Theory of titration curves : Points of minimum slope on potentiometric strong acid-strong base and precipitation titration curves

Strong acid-strong base and precipitation titration curves, like other kinds of potentiometric titration curves, inherently possess an inflection point where the slope is a minimum as well as one where it is a maximum. In any kind of titration the first of the inflection points can be caused to occur earlier, and can eventually be made to disappear altogether, by adopting certain expedients. For a weak acid-strong base titration, for example, these include decreasing the concentration of the acid titrated and titrating in the presence of an excess of its conjugate base. For strong acid-strong base and precipitation titrations they include decreasing the concentration of the substance titrated and increasing the concentration of the titrant. The conditions under which a physically significant inflection point of minimum slope can exist are defined, and explicit equations are given from which its position can be calculated under various experimental conditions, for strong acid-strong base and for both isovalent and heterovalent precipitation titrations.     

Theory of titration curves: Locations of points of maximum slope on potentiometric heterovalent (“asymmetrical”) precipitation titration curves

By a wholly rigorous general treatment of the potentiometric titration curve representing the titration of B^m+ with A^n- to give the precipitate B_nA_m, where n ≠ m, it is shown on taking into account the effect of dilution that: (1) If $\text{n}\text{m}$ > 1, the inflection point must precede the equivalence point, but there is no inflection point if the initial concentration of the ion titrated is smaller than a certain value, which is determined by the values of m, n, and the solubility product of the precipitate. (2) If $\text{n}\text{m}$ < 1, the inflection point may follow the equivalence point — though not by more than a certain definite amount — or may precede it or coincide with it. In every such case there is a concentration of A^n- that will cause the equivalence point to coincide with an inflection point; this concentration depends on the values of m, n, and the solubility product of the precipitate. (3) Unless the difference between the equivalence and inflection points is negligibly small, the traditional treatment in which dilution is neglected gives seriously erroneous estimates of the location of the inflection point.     

Determination of the composition and the stability constants of complexes of mercury (II) with amino acids

The execution of a titration rarely performed, or of one with a reagent of limited stability, may be facilitated by applying multiparametric curve-fitting to titration-curve data in such a way that the amount of the substance titrated and the concentration of the reagent are evaluated simultaneously; the necessity for prior standardization of the reagent is thereby eliminated. In the potentiometric titration of acetate ion, a very weak base, with unstandardized hydrochloric acid, the accuracy and precision of the concentration of acetate thus obtained are approximately five times better than those which can be secured by titration with standardized acid and location of the point of maximum slope. The accuracy and precision of the concentration of acid are comparable to those that can be secured in very careful standardizations against a primary standard that gives a titration curve of far more favorable shape.     

Redox Titrations-II Location of inflection points of titration curves for homogeneous redox reactions

The relative positions of the inflection points and equivalence point of a homogeneous redox reaction have been studied by using the redox buffer capacity to derive an equation for the titration curve. The position of the inflection point corresponding to the maximum slope of the titration curve relative to the equivalence point depends on the electron transfers of the analyte and the titrant (the stoichiometric coefficients of the reaction equation) and on the difference between the formal potentials of the redox couples in a more complicated way than has been described previously.     

Titrimetric applications of multiparametric curve-fitting : Part 1. Potentiometric titrations of weak bases with strong acids at extreme dilutions

A multiparametric curve-fitting procedure is described for locating the equivalence point of a potentiometric titration and is applied to data obtained in titrations of acetate ion with hydrochloric acid over a wide range of concentrations (down to 6.5 · 10^-5M acetate). It does not depend on the existence of a point of inflection on the titration curve, and therefore yields useful results in titrations of this very weak base at concentrations well below that at which the point of maximal slope disappears. Three parameters are involved: the concentration of the base being titrated, the concentration dissociation constant K_a of its conjugate acid in the medium employed, and the apparent activity coefficient y_H⁺ of hydrogen ion in that medium, and all three of these must be evaluated when a new supporting electrolyte is employed. In 3.0 M potassium chloride at 24°, the apparent activity coefficient of hydrogen ion is 1.746₀, and this value permits data obtained in titrations of other or unknown bases in this medium to be interpreted by two-parameter fits. The value of K_a for acetic acid in this medium is 1.313₄ · 10^-5M; by means of these two numerical values routine titrations of acetate in this medium can be interpreted by one-parameter fits. It is possible to locate the equivalence point with an accuracy and a precision that cannot be approached by other techniques and even to obtain useful and reliable results under conditions so unfavorable that other techniques fail completely.     

Use of a digital computer in equilibrium calculations:the effects of dilution and ionic strength of the buffer index and sharpness index in the titration of a monoprotic acid with a strong base

The effects of dilution and ionic strength on weak acid-strong base titration curves and on the buffer index, β, and sharpness index, η, for these titrations have been calculated. It has been shown that the approximate equations generally used to calculate β and η are often misleading.     

Improved linear titration plots for weak-acid titrations

Linear titration plot functions are derived for the following types of titration: weak monobasic acid-strong base, weak dibasic acid-strong base weak acid-weak base. These functions are compared with the corresponding Gran functions, where possible, and the effects of the following sources of error are discussed: the neglect of activity coefficients, errors in equilibrium constants and the presence of a hydrolysable salt.     

The coulometric titration of acids and bases in dimethylsulfoxide media

The coulometric titration of 20–200 μeq of acids and bases in DMSO media is described. In the titration of bases, the electro-oxidation of hydrogen at a platinized platinum electrode is used as the source of protons. The conditions for 100 % current efficiency at this electrode are low current density to avoid passivity and regular treatment of the electrode with potassium dichromate—sulfuric acid to remove a poisoning sulfide layer. The accuracy of the titrations is better than ±1 %. Very weak acids like phenols (pK_a (DMSO) ≈16) can be titrated successfully. Tris(hydroxymethyl)aminomethane is the weakest base titrated.     

Minima on radiometric precipitation titration curves

A mathematical analysis of the equations describing the radiometric titration curves using precipitation reactions is given for the case of constant (or only slightly changed) volumes of the solution. The conditions for the existence of a minimum on the titration curve have been determined and the equations of the coordinates of the minimum are given. The conditions for a minimum located in the neighbourhood of the equivalence point have also been established. In this case, the minimum may be identified with the equivalence point within the admitted error of titration. Criteria have been found for determining the sign of this error and for the conditions where this error does not appear.     

Automatic coulometric titration of acids

An apparatus is described for the automatic titration of acids by the constant current coulometric technique. The generator electrodes comprise a platinum cathode and a silver/silver bromide anode. The increase in pH resulting from the reduction of hydrogen ion at the cathode is indicated by a glass electrode, in conjunction with a Beckman Automatic Titrator which automatically monitors the titration and interrupts the generating current when the equivalence point pH is reached. Quantities of hydrochloric acid in the neighborhood of 0.12 millimole in 50 ml were titrated with a mean error of ^-0.07% and an average deviation from the mean of ±0.15%. The technique is applicable to any strong or weak acid, and to acid mixtures, provided that no substance is present which is either reducible at the platinum cathode or reactive in any way at the silver anode.     

Titration of strong and weak acids by sequential injection analysis technique

A sequential injection analysis (SIA) titration method has been developed for acid-base titrations. Strong and weak acids in different concentration ranges have been titrated with a strong base. The method is based on sequential aspiration of an acidic sample zone and only one zone of the base into a carrier stream of distilled water. On their way to the detector, the sample and the reagent zones are partially mixed due to the dispersion and thereby the base is partially neutralised by the acid. The base zone contains the indicator. An LED-spectrophotometer is used as detector. It senses the colour of the unneutralised base and the signal is recorded as a typical SIA peak. The peak area of the unreacted base was found to be proportional to the logarithm of the acid concentration. Calibration curves with good linearity were obtained for a strong acid in the concentration ranges of 10(-4)-10(-2) and 0.1-3 M. Automatic sample dilution was implemented when sulphuric acid at concentration of 6-13 M was titrated. For a weak acid, i.e. acetic acid, a linear calibration curve was obtained in the range of 3x10(-4)-8x10(-2) M. By changing the volumes of the injected sample and the reagent, different acids as well as different concentration ranges of the acids can be titrated without any other adjustments in the SIA manifold or the titration protocol.     

pH-stat techniques in titrimetric analysis: Part 3. The principles of pH-stat chelatometric titrations

Chelatometric titrations can be effectively monitored by means of the pH-stat technique. During the addition of the primary titrant (the chelating agent), the pH of the titrated solution is kept at a pre-selected value by adding a strong base as the auxiliary titrant. The equivalence point is characterised by a sudden change in the ratio between the added volumes of auxiliary and primary titrant solutions, i.e. in the slope of the linearly segmented titration plot representing the volume of the base against the volume of the chelating agent. The slope of the linear segments depends on the reactions involving proton exchange, which accompany the primary complexation reaction. The dependence of the shape of the plots on the imposed pH for typical titrations (magnesium(II) and copper(II)) is examined and the conditions for the feasibility of pH-stat chelatometric titrations are discussed.     

Determination of equivalence points of sigmoidal potentiometric titration curves

The equivalence point of potentiometric titration curves coinciding with the point of inflection is calculated from the maximum of the first derivative or the zero of the second derivative. The approximation of the curve is carried out by means of cubicB-splines. The quality of the fitted curve depends on the number and position (proportionally spaced or equidistant) of knots. Series of simulated and measured titration curves were evaluated. Optimum fits were obtained with knot numbers about 20–35% of the number of data points. Only proportionally spaced splines are applicable for all types of curves and produce results of high accuracy.     

Substitution titration: a new approach to automatic titration : Determination of Acidity in Precipitation Samples

In substitution titrations, a small portion of the sample is repeatedly replaced by an exactly equal volume of titrant, by means of a rotary injection valve. The titration curves obtained in this manner can be linearized by using suitable functions, and hence equivalence volumes can be calculated. The substitution process means that there is no increase in volume during a titration. It is therefore possible to use a completely closed system, which is advantageous for samples that are sensitive to the ambient atmosphere. The principle is demonstrated for the potentiometric titration of hydrochloric acid containing varying amounts of weak acid (acetic acid or ammonium ion) with sodium hydroxide solutions. By the substitution titration, it is possible to determine the strong and weak acid in HCl/NH₄Cl mixtures with an accuracy of ca. 2%, but acetic acid is difficult to determine in the mixture because of partial protolysis at the concentration level examined. Sample concentrations are chosen in the range 125–1000 μM total acidity, to conform with the concentrations normally found in precipitation samples. The development of a procedure for determining the acidity in rain-watet and throughfall samples is described. The substitution titrations are easily automated and provide adequate sensitivity for routine work.     

Determination of water samples and ene-diols or thiols in samples inaccessible for direct K.Fischer titration

Novel reagents and the rapid technique were developed for the simultaneous determination of water and ene-diols or thiols in chemical products, drugs and other materials which are inaccessible for direct K.Fischer titration. The reagents consist of iodine, tetramethylammonium iodide or potassium iodide, base (diethanolamine, triethanolamine, sodium acetate and/or urea) in methanol mixed with N,N-dimethylformamide, or with formamide, or with dimethyl sulfoxide and N,N-dimethylformamide mixture as a solvent. The use of the reagents is based on the consecutive titration first of an ene-diol or thiol by the novel reagents and then of water by a conventional K.Fischer reagent in the same cell in a titration system protected from water vapour and oxygen, with a double burette and electrometric location of the end point in both titrations. The time for both titrations is 8-15 min.     

Effect of the concentration of solutions being titrated on the shape of conductometric titration curves

The influence exerted by the concentration of solution being titrated on the shape of the theoretical curves of conductometric titration was studied. The concentrations at which the equivalence points are clearly observed in the conductometric titration curves were determined.     

Eine allgemeine theorie der massanalytischen umschlagsschärfe

The buffer capacity of the final solution (exactly titrated to equivalence) of a titration can be calculated from. the equilibrium constant of the titrating reaction. It enables us to establish a theory of the “end point sharpness” applicable universally to the different classes of titrations (acidimetric, redox, precipitations, etc.). The results are more accurate than the approximations derived up to now from the study of the titrated solution further off the equivalence point.The use of an inproper indicator will not only produce erroneous values (high or low) in the titration but at the same time lessen considerably the sharpness of the color (etc.) change at the end point as the buffer capacity of the titrated solution is greater at any than the equivalence point.     

Potentiometric titration of plutonium

In the potentiometric titration of plutonium(III), it has been customary to take the equivalence point as the inflection point on a plot of potential vs. volume of titrant.¹ That the stoichiometric end point corresponds to the inflection point does not seem to have been theoretically demonstrated, however. The purpose of this letter is to suggest that these points may not correspond in the potentiometric titration of plutonium(III). Mound Laboratory is operated by Monsanto Research Corporation for USERDA under contract No. EY-76-C-04-0053.