Theory of titration curves : The locations of inflection points on acid-base and related titration curves

Consideration of the effect of dilution on the locations of the inflection points of potentiometric titration curves for titrations of monobasic acids or bases and for precipitation titrations in which the ions of the precipitate have numerically equal valences shows that;(I) In a strong acid-strong base or isovalent precipitation titration, the inflection point always precedes the equivalence point. No physically meaningful inflection point exists, regardless of the concentration of the substance titrated, if the concentration of the reagent is smaller than a certain value or if, when the concentration of reagent exceeds this limit, the concentration of the substance titrated is smaller than another limiting value.(2) In a weak acid-strong base or weak base-strong acid titration, the inflection point at which the slope is greatest also precedes the equivalence point, and vanishes under certain conditions. Earlier calculations are shown to have given incorrect information regarding the location and existence of this inflection point. The location of the inflection point at which the slope is smallest— the “point of maximum buffer capacity”—is shown to depend on the concentrations of the reagents employed.     

Photometric titrations with indicators

Various types of photometric titration curves are discussed. If a metal M is titrated conipleximetrically using a metal indicator and the absorbance is plotted vs. the titrant consumed, the inflection point appears at a pM value defined by the equation 3 pM_infl = pM_trans + 2 pM_eqThis expression is valid when M combines in a 1 : 1 ratio with the complexing agent and the indicator and when the indicator concentration is small compared to the total metal concentration.The difference between the pM values at the inflection and equivalence points can be calculated from the equation ΔpM = pM_infl — pM_eq = $\text{1}\text{3}$(pM_trans — pM_eq) = $\text{1}\text{6}$log(C_MK²_MI/K_MY)If the inflection point is taken as the equivalence point, the error arising can be calculated from ΔpM, or more simply, read from a diagram.If transmittance, instead of absorbancc, is plotted as a function of the titrant volume, the inflection point depends on the added amount of indicator. However, at high transmittance values, i.e., at low indicator concentrations, the inflection point of a transmittance curve occurs practically at the same volume of added titrant as the inflection point of an absorbance curve. Rules are given for applying an indicator correction for the amount of metal bound to the indicator at the end-point.The derived equations and discussions can also be applied to acid-base titrations.     

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.     

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.     

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.     

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.     

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.     

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.     

End-point construction and systematic titration error in linear titration curves-precipitation reactions

The systematic titration error which is introduced by the intersection of tangents to hyperbolic titration curves is discussed for precipitation reactions. A simple expression for the systematic titration error is derived, and S/C_x² is proposed as a measure of the sharpness of the titration curve. The effects of the conditional solubility product (S), the concentration of the unknown component (C_x), and the ranges used for the construction of the end-point, are considered. A graphical method is presented for the selection of pairs of ranges which result in small systematic titration errors. For equal values of S/C_x² and 1/KC_x, the optimum combinations of ranges are different for precipitation and complexation titrations. The differences are not large for values smaller than about 0.002. For titration curves with a reversed L-shape, the error is calculated when the end-point is constructed by the intersection of the tangent to the second branch of the curve with volume axis; in this case equal ranges result in the same titration error for equal values of S/C_x² and 1/KC_x. The systematic titration error is equal to -S/C_x² when the tangent to the curve is taken at f_a = 3.0.     

Spectrophotometric titration of calcium and magnesium with complexon-III and metal-specific indicators

The equivalence points of the Ca and Mg titrations with complexon-III, using metal-specific indicators, can be determined more accurately by the spectrophotometric than by the visual method. In comparison with the spectrophotometric method in the ultraviolet⁷, where no indicator is required, the end point can be determined more sharply from our titration curve.The accuracy is of the order of 0.5% ; the repeatibility is even more favourable.     

Elimination of equivalence point errors in the potentiometric titration of chloride, bromide and iodide mixtures with silver nitrate

When two or more halides are determined in solution by precipitation titration with silver nitrate as the titrant, significant errors can occur at the first equivalence point as a result of coprecipitation. Errors of up to 33% were found for the first equivalence point for solutions containing mixtures of halides at micromolar levels. The addition of a flocculating agent to the solution reduced coprecipitation by increasing the rate of exchange between the precipitated silver halide and the halide ion remaining in solution. A logarithmic relationship was observed between the charge of the flocculating agent and the logarithmic concentration of the agent needed to minimise coprecipitation. Although flocculating agents reduced coprecipitation, they do not, however, completely eliminate equivalence point errors. Here a new method is presented which effectively eliminates the problem of coprecipitation during precipitation titrations for solutions containing two halides. In order to decrease the possibility of coprecipitation, we used selective complexation of the precipitation ion Ag⁺ in order to control the AgX solubility. For example, in the case of CF⁻ plus X⁻ (X=Br⁻ or I⁻), we added sufficient NH₃ to form Ag(NH₃)⁺ so that the free Ag⁺ activity was reduced below that required for theoretical AgCl precipitation in the absence of the other halides. Once the titration of the less soluble halide was completed and the first equivalence point determined, the Ag(NH₃)⁺ complex was destroyed by acidification of the solution to a pH less than 6. The titration is then continued and the second equivalence point determined. Equivalence point errors were reduced to less than 1.5% with careful application of the method.     

Coulometric titration of iodide and iodine with electrolytically generated hypobromite

Conditions for the quantitative coulometric titration of iodide and iodine with electrolytically generated hypobromite in the presence of borax buffer have been established. Iodide and iodine are oxidized to iodate. The method, with biamperometric indication of the equivalence point, was successfully applied for a wide range of iodide concentrations (6.21–2115μg with reliability intervals of ±0.21–±11μg) and iodine concentrations (24.26–3311μg with reliability intervals of ±0.36–±11.7μg). The determinations are accurate and sensitive even in the presence of large amounts of bromides and chlorides ($\text{Br}{}^{\mathrm{?}}\text{I}{}^{\mathrm{?}}$= 1.2·10⁶ and $\text{C}\text{l}$^?I^?=4.0·10^{3), as well as in the presence of oxidizing agents such as IO₃?, BrO₃? and CrO₄2? ($\text{I}\text{O}$₃?/I₂)=3.2·105, $\text{I}\text{O}$₃?/I₂=3.1·103, $\text{Br}\text{O}$-₃/I₂}=1.1·10⁴ and $\text{Cr}\text{O}$^2-₄/I₂=1.0·10⁴, as was confirmed by statistical tests. The oxidation mechanism under the conditions of coulometric titrations is discussed.     

Adsorption from mixtures containing mono- and bivalent cations on insoluble oxides and a revision of the interpretation of points of zero charge obtained by titration

Acid—base potentiometric titrations of haematite (α-Fe₂O₃) suspensions in mixtures of the non-specifically adsorbing electrolyte KNO₃ and the nitrates of the specifically adsorbing ions Ca²⁺, Cd²⁺ and Pb²⁺ show that the cross-over point of the titration curves, plotted as σ₀(pH), can be interpreted as the point of zero charge (pzc) only if there is no specific adsorption or if the amount of specifically adsorbed cations is so small that all of it is quantitatively adsorbed. This finding implies that the usual procedure for obtaining the pzc from cross-over points of titration curves must be reconsidered in the case of specific adsorption. In reality an Esin—Markov coefficient is measured, the relation of which to the surface charge requires further thermodynamic analysis.     

Amperometric argentimetric and mercurimetric titration of sulfhydryl: Argentimetric titration

Results of amperometric argentimetric titrations at the rotated platinum wire electrode in ammonia buffers of cysteine, glutathione, thioglycolic acid and 2-mercaptoethanol are presented. The method gives satisfactory results with glutathione and 2-mercaptoethanol. Thioglycolic acid gives good results in the presence of sulfite but high results in its absence. Its disulfide can be titrated accurately in the presence of sulfite. In agreement with other workers it was found that cysteine yields high results. The error increases with increasing cysteine concentration and is reduced in the presence of 0.1 M sulfite. It becomes equal to zero at cysteine concentrations equal to or smaller than 5.10^-5M.     

Direct potentiometric titration of chloride and fluoride ions in reversed micelles of an ethoxylated surfactant

The possibility of the potentiometric titration of Cl^– and F^– ions directly in reversed micelles of the ethoxylated surfactant (Neonol APh₉-4) in n-decane is shown. The potential change of the indicator electrode (silver chloride and lanthanum fluoride) only depends on the ion concentration in the aqueous pseudophase of the reversed micelles and is independent of the aqueous phase concentration in n-decane in the region of concentrations from 4 to 0.2 vol %. The determination of the micelle size of Neonol APh₉-4 in the precipitation titration of Cl^– and F^– ions by photon correlation spectroscopy showed the formation of nanoparticles of AgCl and LaF₃ of dimensions limited by original micelle size (r_hd = 6 to 11 nm). The growth of AgCl and LaF₃ nanoparticles was studied at a shortage and excess of the titrant and in the point of equivalence.     

Hydration Structure Around the Carboxyl Group Studied by Neutron Diffraction with 12C/13C and H/D Isotopic Substitution Methods

Time-of-Flight (TOF) neutron diffraction measurements have been carried out on aqueous 8 mol% sodium acetate solutions in D₂O. Scattering cross sections that were observed for sample solutions involving ¹²C/¹³C and H/D isotopically substituted acetate ions were used to derive the first-order difference functions, Δ_H(Q) and Δ_C(Q), and corresponding distribution functions, G _H(r;r) and G _C(r;r), which describe the environmental structure around the methyl and the carboxyl groups within the acetate ion, respectively. Structural parameters concerning the first hydration shell of the carboxyl group within the acetate ion were obtained through the least squares fit to the observed intermolecular difference function, Δ_C ^inter(Q). The nearest neighbor C _O...D _W1 (C_O: carboxyl carbon atom, D_W1: water deuterium atom) distance, r(C _O...D _W1 ), and the angle, ∠ C _O ...D _W1 -O _W (O _W : water oxygen atom), were determined to be 2.63(1) Å and 120(1)°, respectively. The coordination number, n(C _O ...D _W1 ), was obtained to be 4.0(1). These results are consistent with the hydration structure in which water molecules in the first hydration shell of the carboxyl group are hydrogen-bonded with oxygen atoms of the carboxyl group.     

Synthesis and crystal structure of mixed metal(III) tungstenyl(VI) ortho-pyrophosphates

The series of isotypic anhydrous ortho-pyrophosphates M^III(W^VIO₂)₂(P₂O₇)(PO₄) (M: Sc, V, Cr, Fe, Mo, Ru, Rh, In, Ir) was obtained via vapor phase moderated solid state reactions in sealed ampoules. The crystal structure of the phosphates M^III(W^VIO₂)₂(P₂O₇)(PO₄) (M: V, Ru, Rh) was solved from single crystal X-ray data (C2/c, Z = 16). Fairly regular MO₆ and distorted WO₆ octahedra share vertices with PO₄ and P₂O₇ units to form a 3D network. For the ortho-pyrophosphates with M: V³⁺, Cr³⁺, and Fe³⁺ the oxidation state of M is confirmed by magnetic measurements. ³¹P-MAS-NMR spectra of the diamagnetic phosphates M^III(W^VIO₂)₂(P₂O₇)(PO₄) (M: Sc, In, Ir) show surprisingly different isotropic chemical shifts for the seven phosphorus sites. V^III(W^VIO₂)₂(P₂O₇)(PO₄) occurs as equilibrium phase in the quasi-binary system (V_1–xW_x)OPO₄ at x = 0.67 and exhibits a small homogeneity range 0.60 ≤ x ≤ 0.67. The scandium compound shows a fully inverted occupancy of the M sites according to the formulation W(Sc_1/2W_1/2O₂)₂(P₂O₇)(PO₄).     

An oxacalix[2]arene[2]pyrimidine-bis(Zn-porphyrin) tweezer as a selective receptor towards fullerene C70

An oxacalix[2]arene[2]pyrimidine-bis(Zn^II-porphyrin) conjugate was readily prepared via nucleophilic aromatic substitution of a phenolic AB₃-Zn-porphyrin on the upper rim of a (1,3-alternate) 5,17-bis(methylsulfonyl)oxacalix[4]arene precursor. Efficient 1:1 complex formation between the ‘jaws’ bisporphyrin tweezer and fullerene C₇₀ was evidenced by ¹H NMR titrations (K = 3.0 × 10⁴ M⁻¹), while no detectable complexation could be observed with C₆₀. On the other hand, an analogous oxacalix[4]arene-bis(Cu-corrole) conjugate did not show any measurable (C₆₀ or C₇₀) fullerene binding.     

Defect clusters and precipitation/oxidation of MgO-Co1−xO solid solution

MgO and Co_1−xO powders in 9:1 and 1:9 molar ratio (denoted as M₉C₁ and M₁C₉, respectively) were sintered and homogenized at 1600°C followed by annealing at 850°C and 800°C, respectively to form defect clusters and precipitates. Analytical electron microscopic observations indicated the protoxide remained as rock salt structure with complicated planar diffraction contrast for M₉C₁ sample, however with spinel paracrystal precipitated from the M₁C₉ sample due to the assembly of charge- and volume-compensating defects of the 4:1 type, i.e., four octahedral vacant sites surrounding one Co³⁺-filled tetrahedral interstitial site. The spacing of such defect clusters is 4.5 times the lattice spacing of the average spinel structure of Mg-doped Co_3−δO₄, indicating a higher defect cluster concentration than undoped Co_3−δO₄. The {111} faulting of Mg-doped Co_3−δO₄/Co_1−xO in the annealed M₁C₉ sample implies the possible presence of zinc blend-type defect clusters with cation vacancies assembled along oxygen close packed (111) plane.