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.     

Experimental determination of single solute and competitive adsorption isotherms

In order to design and to optimise preparative liquid chromatography, the knowledge of the underlying thermodynamic functions, i.e. the adsorption isotherms, is of large importance. Usually these functions can not be predicted and various techniques have been suggested to determine them experimentally. In this paper, several important methods to measure adsorption equilibrium data are discussed and evaluated. The main focus is set on dynamic methods analysing concentration profiles that could be detected at the outlet of fixed-beds packed with the stationary phase of interest. The theoretical background of the different methods is explained using classical equilibrium theory and the equilibrium dispersion model. Each method is illustrated based on experimental data collected in our laboratory. Based on these personal experiences recommendations are given regarding the potential and the applicability of the methods discussed.     

Band splitting in overloaded isocratic elution chromatography II. New competitive adsorption isotherms

The equations of two new binary competitive isotherms models are derived. The first of these models assumes that the isotherms of the two pure, single compounds have distinct monolayer capacities. Its derivation is based on kinetic arguments. The ideal adsorbed solution (IAS) framework was applied to derive the second model that is a thermodynamically consistent competitive isotherm. This second model predicts the competitive adsorption isotherm behavior of a mixture of two compounds that have single-component adsorption behavior following a BET and/or a Langmuir isotherms. Both models apply well to the binary adsorption of ethylbenzoate and 4-tert.-butylphenol on a Kromasil-C18 column (with methanol-water, 62:38, v/v, as the mobile phase). The best single-solute adsorption isotherms of these two compounds are the liquid-solid extended multilayer BET and the Langmuir isotherms, respectively. The kinetic and thermodynamic new competitive models were compared, regarding the accuracy of their prediction of the elution band profiles of mixtures of these two compounds. A better agreement between experimental and calculated profiles was observed with the kinetic model. The IAS model failed because the behavior of the ethylbenzoate/4-tert.-butylphenol adsorbed phase mixture is probably non-ideal. The most striking result is the qualitative prediction by these models of the peak splitting of 4-tert.-butylphenol during its elution in presence of ethylbenzoate.     

Numerical modeling of the elution peak profiles of retained solutes in supercritical fluid chromatography

In supercritical fluid chromatography (SFC), the significant expansion of the mobile phase along the column causes the formation of axial and radial gradients of temperature. Due to these gradients, the mobile phase density, its viscosity, its velocity, its diffusion coefficients, etc. are not constant throughout the column. This results in a nonuniform flow velocity distribution, itself causing a loss of column efficiency in certain cases, even at low flow rates, as they do in HPLC. At high flow rates, an important deformation of the elution profiles of the sample components may occur. The model previously used to account satisfactorily for the retention of an unsorbed solute in SFC is applied to the modeling of the elution peak profiles of retained compounds. The numerical solution of the combined heat and mass balance equations provides the temperature and the pressure profiles inside the column and values of the retention time and the band profiles of retained compounds that are in excellent agreement with independent experimental data for large value of mobile phase reduced density. At low reduced densities, the band profiles can strongly depend on the column axial distribution of porosity.     

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.     

Prediction of single component adsorption isotherms on microporous adsorbents

The adsorption of gases on microporous solids is a fundamental physical interaction which occurs in many technical processes, e.g. the heterogeneous catalysis or the purification of gases. In this context the adsorption equilibrium can determine the velocity and/or the capacity of the process. Therefore, it has to be known for designing purposes. The aim of this work has been the a priori prediction of the adsorption equilibria of arbitrary gases on microporous solids like zeolites and active carbon based only on the molecular properties of the adsorptive and the adsorbent. The adsorption isotherm is described completely from the Henry region over the transition zone to the saturation region. The quality of the model permits a first approximation of the planned process without further experimental effort.     

Theoretical study of the accuracy of the elution by characteristic points method for bi-langmuir isotherms

The bi-Langmuir equation has recently been proven essential to describe chiral chromatographic surfaces and we therefore investigated the accuracy of the elution by characteristic points method (ECP) for estimation of bi-Langmuir isotherm parameters. The ECP calculations was done on elution profiles generated by the equilibrium-dispersive model of chromatography for five different sets of bi-Langmuir parameters. The ECP method generates two different errors; (i) the error of the ECP calculated isotherm and (ii) the model error of the fitting to the ECP isotherm. Both errors decreased with increasing column efficiency. Moreover, the model error was strongly affected by the weight of the bi-Langmuir function fitted. For some bi-Langmuir compositions the error of the ECP calculated isotherm is too large even at high column efficiencies. Guidelines will be given on surface types to be avoided and on column efficiencies and loading factors required for adequate parameter estimations with ECP.     

Expanding the elution by characteristic point method for determination of various types of adsorption isotherms

Important improvements have recently been made on the elution by characteristic point (ECP) method to increase the accuracy of the determined adsorption isotherms. However, the method has so far been limited/used for only type I adsorption isotherms (e.g. Langmuir, Tóth, bi-Langmuir). In this study, general strategies are developed to expand the ECP method for the determination of more complex adsorption isotherms including such containing inflection points. We will exemplify the methodology with type II, type III and type V isotherms. Guidelines are given for how to determine such isotherms using the ECP method and for the experimental considerations that must be taken into account or that may be eliminated in the particular case.     

Experimental studies on the adsorption of two surfactants on solid-aqueous interfaces: adsorption isotherms and kinetics

A quartz crystal microbalance with dissipation (QCM-D) was used to measure the adsorption from aqueous solutions of CTAB (cationic) and C(12)E(6) (nonionic) surfactants on gold and silica surfaces. QCM-D allows for the determination of adsorption isotherms and also the monitoring of the dynamics of adsorption in real time. By considering the atomic-scale roughness of the solid surfaces and the surface area per head group at the air/water interface, our experiments indicate that at bulk concentrations above the critical micelle concentration adsorbed C(12)E(6) forms a monolayer-like structure on both surfaces and CTAB yields a bilayer-like structure. Although our measurements do not allow us to discriminate between the morphology of the aggregates (i.e., between flat monolayers, hemicylinders, or hemispheres in the case of C(12)E(6) and between flat bilayers, cylinders, or spheres in the case of CTAB), these results are particularly significant when compared to recent QCM-D data reported by Macakova et al. (Macakova, L.; Blomberg, E.; Claesson, P. M. Langmuir 2007, 23, 12436). These authors reported that QCM-D overestimates the amount of CTAB adsorbed on silica by as much as 30-40% as a result of entrapped water. Our analysis suggests that the effect of entrapped solvent is not as important as previously assumed and, in fact, QCM-D may not overestimate the amount of CTAB adsorbed when roughness is considered. Results for the kinetics of adsorption suggest that the aggregate structure as well as whether micelles are present may influence the adsorption mechanism. We discuss our results in the perspective of molecular theories for both the equilibrium and kinetics of surfactant adsorption.     

Quantification of single solute and competitive adsorption isotherms using a closed-loop perturbation method

A modification of the classical method of perturbation chromatography for measuring isotherms of the adsorption of dissolved components is suggested. The general principle of the method consists in analyzing responses of the chromatographic system to small perturbations at different equilibrium concentrations. Essential advantages of the method are: (a) only retention times or volumes have to be measured and no detector calibration is required and (b) experiments with mixtures can be performed and analyzed efficiently. The modification suggested in this paper is the application of a closed-loop arrangement allowing the efficient exploitation of the sample. Experimental data for four different chromatographic systems are presented to illustrate the method. With the determined adsorption isotherms elution profiles could be predicted satisfactorily.     

Lattice gas 2D/3D equilibria: chemical potentials and adsorption isotherms with correct critical points

A priori information is used to derive the chemical potential as a function of density and temperature for 2D and 3D lattice systems. The functional form of this equation of state is general in terms of lattice type and dimensionality, though it contains critical temperature and critical density as parameters which depend on lattice type and dimensionality. The adsorption isotherm is derived from equilibrium between two-dimensional and three-dimensional phases. Theoretical predictions are in excellent agreement with grand canonical Monte Carlo simulations.     

“Sharp convexity” of the adsorption isotherms and position of the breakthrough curves at low concentrations

Conclusions A mathematical model for adsorption dynamics was used to show that the position of the linear segments of the breakthrough curves at low concentrations for adsorption isotherms with different extents of sharp convexity is a function of parameters or the curvature of the adsorption isotherm over the entire range of measured concentrations in addition to kinetic factors.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 10, pp. 2333–2336, October, 1987.     

A parallel pore and surface diffusion model for predicting the adsorption and elution profiles of lispro insulin and two impurities in gradient-elution reversed phase chromatography

Lispro insulin (LPI), a widely used insulin analog, is produced on tons per year scale. Linear gradient reversed phase chromatography (RPC) is used in the production to separate LPI from two impurities, which differ from LPI by a single amino acid residue. A chromatography model for the ternary separation in this RPC process is unavailable from the literature. In this study, a parallel pore and surface diffusion model is developed and verified for LPI and the two impurities. The LPI can be recovered with high yield (≥95%) and high purity (>99.5%). A new method, which requires a small amount of materials and an order of magnitude fewer experiments, has been developed to estimate the solvent-modulated isotherm parameters. A modified reversed phase modulator model is developed to correlate the adsorption isotherms of LPI and impurities. A strategy has been developed for estimating the intrinsic pore diffusivity and surface diffusivity. Since the adsorption affinities decrease by more than three orders of magnitude as organic fraction (φ) increases from 0.19 to 0.40, the apparent diffusivities based on a pore diffusion model or a surface diffusion model can also vary by several orders of magnitude. For this reason, a pore diffusion model or a surface diffusion model with a constant apparent diffusivity cannot predict closely the chromatograms over the same range of organic fractions, concentrations, and loadings. The parallel pore and surface diffusion model with constant diffusivities can predict closely the frontal and elution profiles over a wide range of organic fractions (0.19-0.40), LPI concentrations (0.05-18 g/L), linear velocities (<10 cm/min), and loading volume (0.0004-13 CV). For large loading stepwise and linear gradient elution, the peaks of LPI and the impurities are strongly focused by self-sharpening and gradient focusing effects as a result of the steep decrease of adsorption affinity from the loading φ (0.19) to elution φ (≥0.27). When the ratio of diffusion rate to convection rate is greater than 10, spreading due to diffusion is largely compensated by the focusing effects. As a result, a pore diffusion model with a constant pore diffusivity can predict closely the elution profiles in stepwise and linear gradient elution. The experimental yield values (≥95%) can be predicted to within ±1% by the model.