c bridge techniques to accurately obtain conduCtance. This is the d-c technique. introduced by Newberry (1) in 1918. He showed some of the disadvantages of a-c methods and presented a cell arrangement for d-c con- ductance measurements. The d-c technique consists of applying a constant current through a cell and measuring the potential drop across a certain portion of the solution with two elec- trodes which carry no current. Newberry used calomel elec- trodes for all four electrodes. and made sure that each pair had exactly the same open circuit potential. He compared his measurements of the specific conductance of KCl solutions to those of Kohlrausch, and found the Kohlrausch values to be about 0.5% lower. Eastman (2) extended this d-c technique and found the Kohlrausch values to be about 0.2% low. He built a bridge arrangement which could be used with either d-c or a-c tech- niques. He concluded that conductivities at zero frequency are about 0.02 - 0.03% higher than those at frequencies between 1 - 2 kHz. Fuoss and Kraus (3) found the d-c tech- nique necessary for the measurement of the conductance of solutions with resistances over 100 kohm. In their 3 :onumenta constant 103 Re tr“ Pl 1 '1 m (D H (.2. F10 N fl 1" a monumental work to determine the influence of dielectric constant on conductance. a-c techniques were used up to 100 kohms. and d-c techniques above that value. Gunning and Gordon (4) showed that the d-c technique can be a serious competitor to the a-c bridge techniques (0.02% accuracy). They pointed out the necessity for small. reversible electrodes. and they improved the cell design. Elias and Schiff (5) modified this technique even further by using a liquid junction to remove the measuring electrodes from the solution. This solved the problem of finding rever- sible electrodes for some non-aqueous solvents. Gavis (6) rigorously derived the current-voltage rela- tionships in d-c conductance work. He gives equations for the proper cell design. and shows under what conditions a-c methods may be used. The determining factors include dielec- tric constant. resistance. parallel cell capacitance. and frequency. Others (7. 8) have also had success using d-c techniques. especially with very low conductance solutions. B. A-C Techniques 1. Traditional Bridge Techniques a. Typical Solutions From the very beginning of conductance measurements. it was realized that an a-c signal was necessary in order to reduce polarization of the electrodes. If a d-c potential were applied to the cell. the electrodes would become polar- ized. and would quickly reach a potential at which a reaction would take place (e.g. - hydrogen liberation). Applying an c h '1. ith t .;U'. half-c W a flu b... V O a .l R Q s u e H... Ox; “I Cc m... a . 5 a-c signal to a Wheatstone bridge with the cell in one arm did cause a problem: the bridge could not be brought to a null. The electrodes were slightly polarized during each half-cycle. giving rise to an effective capacitance (the double-layer capacitance) in series with the solution resistance. Kohlrausch (9) was first to put a capacitor in parallel with the standard resistor in order to compensate for the reactance caused by the double layer capacitance. For con- ductance work prior to this. see the literature cited in his book (10). Figure 1 shows a typical conductance bridge where Rx is the solution resistance. Cx is i the double layer capacitance (for identical electrodes), and Cp is the inter- electrode and stray parallel capacitance. The Kohlrausch equivalent circuit is shown in Figure 1 with C O. This p: method introduces a relative error of (RXCXu)'2, where a is the angular frequency (27f), as will be shown later. Kohl- rausch recommended several precautions to cut down on this error which is due to polarization of the electrodes: 1) platinize the electrodes 2) use frequencies > 1000 Hz 3) use large electrodes 4) design the cell so that RX > 100 ohms Washburn and Bell (11) extended the Kohlrausch method by using a "pure" signal source and a tuned telephone detec- tor. They also stressed more careful capacitance balancing. Washburn (12) discussed the theory of cell design and showed \ that at high frequency. Cx can be neglected, but that Figure l A—C Conductance Bridge. 1191 < I‘d “F ~- 0 mi 3110... flag? y VA m) alsc (h cOI 'rr‘fi e U 3 U. gun v conduc ta IRE TERI a V phase 2: ‘wo pan . U 9.. .e L. 03011‘ ‘ v0 ‘8. ‘3‘ P 9. S 9 b. l e l .3 F 9 h. 3 h. A: nu 8 parallel cell capacitance causes interference. Hall and Adams (13) extended the telephone null detector by using a thermionic amplifier to increase the sensitivity. Gieringer (1#) also developed an improved a-c null indicator. Jones and Josephs (15) developed a direct-reading a-c bridge technique using a modified Wagner ground that extended conductance measurements to solutions of low conductance (up to about 60 kohm). They demonstrated that for bridge balance, the reactance in any arm must be balanced in another arm. Thus. the relationship R Rx = R R3. is valid only if the 1 2 phase angle between the voltage and current is the same in two pairs of adjacent arms of the bridge. They concluded that the method of Kohlrausch was the best approximation to this balance. and that R1 and R2 should be equal to minimize the chore of balancing reactances. They discounted the method of Taylor and Acree (16), who used an inductor in series with the cell. They also discarded as impractical the idea of putting a large capacitor in series with R8. Jones and Bollinger (17) made refinements in both the oscillator and detector portions of the a-c bridge. Their modifications reduced polarization because of a lower signal voltage. They were first to realize the importance of the parallel cell capacitance (18). Their model consisted of a series resistance (Rt) and capacitance (Ct) across a cell of pure resistance (Rx)' They were working with a capillary cell which had large filling tubes (the source of Rt and Ct)’ and by its nature, a large Rx' which allowed CX to be ig- nored. They used R and Ct to eXplain the Parker effect t which 0 censtan problem imity c (I) ideas a itprove I. :D ’ 1 :3 (D (D m I 0 H0 :T H' 73 :1 a H 0 9 which other authors (19, 20, 21) 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 prox- imity of parts of opposite polarity would be avoided. Shedlovsky (22) was able to obtain very accurate meas- urements (0.001% sensitivity) by using a screened bridge to measure conductance. Fey and Martell (23) used Shedlovsky's ideas along with suggestions by other authors to build an improved conductance bridge. Cole and Cross (24) developed a bridge that could accurately measure both conductance and capacitance over a fairly wide range. Hladek (25) designed a similar bridge (0.04% accuracy). b. Solutions 32 High Conductivity When working with solutions of low resistance, one must be concerned not only with Cx and Rx' but also with the lead and contact resistance. RC, and the Faradaic impedance, ZF, if there is any possibility for a reaction to take place. A complete conductance cell can be represented by the equiva- lent circuit shown in Figure 2, with the symbols described in Table l. ZF consists of several terms, including the frequency dependent Warburg impedance (26), which varies in- versely as the square root of the frequency. Winterhager and Werner (27) 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 ZF' No technique has been used to compensate for ZF; the experimen- tal conditions are adjusted to minimize its effects (high frequency. low es. etc.). Feates. Ives, and Pryor (28) have 10 Figure 2 Complex Equivalent Circuit of Conductance Cell cl ~ ‘mn 158‘.) ,‘U +9 fly .3 11 TABLE 1 Complex Conductance Cell Symbols Name solution resistance double layer capacitance Faradaic impedance contact and lead resistance contact and lead capacitance inter-electrode capacitance filling tube capacitance filling tube resistance Value in Simplified Model (Figure 1) remains the same forms Cx >> impedance of C d reduced to zero combine to form Cp low or non-existent sta res; o N A 1 call; < l 0"n .A. ’, i 12 stressed the importance of eliminating ZF. They also have devised a method to eliminate the effects of lead and contact resistance by using a special switching arrangement in their bridge. They used a four-lead, double-cell technique. Metcalf (29) points out that the double layer becomes chemi- cally ill-defined if ZF is important. He also points out that ex is normally 10” - 106 times Cp. Probably the best method to date for measurements on solutions of low resis- tance that have low Cd is that recently developed by Robbins and Braunstein (30). They use a specially designed bridge with a series (instead of parallel) RS and CS. and have ob- tained an accuracy of better than 0.5% (RX = 100 ohm, Cx 8 20 pF). 2. Unigue Bridge Techniques Bertram and Cruse (31) have developed a unique R-C method for conductance measurements. The basis of the method is that the frequency and magnitude of an R-C signal genera- tor is dependent on the characteristics of the feedback loop. They point out the frequency and sensitivity limits of their method over its entire range of conductivity measurement (10'12 - 10 ohm'lcm'l). Also. the power of this method is pointed out (32) and the method is compared to d-c methods. Cruse and Bahr (33) use the R-C method with a three-electrode system to follow reactions with half-lives of several hun- dred seconds. Wershaw and Goldberg (34) used the equivalent circuit shown in Figure l for the cell model and devised an ingenious technique using an inductive divider bridge to obtain very must be arsed pr tcrs mus \u ‘- SuppOI‘. 30 ECELE zillion. 13 precise measurements of conductance changes. They have a variable capacitor in each of two bridge arms which is varied to balance the reactance of Cp or Cx' The frequency must be varied to ensure that each cell capacitance is bal- anced precisely. The main disadvantage of this method is the length of time required for a measurement (RS and two capaci- tors 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 parts per million. There have been many other attempts made to obtain the conductance more easily or more accurately using a-c bridge techniques. Warshawsky (35) developed a multiple-bridge circuit. Griffiths (36, 37) deve10ped a contactless, four- electrode system (using a three-terminal transformer as the ratio arms) that uses relatively low frequency. The four- electrode electronic half-bridge system of Ferris (38) and the very low frequency (0.8 Hz) method of Hanss and Guermon- prez (39) are just two more examples. 3. Phase-Sensitive Bridge Techniques Recently. several phase-sensitive devices have been applied as null detectors in a-c bridge arrangements. Janz and McIntyre (40) used a modified impedance comparitor to separate the difference signal into in-phase and quadrature components. They were able to achieve very precise bridge balance (0.002%) over a wide resistance range (2 ohm - 20 Mohm) using an inductively coupled ratio arm and four cell leads. Wal observe a resistive an accura nique (#2 conductor half-life bfidge(( Sc! can be us fully wi‘ greatly 1 hsted 1; later). L'o Tech \' 3. Cf l4 Walisch and Barthel (41) have developed a technique to observe a rapid phase-shift in the difference signal near resistive balance. They claim a sensitivity of 0.0001% and an accuracy of 0.03% for their method. They used this tech- nique (42) to follow the titration of Fe2+ with Cr2072‘ conductometrically. They also studied a reaction with a half-life of approximately one minute using a self-balancing bridge (0.1% accuracy). Schmidt (43) has shown that a phase-angle voltmeter can be used in a conductance bridge circuit quite success- fully without any capacitance balancing. Although this greatly simplifies bridge balance. the frequency must be ad- justed if one expects accurate results (as will be shown later). 4. Techniques Eggg 32 Follow Conductance Changes a. §l2g Changes Often, it is not necessary to know the absolute con- ductance, but merely to follow a change in conductance. Most often. under these circumstances. it is desirable to be able to follow the change without further manual intervention (such as balancing the capacitance and resistance). Some- times there is time to allow semi-balance or even complete balance while following such changes. For example, Hladek's method (25) allows separate capacitance and resistance balance with little interference between them. Robertson (44) used a different method to obtain separate resistance and capaci- tance balance. He studied some reactions (44, 45) with half- lives on the order of 1000 secs and obtained an accuracy of 4 a 8057‘: 1n 4593 18 to withi Si afiplifie: method h; total) 0' uEVt (l - 2,4 < 1’10 Lepra‘ 100 k‘z . as the o._ S‘ 4 Eu: WOrk. fluetifl _ 15 0.5% in k (pseudo first-order rate constant). The method used is unique in that two arms of the bridge consist of identical cells: one for the reaction and the other for a titration. In this way. he could titrametrically determine the concentration Kg; time. Murr and Shiner (46) developed an improved technique. and were able to obtain rate constants to within 0.03%. b. Rapig Changes Sirs (47) used a 50 kHz oscillator with a rectified and amplified difference signal to obtain a d-c output. His method has an estimated relative accuracy of 5% (0.5% of total) over a 10% conductance change. and is capable of fol- lowing reactions with half-lives of about 30 msec. Prince (48) developed a similar method using a 20 kHz signal and was able to follow reactions with half-lives as small as 10 msec (l -‘2% conductance accuracy). Strehlow and Wendt (49) were able to follow very fast reactions with relaxation times as short as 50 psec, using a "pressure-step" technique. They photographed the entire relaxation curve envelope (50 or 100 kHz signal observed on the time scale of the experiment) as the system returned to bridge balance. Eurato and Leimu (50) used a system that was based on a working curve for bridge off-balance in order to obtain kinetic results with half-lives in the seconds range. Dusek and Lesek (51) developed an improved four-electrode technique for measuring fast conductance changes. Tregloam and Laurence (52) have devised a very sensitive bridge arrangement using a tuned a-c detector followed by an a-c to d-c converter. 16 They can follow a 1% conductance change within 0.5% (0.005% of total) with a response time of 1.5 msec. c. Conductivitnyetectors In addition to their use in kinetic studies. there has been much interest in the past few years in developing good detectors for ion-exchange and liquid-liquid chromatography. Conductivity methods have even been applied to paper chroma- tography (53). Duhne and Sanchez (54) developed a four-cell conductivity bridge for use as an ion-exchange detector. They used a 20 V, 20 kHz signal source. Pecsoc and Saunders (55) developed a very sensitive, low-volume conductivity detector for liquid chromatography. They used this method to determine the distribution coefficients in inorganic gel chromatography (56). Knudson, Ramaley, and Holcombe (57) suggest two circuits for general conductivity detectors (es- pecially for chromatographic use) using operational amplifier bridge networks. One circuit is extremely sensitive (0.001%) and is linear for small changes. The other circuit is not as sensitive but is linear for even large conductance changes. Additional information on conductance methods is given in several review articles. Gerischer (58) has written an excellent article with 203 references. He critically ana- lyzes various bridge networks for conductance measurements. His article includes balance equations. bridge equivalent circuits. effects of Z Cp. Cx' lead inductance, and cell FD design. Cruse (59) also has a very good article, with a review of the classical bridge techniques, and with Special I l emphasis c consulted. 17 emphasis on high frequency methods. A review of the theory of electrolytic solutions is given by Barthel (60) with 248 references. There are also many books (61 - 64) on the subject of conductance theory and methods that can be consulted. n U ~ g OI‘Q .er. dev sated to heref fit an al :siel i y. b as; o‘c‘ III. THEORY OF A-C BRIDGE TECHNIQUES A. General Bridge Equation A complex model (Figure 2) of the conductance cell has been developed in Chapter II. This model is far too compli- cated to use realistically in practical applications. Therefore. the experimental parameters must be adjusted to fit an acceptable simplified model. The best simplified model is shown in the bridge network of Figure 3. This model assumes that the lead and contact resistance is negligible and that the frequency is high enough so that ZF can be con- sidered infinite. These are reasonable assumptions since they are necessary conditions for accurate measurements. When the experimental parameters make this model inappropriate accurate conductance data is extremely difficult, if not im- possible. to obtain. Therefore. the following mathematical theory of a-c conductance measurements will be based on this simplified model. Assuming a sinusoidal signal source for the bridge net- work in Figure 3, eS = V sin at (all voltages with respect e2 = A sin at + B cos at to the reference point) e = D sin at + E cos wt 18 19 Figure 3 A-C Bridge Analysis Cell consists of R . C , and C x x p where y D . . 03‘ C p = C . ' I Rd _ F wastitu‘t and Solvi and 21 From Figure 3, 1 eZ-eC - C deC 2 ‘ 'R;' ‘ x dt or (A-D) sin wt + (B-E) cos at = RXCxw(D cos wt-E sin wt) (1) Solving Equation 1 for D and E and letting Ky = wRXC where y = x. s. or p. yields: D (A + ka)/(1_+ 183') (2) E . 2 (B - KXA)/(l + KX ) (3) If no current flows through the measuring device between e1 and e2, . dez e2-eC deS de2 eS-e2 ..det + RX =Cs
> Cp/Cx. this becomes 1 .1. .. kX a (ox/cp)[(1 + 122/121); — 1]2 (26) Equation 23 can be rearranged to get: a 2 2 2 Kx (op/0x) + kX [(Cp/CX) + zap/cx - Q(1+R2/R1+Q R2/R1)] —(1+0)2 RZ/Rl = 0 (27) which is quadratic in sz. Table 2 shows Kx as a function of RZ/Rl' Cx/Cp. and Q. The estimated value of XX is calcu- lated using Equation 26, the exact value using Equation 25, and the Kx for the errors indicated using Equation 27. 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For example, with Rx = l Mohm and Cp = 1000 pF, the relative scatter was 0.2%. This is, however, a condition which has no physical significance because Cp would never be that high, and especially not for a l Mohm solution resistance. 3. Dependence 23 Series Cell Capacitance A dummy cell consisting of a capacitor (0.05 - 10 uF) and a resistor (l kohm - l Mohm) in series was made to test the dependence of the conductance measurement on the series capacitance. It should be noted that the largest capacitor (10 pF) used is still much smaller than t the typical double layer capacitance. The results of the experiment are shown in Table 6. From Table 6, it is obvious that there is a trend toward larger errors as Rx and Cx decrease and as t increases (a completely expected result). For large resistances, the pulse length is increased to allow the amplifier (A2) to reach its final voltage before making a measurement. For the low conductance ranges, the feedback resistor (R2) is either 100 kohm or 1 Mohm, and 20 usec may not be enough time to achieve the final voltage. It should be noted that in these conductance ranges, very little current flows through the cell, so Cx still does not charge significantly. If one uses a typical conductance cell in which Cx > 100 pF, the errors would all be negligible, even for lower resis- tances. In a situation in which unplatinized electrodes must be used (such as in some non-aqueous work), the resistance is 52 TABLE 6 Dependence on Cx 10 50 100 50 100 500 kn kn kn kn kn ka kn k0 Mn % Error for Cx in uF 10 1 0.5 0.2 0.1 0.05 0.050 0.861 1.930 0.007 0.059 0.091 0.197 0.005 0.046 0.080 0.163 0.423 t = 20 usec 0.001 0.020 0.042 0.103 0.206 0.000 0.003 0.010 0.036 0.085 0.252 0.006 0.024 0.042 0.137 t = 0.2 msec 0.005 0.008 0.028 0.069 0.001 0.004 0.026 0.058 0.122 0.300 0.000 0.001 0.005 0.017 0.042 0.133 53 normally high so that the decrease in Cx is compensated for by an increase in Rx' Therefore, the error remains small. In any case, the error involved using this technique is smaller than the error involved when using a bridge tech- nique on the same system. 4. 5:9 Bridgg Comparison The conductance of three solutions was measured using both this instrument and a modified Wien bridge (65). Below 10 kohm the two methods agreed to better than 0.1%, but at 60 kohm, the bridge could not be nulled and was about 2% high. E. Instrument Tests 93 Chemical Systems 1. Acid - Base Titration As a first test, the chemically uninteresting reaction between an acid and base was followed conductometrically. A solution that was 10"3 M in HCl was titrated with 5 M NaOH using a constant delivery burrette. Delivery was made through a capillary tube with a 15-foot hydrostatic head to provide constant pressure. A typical curve is shown in Figure 9. The experiment was repeated several times, and in each case. the same shaped curve was obtained. When the amount of HCl was varied, the endpoint was proportional to concentration, but the magnitude of the minimum remained unchanged. From this minimum, one can calculate the ratio of the equivalent conductances of NaCl to HCl of 0.296. This compares very favorably with the Kohlrausch ratio of 0.297. Figure 9 Acid - Base Titration 0.001 M HCl with 5 M NaOH SS 56 2. Reaction Rate Studies The pseudo first-order reaction between ethanol and acetyl chloride was studied at three different temperatures. The ethanol was dried by the method of Lund and Bjerrum (66), and the acetyl chloride was used without further purification. Approximately 3 p1 of acetyl chloride was injected into 125 m1 of ethanol for each experiment. Several experiments were run in the same solution (using the current offset to keep the same sensitivity), and no significant deviation was found between runs. The conductance change was measured using a Heath Universal Digital Instrument (UDI) in the digital voltmeter mode (0.1 sec aperture), with a period of 1.500 sec between measurements. Every fourth measurement was manually recorded (with another UDI keeping track of counts) so that a data point was obtained every six seconds. The rate con- stants were obtained using the three parameter least squares curve-fit computer program, which is included in the appendix. This program was written for the PDP 8/i computer, but could easily be revised to adapt to other systems. It minimizes the general equation: E = i "MS 1 [Y1 + B'exp(-kXi) -A]2 (7) where Xi and Yi are the data points, and A, B, and k are the curve-fit constants (k = rate constant). To guarantee con- vergence, a modification of the Gauss - Newton successive approximation method was used. It is similar to the modifi- cation by Hartley (67), which Daum (68) used in his treatment of current impulse data. The results are given in Table 7. TABLE 7 Reaction Rate Studies -- Acetyl Chloride and Ethanol Temperature k (sec'l)i std. dev. Number of Runs 4.740C‘ 0.03oéio.oooz 7 6.4o°c 0.0346i0.0002 5 15.01°c 0.07ooto.ooo7 15 Typical plots of (n (G. - G)/G. lg; time are shown in Figure 10. The standard scatter for a typical experiment is less than 0.1%. The data obtained agree very well with data obtained by Euranto and Leimu (50), who used a rather com- plicated, fast-quenching titrametric method. Their data and these data obtained conductometrically are shown in the Arrhenius plot in Figure 11. It should be mentioned that constant temperature was assumed during the reactions, although this is not strictly true. The temperature of the bath was constant, but the heat of reaction, and to some extent, the original higher tempera- ture of the acetyl chloride, caused the temperature to increase by about 0.01°C during the reaction. A temperature profile is shown in Figure 12 for a 5 - 6 times greater concentration of acetyl chloride than was used to determine the reaction rate. These data were obtained by using the bipolar pulse conductance instrument to measure the conduc- tance of a calibrated thermistor in place of the conductance cell. This instrument permits a rapid, accurate measurement Figure 10 Reaction Rate Studies Acetyl chloride (3 pl) and ethanol (125 ml) Consecutive reactions at 6.4000 ._ 1——— 59 938-99 5- Time (sec) 60 Figure 11 Temperature Dependence of Reaction Rate Circles obtained conductometrically Dots obtained by another method -ln K 61 l05/T L— F' J.I I...Jil. 330 340 350 360 62 Figure 12 Temperature Profile During Reaction 18 pl acetyl chloride into 125 m1 ethanol at 6.420C 63 '6A7r r I 6.46 '- 6.45 " T°C 6A4“- 6.43 '" 6.42% l 1 l 4 1 l 1 j I I00 200 Time (sec) 300 64 of temperature with extremely little solution heating. The half-lives of these reactions were between 10 - 20 seconds. Much faster reactions could be studied if appropri- ate readout and recording were used. Reactions with half- lives of less than 1 msec could easily be followed using stopped-flow or other techniques. Since the output is pro- portional to concentration, the results are very easy to manipulate. 3. Conductometric EDTA Titration 9f Zinc Known volumes of 0.04 M Zn(N0 and pH 10 buffer 3’2 (0.55 M NH3 and 0.10 M NHuCl) were pipetted into a beaker and diluted to 125 ml. The resulting solution was titrated with a 0.1007 M EDTA solution using a Sargent Automatic Burrette (10 ml). The conductance cell consisted of two unplatinized platinum disks 1 cm in diameter and 1 cm apart. The solution resistance varied between 21 and 35 ohm, depen- ding on initial concentrations of zinc and buffer, and the change at the endpoint was only about 0.02 ohm (0.1% of total). This is truly a worst case situation since the conductance instrument was not designed to operate below 200 ohm! The noise level was under 0.01% of the total con- ductance, even at this very high conductivity. The change in conductance was recorded on a strip chart recorder at 5 in/min with a full-scale sensitivity of 175 pmhos (about 0.5% of the total conductance). The offset current source was used to "buck out" about 99.5% of the total cell current. Typical endpoints are shown in Figure 13 for 10 ml of the zinc solution and varying amounts of buffer. Notice that f...“ “r.- (:01 .i 65 Figure 13 Conductometric EDTA Titration of Zinc (a) 10 ml buffer (0.0281 0‘1) (b) 15 ml buffer (0.0377 n'l) (c) 20 ml buffer (0.0476 0'1) Full scale = 175 “0‘1 Arrows indicate endpoint Sharp breaks due to changing the offset current INCREASING CON DUCTANCE I4—Iml —->I _ — INCREASING VOLUME mi 67 this is not a "typical" conductance titration curve in that there are actually two breaks near the endpoint. The first break is taken as the endpoint for the following reasons: (1) ‘The first break is proportional to the amount of zinc added (whereas the second break is not.) (2) There is a sharp conductance increase which levels off to a more moaerate increase even when no zinc solution is added (only 15 m1 of pH 10 bUffer) e (3) The first break agrees within 0.2% with an Eriochrome Black T titration on the same system. While it is not the purpose of this thesis to enter into a detailed study of equilibria, a few observations of the behavior near the endpoint should be mentioned. The pH is constant to within 0.001 pH unit throughout the titration, as recorded using a Heath eXpanded scale pH meter/recorder. It is not a slow, rate-controlled process, but rather an equi- librium situation. The same type of endpoint behavior is obtained when calcium is used instead of zinc. The difference between the initial and intermediate (between breaks 1 and 2) slopes is independent of both the zinc and buffer concentra- tions. This slope is: 3.2to.2 ver div/hor div x 5 hor div/m1 x 17.5 pmhos/ver div = 270 nmhos/ml of EDTA. Assume that the conductance is proportional to the total normality, then G = K'N where K is a constant. Before the endpoint, the reaction is: 2- 4. 2+ 2 Na , HZY 2 NH3 + Zn 7‘ ZnY2-+2NH1++ + 2 Na+ (8) yielding an increase in ions of 0.4 meq/ml of EDTA. 1n the case of 15 m1 of buffer, the conductance is practically 68 constant before the endpoint. Thus, the increase in ions is almost exactly offset by the dilution with titrant. Therefore, in this case, N 0.4 meq/ml and -1 _. 000377“ - '1 “1 After the endpoint, the reaction (at pH 10) is 2- 4 Na+, 2 H Y _ _ 3 NH3 2 as 3 NH4+ + HY3 + Y4 + 4 Na+ (9) yielding an increase of 0.7 meq/ml of EDTA. Of this, 0.4 meq/ml is needed to offset dilution and the remaining 0.3 meq/ml increases the conductance of the entire solution. Thus, the slope after the endpoint should be: 0.3 meg(ml EDTA 129 m1 x 0'1 ml 9.1 = 230 uthS/ml EDTA which is very nearly the experimental result. These data tend to point toward the formation of a complex between EDTA and NH4+ after the EDTA concentration is built up following the endpoint. The endpoint is easily detectable, even though it must be continuously recorded rather than obtained by making a few measurements and drawing intersecting lines. The result of seven runs with different volumes of zinc and buffer solutions is: 2.13210.003 inches/m1 of Zn2+ A volume reading of 6095 on the burrette was found to be equivalent to 42.31 inches of chart paper. The eXperiment 69 was repeated using 10 ml of Zn2+, 15 ml of pH 10 buffer, and 5 drops of Eriochrome Black T indicator. The result of four such titrations is: reading = 310614 blank 2 0029 volume = 307714 To correlate the results: + . 31%Zfi1 X Egggégifl = 2.136io.oo3 in/ml Thus, there is excellent agreement between the two ,1 methods. It should be obvious that since this method is capable of precisely following very small changes even in highly conducting solutions, it makes possible conductometric analysis on other systems (such as redox reactions and ion exchange monitoring) in whiCh it has previously been im- possible or very impractical to make conductance measurements. V. A NEW INSTRUMENT A. Improvements The prototype instrument worked so well that it was decided to build another instrument with many added features. In the new model, the pulse width is controlled by a 1 MHz crystal oscillator. This makes the pulse widths much more stable than the monostable multivibrator pulses used in the prototype. This, in turn, results in higher accuracy, since the pulses are much closer to being complementary. In addi- tion, the pulses are only 10 usec long for the shortest pulse width. Thus, the error due to polarization will be only half that of the prototype. The same crystal oscillator also controls the pulse repetition rate so that experiments may be carried out on a very accurate time scale. As an additional improvement, 0.005% precision resis- tors are used so that a high absolute accuracy may be ob- tained. The offset current along with the sensitive null meter on this instrument allow it to make accurate measure- ments as a self-cantained conductance ”bridge" (with a linear, calibrated offset). The circuit was completely re-designed so the solid-state switch resistances no longer affect the accuracy. It can also operate with four cell leads so that all lead and contact resistances are unimportant. 70 71 Finally, an automatic temperature compensation circuit was added. This allows the temperature of the solution to vary by as much as 1°C without having a noticeable effect on the accuracy of the conductance readout. The output is automatically corrected to the value it would have had if the temperature of the solution had not changed. B. The New Circuit 1. Analog Circuit The block diagram of the new instrument is shown in Figure 14. The switches are in the positions as shown until condition A, B, or 0 exists, in which case, the appropriate ~ switch changes state. The conditions occur as follows: A - during the first pulse, B - during both pulses, and C - during the sampling time (the last 8/10 of the second pulse). ,The function of the operational amplifiers is given below. A; controls the top electrode of the cell to be precisely at +5 RZV/lo kS'l during A and -5 RZV/lo k9. during the second pulse; iggé A-B ("A“ means “not the condition A” and ”a“ is the logical ”and” symbol). Ag furnishes an offset current to compensate for the cell current. R1 is a lOOlohm, 5 decade resistance box (0.005%). If R1 and R3 are adjusted so that the current from A2 is exactly opposite from the cell current at the end of condition B, then a ”balanced-bridge" condition exists. 5} provides an output voltage proportional to the conductance (with intercept determined by the offset current). 72 Figure 14 The Improved Bipolar Pulse Conductance Instrument 74 During C, A3 controls the bottom electrode to be at virtual ground and it charges the holding capacitor (Ch) so that at the end of the second pulse, the voltage across Ch is proportional to conductance. During 0, Ch is connected to virtual ground, so that eo remains proportional to the conductance. 54 keeps the bottom electrode at virtual ground during 0. This further ensures complementary pulses, as was pointed out as necessary on page 46. A4 must sink the current supplied by A1 and A2 during C. 55 yields an output which is proportional to the total conductance (with an intercept of zero). This is used for temperature compensation. A6 yields an output which is approximately proportional to the temperature deviation. RT is a thermistor and RA is the temperature adjustment. At the start of an experiment, RA is adjusted to equal RT (ed 2 0). If the temperature changes, ed will no longer be zero. This voltage and the output from A5 are multiplied together so that a temperature correcting current (which is proportional to the product of the conductance times the temperature deviation) is supplied to the summing amplifier, A3. It is important to observe that the electrodes them- selves are at precisely controlled potentials. The top electrode is connected to the inverting input of the voltage follower A1, and the lower electrode is connected to virtual ground of either A4 (during 0') or A3 (during C). The outputs of the operational amplifiers control these inputs 75 (gig the feedback loop) to be at the same potential as the non-inverting inputs (ground potential in the case of A3 and A4). Thus, both electrodes are controlled to be at precisely defined potentials regardless of switch, lead, contact, or cell resistance or inductance. Amplifiers A1, A3, and A4 are very fast, settling to Within 0.01% in 1 usec (69), so that rapid_measurements can be made. The other amplifiers are not as fast, but since they are not involved with the pulses, speed is not a re- quirement.. They do have low temperature drifts, so accuracy may be retained over a reasonable ambient temperature range. The multiplier has a rated accuracy of 1% and an output of exey/10V. 2. Digital Circuit The digital logic circuit is shown in Figure 15. The output of a 1 MHz crystal oscillator is scaled down to con- trol both the pulse width (PW) and the pulse repetition frequency (PRF). The first three decades are used for the PW, and are gated so that the output of N1 is high during the last 8/10 of a period (the pulses are each one period long). 81 and 82 are LATCHED bistable miltivibrators (J-K type), with the Q outputs going low only by resetting. The first law transition following resetting causes B1 to change states, but since the J input of B2 is connected to the Q output of B1, B2 will not change states until the second pulse. Since the K inputs are connected to the Q outputs, the Q outputs cannot return to a low state (until reset). The logic levels A, B, and C are high during the first pulse, both pulses, 76 Figure 15 Digital Logic Circuit for Improved Instrument 77 lMHz SCALER RESET- ,‘qO -I-lOO -;-|ooo +l0+ -i-l<35 -E- to6 €- l07 d. d. ct d. .L____. M: 2.8733 . ____., 5‘ 7 C ‘1] $5 cxs m N6 LED‘H ' ' N8 N9 N” ' LL [ I J' R a T EL ._ K Q )Av Arl I £2 Q 7' £3 -+K 2 " B 3.45 78 and sampling time, respectively. These signals are con- verted to +5V/-15V levels in order to drive the switches. N8 and N9 functknias a gated reset monostable so that if the input from SS to N8 goes low while logic 8 exists, the reset cycle will start. The gated reset flip-flop (consisting of N10 and N11) then causes the sealer, B1, and B2 to be reset. Notice that a new cycle cannot be started until the previous cycle is completed. When in the exter- nal trigger mode, N6 — N11 synchronize the trigger signal to the PW. Typical waveforms are shown in Figure 16 for the three trigger modes. A complete circuit diagram, along with a further explanation of the components' functions and characteristics, is presented in the Appendix. C. Temperature Compensation Assume that over a narrow temperature range the con- ductance of a solution and the conductance of a thermistor vary approximately linearly with temperature; i.e. ' _ 1 _ 1 for the thermistor, GT - fi- _ RU (1 + FTAT) (1) T T ' = 4L.= and for the solution, Gx Rx fit (1 + FXAT) (2) x where FT and Fx are the temperature coefficients of con- ductance. From the block diagram in Figure 14, d- (g1 .. g1) 500kfl A T (D I 01' ed ( o . _ o —)v FTAT 500 kQ/RT If R _ RT (3) P.1L’A “ m1 - _ 79 Figure 16 Waveforms of Logic Circuit Logic Levels in Improved Instrument Pulse width setting is 10 psec Mode l 2 Pulse repetition on even decades Mode 2 = Pulse repetition every 2.8 periods (dashed lines) External triggering mode ll Mode 3 80 .SggscmsmmaommsememmHommsememmfiosa 81 edet 5v R 5v R2 Also. 90 = m + 3 - TUE-W; R147 (’4') d 60 5V R1 in et = " R; * item-ii; R5 e e 5V R _ d t - 2 °r et ‘ [lav—Kg “ram-R;le (5) Combining Equations 2, 3, and 5: -5v FTAT 500m et 5v R2 6 et = a; 10V R6 " m (“PX”) R5 ( ’ In order to have correct temperature compensation, et=e.c = W 2 (7) X Therefore, in order to have correct compensation, from Equations 6 and 7, o 'I 5V FTébT SOOKQ et + 5V R2 bxiT = 0 (8) RT 10V R6 10kfl Rx Combining Equations 7 and 8 again yields: 2R°R F R = 5 500 kQ.FT Since R6 = 5 kfl, 0 RF ReTX (10) 5 3U'F; Therefore, from Equation 10 it is obvious that temperature compensation is possible providing that either the tempera- ture coefficients are constant or that they vary in exactly 82 the same way. They should be nearly constant over a 1°C temperature range (the range for which this instrument was designed). The temperature adjustment is made by varying RA (a 10 turn, 100 kflpot) until e is zero. At this point, d RA = R;, so if the ratio of the temperature coefficients is known, R5 (a 10 turn, 2 k0 pot) can be calculated directly. Armitage and French (70), however, have shown that it is very difficult to make an a priori temperature—coefficient prediction. In practice, therefore, R5 will be experi- mentally set by varying the temperature of the solution and adjusting R5 until eo = eg. D. Obtaining Absolute Conductance If proper compensation is made, the combination of Equations 2, 4, and 4 yields: 5V R4 10 82 R1 eo=ioomT-§; (11) X Equation 11 is the main equation for this instrument, as it allows the accurate determination of conductance. The voltage eo is available at the front panel and also to the null meter. It is evident that if eO = 0, then: 3156 - Tar-n— <12) X Therefore, this instrument can be used as a conductance “bridge“ with no capacitance balancing. If the offset cur- rent is not used, the output, e0, is a linear function of conductance with a zero intercept. If R3 f-, the intercept will change and a "bridge“ with a linear off-balance is '73 F- ' I 1 6. .‘II’I- ‘1 : 83 obtained. Equation 11 can be written: 100k0 e° R T‘SV‘Q+R'1' 4 3 1 1 1 fit we; (13) X Equation 13 suggests an extremely simple readout. Define five dimensionless variables: a = R /10 k0 (read one decade to the left of 1 decimal point) b = 100 RSI/R3 = 100, 10, 1, 0 for R3 = 1, 10, 100;" kg c = 10 kQ/RZ = 100, 10. 1 for R2 = 0.1. 1, 10 k9 d = 200 kQ/Ru = 100, 10, 1 for R4 = 2, 20, 200 k0. e = eo/IV (read output in volts) Equation 13 then reduces to: %a = (ab + de)c “9‘1 (14) X E. Instrument Tests 1. Absolute Accuracy The absolute accuracy of the instrument was tested over its entire conductance range, using precision resistors in place of the cell. The results are summarized in Table 8. There appears to be a systematic error that leads to a measured conductance which is approximately 0.02% low. The tolerance of the input resistor of A2 is temporarily only 0.01% (a 0.005% resistor will replace this resistor as soon as it arrives). The rest of the error can be attributed to the lack of optimization of the circuit. In the present work, for example, the amplifiers are over-compensated with 1 ‘4.'l:'. . L 84 TABLE 8 Absolute Accuracy Tests* Resistance Measured G Rx(ohm) Tolerance (%) (p971) % Error 10 0.05 99,906 -0.094 50 0.01 19.998 -0.010 100 0.01 9.995-5 —0.045 500 0.01 1.999.“ -0.030 1 k 0.01 999-7 -0.030 5 k 0.01 199.96 -0.020 10 k 0.01 99.98 -0.020 50 k 0.01 19.991 -0.045 100 k 0.01 9.9980 -0.020 500 k 1 2.0057 +0.285 1 M 1 1.0038 +0.380 5 M 1 0.1990 -0.500 10 M 1 0.0990 -l.000 30 M 1 0.0332 -0.400 I: Unless stated otherwise, all measurements were made with a pulse width of 1000 usec and with the smallest "c" possible. 85 respect to high-frequency oscillation. This inhibits their response times and can cause error (especially for high R4 and short pulse widths). There is also the switching charge (the switching signal is coupled through the FETS’capacitance to Ch) which has not been fully compensated for in the sample and hold portion of the circuit. These problems are solvable, but their solution will take somewhat more time. Until the circuit is optimized, 0.05% absolute accuracy is all that can n“ be Claimed e 2. Sensitivity The sensitivity of this instrument was very good over its entire range. The limit of detection is shown in Table 91(limit is 5 0.1 scale division). TABLE 9 Sensitivity Kg. Resistance Resistance “1) Detection of 1 part in 10 60,000 100 200,000 1 R 1,000,000 10 k 100,000 100 R 20,000 1 M 8,000 10 M 1,000 The detection below 1kg is limited by noise (the 50 mv and 0.5V pulses have relatively more noise than the 5V pulse). Above lld2the detection limit is due to the closed-loop gain of A3. 86 3. Dependence pg the Parallel Cell Cgpacitance The dependence on the parallel cell capacitance is shown in Table 10. These data were obtained using a resis- tance substitution box for RX and placing capacitors across the terminals. From Table 10 it is obvious that the output is quite independent of Cp. There is some error as Cp increases, especially for large Rx' 4. Dependence pg Series Cell Capacitance A series combination of a resistor and capacitor was used to determine the influence of the double-layer capaci- tance using this instrument. An ESI 0.01% decade resistance box was used for RX. The results of this test are given in Table 11. From these data, it can be seen that the dependence on Cx is not large for small t/RXCX. It should be pointed out that the errors for small t/RxCx are probably due to the error in the measurement of Rx (see Table 8) rather than the error caused by Cx' Again, it should be pointed out that the largest capacitor tested is still much smaller than the typical Cx of real cells. 5. Egg pf Four Cell Leads The use of four cell leads to eliminate the influence of contact and lead resistance on the conductance measurement was explored. The conductance of two resistors was measured in three configurations: (1) with two leads, (2) with four normal leads, and (3) with a one ohm resistor in series with each lead. The results are listed in Table 12. 87 TABLE 10 Dependence on C "a” term for balance with Cp in pF Rx(0¢10%) 0 100 570 1070 1 k 9.0324 9.0321 9.0322 9.0324 2.2 k 4.3352 4.3352 4.3354 4.3358 4.7 k 2.0529 2.0528 2.0528 2.0529 10 k 1.0253 1.0253 1.0254 1.0259 22 k 4.1332 4.1334 4.1335 4.1341 47 k 2.0137 2.0118 2.0097 2.0110 100 x 1.0370 1.0391 1.0380 1.0398 220 k 4.8760 4.8770 4.8784 4.8811 470 k 2.0280 2.0250 2.0272 2.0299 1 M 1.1071 1.1036 1.1064 1.1085 2.2 M 0.4330 0.4336 0.4350 0.4380 4.7 M 0.2290 0.2299 0.2310 0.2340 10 M 0.1089 0.1099 0.1110 0.1133 88 TABLE 11 Dependence on Series Capacitance measured G for Cx Theoretical C Rx (nflrl) l uF’ 10 pF .1 10 100,000 86,100 99,070 50 20,000 19,220 19,996 100 10,000 9,906 9.999 t = 10 usec 500 2,000 1,998.3 1.999.8 1000 1,000 999.8 999.91 5000 200 199.96 199.92 50 20,000 10,100 19,187 100 10,000 8,200 9,910 500 2,000 1,918.0 1,998.3 t = 100 psec 1000 1,000 990.1 999.8 5000 200 199.82 199.96 89 TABLE 12 Elimination of Contact and Lead Resistance Measured G (ma-1) Rx Tolerance of Rx 2 Leads 4 Leads 4 Leads + 29 10 0.05 99.904 99.907 99.9 50 0.01 19.998 19.999 20.0 Although the insertion of resistors in series with the leads reduced the sensitivity (because of noise), the fact that an accurate measurement can be made with a lead resis- tance that is 2/10 the value of Rx proves the value of the four-lead technique. It should be mentioned that long leads should be avoided unless proper compensation is provided to Al, A3, and A4; otherwise they are unstable. 6. A;Q Bridge Comparison The conductance of two solutions of K01 and of some conductivity water was measured using the bipolar pulse instrument and a Wayne - Kerr Bridge (Model # B 221). An experienced operator obtained the bridge measurements. The results are given in Table 13. TABLE 13 A-C Bridge Comparison Bipolar Pulse Technique A-C Bridge Techniqpe 2.366 1.04 (PW = 1000) 2.36 110'1 -1 _ 11.295 n0. (PW = 1000) 11.30 p9 1 2.8000 ma'l (PW 100) 2.795 mn'l 90 7. Temperature Compensation A variable temperature bath was set up to test the automatic temperature compensation. The temperature was varied by adjusting the ratio of hot to cold water flowing into the bath. The conductance of a KCl solution was followed Kg. temperature with and without compensation. Without changing the temperature adjustment or the T.C. adjustment, more KCl was then added to the solution, and the experiment was repeated. The results are shown in Table 14. From Table 14, it is obvious that temperature compen- sation by this method does work. The accuracy is increased by 1 - 2 significant figures. It should be noted that the zero temperature adjustment is slightly inaccurate in these examples. The temperature of the bath was easy to vary, but was very difficult to maintain at a constant temperature. Therefore, all adjustments (balance of A5 and the multiplier, adjustment of RA' and the reading of the conductance with and without compensation) were not made with AT = 0. This would not be a problem in a realistic situation, since the tempera- ture would be constant. Although compensation was excellent for negative temperature deviations, it was not very good above a positive deviation of 0.2500. Most likely the multi- plier was not properly adjusted to obtain the best linearity for a large negative input signal (especially since et is a small negative signal). It is important to observe that the conductance and the temperature can both vary once proper adjustments are made. This is very important when following a reaction or a titra- tion conductometrically. Ani‘a‘fi . it 91 TABLE 14 Temperature Compensation in KCl Solutions Measured G (pa-1)(PW : 100 usec) thjoC) With Comp. Without Comp. -0.90 4,851 4,680 —0.75 4,853 4,704 -0.63 4.853 4.726 -o.50 4.853 4.751 -0.25 4,854 4.802 T = 14.5°c 0.00 4,852 4,847 0.25 4,853 4,896 0.50 4,867 4,940 0.60 4,887 4,988 -1,25 5,492 5.195 -1.00 5,491 5,271 -0.75 5.49“ 5.323 -0.50 5,496 5,376 -0.25 5.493 5.438 -0.12 5,493 5,456 Temp. adj. and 0.00 5,492 5,484 $02.5288: remain 7.25 5.993 5.540 More KCl added. 0.50 5.503 5.590 0.75 5.525 5.646 0.90 5,542 5,680 VI. CONCLUSION The bipolar pulse technique makes many experimental parameters non-critical or at least not as critical as when using a-c or d-c techniques. The method is unique in that it combines all of the desirable features of high and low frequency a-c techniques while making the actual measurement using a d-c potential. Most of the limitations of the tradi- tional techniques are eliminated or reduced to a tolerable level. The most important advantage of this technique is its relative independence of all cell capacitances. Since this technique is independent of Cp, accurate conductance measurements can be made in solutions of low conductance. The cell design and lead length are no longer critical. Since the Parker effect is eliminated, oil baths are not needed. This techniQue should prove extremely useful when working with non-aqueous solutions. Since this technique is not strongly dependent on Cx’ many experimental difficulties are eliminated. Platinization is not necessary. Therefore, many new chemical systems, in which surface adsorption on platinized platinum is a problem, may be studied conductometrically. It is also possible to use small electrodes. This makes possible conductometric determinations using small cells (such as would be used for a conductivity detector in ion-exchange). This also has 92 93 application in the bio-medical field, such as in the study of Na+/K+ exchange in nerves and blood. The lack of dependence on Cx also permits accurate measurements on very high con- ductance solutions (such as molten salts). Independence of the cell capacitances has several other advantages. The time-consuming bridge balance is not necessary (bridge balance also requires a pure signal with no harmonics, as the capacitive reactances are only balanced at one frequency). This permits very rapid conductance measurements. Rapid measurements with small electrodes are very useful for fast kinetic studies (such as stopped-flow). The bipolar pulse technique is also readily applicable to other flow systems which require continuous monitoring, such as ion—exchange or liquid-liquid chromatography. It may even be possible to detect non-conducting liquids by injecting a continuous flow of ions into the effluent stream. As the non-conducting liquid comes off the column, it may change the dielectric constant of the effluent and/or the solvation spheres of the ions enough to detect a change in conductivity (recall a sensitivity of up to 1 ppm). The circuit of the bipolar pulse instrument permits compensation for various experimental parameters. The tem- perature compensation circuit of the new instrument is just one example. Another example is the application of a current ramp to the summing point of the output amplifier in order to compensate for a linearly increasing (or decreasing) conduc- tance. This was done using the prototype instrument to follow the change in slope during a titration. This 94 technique may allow conductance to be used to follow titra- tions in which the number of ions is unchanged, but in which the ions undergo a change in salvation number (or in which ions of similar, but not equal, equivalent conductance are exchanged). In summary, the bipolar pulse technique can be used under any conditions that traditional techniques can be used (including the accurate determination of cell constants and equivalent conductance). In addition, it can be used in many circumstances that the traditional techniques cannot be used. This technique is extremely simple to use, as it requires no balancing, and the output is directly proportional to the conductance. In addition, very rapid measurements may be made. Thus, this technique should prove to be very useful for the accurate determination of conductance under a wide variety of experimental conditions. REFER ENC ES REFERENCES l. Newberry. E. J. Chem. Soc. 113. 701 (1918). 2. Eastman, E. J. Am. Chem. Soc. 3;. 1648 (1920). 3. Kraus. C. and Fuoss. R. J. Am. Chem. Soc. 55, 21 (1933). 4. Gunning, H. and Gordon. A. J. Chem. Phys. ‘19. 126 (1942). 5. Elias, L. and Schiff. H. J. Phys. Chem. 59. 595 (1956). 6. Gavis, J. J. Chem. Ppys. 41. 3787 (1964). 7. Jaffe. G. and LeMay, C. J. Chem. Phys. 21, 920 (1952). 8. Douwes. C. and van der Waarder, M. J. Inst. Petrol. 9. Kohlrausch. F. Wied. Ann. 49, 249 (1893). 10. Kohlrausch. F. and Holborn. L. Das Leitvermdgeg Der Elektrolyte Insbersondere Der Lbsungen. Leipzig: B. G. Teubner, 1898. ll. Washburn. E. and Bell. J. J. Am. Chem. Soc. 35. 177 (1913). 12. Washburn, E. J. Am. Chem. Soc. 38. 2431 (1916). 13. Hall. R. and Adams, L. J. Am. Chem. Soc. 41. 1515 (1919). 14. Gieringer. C. Rev. Sci. Instr. Z. 414 (1936). 15. Jones, G. and Josephs, R. J. Am. Chem. Soc. 50. 1049 (1948). 16. Taylor, W. and Acree, S. J. Am. Chem. Soc. 38. 2403 (1916). 17. Jones, C. and Bollinger, C. J. Am. Chem. Soc. 5;. 2407 (1929). 95 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30° 31. 32. 33- 34. 35. 36. 37. 38. 39. 96 Jones. C. and Bollinger, G. J. Am. Chem. Soc. 53. 411 (1931). Parker. H. J. Am. Chem. Soc. 45. 1366. 2017 (1923). Randall. M. and Scott, G. J. Am. Chem. Soc. 49. 636 (1927). Smith, F. J. Am. Chem. Soc. 42. 2167 (1927). Shedlovsky. T. J. Am. Chem. Soc. 53. 1793 (1930). Roy. W. and Martell. A. Rev. Sci. Instr. 19, 628 (1948). Cole. R. and Gross. P. Rev. Sci. Instr. 29. 252 (1949). Hladek. L. Chem. Listy 69. 238 (1966). Warburg. E. Wied. Ann. PhXSik. éZ. 493 (1899). Winterhager, H. and Werner. L. Forschungsber. des Witschafts. No. 341 (1956). Feates. F.. Ives, D.. and Pryor, J. J. Electrochem. SOC. 10:. 580 (1956). Metcalf. W. J. SCi. InStr. fig. 742 (1965). Robbins. G. and Braunstein, J. J. Electrochem. Soc. 116. 1218 (1969). Bertram. R. and Cruse. K. Ber. Bunsenges. Physik. Chem. 92. 98 (1963). Bertram, R. and Cruse, K. 2. Anal. Chem. 86, 213 (1962). Cruse. K. and Bahr. E. Z. Anal. Chem. 192. 141 (1963). Wershawé R. and Goldberg. M. Anal. Chem. 31. 1180 19 5). Warshawsky. 1. Rev. Sci. Instr. g5. 711 (1955). Griffiths. V. Talanta. g, 230 (1959). Griffiths. V. ApalLChim. Acta. l8. 1?“ (1958). Ferris. C. Rev. Sci. Instr. 33. 109 (1963). Hanss M. and Guermonprez R. J. Chim. Phys. 6 . 663 (i966). ’ ‘1 40. 41. 42. 43. an. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 97 Janz. G. and McIntyre, J. J. Electrochem. Soc. 1 8, 272 (1961). ”- Walisch. W. and Barthel. J. Z. Physik. Chem. 4, 38 (1962). Walisch. W. and Barthel, J. Z. Physik. Chem. 32. 235 (1963). Schmidt. K. RevL Sci. Instr. 32. 671 (1966). Robertson, R. Can. J. Chem. 33, 1536 (1955). Robertson. R. and Hyne. J. Can. J. Chem. 35. 623 (1957). Murr. B. and Shiner. V. J. Am. Chem. Soc. 84, 4672 (1962). Sirs, J.w Trans. Faraday Soc. 4, 201 (1958). Prince, R. Trans. Faraday Soc. 54. 838 (1958). Strehlow, H. and Wendt, H. Inorg. Chem. g. 6 (1963). Eurato. E. and Leimu, R. Acta. Chem. Scand. 29. 2029 (1966). Dusek. K. and Lesek. J. Chem. Listy. 52. 1353 (1965). Tregloam, P. and Laurence, G. J. Sci. Instr. fig. 869 (1962). Mori. I. and Chikui. S. J. Chrom. 29, 627 (1965). Duhne. C. and Sanchez, 0. Anal. Chem. 53. 1074 (1962). Pecsoc. R. and Saunders, D. Anai. Chem. 39. 1756 (1968). Pecsoc. R. and Saunders, D. Anal. Chem. 49, 44 (1968). Knudson. G., Ramaley, L. and Holcombe, W. Chem. Instr. 1. 325 (1969). Gerischer. R. Z. ElektrochemA 58. 9 (1954). Cruse. K. Z. Chem. 5. 1 (1965). Barthel. J. Angew. Chem. Int. Ed. 2, 260 (1968). Fuoss. R. and Accascina, F. Electrolytic Conductance. New York: Interscience Publishers, Inc.. 1959. Daviesézc. Ion Association. London: Butterworths, 19 . 11' I! i l 1‘ ‘ .II III . Ill! 63. 64. 65. 66. 67. 68. 69. 70. 98 Monk. C. Electrolytic Dissociation. New York: Aca- demic Press. 1961._ Nancollas. G. Interactions in Electrolyte Solutions. New York: Elsevier Publishing Co.. 1966. Thom€soné)H. and Rogers, M. Rev. Sci. Instr. 32, 1079 195 . . Lund, H. and Bjerrum, J. Ber. B64 210 (1931). Hartley. H. Technometrics 3, 269 (1961). Daum. P. Ph. D. Thesis, Mich. State Univ.. East Lansing, Michigan (1969). Analog Devices. §_Selection Handbook and Catalog Guide 59 Operational AmElifiers. (1969). Armitage, P. and French. C.) J. Chem. Soc. 1963, 743. A PPENDI CES APPENDIX A PROGRAM FOR CALCULATING PSEUDO FIRST-ORDER RATE CONSTANTS The general equation for the pseudo first-order rate constant is: C“ - Gt 0 where C is the concentration at time 0. t, and... and k is the rate constant. If the conductance is proportional to concentration, Equation 1 becomes: 0‘ - Gt m— : exp(-kt) ‘ O or Gt + (G —Go)exp(-kt) - G. = 0 (2) A computer program was written to minimize the equation: n 2 E = ‘Z [Y. + B’exp(-kX.) - A] (3) . 1 1 1:1 where Yi = the ith conductance reading X.1 = the 1th time reading A = G O B =Go-Go Equation 3 is differentiated with respect to each curve-fit variable. A. B. and k. The partial derivitive in each case is set equal to zero and a value for AA. AB, or Ak is 99 100 calculated: AA = n (4) [Y1 + B'exp(-kXi) - A] gflflfl 1 OE eXp(-2kXi) AB - n 1‘1 f (5) i§1 eXp(-kXi)‘[Yi + B-exp(-kXi)-—A] 2 igi Xi exp(-kXi)[Yi + 2B-exp(-kxi)-A] Ak ‘ n (6) iii Xi exp(-kXi)[Yi + B'exp(-kXi)-A] Since A. B. and k are not independent variables. the whole difference cannot be used as it can with the simple Newton successive approximation. -It is assumed that the least squares error. E, as calculated from Equation 3 is a parabolic function of the fraction, v, defined by the equations: Al+1 = A5 + vAA (the (j+1)th iterative value of A) (7) ‘ (v is a fraction between 0 and 1) B3+1 = Bj + vAB (8) kj+1 = kj + vAk (9) The optimum v occurs at the minimum of the parabola and is: (4 E8 - 3 E0 - E1) " - 2E Op 4 (E1 + E0 %) where E0 = E when v = 0 E% = E when v = % E = E when v = 1 I IIIIII.‘ (1‘1: 101 The new values of A, B, and k are then calculated from Equations 7 - 9 with v = vop; As soon as the values do not change by more than the quitting tolerance, the iterative procedure halts. Table A-1 shows the three-parameter, least-squares, curve-fit program which was written to obtain the results listed in Table 7. It was written for a PDP 8/i computer, but could easily be modified for other computers. As written, there are three independent options for operation, depending on the code that is typed: odd code type log plot using final values code < 5 print values after each iteration 2 < code < 7 = pause after each set of data Thus. the input data necessary is: (1) the experiment name or number (2) the number of seconds in each time interval first line (3) the infinite conductance (4) the iteration stopping tolerance (%) (5) time succeeding lines (6) conductance (7) time = -l.0 end of data Typical input and output is shown in Table A-2. THREE-PARAMETER. LEAST-SQUARES. CURVE-FIT PROGRAM 102 TABLE A-l C THIS PROGRAM CALC C PROPERTY WHICH I C 108 21 22 101 12 103 13 15 S PROPORTIONAL T0 CONCENTRATION. A J-PARAMETER LEAST SQUARES CURVEFIT IS DONE. DIMENSION X(25).Y(25).Q(3) READ (1.108) 111 FDRHAT ('CODE : '12) K.:1 IF (K1)21.21.22 K4:0 R3:(K1-3)t(R1-6) K2:R1-2.(Kl/2) K1:K1-5 READ (2.101)ALPH.TH.FL.TOL FORMAT (A6.2X3F8.3) TOL:(.01#TOL>#*2 1:0 I:I+I READ (2.103)X(I).Y(I) FORMAT (2F8.3) X(I):X(1).TH IF (X(I)) 13.12.12 Rzl-l FN:N VX:0 IF (FL)3.4.4 1:0 X2:(X(R)-X(1))/2. 1:1+1 IF (X2-X(I))6.6.5 12:1-1 Y2:Y(I)-(Y(I)-Y(12)).(X(I)-x2)/(X(1)-X(12)) FL:Y2+(Y2-Y(1)):(Y(N)-Y2)/(2.tY2-Y(1)-Y(N)) G=ALOG((FL-Y(1))/(FL-Y(N)))/(X(N)-X(1)) 8:(FL-Y(1)).EXP(G*X(1)) 81:0 7 52:0 53:0 54:0 85:0 C SUMHING ROUTINE DO 50 1:1.N T:X(1) A:EXP(-Gtt) R=Y(I)+BtA-FL $1=SI+N ULATES A FIRST ORDER RATE CONSTANT FOR ANY 50 32 200 53 103 TABLE A-1 (continued) SZ:SZ+A¢R $5:SS+A..2 Sa:84+TttZ.A.(R+A.B) SS:SS+TtAtN IF (K1)32.33.33 HRITE (1.200) FL.G.B.VX FORMAT (3E12.6.F5.3) DL=.5t51/Fl DB:-.5¢52/53 ox:.5:ss/sa C CONVERGENCE GUARANTEE 39 40 3A 35 56 26 59 106 60 101 61 TEST DO 40 J:1.3 55:0 DO 59 13'." SS:SS+(Y(I)+B.EXP(-GtX(I))-FL)..2 0(J):SS B:B+D8 G:G+DN FL:FL+DL V:-.S*(4.t0(2)-3.*0(1)-O(3))/(Q(3)+O(1)-2.t0(2))-3. VX=(V+3.)/2o G:G+V#Dx FL:FL+DLtV¥tK4 B:B+V.D8 FOR CONVERGENCE IF ((DK/G)..2-TOL)34.15.15 IF ((DL/FL)t#2-TOL)35.15.15 IF ((DB/B)..2-IOL)36.15.15 55:0 DO 26 I:1.N SS:SS+(Y(I)+B¢EXP(-G*X(I))-FL)#:2 SC:IO0.t(SS/(Fl-l.))tt.5/(Y(N)-Y(1)) URIIE (1.100) ALPR.G.FL.SC FORMAT (A6.' N: ' E12.6.' INF G = ' F10.6.' SCAT = 'FT.3.° 2') IF (N2)61.51.59 WRITE (1.106) FORMAT (' TIME LN(GI-G)') DO 60 I=I.N FN:ALOG(FL-Y(I)) WRITE (1.107)X(I).FN FORMAT (2(1XF10.A)) IF (N3)l.|.62 PAUSE GO TO 1 END A1.1 K: 1 T INPUT AND OUTPUT OF LEAST-SQUARES PROGRAM INFIN .707209E-01 INFIN G IME 12.0000 18.0000 24.0000 30.0000 36.0000 42.0000 48.0000 54.0000 60.0000 66.0000 72.0000 78.0000 84.0000 104 TABLE A-2 Al.1 2.0 6. 4150. 9. 5297. 12. 6041. 15. 6529. 18. 6847. 21. 7056. 24. 7194. 27. 7282. 30. 7339. 33. 7377. 36. 7403. 39. 7420. 42. 7430. “lo 0. I LN(GI-G) -.8150 - .— -1.2419 ”1.6657 -2.0905 ‘2.5135 -209383 “3.3683 ‘3.7878 -4.1996 -4.6146 -5 . OLI'8L“ ’50n87l ”5.8785 7457. 7463.2 SCAT r 0.01 .052% 1.111 ‘I I111 11‘. 1|] Ill!‘ 11"}! '11} I'll APPENDIX B CIRCUIT DESCRIPTION OF IMPROVED BIPOLAR PULSE CONDUCTANCE INSTRUMENT The circuit diagram of the analog portion is shown in Figure 8-1. A description of each critical com- ponent is given below. The -5V reference supply is an integrated circuit voltage regulator (LM 304, made by National Semiconductor). Its rated accuracy is 0.01%. The resistors from -5V to ground and to A2 have tolerances of 0.005%. as do the resistors from A2 and A3 to the summing point (bottom electrode of cell). The feedback resistance of A2 is a 5 decade. 100 kCZresis- tance sub-assembly with a rated accuracy of 0.005%. All of these are metal film resistors and have temperature coefficients of l ppm/Co. These resistors have very low capacitances, having rise-times of less than 1 nsec. They are the only resistors involved with the actual pulses. A3 and A4 are very fast (to 0.01% in 1 usec) differential amplifiers (Analog 1488). Their common-mode rejection ratio (CMRR) is not very high. but since the non-inverting input is grounded. the CMRR is unimportant. 105 106 8 0 too 10 I DO '\.. I .OIAF Figure B-1 CIRCUIT DIAGRAM OF THE IMPROVED CONDUCTANCE INSTRUMENT (A11 resistances in kohm) 107 They have low input currents (necessary for sample and hold) and up to 20 mA of output current. A4 must sink all of the current coming to the summing point during 5. A1 (1498) has similar specifications to A3 and A4. but it also has a high CMRR (15.000). This is necessary since it functions as a voltage follower. A2. A5. and A6 (119K) are low drift (5 uV/Co) ampli- fiers. They have relatively high input currents (35 nA), but they are not used in any portion of the circuit that would be affected by this current. A7 (1428) is used simply as an inverter and is not at all critical. The multiplier (Hybrid Systems 1070) has a rated accuracy of 1% over four quadrants and has an output of exey/IOV. The 100 k9. pot (10 turn) to A6 has a linearity of 0.1%. There is a 10 kfll% resistor in series with it. and the dial reads from 10 kIZto 110 k9. This potentiometer is used to adjust for temperature during temperature compensation. RT is a thermistor (YSI, 1%) with a resistance of 30 Id? at 25°C. Without adding any more resistance to the 100 kfl pot. this thermistor permits temperature balance from -4 to +520C. The feedback resistance of A5 is a 2 RR. 10 turn pot (0.1% linearity). It is used to adjust the temperature coefficient (see Equation 10 on page 81). The feedback re— sistance of A6 and the output resistance (5 k9) of the multi- plier have a 1% tolerance. A11 previously undefined resis- tors are 0.1%. The 10.2V zener diodes across A6 prevent it from limiting when the temperature is unbalanced. 108 Switch 1 is a non-critical three-position switch (it may have high contact resistance since no current flows through it). Switches 2 and 3 are high-quality switches with silver contacts since they must have resistances which are negligible with respect to l kfl.(S3) or 2 kg (82). The transistor switches (a~1) are driven by the output of a Fairchild 9624 interface gate (which converts a + 5V input to -15V, and a CV input to +5V as used in this instru- ment). All six of the N-channel junction FETs are T1873 transistors." A gate signal of -3V turns them ”off". The P-channel. enhancement mode MOS-FETs (a, c, f, g, i. and k) do not turn "on" until the gate signal is -6V. Since all of the gates in each set of transistors are driven by the same signal, a "break-before-make", single. double, or triple pole, double-throw switch is obtained when driven by "A", "B", or ”C“. respectively.- During condition A (see page 75), "A" is -15V, during condition 8. "B“ is at +5V, and during condition C. ”C" is at -15v,. The choice of logic level to use is explained as follows. Transistor “a" is further ”on" if connected to +5V rather than -5V. and A1 has to be connected to the minus voltage during the second pulse. Therefore. “A" must be -15V. The N-channel FETs have a much lower "on" resistance than do the P-channel FETS. It is desirable. therefore, to have transistors d and e "on" during condition 8. Thus, "B" = +5V. Amplifier A4 must sink considerable current (up to 10 mA) during 5. Therefore. the P-channel transistors are 109 ”on" during 0 (they have an 8 mA rating). Notice that the “on“ resistances of transistors a. b, c. e. i. j. and k are not critical since no current flows to the inputs of the operational amplifiers. As long as the amplifiers are functioning, the electrodes themselves are the controlled points. since they are connected to the inverting inputs of Al and eitherA3 (during C) or A4 (during 0) by transistors e. i. and j, respectively. The 0.01 nF holding capacitor is connected to ground during C and to virtual ground during C. Thus. eO remains (during C) at the voltage that is proportional to the con- ductance. et is proportional to the total conductance (with zero intercept). This value is multiplied by the temperature~ deviation in order to feed a temperature-correcting signal to A3 that is_proportional to the product CAT. It should be mentioned that separate signal and power grounds (connected at a common point) were used. Also. separate power supplies were used for the amplifiers involved with the pulses and those working at d-c. The analog portion is covered with a metal shield and is completely separate from the +5V digital logic supply. The digital portion of this instrument consists of: (1) a Heath EU 800 27 crystal oscillator card (2) a Heath EU 800 28 7adecade scaler card (3) a card consisting of: (a) two SN7400N quad. 2-input NAND gates (b) one SN7410N triple, 3-input NAND gate (c) a SN7476N Dual J-K Flip-flop (d) the interface gates (switch drivers) 110 A picture of the front panel is shown below. Figure B-2 The Bipolar Pulse Instrument Front Panel The instrument panel has been arranged for the convenient measurement of conductance. The pulse width should be adjus- ted for the conductivity of the solution with a rough guide being: PW = 10 usec for G > 1 m971 PW = 100 usec for 1 ma"1 > G > 0.1 m9'1 PW = 1000 usec for _ G < 0.1 m9'1 Normally the pulse repetition frequency is at least ten times the PW. The maximum cell current is 5 mA. The "c" switch (lower right) controls the pulse amplitude to be consistent with this current. 111 c = 100 (v = 50 mV) for 10 m RX < 1000 c = 10 (v = 500 mV) for 100 11 < Rx < 1 1:11 C = l (v = 5V) for Rx > 1 k9 The lower the pulse voltage. the higher will be the relative noise-level. so that unless Cx is small (so that ZF might be important), the highest voltage that provides under 5 mA should be used. One should begin with c = d = 100 and decrease c until "e“ is between one and ten volts. Then ”a" (the feedback resistor of A2) and "b" (the offset multiplier) are adjusted until e = 0. The output amplifier (A3) gain may then be increased (d decreased), and "a" readjusted. The meter may also be made more sensitive in order to achieve very sensitive balance. .er I1“ " H H H II, Mm II I 3 03047 0631 1 3 129 11111111111111