Comparison of electrical double layer models postulating the linear or quadratic dependence of the anion adsorption energy on the electrode charge

Equations for calculating the concentration of a singly charged specifically adsorbed anion, the electrode potential, the differential capacitance, and interface tension are derived under the condition that the anion transition from the outer to the inner Helmholtz plane is described by the Frumkin isotherm and the energy of this transition (βG _A) is a quadratic function of the electrode charge q. By the example of the Hg/[H₂O + mc NH₄NO₃ + (1 − m)c NH₄F] system, it is shown that in contrast to the linear ΔG _A vs. q dependence, the quadratic dependence allows the differential capacitance curves to be adequately described throughout the m range. However, this model gives no way of describing the transition from the quadratic δG _A vs. q dependence for small m to the linear dependence observed experimentally in binary solutions with m = 1.     

Determination of the heterogeneity factor in the Frumkin adsorption isotherm from the width of capacity peaks

An appraisal of the various known surface preparations for the single crystal TiO₂ as used in solar photoelectrolysis cells is presented. The results presented are derived from photocurrent-voltage characteristics and interpretations of the surface behaviour are made on the basis of these measurements and their deviations from theoretically “ideal” behaviour. A novel preparation procedure is also introduced, producing results that represent the closest approach to the theoretically ideal surface known to date.     

Adsorption parameters in the Alekseev-Popov-Kolotyrkin model complemented by the Frumkin isotherm

The method of a nonlinear regressive analysis of nonequilibrium differential capacitance curves in terms of the Alekseev-Popov-Kolotyrkin (APK) model complemented by the Frumkin isotherm is used to determine adsorption parameters of an organic cation of tetrabutylammonium, an inorganic nitrate anion, and a butanol molecule at an interface with a mercury electrode. The adsorption behavior of the nitrate anion is quantitatively described by the APK model, whereas the Frumkin model is preferable in the case of butanol. With the [C₄H₉]₄N⁺ cation adsorbed on a mercury electrode, the EDL structure is probably strongly affected by the formation of ionic pairs. Dedicated to the ninetieth anniversary of Ya.M. Kolotyrkin’s birth.     

On stability of model electrocatalytic process with frumkin adsorption isotherm occurring on spherical electrode

The method of impedance spectroscopy was used for theoretical studies of the conditions of appearance of Hopf instability in a model electrochemical system with a preceding homogeneous chemical reaction in the Nernst diffusion layer and electrocatalytic reaction on the spherical electrode surface under potentiostatic conditions. It is shown within the suggested electrochemical instability model based on the potential-dependent adsorption/desorption that the effective rate of the preceding homogeneous chemical reaction may affect the system stability. The effect diminishes at a decrease in the electrode radius. The instability region grows at an increase in the thickness of the Nernst diffusion layer.     

Calculation of the sticking coefficient in the case of the linear adsorption isotherm

The expression of the sticking coefficient for the linear adsorption isotherm was derived. The sticking coefficient is an increasing temperature function and directly related to the entropy factor. The limits of applicability of the derived equation are discussed.     

pH dependence of the crystal violet adsorption isotherm at the silica-water interface

The pH-dependent adsorption isotherms for the charged chromophore crystal violet, CV(+), have been measured with three different bases by a free-running cavity implementation of evanescent wave cavity ring-down spectroscopy. The ratio of the maximal absorbance measurements at pH 5.10 and 9.05 is consistent with a Q2:Q3 silanol site ratio of 72.8:27.2. The adsorption isotherms have been interpreted in terms a cooperative binding adsorption allowing more than one ionic species to bind to each silanol group. The surface concentration is consistent with a silanol charge density of 1.92 +/- 0.55 nm(-2) and a total neutralized interface layer structure extending 9 nm from the surface. Binding constants and stoichiometric coefficients are derived for CV(+) to both the Q2 and Q3 sites. A variation of the adsorption isotherm with base is observed so that the isotherm at pH 9.05 adjusted with ammonium hydroxide sets up a competitive acid-base equilibrium with the SiOH groups with only 49% of the surface silanol sites dissociated. The implications for functionalized surfaces in chromatography are discussed.     

Molar volume and adsorption isotherm dependence of capillary forces in nanoasperity contacts

The magnitude of the capillary force at any given temperature and adsorbate partial pressure depends primarily on four factors: the surface tension of the adsorbate, its liquid molar volume, its isothermal behavior, and the contact geometry. At large contacting radii, the adsorbate surface tension and the contact geometry are dominating. This is the case of surface force apparatus measurements and atomic force microscopy (AFM) experiments with micrometer-size spheres. However, as the size of contacting asperities decreases to the nanoscale as in AFM experiments with sharp tips, the molar volume and isotherm of the adsorbate become very important to capillary formation as well as capillary adhesion. This effect is experimentally and theoretically explored with simple alcohol molecules (ethanol, 1-butanol, and 1-pentanol) which have comparable surface tensions but differing liquid molar volumes. Adsorption isotherms for these alcohols on silicon oxide are also reported.     

Determination of the average heat and characteristic energy from the adsorption isotherm

The isotherms of the total content, isosteric and average heats of adsorption, as well as characteristic energies of adsorption were determined from the isotherms of excess adsorption of carbon dioxide on six different carbon adsorbents at temperature ranging from 293 to 423 К at pressures up to 6 MPa. The average isosteric heats are in agreement with the average heats of adsorption, which were determined from the equation relating the heat of adsorption with the characteristic energy of adsorption.     

Limits of the numerical estimation of the adsorption energy distribution from adsorption isotherm data using the expectation-maximization method

The limits of the use of the expectation-maximization (EM) method for the study of the heterogeneity of adsorbent surfaces were tested by calculating the adsorption energy distribution of systems having known degrees of heterogeneity. Connecting on-line two different columns allows the simulation of a heterogeneous system. The two columns used were endcapped, C(18)-bonded silica used as stationary phases and having different degrees of C(18) chain coverages (0.42 and 2.03 micromol/m(2)). The adsorption constants of phenol measured by frontal analysis (FA) are significantly different on these two columns. On each column, the adsorption behavior was best accounted for by a bi-Langmuir isotherm model, corresponding to a heterogeneous surface with a bimodal energy distribution. The difference between the adsorption energies on the weak adsorption sites of the two columns is 1.5 kJ/mol. The energy difference of their high energy sites is 2.2 kJ/mol. The EM method can readily distinguish between adsorption sites having energies that differ by more than 5 kJ/mol after more than 10 million iterations, but it cannot distinguish between adsorption sites for which this energy difference is less than 2 kJ/mol, even after 100 million iterations. For highly heterogeneous systems, (e.g., those with more than three different types of adsorption sites), the EM program does not converge necessarily towards the actual energy distribution function but toward a simpler one, having fewer adsorption sites that are almost equally spaced in the energy space. This failure of the EM program is related to the fact that, despite the excellent precision of the FA measurements (<1%), any series of adsorption data can be represented by several distinct AEDs. Thus, the degree of heterogeneity of RPLC adsorbents determined with the EM method might often be minimized, resulting in erroneous values of the isotherm parameters.     

The adsorption of RNAse A,BSA and cytochrome c at the graphite powder|liquid interface using in parallel the adsorption isotherm plot and linear sweep voltammetry on graphite paste electrode

The adsorption of RNAse A, BSA and cytochrome c on graphite powder has been investigated using in parallel the adsorption isotherm plot and linear sweep voltammetry on a graphite paste electrode (g.p.e.). The principle of the latter depends on the determination by electrochemical oxidation of tyrosyl or tryptophan residues having access to the interface. Adsorption isotherms exhibit bimodal adsorption features with a cooperative phenomenon leading to the formation of a close-packed two dimensional surface phase. Protein molecules are adsorbed in a monolayer when the pH is different from the isoelectric point. This layer is built up by the entanglement of a double network:

• •A network of molecules irreversibly adsorbed side-on, where hydrophobic residues move from the core to the surface. The tertiary structure is likely altered. The new dimensions of these adsorbed molecules have been determined (surface area per molecule, thickness).
• •A network of molecules adsorbed end-on, lying between the irreversibly adsorbed molecules.

     

Modeling of co-adsorption of cations and anions localized in different layers of the dense part of electrical double layer under the conditions of linear charge dependence of the adsorption energy

Effects of interference of the adsorption of cations and anions located in different layers of the electrical double layer dense part (under the conditions of linear dependence of their adsorption energy on the electrode charge) analyzed in the context of a system comprising two Frumkin mixed isotherms. The model was shown to suit for the explaining of the retraction of weakly adsorbed cations into the dense layer in the case of practical independence of the total surface excess of the cations on their localizing plane in the electrical double layer. The effect of co-adsorption of cations and anions in such systems on the total differential capacitance of electrode is analyzed.     

A site energy distribution function from Toth isotherm for adsorption of gases on heterogeneous surfaces

A site energy distribution function based on a condensation approximation method is proposed for gas-phase adsorption systems following the Toth isotherm. The proposed model is successfully applied to estimate the site energy distribution of three pitch-based activated carbons (PA, PFeA and PBA) developed in our laboratory and also for other common adsorbent materials for different gas molecules. According to the proposed model the site energy distribution curves of the activated carbons are found to be exponential for hydrogen at 77 K. The site energy distribution of some of the activated carbon fibers, ambersorb, Dowex optipore, 13X Zeolite for different adsorbate molecules represents a quasi-Gaussian curve with a widened left hand side, indicating that most sites have adsorption energies lower than a statistical mean value.     

The effect of heat treatment on the adsorption properties of solids. II. The effect of heat treatment on the character of the adsorption isotherm

In this study the nitrogen adsorption isotherms of heat-treated zinc oxalate, sintered magnesium oxide, and some oxidised pitch resins are considered. It is shown that characterisation of the adsorption isotherm can be via the monolayer capacity, the BET constant C, or plots of the degree of coverage of the surface at various relative vapour pressures. These parameters are critically assessed and shown to be dependent upon the closeness to the manner in which the complete adsorption isotherm is described by the BET equation. In considering the complete adsorption isotherm it is considered best to characterise the adsorption data by quoting the statistical monolayer capacity and the value of C at this point on the adsorption isotherm. The adsorption isotherms are then best compared by plotting as the number of statistical layers against the relative pressure. The further characterisation by plotting the apparent variation in C or the degree of coverage of the surface against the relative pressure has a usefulness if the limitations of the method are noted.     

Controlling energy and charge transfer in linear chlorophyll dimers

A series of linkers constructed from combinations of phenyl and ethynyl groups is shown to permit ultrafast energy transfer between two chlorophylls, while allowing control over radical cation migration between them.     

Insight into the extractability of metals from soils using an implementation of the linear adsorption isotherm model

This paper deals with an often overlooked artifact in sequential and single extraction of metals from soils, viz. the volume to mass (V/m) ratio as a potential source for inadequate extraction yields. We offer a theoretical framework to get a grip on this intricate parameter and came up with a model based on a linear adsorption isotherm to derive the correct maximal metal extractability for a certain extractant. We verified the model experimentally using 0.1 mol l⁻¹ nitric acid for extraction of seven metals (Cr, Co, Cu, Cd, Pb, Ni and Zn) from an urban soil sample, and concluded that commonly used V/m ratios in the range of 10-40 ml g⁻¹ may give as much as 50% too low extraction yields. Thus, a strong caveat is in place as to be very critical what V/m ratios to use and preferably apply the model derived to obtain the correct maximal extractability using a variable V/m ratio method.     

Estimation of adsorption isotherm parameters with inverse method--possible problems

In recent years the inverse method (IM) has been frequently applied to estimate of isotherm parameters. The IM has been used for adsorption process modeling for one, two and even three component chromatography. This method requires only a few injections with various sample concentrations, so the solute consumption and time requirements are very modest. The successful estimation of isotherm parameters with IM depends on applied chromatography column model and a numerical method used to solve the model. For HPLC column the classical equilibrium-dispersive (ED) model can be used. This model is solved frequently with very fast Rouchon finite difference method. However, the accuracy of computations with Rouchon method is decreasing with increase of the number of analyzed components. The aim of this work is the comparison of the results obtained with inverse method when ED model was solved with Rouchon or orthogonal collocation on finite element (OCFE) scheme. Assuming that solution of ED model with OCFE method can be regarded as real a solution, it was found that the Rouchon scheme may not give satisfactory results even for column with 10,000 theoretical plates for three component chromatography. Moreover, the optimal conditions for separation, calculated with Rouchon method, can be remarkably different from that obtained with the OCFE method. The next aim of this work is the presentation of Craig method application to estimation of model parameters with IM.