LFPPAPY This is to certify that the thesis entitled A FEASIBILITY STUDY OF THE SILVER SULFIDE ION-SELECTIVE ELECTRODE AS A GEOCHEMICAL EXPLORATION TOOL presented by MICHAEL REED SCHOCK has been accepted towards fulfillment of the requirements for Masters degree in Geology Major professor Date 11/8/78 O") 639 A FEASIBILITY STUDY OF THE SILVER SULFIDE ION-SELECTIVE ELECTRODE AS A GEOCHEMICAL EXPLORATION TOOL by Michael Reed Schock A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Geology 1978 (3 t. .1- I) i ‘1’} .5!" ABSTRACT A FEASIBILITY STUDY OF THE SILVER SULFIDE ION-SELECTIVE ELECTRODE AS A GEOCHEMICAL EXPLORATION TOOL by Michael Reed Schock A literature review was undertaken and the theory of electrode response, interferences, and buffer solutions is described. Standard addition techniques are surveyed and discussed, and a general method derived for volume change corrections in multiple addition potentiometry. Following investigations of pre-extraction by dithizone, a sulfite-bisulfite buffer system, and a method for silver in photographic fixing solutions, the CAOB solu- tion previously employed for electrode measurements of cupric ion in soils and natural waters was adapted for use. Syn- thetic solutions 8 N in HNO3 containing 1.92 ppm silver ion were analyzed with an absolute error of +3.5% and a relative precision (95% c.1.) of ~6%. It was determined that additional work is necessary to perfect an extraction/pre-filtration step to eliminate unstable potential readings due to solid. particles in suspension. Detailed instructions are given for preparation and modification of the CAOB solution. An appendix of ion-selective electrode applications in geochemistry includes 59 entries, along with a bibliography of 76 references. ACKNOWLEDGEMENTS I would like to extend special thanks to my advisor, Dr. John Wilband. In addition to supplying the primary electrodes, he showed monumental faith and patience in allow- ing me to pursue this investigation. I am very appreciative of the interest and enthusiasm of Dr. Stanley Crouch of the Department of Chemistry, who also served on my thesis committee. Dr. Jim Trow was a special friend, and I am greatful for his constant encourage- ment and attempts to extract some outside financial support. Enabling this work to be finished was Dr. Philip Malone, Department of Geology, Wright State University, who opened his laboratory facilities to me, provided inspiration for this study, and many discussions of problems. I am also indebted to Dr. Michael Smith of the Wright State Department of Chemistry, who first badgered me into taking analytical chemistry courses as an undergraduate, thus awakening my interest in truly interdisciplinary geochemistry as a career. He also gave valuable suggestions from his own personal experience with electrodes, as well as "super water" and other miscellaneous items. Last, but not least, are my close friends Sue Leo, Don Carpenter, John Ansted, Jim Dalman, Fred Lawrence, Russ ii Harmon, and the local Coca-Cola Bottling Company, who became a bottomless well of moral support and good humor when the skies were the stormiest. iii TABLE OF CONTENTS LIST OF FIGURES Chapter I. II. III. IV. VI. VII. INTRODUCTION AND SCOPE OF STUDY . PREVIOUS WORK . THEORY OF ELECTRODE RESPONSE. Glass Membrane Electrodes Solid- State Membrane Electrodes . . Construction of Solid-State Electrodes. Reference Electrodes. . Solid-State Electrode Interferences STANDARD ADDITION TECHNIQUES. Introduction. . . Single Standard Addition. . Multiple Standard Addition. ELECTRODE BUFFER SYSTEMS. General Considerations. Previous Applications Present Application . EXPERIMENTAL RESULTS. Instrument and Equipment Notes. . Initial Electrode Response Check. Thiosulfate Complexing Buffer Medium. Dithizone Pre-Extraction Investigation. Modified CAOB Investigation with Silver Ion . . . . . . . . . . . . . . . RECOMMENDATIONS FOR FURTHER RESEARCH. iv iv 12 12 23 24 26 33 33 4O 47 47 49 51 55 55 55 56 63 67 72 APPENDICES 1. ELEMENTARY pH BUFFER EQUILIBRIA EXPRESSIONS . 2 CALCULATION OF BUFFER CAPACITY. 3 PREPARATION AND MODIFICATION OF THE CAOB. 4 PREPARATION AND STORAGE OF SILVER SOLUTIONS . 5. CLEANING PROCEDURES . 6 SOME CONCENTRATION EQUIVALENTS FOR Ag+. 7 APPLICATIONS OF ION- SELECTIVE ELECTRODES TO GEOCHEMICAL SYSTEMS . . . . . . . BIBLIOGRAPHY. 76 8O 84 91' 95 98 100 108 LIST OF FIGURES Observed potential vs. Ag activity for AgNO3 in pure water Spike recovery from a 0.06 M sulfite-bisulfite and 0.02 M thiosulfate buffer . Spike recovery from a 0.8 M sulfite-bisulfite and 0.02 M thiosulfate buffer Effect of sulfite-bisulfite buffer concentra— . + tion on electrode response to Ag vi 57 59 6O 62 CHAPTER I INTRODUCTION AND SCOPE OF STUDY In view of the absence of articles describing the use of a silver sulfide solid-state membrane electrode for geo- chemical investigations, this study was devised with the intention of exploring the feasibility of its use in that capacity. Since reasonably accurate and precise laboratory methods presently exist (Rubeska et al., 1967; Slavin, 1968; Ward et al., 1969; Perkin-Elmer, 1971; Nakagawa, 1975; and Minkkinen, 1975, to choose but a few), this study would emphasize the potential of the electrode for use in or near the field, in place of colorimetric techniques. There are several areas that need to be investigated to determine the potential usefulness of the electrode. One is the linear response range, value of the slope, and the reproducibility of the slope for the buffer medium selected. A suitable decomposition method must be found to liberate the silver, and a buffer must be designed to homogenize var- iations in the solution matrices from sample to sample, and to eliminate any significant chemical or electrode inter— ference. 2 Finally, the detection limit of the electrode/buffer system must be investigated. Prior studies have indicated a lower detection limit on the order of 0.01 ppm (0.01 ug/g) under ideal laboratory conditions, for-samples which are not buffered with respect to silver activity. The laboratory samples at the low concentrations have also all been at very low ionic strengths. The general range in silver content to be expected in soils has been reported to be 0.01 to 5 parts per million by Siegel (1974). Without dilutions resulting from decompos- ition and buffering, the experimentally determined lower detection limit is already nearly reached, so this may pos- sibly provide a very real limit on the applicability of the electrode. The effect of higher ionic strength on the detection limit and the slope needs to be investigated, and could also conceivably be a limitation. The development of a successful field method for silver analysis using an ion-selective electrode would be a tremendous aid to the exploration geochemist, even though it would by no means replace such laboratory methods as atomic absorption or neutron activation. Due to the scarcity of information of the application of cation-selective electrodes in general to geochemical systems, and the large number of potentially useful ones, this investigation hopefully will provide useful information for future analytical geochemists working in this field. CHAPTER II PREVIOUS WORK Since the'mid-1960s, there has been a substantial rise in applications and use of ion-selective electrodes. This, in part, was due to the development of solid crystal, liquid, and gel exchange membranes, with much greater selectivity than most of their glass forerunners. Electrodes have been especially useful for soil scientists, since they primarily respond to activities of ions in solution, and can be fairly directly applied to cation exchange equilibrium studies (Carlson and Keeney, 1971). Water quality parameters that are defined by activity terms can be more efficiently determined (theoretically) using direct measurement by electrodes (Durst, 1969; Carlson and Keeney, 1971). Con- tinuous and/or automated monitoring of certain chemical con- stituents in water supplies such as the F-/Cl- ratio studies cited by Durst (1969, p. 393), is facilitated by the selective-ion electrode and chart recorder. Though many cation and anion-sensitive electrodes presently exist, and can sometimes be fabricated upon demand, only several have seen widespread usein geochemical appli- cations. 4 Probably the most documented application is that of the fluoride ion-selective electrode. This electrode is of the solid variety (using a Eu3+-doped LaF3 crystal), and was applied to water analysis by Frant and Ross, Jr. (1968). Many water studies have followed (see Appendix G for examples). Application to fluorine determinations in rocks and minerals has also been greatly studied. Van Loon (1968) fused powdered fluoride-bearing minerals, decomposed the bead, and determined the fluorine concentration using direct potentiometric comparison with standard solutions. Jagner and Pavlova (1972) determined fluorine concentrations in silicate rocks by first sintering powdered samples with sodium carbonate and zinc oxide, and then the sinter was digested for 12 hours in a steam bath. Following leaching with sodium carbonate solution, rinsing, and filtration, a complexing buffer solution was added to adjust the pH, and then the actual concentration in the final solution was determined by multiple standard additions and a Gran plot. Crenshaw and Ward (1975) report a similar procedure, which is an extention of methods of Edmond (1969), Ficklin (1970), and Ingram (1970). They used a sodium carbonate/potassium nitrate flux for sintering, and employed a pH and ionic strength adjustment buffer (TISAB) adapted from Frant and Ross, Jr. (1968). The concentration was determined directly from a known—increment scale on an Orion 407 millivolt meter, following a standard addition. The reported deviation from a recommended standard values for the samples tested is substantially greater for the Crenshaw and Ward procedure than that of Jagner and Pavlova. This may be due to the implicit assumption of true theoretical Nernstian slope in reading directly off of the meter scale, as well as any solution variables. A slightly different approach to the fusion-analysis of soil and vegetation fluorine has been taken recently by McQuaker and Gurney (1977). Following fusion with NaOH, the pH was adjusted to 8-9 and the samples were filtered to remove most of the Al, Fe, Ca, and Mg as insoluble silicates. Following filtration, the F- concentration was determined using a TISAB and a single solution calibration procedure. The precision was reported to be 4.1% for soil samples (mean relative standard deviation), and recovery studies showed better than 95% accuracy. A versatile and simple technique for silicate materials has been presented by Bodkin (1977) for samples of up to 0.4% F in the sample solution. Samples are decomposed by lithium metaborate fusion in a muffle furnace, and the beads are dissolved in 4% (v/v)HNO'3. The buffer system employed is 0.5M in citrate, 0.5M in NaCl and 0.03M in DCTA at a final pH of ~5.5. Numerous samples were analyzed with excellent agreement to ”accepted" values. Volatile loss of fluorine was observed when fusions were carried out over gas burners, adding an additional consideration to sample preparation pro- cedures. 6 Sulfide determinations can be made in water (Orion, 1969), and have been reported by Allum, Pitts, and Hollis (1971) for submerged soils. Many calCium determinations using glass as well as liquid and gel membrane electrodes have been reported, though they are normally subject to some selectivity and pH problems. Woolson, Axley, and Kearney (1970) compared calcium in soil by electrode to atomic absorption values with some success, and Thompson and Ross, Jr. (1966) used an electrode to determine calcium in sea water. Thompson (1966) described a method for magnesium determination in sea water, and another method for calcium analysis in natural water samples was developed by Hulanicki and Trojanowicz (1974) using a constant complexation buffer. Haynes and Clark (1972) used a combination standard addition and potentiometric titration method to determine chlorine in silicate rocks, following fusion of the samples. Soil scientists have developed quite a few methods using electrodes selective for K+, NOS, B (as tetrafluoroborate), and Na+, and several examples are cited by Carlson and Keeney (1971). Though electrodes specific to many more ions, including 2*. Ag+. Pb2+. potentially geochemically useful ones Cd2+, Cu Zn2+, and Hg2+ exist at present, applications or attempts at applications to other geochemical systems are conspicuously absent from the literature. Suggestion of the applicationcxfthe Cu2+ electrode to soil analysis was made in Carlson and Kenney (1971). The 7 cupric ion—selective electrode was first applied to the analysis of natural waters by Smith (1972) and Smith and Manahan (1973). This was accomplished by the design of a special complexing antioxidant buffer (CAB or CAOB) which simultaneously adjusts pH, ionic strength, keeps dissolved copper ion in the divalent state, ties up iron which might precipitate as hydroxides and oxyhydroxides, and provides a constant weakly-complexed fraction of the cupric ion in the solution. Friederick et a1. (1972), suggested the feasibility of the cupric ion-selective electrode for water analysis in geochemical stream surveys in conditions of low interfering ions, but the Smith and Manahan method overcomes virtually all major interferences that would likely be found. Potential interferents Ag+ and Hg2+ would not normally be found in high concentrations in natural waters, nor would interfering anions Cl- and Br_. 2 Schuller (1976) was the first to develop a system to apply the cupric ion-selective electrode to geochemical reconnaissance of soils. This method is a straightforward adaptation of the Smith and Manahan (1973) procedure, and uses the residual acid from the soil digestion as a part of the acetate buffer system. The copper concentration was calculated by standard addition, from a value for the response slope determined experimentally beforehand by a fairly simple multiple addition technique. Though copper concentration trends could be precisely followed compared to atomic absorp— tion values, the electrode values were significantly lower 8 (in the neighborhood of 30%) in most cases. Possible explanations include sample inhomogeneities, the poorer efficiency of the cold 6 M HCl digestion, poor precision and/or reproducibility of the experimentally determined slope value, and surface adsorption on particles in the unfiltered samples. There may also have been problems due to a "suspension effect" at either of the electrode/solution interfaces, analogous to that reported for pH measurements by Feldman (1956). The silver ion-selective electrode is usually con- structed with an AgZS solid crystal membrane, and has been used in several chemical studies. Hseu and Rechnitz (1968) first evaluated the Orion 94-16 Ang solid membrane electrode in alkaline solution, at 25 C. This electrode was found to respond in good agreement with the theoretical Nernstian pre- diction, the experimental slope being determined to be 59.0 mV per decade of Ag+ activity over a range of pAg+ of 1 to 4. The pH of the tested solutions was not specified beyond "alkaline" for aAg+ (Ag+ activity) measurements. The electrode was found to respond linearly to 032- with a slope of 29.7 mV (at u = 0.3 and 0.5 M) per decade asz‘ at a pH between 11.4 and 11.8 and for pAg+ from 2.2 to 7.0. The theoretical Nernst slope is 29.58 mV per decade activity change for divalent ions and 59.16 mV per decade activity change for monovalent ions at 25 C (see Chapter III). Light and Swartz (1968) extended the investigation of the Orion model 94-16 Ag2S solid membrane electrode to 9 include temperature coefficients, response time, the effect of oxidizing conditions, and other possible interferences. Buffering the solution for ionic strength and pH (using a background of 1.0 M NaOH), a response slope of 29 mV per decade S2- was determined at 25 C over an unspecified con- centration range. No detailed results for detection limits or response slope to dAg+ were given, but a summary table stated Nerstian response to aSZ- over a p82’ range of 0 to 5 for total sulfide, and a p82- range of 0 to 20 for free sulfide. The slope was said to be Nerstian toward aAg+ over the pAg+ range of 0 to 5 for total silver ion, and 0 to 23 for free silver ion. The electrode was further stated to be pH-independent from pH 0 to 14, and the only interference to either ion analysis was ng+ to silver ion. This inter- ference was somewhat predictable from comparison of the solubility product contants at u = 0.1 (Ringbom, 1963) for figs (sz0 = 51) and AgZS (pKS = 48.1). The nature of this 0 interference is most likely reaction between the Hg2+ ions and the sulfide component of the solid membrane, along the lines of the example given by Ross (1969, pp. 82-86). Ross further ascribed the lower detection limit of approximately 10'.8 m Ag+ for "total" silver ion to difficulties arising from adsorption and desorption phenomena at the container and electrode surfaces (ibid., p. 77), and freedom from interference due to strong oxidizing conditions was reaffirmed. The Light and Swartz (1968) and Hseu and Rechnitz (1968) studies are also summarized in Butler (1969, pp. 177-80). 10 Durst (1969, pp. 403-6) described Nernstian response down to pAg+ = 25 M on sample volumes of only 5uL, and suggested that since this was equivalent to a probability of substantially less than one free silver ion in the solution (which was continually buffered in silver-complexing agents), the electrode must have at some point responded to complexed silver ion. He further pointed out that usability as a silver sensor would be limited in basic solutions due to the precipitation of Ag20. If silver-complexing agents are used to prevent this precipitation, the use of a standard addition technique is necessary to determine total silver ion con- centration. There have been only a few subsequent applications of theAgZS solid membrane electrode, and none of them par- ticularly useful geochemically. Mfiller, West, and Mfiller (1969) described the feasibility of determination of silver concentrations down to approximately 10-7 M (~13.5 parts per billion) by direct potentiometry, once again with the Orion 94-16. Their response slope determination was done by addition of small aliquots of AgNO3 solution to 100 ml of 0.1 M NaNOa, and adjusting the sum of added silver ion for volume change. Their electrode showed virtually theoretical response over the range of pAg+ from 5.92 to 6.92 (about 13 to 130 ppb), at temperatures between 22 C and 28 C. Mal- functioning of the electrode was observed when the Cl- con- 3 centration exceeded about 2.8 x 10- M (100 ppm) for an unspecified silver condentration (<4.6 x 10‘7 M). 11 Durst and Duhart (1970) used the Orion 94-16 electrode to monitor silver loss from solution due to adsorption on the container walls. Silver loss was determined by direct potentiometry, comparing the observed potentials to a calibration curve in the silver concentration range of 10’6 M to 10‘7 M. The solutions were simply dissolved AgNo3 in distilled water, with approximately neutral pH. A method has been published (Orion Applications Bulletin No. 14) that describes a procedure useful for determining total silver ion concentration in photographic fixing bath solutions from 10‘1 M to lo"4 M in silver ion, by standard addition potentiometry. This can be accomplished even though almost all of the silver ion would be tied up in 8.82 13.5 a strong (81 = 10 = 10 , Ringbom, 1963) thiosulfate 82 complex. Details of the reasons for the concentration limits of the method as well as the precise nature in terms of chemical composition and pH of the fixing solution are not given, and must be inferred. Also, no accuracy or precision data are presented. Lapatnick (1974) used the 4. silver ion-selective electrode to monitor Ag concentration in silver plating baths, with an average error of 2%. CHAPTER III THEORY OF ELECTRODE RESPONSE Glass Membrane Electrodes There are two main types of ion—selective electrodes quantitatively sensitive to silver ion. One is a glass membrane electrode, which is rendered selective to silver ion by the composition of the glass. Though each manufacturer has its own particular recipe of additives, generally the silver-selective glass membranes have a high Na20 (ll-29%) O and high Al (18-19%) content, according to Rechnitz (1967). 2 3 A reliable method to predict properties of a glass given its composition has yet to be found, and development is still largely a matter of trial and error. Ilselectivity constant can be defined for glass electrodes, which is generated simultaneously by the relative mobilities in the glass of the ions under consideration, and their relative ion-exchange constants for aqueous phase/ glass interaction (Eisenman, 1969). For convenience, the selectivity ratio of glass membrane electrodes is often expressed for each cation relative to the hydrogen ion. Eisenman has defined a selectivity ratio K for M: over M5 using the following equation (Rechnitz, 1967): 12 13 ll 0 H a a ll 0 a RT = - -— 1n + + F KMl/Mz II 0 a K ll 0 .1 M H-r m-+ where E is the measured cell electromotive force (emf), R is the universal gas constant, F is the Faraday, T is tempera- ture in degrees Kelvin, and MI and M; are the only potential- determining cations. In terms of construction, a glass electrode is made from a mixture of oxides of elements of oxidation state +3 or greater with oxides of elements of oxidation state +1 or +2 (Eisenman, 1969). After the mixture is melted and then cooled, a solid is formed in which the most mobile charged species are monovalent cations. When the bulb of a glass electrode is inserted into an aqueous solution, a layered structure is formed between the external solution to be measured and the internal filling solution. Schematically, it may be represented by (modified from Skoog and West, 1971): hydrated gel layer ' — m external solution N10 4mm {dry glass 0.1 mm hydrated gel layer ’VlO-finfll internal solution though there may be in reality more intermediate layers. As the hydrated layers dissolve, additional hydration of the glass takes place, such that the hydrated gel layer maintains a reasonably constant thickness. The rate of hydration is 14 largely a function of the hygroscopicity of the glass and the sample solution (or filling solution), and is mainly responsible for determining the practical lifetime of the electrode. Metal ions are taken up by the glass in a cation- exchange process, and diffuse into the hydrated layer to some extent. Since tracer experiments failed to show complete migration of tritium from the labeled to unlabeled solutions (Rechnitz, 1967), it was concluded that the current generated must be the rate of some charge-transfer process across the membrane. Further experimentation revealed that the charge is carried by the cationic species of lowest charge available through an interstitial mechanism, wherein each charge carrier only has to move a few atomic diameters before passing on its energy to the next carrier. When a membrane made of one of these specially- prepared glasses is used to separate an aqueous solution to be measured from an internal filling solution, both of the same salt, a cell potential is developed. This potential can be related to the activity of arbitrary monovalent ion M (aM) in solution. First, on each side of the glass there will be a half- cell reaction concerning aM at each interface, which may be expressed as: = .31 0‘10 E1 k1+F1naM1g _ RT aMli Ez"k2+ F luau 15 wherein R, T, and F have their usual thermodynamic identities; aM and “M11 are the activities in the solutions on the out- 10 side and inside of the glass respectively; M1g and M1h the activities in the corresponding gel layers; and k1 and k2 are constants relating the respective available sites for M at the gel surfaces (Skoog and West, 1971). The concept of activity and methods for the calculation of activity coef- ficient is discussed in detail in Hem (1961), Butler (1964), Garrels and Christ (1965), Stumm and Morgan (1970), and Leyendekkers (1971) to cite but a few. Since the calculation methods used in this study solve directly for concentration, the discussion of activity is not undertaken here. If the exchange equilibrium goes so far to the right that virtually all sites available to it are occupied by M, then “Mlg and aMlh should be equal. If, also, there are the same number of exchange sites available in each gel layer, k1 = k2. Since the boundary potential, Eb for a membrane is given by E = E - E b 1 2 where E1 represents the component at the external solution and gel interface, and E2 at the internal interface, these two full equations may be substituted, yielding 1. l along with the discussed simplifications. Restating, if the two gel surfaces are identical, the potential developed will depend only on M in the external and internal solutions. If the internal filling solution has a constant composi- tion (and therefore, aMli), then that term can be removed as a constant, and the overall expression then becomes E = Constant + RT ln aM b “r“ 10 From this equation it can be seen that it is possible under these circumstances to obtain a direct measure of the activity of ion M in an external solution, as soon as a suitable reference electrode is connected to complete a circuit. In actual practice there are normally several ions which interact in the solution, and which ellicit some electrical response from the membrane. The contributions of these other ionic species are weighted through their selectivity constants. An example of the adjustment necessary for the case of two cations is displayed in the expression aMlo + 112/u )K1,2 “M20 E = E + R: In 1 b c .1 F 0”‘11 +K‘z/ )K1,2 IGMZi u1 for the ideal case where n, the ion exchange exponent, = 1. u The weighting factor, B 2/u1)K1 2] contains terms (u2/u ) ’ 1 and K which are the mobility ratio of ions M2 and M1 in 1,2’ the membrane, and the ion exchange factor characteristic of the exchange of M1 and M2 between the aqueous phase and 17 membrane, respectively (Eisenman, 1969). This weighting factor is sometimes referred to as the "potentiometric selectivity ratio.” The mobility may be combined with the ion-exchange equilibrium constant into a ”selectivity coef- ficient." and [(uZ/u1)K1,2] may be replaced by KaMzzz/ZI. where K is the new selectivity coefficient, 22 is the charge of the interfering ion, and z1 is the charge on the primary ion of interest. The addition of more activity and selectivity terms enable the equation to be extended to more complicated cases. The above equation is also independent of time, and is there- fore applicable as soon as equilibrium at the membrane/ solution interfaces is established (Eisenman, 1969). Since for these glass membrane electrodes (and any solid ion-exchange types) the overall selectivity is a combina— tion of the ion exchange equilibrium selectivity and the relative mobilities of the ionic species within the exchanger, only several have a combination of these two properties which permit operation with few interferences. The most commonly 4. used glass ion-selective electrodes are for H , Ag+, Na+ and Li+. If the selectivity ratio is very high for one particular ion over the others, the terms for these other ions may be assumed to be negligible, and they may be eliminated from the overall equations. Also, under some cir- cumstances, if their activities are not negligible but are a constant background to the activity change of a single species, the more convenient form = z Eobs Econst + %% ln GM may be used, where EObs is the observed potential, Econst is a constant term combining effects of the internal filling solutions, the reference electrode used, liquid junction potentials, the other ionic species in solution, and z and Z are the charge on the species of interest (including sign). The value of this constant tends to very with time, frequently called "potential drift,” which makes direct potentiometric measurements more difficult to do accurately than addition techniques, discussed later. It should also be pointed out that for many reasons, various electrodes will deviate to some degree from the ideal Nerstian slope of RT/ZF. In such cases, an experi- mentally determined slope, S, may be inserted instead, giving - z EObs - EC + 2.3038 log aM under specified temperature, pressure, and solution composi- tion conditions. Curvature of the ideal Nerstian line may also occur due to variations of the liquid junction potential of the reference electrode over wide concentration ranges, as pointed out by Johansson and Edstrdm (1972). Solid-State Membrane Electrodes Solid membranes can be made from cyrstalline materials that are ionic conductors at the temperature of use of the 19 electrode. The crystal may be used in the form of a thin disk, providing it is of low solubility in the sample solu- tion, mechanically stable, and chemically inert in the sample solution (Ross, Jr., 1969). The mechanism of electrical conduction is facilitated by lattice defects. Mobile ions adjacent to the defects are induced to move into the vacancies, which are ideally tailored for specific ions with regard to charge distribution, size, and shape. Normally only one lattice ion is considered to be the mobile one, and usually it is the one with the smallest charge and ionic radius. The very high selectivity of the solid-state electrode is mainly due to this ability to restrict the movement of ions other than the one to be sensed. In some cases, resistivity to electrical conduction can be lowered by "doping" with a divalent cation. In the highly successful LaF3 membrane electrode, the divalent cation is Eu2+, though the charge is still carried by the F- ions (Rechnitz, 1967). The membrane of the silver-sulfide electrode used in this study is a homogeneous AgZS solid, in which the only mobile ion is Ag+. This electrode responds directly to Ag+ ion in external solution, and responds to S2- activity in the external solution indirectly. The mechanism for $2— sensing appears to be the fixing of Ag+ activity at the membrane surface by the 82- in solution (Carlson and Keeney, 1971). Cammann and Rechnitz (1976), however, present new evidence 20 that 82- reacts with interstitial silver ion and is also involved in the charge-transfer process. Certain other solid membrane electrodes, notably the cupric ion—sensitive electrodes, have their membrane pellets made out of a finely divided mix of MS (where MS is the sulfide of any divalent cation, M2+) dispersed in a matrix of AgZS, and pressed into a disk. The charge is thought to be carried by the Ag+ ions. The presence of the considerable amount of AgZS means that in solutions containing Ag+ ions, the electrode also functions as a silver detector, creating an interference. The solubility of many of the metal sulfides in acid solutions provides a lower pH limit to their Nernstian response (Ross, Jr., 1969). Since no foreign ions can enter the lattice structure, interferences result only from chemical reactions of the ions in solution (method interferences) or at the surface of the crystal (electrode interferences). Because of the different nature of this electrode from the glass-types, the selectivity is much greater, and response should theoretically always be Nernstian. This can be expresssed in fairly simple equations as long as selectivity factors are excluded. A more rigorous look at specific interferences of interest will be taken later. The observed potential EObs can be related to activity + of monovalent ion M in the solution through: _ 2.303 RT + EObs - BC + ———§———— 10g aM 21 It should be recognized that the concentration of M+ in the external solution is the sum of that originally present plus those M+ ions resulting from the dissolution of the membrane. In most cases the latter contribution is negligible (Parthasarathy et al., 1974). This assumes as before that the junction potential, if not negligible, can be described within the constant, and that the aM+ in the internal filling solution (if present) is also fixed and is contained in the constant term. Deviations from the theoretical Nernst factor of §;§%%_§I, which is 59.2 mV at 298 K, do often occur in practice, in which case an experimentally determined slope value is sub— stituted. For the CuS solid state electrode, representative of the mixed—sulfide type, the expression can be derived as follows, based on Ross, Jr. (1969). If we start with a solution containing cupric ion but not silver ion, the potential at the crystal/solution inter- face is determined by both 2- fl 2Ag- + s AgZS(S) and Cu2+ + S2— ++ CuS(s) for which KsO(AgZS) = (aAg+)2(aSZ-) 22 KsO(CuS) = (aCu2+)(dS2—) if AgZS and Gus are pure solids. Solving for aAg+ % . Ks0(Ag23) = uAg+ aSET and substituting for (aSz-), we get KSO(AgZS) aCu2+ é = aAg+ Kso(CuS) This may be substituted in the previous full Nernst equation for aM+, yielding: = 2.303 RT Eobs Ec + 2F Since KsO(Ag28)/KSO(CuS) 1S a constant, we can further simplify to _ 2.303 RT 2+ EObs — Ec +-—-—§F——— log aCu where the solubility ratio becomes part of the new constant E . 0 Because Z = 2 for divalent cations, the theoretical Nernst factor becomes 29.6 mV. The AgZS membrane electrode responds to aSZ- in solution in a similar indirect manner. We can begin to derive this sensitivity if we start with a sample solution initially containing no silver (I) ion. The Ag2S membrane 23 will then provide a few Ag+ ions through its dissolution. This provides a finite silver ion activity, however small it may be. This activity is governed, in the absence of complexing agents, by the sulfide ion activity through the solubility product constant, and can be calculated from + KsO(Ag2S) * Ag = d82- as before. Substituting again for aAg+, K E = E + 2.303 RT log 8°(Agzs) obs c 2F o‘82-- Since KsO(Ag2S) is a constant, it can be incorporated into Ec’ and moving (aSz-) to the numerator, the form becomes: _ 2.303 RT 2— Eobs - Ec - -——§Fr—— log as It has been suggested by Ross, Jr. (1969, ibid., p. 435) that the lower limit of response 'mo dAg+ of the silver sul— fide electrode may be raised by adsorption of silver ion.on the membrane surface, but experimental data have not been presented in substantiation for this particular electrode. Construction of Solid-State Electrodes Normally, a solid—state electrode consists of an epoxy or inert plastic hollow cylindrical body, in which a sensing wire is passed and connected to the inside of the crystal membrane (Fritz and Schenk, 1974). 24 It is also possible to have a built-in internal reference electrode immersed in a filling solution. A cation- responsive internal system may use an AgNO3 filling solution (Covington, 1969, and Light and Swartz, 1968). This par- ticular arrangement is used in the popular Orion model 94-16 employed in the studies discussed in Chapter II. In many cases, the exact mode of internal construction is proprietary information, unless one disassembles the product to find out. Reference Electrodes In order to complete an electrical circuit, a reference electrode must be employed having a fixed potential in the analyzed solutions. There are many different kinds described in the chemical literature and manufacturers' brochures, especially with regards to pH measurements (which, in itself, is a cation-selective electrode process). Only practical considerations relating to their use in this investigation will be discussed here. A conventional single-junction calomel or silver chloride reference electrode is unsuitable for use in analyses for Cu2+, Ag+, and S2". There are several require- ments they do not fulfill, mainly in association with problems stemming from solution leakage. 1. The filling solution should not contain the ion to be measured. 25 2. No ion in the filling solution may form a complex with the ion to be measured. 3. No ion in the filling solution may form a precipitate with the ion to be measured. 4. No ion in the filling solution may form a precipitate with any ion in the sample solution. It is likely that in silver, copper, or sulfide determinations in natural waters or geochemical samples, all four of the requirements would be violated. Chloride complexes or precipitates cupric ions and silver ions, which not only remove the ions from availability to be measured, but the precipitates may also clog the electrode membranes, alter the junction potentials, or react with the membranes themselves. Silver (I) ions can leak from the electrode, contaminating the solution to be measured. This gives erroneous concentration information if silver is being determined, and an electrode interference in cupric ion analysis. Mercury (II) ions would possibly form insoluble chlorides or sulfides, clogging the membrane, creating a chemical interference in the solution, or an electrode inter— ference, for both the Ag+ and Cuz+ probes. These problems can be avoided by employing a double- junction reference electrode, with an inert bridging electrolyte. In this study, double junction electrodes using the original manufacturers' filling solutions were employed. In both cases the internal filling solutions were of K01, and 26 the external solutions were of KNO3, which would not inter- fere with the ions of interest in solution. Solid-State Electrode Interferences Interferences fall into two broad types, which are defined by Riseman (1970) as follows. "Method interferences” are those which mask the ion that is sought by making it unavailable to the electrode. "Electrode interferences" are chemical species which either destructively react with the membrane, or which are mistaken by the electrode for the ion being measured. The latter includes common ion(s) adsorbed onto the membrane surface. Method interferences will not be considered here, since they are so highly variable with the scheme of analysis. They will be considered in the experimental section as they present themselves. The first set of interferences to be considered is that for the AgZS membrane electrode during silver ion analysis. In order for Nernstian response to hold, the sample solution must be free of species that can react with either component of the membrane. With many types of electrodes, there are several inorganic ions that combine to form more insoluble salts. An example is the interference of SCN- in solution to the Br- sensing electrode (AgBr membrane). If (aSCN-) exceeds (aBr-) to the same ratio as the ratio of their solubility products, then the AgBr in the 27 membrane will begin to convert to AgSCN, and the electrode will become a thiocyanate sensing device (Ross, Jr., 1969). A sharp break is seen in a plot of potential versus anion (Br- or SON-) activity. For the AgZS membrane electrode, there are few common interferences of this type, since K 8 s0(Ag2S) ls extremely high (~10-4 ), and the membrane is quite stable. Mercury (II) ion is an interferent when present, since sz0(HgS)m51’ but no other cations form sulfides as insoluable as those of silver (I) and mercury (II). The onset of mercuric ion interference may be predicted in general terms from the fol- lowing proposed reaction: 2+ Ag2S(s) + Hg 2 HgS(s) + 2Ag+ _ 2+ 2- _ + 2 2— 2+ + 2 _ (aHg )/(aAg ) ‘ Kso(Hgs)/KSO(Ag2S) (aHg2+)/(aAg+)2 m 10'3 For the copper (II) ion-selective electrode, an analogous procedure will yield the following cation inter- ference ratios, using data from Ringbom (1963): (aAg+)2/(dCu2+) m 10'14 (aHg2+)/(aCu2+) m 10‘17 28 Smith and Manahan (1973) also report that due to the ”relative solubilities of copper (II) sulfide and iron (III) sulfide" ferric ion may be expected to interfere when [Fe3+]/ [Cu2+] = 0.1. Using solubility ratios is a highly simplistic approach, and should normally be considered to be a rough predictive tool, since actual solubilities are controlled to a great extent by the common ion effect and complexation equilibria involving other species in the solution. The reported KSO values and those calculated from free energy data also vary by several orders of magnitude, making experi- mental and theoretical quantitative agreement highly fortuitous. Adsorption of sensed ions onto the membrane as previously cited may also be responsible for raising the estimated detection limit based on these KS considerations by as much 0 as 10-12 orders of magnitude. A variation of this electrode interference is the creation of a solid precipitate that can clog the membrane. An example would be the formation of AgCl precipitate when making silver additions to solutions containing free chloride ions in the appropriate amounts. In addition to removing the silver from the solution, surface adsorption on the membrane inhibits the accurate measurement of ionic potentials at the solution/membrane interface, or impedes the movement of ions in the membrane itself. Tarnishing was observed by Mfiller, West, and Mfiller (1968) for solutions of unspecified 3M [Ag+](<4.6 x 10‘7M) having [01'] = 2.8 x 10’ at u = 0.1. 29 The CuS-AgZS membrane electrode is also subject to a form of interference due to conversion of part of the AgZS phase to 2AgCl by the following reaction (Ross, Jr., 1969): 2+ Ag2S(s) + Cu + 201' I 2 AgCl(s) + CuS(s) For the above reaction, (aAgCl)2(aCuS) KRXN = (uAg23)(oCu2+)(dCl‘)2 Substituting the K30 expressions + 2 - 2 2+ 2- K = (dAg ) (aCl ) (aCu )(aS ) KsO(Ag2$) RXN 2 2- 2 K 80(Ag01) KSO(CuS)(aAg+)2(aS ) (aCu2+)(dc1')2 which simplifies to KsO(Ag28) K :- RXN K2 sO(AgCl)KSO(CuS) This implies that, to prevent the reaction from proceeding to the right, K < sO(Ag2S) (aCu)(aCl)2 2 K sO(AgCl) Kso(CuS) Should the activity product exceed the value on the right, the silver sulfide at the membrane surface would theoretically become converted to silver chloride, and the electrode would abruptly change function to a chloride ion detector. Similar problems with other ions are possible. Another major type of electrode interference to be considered is that due to the oxidation or reduction potential 30 existing in the solution. While using the cupric ion- selective electrode, sulfide or sulfide species in equilib- rium at the electrode surface may become oxidized, resulting in electrode instability (Smith, 1972).. The ferric ion is thus a dual interferent; first, because of its ability to form an insoluble sufide at the electrode surface, and also because of its behavior as a moderately strong oxidizing agent. This is particularly important in natural water samples, in which Fe2+ is easily oxidized through atmospheric contact. The mixed CuS-AgZS membrane is also easily oxidized in 0.1 M HClO according to Johansson and Edstrom (1972). 4, The Ag2S membrane is reportedly not sensitive to redox couples in solution, including Fe3+ and nitric acid, according to both Butler (1969) and Swartz and Light (1968). Some membranes with higher solubilities may also show a pH limitation. This is due to an increased rate of membrane dissolution attributed to the formation of HST and H28 in the solution under oxidizing conditions (Ross, Jr., 1969). The usable pH range is reported to be 0 to 14 for the AgZS membrane by Light and Swartz (1968), but this does not con- sider the effects of the pH on the silver species in the solution. The theoretical pH range for the mixed CuS-AgZS membrane is also reported to be 0 to 14, but is subject to the limitations imposed by the formation of insoluble copper hydroxides, phosphates, carbonates, etc. (Orion, 1968). Strong complexing agents in solution (e.g., formation or stability K values very large) for one of the membrane 31 components have a tendency to raise the lower detection limit of the electrode. This is done mainly by increasing the solubility of one or more of the membrane components in the test solution. The electrode "sees" this as an increase of that species in the solution. This has been documented for F- analysis using the LaF3 membrane electrode in solutions containing citrate by Ross, Jr. (1969), and for the cupric ion-selective electrode in solutions containing NazEDTA by Smith (1972). Butler (1969) states that the AgZS membrane electrode is free from interference by strong complexing agents such as thiosulfate and iodide. Durst (1969) shows that that particular electrode may to a degree respond to complexed silver ion, and laboratory results presented later in this report indicate that contrary to Butler (1969), the lower level of detection for Ag+ is raised by high levels of thiosulfate in the solution. By definition, this should constitute an electrode interference. A problem with almost all electrode measurements, though possibly not technically always an interference, is that of long-term potential drift. This is likely due to changes in components of the ”constant” term of the response equation, rather than in sensitivity to the ion being measured (Smith, 1972). Changes of :10 mV are not uncommon from day to day in identically prepared standard solutions. The length of time for these changes to take place is highly variable, and it is gradual rather than abrupt. Frequent 32 restandardization is often necessary when direct potentio- metric analyses are being conducted. Colloids and suspensions in the solution may cause a variable modification of the liquid junction potential or the true membrane potential of both the indicator electrode and the reference electrode when the particles are highly charged. This effect was pointed out for pH measurement by Feldman (1956), and may be extended to ion-selective electrode operation by analogy. Using the ion-selective electrode in organic solutions may necessitate changes in the internal and reference electrode filling solutions. Johansson and Edstrdm (1972) extensively tested the Orion model 94-29 cupric ion-selective electrode to see the effects on the theoretical Nernst response and electrode stability of the topography of the membrane surface. Their results showed that deviations from the Nernst slope could result from spots, cracks, and pits on the crystal surface. Membranes with pits produced calibration lines having slopes of less than the theoretical values by approximately 1-13 millivolts. The potential readings were also less stable, and the electrodes had longer equilibration times. Dis- locations on the surface were also found to be areas of rapid corrosion and etching. The authors suggested that to maintain nearly theoretical response, the electrode should be frequently diamond-polished and treated with silicone oil. These results should prove applicable to other solid-state membrane electrodes. CHAPTER IV STANDARD ADDITION TECHNIQUES Introduction There are several important problems with direct potentiometric analysis, which render it highly unsuitable for geochemical application. First, the measured potential is a combination of many factors that are difficult to evaluate at the necessary levels of precision and accuracy. These include the contri- butions from the liquid junction potential, the true potential of the reference electrode in the solution being analyzed, and the potential of the ion-selective electrode itself. The latter is a combination of several internal contributions discussed in Durst (1969) and in many other papers on the developed potentials of glass and solid-state ion-selective electrodes. Secondly, the results are in terms of activities, which means that when concentration is the variable of interest, precise and accurate data on solution composition and activity coefficients must be obtained for each sample solution analyzed. There is additionally an inherent limitation in the precision attainable to direct potentiometric analyses 33 34 stemming from the logarithmic nature of the Nernst equation. This is pointed out by Skoog and West (1971) and Durst (1969, p. 376), who show that the resulting error in reported activity is approximately i4% for univalent ions, and i8% for divalent ions per millivolt uncertainty in observation. These few difficulties render this technique highly unsuitable for all but the most sophisticated laboratory environments, and makes it very time-consuming for routine analyses. Calibration curve techniques can be applied to some natural water situations. Once again, however, the ionic strength of the samples and standards must be the same, and similar electrolyte and complexing agent backgrounds must be assured. This is in addition to simple variables such as temperature, pH, and stirring rate. A total ionic strength adjustment buffer (TISAB) can sometimes be used to overcome most of the difficulties for simple systems, but matrices present from soil, rock, and vegetation samples are highly complex, and often unpre- dictable with the necessary certainty. Because of potential drift, calibration curves should also be prepared frequently, and this adds time to the overall analytical procedure. A variation on this theme is the bracketing of samples with standard solutions, analogous to the commonly accepted practice for pH measurements. McQuaker and Gurney (1977) offer a "single solution" method providing a constant ionic background and temperature. This reportedly allows 35 more accurate blank determinations, linear calibration curves, and enhanced sensitivity and accuracy of the concentration values. The most suitable techniques for samples with complex and/or highly variable backgrounds are single and multiple standard addition, otherwise known as ”known increment" methods. Each of the two have special advantages and dis- advantages, depending on the exact circumstances of the problem. Standard addition techniques are presently the only reliable methods to determine the concentration of an ion when complexing agents are present. Single Standard Addition The major prerequisite for the single standard addition method is that the response slope, whatever it may be, is known accurately and is reproducible with high precision for a given set of conditions. The technique involves adding an aliquot of a standard solution of the ion of interest to a known volume of sample, and using the resulting change in the measured emf of the solution coupled with the known slope to calculate the original concentration of that ion. There are several other practical requirements which, depending on the sample composition, may or may not dictate solution pre-treatment before the actual analysis. 1. The addition of standard must not change the activity coefficient of the species of interest. For 36 practical situations this implies that the ionic strength of the solution must remain constant, and that the junction potentials of the electrodes must not change. 2. If complexing agents are present in the original solution, the fraction of the species of interest complexed must not change after the addition, and the complexing ligand identity should be as constant as possible. 3. The response of the electrode must be linear and equal to the previously determined slope value between the unknown activity and the sensed ion's activity after the standard addition. 4. Electrode interferences are not present in the original sample, or are eliminated by sample pre-treatment. The following is a derivation of the equations that will enable the concentration calculation to be made. For a solution containing the species of interest, X, the observed potential (E0) can be described by _ 2.303 RT Eo ‘ K + ""' 2' F "' l°g ¢xYxCx (IV-l) where R, T, Z, and F retain their previous meanings, 2.303 is the natural logarithm to base-10 logarithm conversion factor, K is the combined constant term (previously Ec)’ C is the total concentration (of X), Y is the activity coef- x ficient for ion X, and ¢x is a term correcting for the availability of X due to complexation. If there is no com- plexation, ¢x is unity. 37 We can substitute an empirically determined value for the slope, S, when the electrode responds in a non-theoretical (but reproducible) manner. This results in E0 = K + S 10g ¢XYXCx (IV—2) When an aliquot of standard is added, the new observed potential (E1) is expressed by Csvs . E1 = K + S log Cx + V;_:_V; Yx ¢x (IV-3) In this equation, ox' and YX' are the same as above (but for the new concentration of X), Vx is the volume of standard added, CS is the concentration of the standard solution, Vx is the original solution volume, and K' is the new catch-all constant. At this point, several adjustments can be made to simplify the situation. First, if the ionic strength of the solution can be assumed to remain constant Yx' = 7 If the sample is pre- x' treated so that complexing agents (if present in the first place) are in excess, the available fraction of ion X is not changed, and the ¢x and ¢x' terms are equal. Combining these simplifying stipulations, the change in observed potential (AE) may be seen to be AE ll [Til | till 1 O (IV-4) or, 38 Csvs —1 AB = S log Cx + W (CX) (IV-5) At this point, if it is possible to make Vs negligible with respect to Vx (say, less than 1%), then the equation can be simplified and solved as follows, with substitution of CSVs V x AC = Cx + AC CX (AE/S) = log antilog (AE/S) = 1 + (AC/CX) Cx = AC [antilog (AE/S)--1]"1 (1v_5) If it is not possible to assume VS is negligible with respect to Vx’ then the general form may be derived from (IV-6)- C V C V -1 AB = 8 log [—3535— + —§-—S-‘-—- (C ) (IV-7) Vx + VS Vx + VS x Vx Csvs antilog (AE/S) = fir—1—V‘ 1'0 V + C V x s x x x s Vx + Vs Vx ] -1 -—————— antilog (AE/S) - = C CSVs [ Vx + Vs x or -1 C V V _ s s . Ag x . CX - —————v + V [antilog S - ——-—-—V + V ] (IV—8) x s x s By evaluation of the above equations, a practical consideration that arises is that the relative uncertainty of the emf measurements is the largest when AE is small. 39 This leads to the analytical objective of the largest possible AC consistent with maintaining constant ionic strength, relative complexation, and staying in a region of constant slope. A rule of thumb often suggested is that the addition solution should be approximately 100X as concentrated in the ion of interest as the unknown solution. This probably comes from the idea of using 100 mL solutions and trying to double (approximately) the concentration of the sought ion. The use of micropipettes facilitates the delivery of negligibly small volumes of standard solutions. When microliter-range additions are made, there is a danger of large errors in concentration being propagated if the delivery device is at all worn. These should be recalibrated before use for the most accurate results. A nomograph has been developed by Karlberg (1971) based on similar equations for single and multiple standard additions, as well as analate addition. The slope need not be theoretical, but must be between 56/|Z| and 62/|Z|. The AE value also must be in the range of 0 to 65 mV, and the volume ratio Vs/Vx must be between 0.5 and 100. It may also be pointed out that for a series of routine analyses, V and Vx may be constant and can therefore be s numerically combined into a factor. In addition, S should be constant. By fixing these values, and setting AC = l, a table of values may be easily generated by computer or programmable calculator so that ratios of Cx/AC may be 40 obtained by direct comparison with AE values for any particular case. ACX can then be easily solved for, since AC is a known quantity. Orion Research Incorporated has generated such a table for 298 K and the theoretical slope, and is available upon request (see Bibliography). The form of equation most useful for the generation of the tables would be: Cx/AC = [antilog (AF./S)-1]-1 (IV-9) Methods for determining S by dilution and multiple additions are described by Orion (1970) and Schuller (1976), and a computer least-squares method is described in Smith (1972). Multiple Standard Addition Multiple standard addition, as pointed out by Orion (1970) relies on the same essential prerequisites concerning ionic strength, complexing agents and absence from inter- ferences as single standard addition, but has an added utility. The slope value need not be known beforehand, though it must be constant throughout the region to be analyzed. When experimental S values are imprecise, this method may be employed instead. The drawback to this pro- cedure, as can be seen from the descriptive equations, is that potential errors can be very large for even small reading errors of 0.1 mV or less. Thus error propagation 41 becomes especially large as AEz/AEl-+2.0. This problem may be somewhat lessened by the use of digital millivolt meters. It is also necessary in the following development to stipulate that the additions be of equal volumes and concentrations, and that the volumes of the additions be negligible with respect to that of the sample solution (Orion, 1970). A later modified derivation will demonstrate a method for volume correction, but solution composition requirements are still the same. Starting as before, E1 is the observed potential after the addition to the unknown solution with potential E0. C V The symbol AC represents 3 S, the amount of the standard x addition. E0 = K + S log Cx (IV-10) E1 = K + S log (Cx + AC) (IV-ll). The difference, AE1 = E1 - E0, is then: Cx + AC AE1 = S log -——E;—— (IV-12) If a second, equivalent addition of AC is made, this E2 will result E2 = K + S log (Cx + AC + AC) (IVsla) and a AE2 may be defined as E2 - E0, giving: C + 2AC AE2 S log Cx 42 It is then possible to compare the ratio R = AEZ/AEl’ which is expressed by: Cx + 2AC Cx + AC R = AEz/AEl = log (-——E--) log (--E-——) (IV—15) X . x This eliminates the slope term S. It introduces the problem that Cx can not be solved for explicitly in terms of R. By successive approximations using a computer (Orion, 1970, and Lawrence, 1976) or a programmable calculator, tables can be constructed of R versus Cx’ One manner in which this can be done is by fixing AC = 1, and then selecting ratios of Cx/AC and inserting that fraction or multiple of AC for Cx in the equations. Two examples follow. For Cx/AC = 0.01, the equation to be solved becomes: [1% (Wlllhg (gist—AH ‘1 R = 1.149 R This is interpreted as meaning that a solution which contains OK at a concentration 0.01 times that of each of the additions will have a ratio AEZ/AE1 of 1.149. For Cx/AC = 10.0, [log (%>l[1°g