Affinity distribution functions in multicomponent heterogeneous adsorption. Analytical inversion of isotherms to obtain affinity spectra

An analytical approach for the interpretation of multicomponent heterogeneous adsorption or complexation isotherms in terms of multidimensional affinity spectra is presented. Fourier transform, applied to analyze the corresponding integral equation, leads to an inversion formula which allows the computation of the multicomponent affinity spectrum underlying a given competitive isotherm. Although a different mathematical methodology is used, this procedure can be seen as the extension to multicomponent systems of the classical Sips's work devoted to monocomponent systems. Furthermore, a methodology which yields analytical expressions for the main statistical properties (mean free energies of binding and covariance matrix) of multidimensional affinity spectra is reported. Thus, the level of binding correlation between the different components can be quantified. It has to be highlighted that the reported methodology does not require the knowledge of the affinity spectrum to calculate the means, variances, and covariance of the binding energies of the different components. Nonideal competitive consistent adsorption isotherm, widely used in metal/proton competitive complexation to environmental macromolecules, and Frumkin competitive isotherms are selected to illustrate the application of the reported results. Explicit analytical expressions for the affinity spectrum as well as for the matrix correlation are obtained for the NICCA case.     

Kinetics and equilibrium of multicomponent adsorption on chiraly templated surfaces

In this contribution we propose a simple model of adsorption of a binary (racemic) mixture on a chiraly templated surface. As an example, the adsorption of a liquid mixture of enantiomers on a chiral stationary phase (CSP) is considered. In particular, we study the effect of the lateral interactions in the adsorbed phase on the kinetic and equilibrium isotherms of the enantiomers. Additionally, we investigate the influence of the composition of the surface on the performance of the CSP in the presence of the lateral interactions. To that end, the adsorption of the mixture is modeled by using Monte Carlo simulations as well as by applying an analytical approach involving rate equations coupled with the Mean Field Approximation (MFA). The predictions of the theory are found to be in good agreement with the results of the simulations.     

Competitive adsorption from multicomponent non-electrolytic liquid mixtures on heterogeneous solid surfaces

Summary A simple model for competitive adsorption from multicomponent non-electrolytic liquid mixtures on energetically heterogeneous solid surfaces is discussed in terms of statistical thermodynamics. Integral equations are derived for the fundamental thermodynamic quantities that characterize competitive adsorption at the liquid-solid interface. Extensive model studies are presented in order to illustrate the influence of adsorbent heterogeneity on the behaviour of these thermodynamic quantities.on leave from Faculty of Chemistry, M. Curie-Sklodowska University, PL-20031 Lublin, Poland     

MALDI mass spectrometric determination of dendritic iron chelation stoichiometries and conditional affinity constants

The iron chelation stoichiometries of a dendritic iron(III) chelator with N(1), N(3), N(5)-trimethylbenzene-1,3,5-tricarboxamide at its core, and containing 3 identical hexadentate tris-hydroxypyridinone branches D was studied by MALDI mass spectrometry. At pH 7.2, the speciation of the system included FeD, Fe(2)D and Fe(3)D species with the respective conditional stability constants of 26.74, 26.03 and 25.36. The differences in the stepwise affinity constants arise from the statistical distribution of iron(III), and there was no evidence for cooperativity between the iron-binding sites.     

Statistical thermodynamics of adsorption from multicomponent liquid mixtures on heterogeneous solid surfaces

The statistical thermodynamics of adsorption from multicomponent liquid mixtures on heterogeneous solid surfaces is discussed by assuming the cell adsorption model and ideal adsorbed phase.Two integral representations for the adsorption isotherm are proposed: one based onn-dimensional energy distribution function (i.e., each adsorption site is characterized by adsorption energies of all components), and the other based on distribution of differences of adsorption energies ofn-1 components in relation to adsorption energy of the chosen component (i.e., each adsorption site is characterized byn-1 differences of adsorption energies of the components in relation to adsorption energy of the chosen component).The expressions for differential adsorption heat for adsorption from binary liquid mixtures have been derived from both integral equations.

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Statistische Thermodynamik der Adsorption aus flüssigen Mehrkomponentenmischungen auf heterogenen festen Oberflächen

Zusammenfassung Die statistische Thermodynamik der Adsorption von aus mehreren Komponenten bestehenden flüssigen Mischungen auf heterogenen festen Oberflächen wird für das Modell der Zellenadsorption bei ideal adsorbierter Phase diskutiert.Zwei Integraldarstellungen der Adsorptionsisotherme werden vorgeschlagen: eine auf einen-dimensionale Verteilungsfunktion der Energie gestützte (das heißt, jede Adsorptionsstelle wird durch Adsorptionsenergien von allen Komponenten charakterisiert); die andere basiert auf der Verteilung der Unterschiede von Adsorptionsenergien dern-1-Komponenten in bezug auf die Adsorptionsenergie der ausgewählten Komponente (das heißt, jede Adsorptionsstelle wird durchn-1-Unterschiede charakterisiert. Formeln für differentiale Adsorptionswärmen für die Adsorption aus binären flüssigen Mischungen sind von beiden Integral-Gleichungen abgeleitet worden.

     

Fixed-bed adsorption of chlorinated hydrocarbons from multicomponent aqueous solution onto activated carbon: equilibrium column model

The results of studies on the adsorption dynamics of light chlorinated hydrocarbons, 1,2-dichloroethane, 1,1,2,2-tetrachloroethane, chloroform, carbon tetrachloride, 1,1-dichloroethene, perchloroethylene and 1,1,2-trichloroethene, from a seven-component solution on to activated carbon are presented. The experimental results were described using the equilibrium model. The application of this model allows to determine the location of the midpoint of the breakthrough profile.     

Single and multicomponent equilibrium studies for the adsorption of acidic dyes on carbon from effluents

The ability of activated carbon to adsorb three acidic dyes, namely, Acid Blue 80 (AB80), Acid Red 114 (AR114), and Acid Yellow (AY117), from wastewater has been studied at 20 degrees C. The three single-component systems and the three binary equilibrium systems have been measured experimentally. The three single-component isotherms were analyzed using the Langmuir, Freundlich, Redlich-Peterson, and Sips equations. The Redlich-Peterson equation gave the lowest errors using the sum of the squares of the errors closely followed by the Sips and Langmuir equations; the Freundlich fits were significantly worse. The three bisolute experimental equilibrium sets of data were analyzed by incorporating the previous four single-component isotherm equations into the ideal adsorbed solution theory (IAST). The solution methods for each of the four isotherm equations are presented in the paper, and the predicted results for the three bisolute systems, using the four isotherm equations, are compared. For the three bisolute systems (AB80 + AR114, AB80 + AY117, and AR114 + AY117), the Redlich-Peterson isotherm gives the best correlation with the experimental isotherm data.     

Complexes of DOTA-bisphosphonate conjugates: probes for determination of adsorption capacity and affinity constants of hydroxyapatite

The adsorption on hydroxyapatite of three conjugates of a bisphosphonate and a macrocycle having C1, C2, and C3 spacers and their terbium complexes was studied by the radiotracer method using 160Tb as the label. The radiotracer-containing complex of the conjugate with the C3 spacer was used as a probe for the determination of the adsorption parameters of other bisphosphonates that lack a DOTA unit. A physicochemical model describing the competitive adsorption was successfully applied in the fitting of the obtained data. The maximum adsorption capacity of bisphosphonates containing bulky substituents is determined mainly by their size. For bisphosphonates having no DOTA moiety, the maximum adsorption capacity is determined by the electrostatic repulsion between negatively charged bisphosphonate groups. Compounds with a hydroxy or amino group attached to the alpha-carbon atom show higher affinities. Macrocyclic compounds containing a short spacer between the different bisphosphonic acid groups and the macrocyclic unit exhibit high affinities, indicating a synergic effect of the bisphosphonic and the macrocyclic groups during adsorption. The competition method described uses a well-characterized complex and allows a simple evaluation of the adsorption behavior of bisphosphonates. The application of the macrocycle-bisphosphonate conjugates allows easy radiolabeling via complexation of a suitable metal isotope.     

On the mechanical equilibrium in anisotropic heterogeneous systems

The conditions for equilibrium of liquid surfaces dividing two anisotropic bulk phases have been deduced by means of variational analysis.It has been shown that the presence of an anisotropic tensor in the bulk requires a gradient of surface tension in a state of equilibrium.     

Spectrophotometric determination of equilibrium constants in solution

This paper reviews critically a number of published methods for using ultraviolet or visible spectrophotometric data for the determination of equilibrium constants in solution. Single-stage or overlapping equilibria are considered, and a number of ways in which the experimental data can be handled by electronic computers are suggested.     

Projection maps for the determination of equilibrium constants

Projection maps are suggested for the construction of model functions to be compared with potentiometric measurements of the free ligand concentration. The theoretical curves are found to be a valuable guide in the choice of the optimal conditions for the evaluation of complex formation equilibria. As an example, measurements of [H⁺], with a quinhydrone electrode, have been planned with the help of model maps for the study of the protolysis of oxalic acid in 1 M(KCl) at 25°C.Projection maps are also proposed to interpret data obtained by the method of competitive reactions.     

Improved model of multicomponent adsorption in reversed-phase liquid chromatography

This work presents a modification to the real adsorbed solution model using a Flory-Huggins type of expression that was reported previously. This modification consists in replacing the Flory-Huggins activity coefficient by the spreading pressure dependent approach. This new model takes into account explicitly the adsorbate-adsorbate interactions taking place in the adsorbed phase. It provides an excellent prediction of the competitive, ternary adsorption equilibrium of benzyl alcohol, 2-phenylethanol and 2-methyl benzyl alcohol observed in a reversed-phase liquid chromatographic system using information merely derived from the single-component adsorption experimental data.     

Cyclization equilibrium constants and the conformations of polydimethylsiloxane chains

Experimental molar cyclization equilibrium constants for dimethylsiloxane cyclics having from 8 to ca 400 skeletal bonds are presented. The results for an undiluted equilibrate, an equilibrate in toluene solution, and an equilibrate in diglyme solution are compared. The increases in the equilibrium concentrations of cyclic oligomers, previously found to be large in good solvents (e.g., toluene) are found to be small in the poor solvent (diglyme). For macrocyclics, the cyclization equilibrium constants in the undiluted equilibrate and in diglyme (theta conditions) are found to be in good agreement with calculations which were carried out by assuming that chains are unperturbed and by using the rotational isomeric state model of Flory, Cresenzi, and Mark for polydimethylsiloxane. However, with chain polymer in dilute solution in a good solvent (toluene), the macrocyclization equilibrium constants are found to be lower than in the other two equilibrates. This result confirms the work of other authors who suggested that in this case the chains were expanded by excluded-volume effects.