On the use of the dual process Langmuir model for predicting unary and binary isosteric heats of adsorption

Analytic expressions for unary and binary isosteric heats of adsorption as a function of the adsorbed phase loading were derived from the dual process Langmuir (DPL) model using the Clausius-Clapeyron equation. Unary isosteric heats of adsorption predicted from these expressions for several adsorbate-adsorbent systems were compared to values in the literature predicted from the well-accepted graphical approach using Toth and unilan models (Adsorption Equilibrium Data Handbook; Prentice Hall: NJ, 1989). Predictions from the DPL model were also compared to rare experimental unary and binary isosteric heats of adsorption in the literature for another adsorbate-adsorbent system. In all cases, very good agreement was obtained, showing that the DPL model can be used in adsorption process modeling for accurately predicting not only ideal and nonideal mixed-gas adsorption equilibria (Langmuir 2011, 27, 4700), but also unary and even binary isosteric heats of adsorption.     

On the validity of Langmuir adsorption on supported nanoparticles

The general form for adsorption on nanoparticles is presented. Conditions for the validity of Langmuir adsorption isotherm for supported nanoparticles are discussed.     

Isothermal phase equilibria for the (xenon + cyclopropane) mixed-gas hydrate system

Isothermal three-phase equilibria of gas, aqueous, and hydrate phases for the {xenon (Xe) + cyclopropane (c-C₃H₆)} mixed-gas hydrate system were measured at two different temperatures (279.15 and 289.15) K. The structural phase transitions from structure-I to structure-II and back to structure-I, depending on the mole fraction of guest mixtures, occur in the (Xe + c-C₃H₆) mixed-gas hydrate system. The isothermal pressure–composition relations have two local pressure minima. The most important characteristic in the (Xe + c-C₃H₆) mixed-gas hydrate system is that the equilibrium pressure–composition relations exhibit the complex phase behavior involving two structural phase transitions and two homogeneous negative azeotropes. One of two structural phase transitions exhibits the heterogeneous azeotropic-like behavior.     

On the Jovanovic model of adsorption

Summary The effect of the hydrostatic pressure exerted by bulk phases on adsorbed phases depends on the mobility of adsorbed phase and on the topographical distribution of adsorption energy. It is shown in this paper that the Langmuir's equation applies also to adsorption of fully mobile hard discs, on heterogeneous surfaces having fully random topographical distribution of adsorption energy. The effects of the hydrostatic pressure are discussed also in the case of heterogeneous surfaces exhibiting patchwise topographical distribution of adsorption energy.

Symbols

Roman Letters a parameter in the Jovanovic isotherm - A fraction of total surface covered - A ₁ fraction covered of sites having a given adsorption energy - A _j fraction of surface covered as evaluated from the Jovanovic adsorption isotherm - b,b _t parameters in various versions of the Langmuir isotherm - B number of adsorption sites on an adsorbent surface - H enthalpy of adsorption - F free energy of adsorption - k Boltzmann's constant - K, K _j ,K _L parameters in various versions of adsorption isotherms - m mass of the adsorbate molecule - N number of adsorbed molecules - N ₀ monolayer capacity in mobile adsorption - N ₁ local adsorption isotherm - N _t overall adsorption isotherm - p pressure of gas phase adsorbate - q canonical molecular partition function for mobile admolecules - q _st isosteric heat of adsorption - R gas constant - S surface area of adsorbent - S ₁ surface area of sites having a given adsorption energy - S ^* minimum change in the external variable S - T absolute tempetature - T _t frozen temperature Greek Letters Greek Letters parameter of the adsorption isotherm - isothermal-isobaric partition function - adsorption energy - area occupied by one adsorbate molecule - area of box of uniform adsorption energy - chemical potential - v frequency of vibrations of admolecules - range of variation of - () distribution of adsorption energy With 2 figures and 1 table     

Immersion enthalpy and the constants of Langmuir model in the 3-chloro phenol adsorption on activated carbon

The adsorption process of 3-chloro phenol from aqueous solution on a activated carbon prepared from African palm stone and which presents a specific surface area of 685 m² g⁻¹, a greater quantity of total acid groups and a pH_PZC of 6.8 is studied. The adsorption isotherms are determined at pH values of 3, 5, 7, 9 and 11. The adsorption isotherms are fitted to the Langmuir model and the values of the maximum quantity adsorbed that are between 96.2 and 46.4 mg g⁻¹ are obtained along with the constant K_L with values between 0.422 and 0.965 L mg⁻¹. The maximum quantity adsorbed diminishes with the pH and the maximum value for this is a pH of 5. The immersion enthalpies of the activated carbon in a 3-chloro phenol solution of constant concentration, of 100 mg L⁻¹, are determined for the different pH levels, with results between 37.6 and 21.2 J g⁻¹. Immersion enthalpies of the activated carbon in function of 3-chloro phenol solution concentration are determined to pH 5, of maximum adsorption, with values between 28.3 and 38.4 J g⁻¹, and by means of linearization, the maximum immersion enthalpy is calculated, with a value of 41.67 J g⁻¹. With the results of the immersion enthalpy, maximum quantity adsorbed and the constant K_L, establish relations that describe the adsorption process of 3-chloro phenol from aqueous solution on activated carbon.     

Application of the vacancy solution theory to describe the enthalpic effects accompanying mixed-gas adsorption

The possibility of utilizing vacancy solution theory (VST) to study the enthalpic effects accompanying mixed-gas adsorption equilibria is presented. Besides heterogeneity, the interaction effects by using the regular adsorbed solution, Flory-Huggins, and Wilson models of nonideality in the adsorbed phase are taken into account. To predict adsorption phase diagrams and calorimetric effects in the mixed-gas adsorption system, only a knowledge of the single-gas adsorption isotherms and accompanying calorimetric effects is required. The possibility of simplification of the obtained theoretical expressions is shown. The obtained agreement between theory and experiment is very satisfactory.     

From Langmuir kinetics to first- and second-order rate equations for adsorption

So far, the first- and second-order kinetic equations have been most frequently employed to interpret adsorption data obtained under various conditions, whereas the theoretical origins of these two equations still remain unknown. Using the Langmuir kinetics as a theoretical basis, this study showed that the Langmuir kinetics can be transformed to a polynomial expression of dtheta t /d t = k 1(theta e - theta t ) + k 2(theta e - theta t ) (2), a varying-order rate equation. The sufficient and necessary conditions for simplification of the Langmuir kinetics to the first- and second-order rate equations were put forward, which suggested that the relative magnitude of theta e over k 1/ k 2 governs the simplification of the Langmuir kinetics. In cases where k 1/ k 2 is greater than theta e or k 1/ k 2 is very close to theta e, adsorption kinetics would be reasonably described by the first-order rate equation, whereas the Langmuir kinetics would be reduced to the second-order equation only at k 1/ k 2 < theta e. It was further demonstrated that both theta e and k 1/ k 2 are the function of initial adsorbate concentration ( C 0) at a given dosage of adsorbent, indicating that simplification of the Langmuir kinetics indeed is determined by C 0. Detailed C 0-depedent boundary conditions for simplifying the Langmuir kinetics were also established and were verified by experimental data.     

Isothermal phase equilibria for the (HFC-32 + HFC-134a) mixed-gas hydrate system

Isothermal phase equilibria (pressure-composition relations in hydrate, gas, and aqueous phases) in the {difluoromethane (HFC-32) + 1,1,1,2-tetrafluoroethane (HFC-134a)} mixed-gas hydrate system were measured at the temperatures 274.15 K, 279.15 K, and 283.15 K. The heterogeneous azeotropic-like behaviour derived from the structural phase transition of (HFC-32 + HFC-134a) mixed-gas hydrates appears over the whole temperature range of the present study. In addition to the heterogeneous azeotropic-like behaviour, the isothermal phase equilibrium curves of the (HFC-32 + HFC-134a) mixed-gas hydrate system exhibit the negative homogeneous azeotropic-like behaviour at temperatures 279.15 K and 283.15 K. The negative azeotropic-like behaviour, which becomes more remarkable at higher temperatures, results in the lower equilibrium pressure of (HFC-32 + HFC-134a) mixed-gas hydrates than those of both simple HFC-32 and HFC-134a hydrates. Although the HFC-134a molecule forms the simple structure-II hydrate at the temperatures, the present findings reveal that HFC-134a molecules occupy a part of the large cages of the structure-I mixed-gas hydrate.     

Cubical equations of state for predicting the phase equilibria of poorly studied substances

A new procedure for determining the coefficients of the Peng-Robinson equation of state is proposed, for which a minimum of information is required. It is shown that using the Morachevskii complexity factor of molecular interaction in the algorithm for calculating the saturation vapor pressure of substances enables us to study the parameters of the vapor-liquid equilibria of substances with various polarities. Based on our validation of the procedure for determining the coefficients of the Ping-Robinson equation, it is concluded that the values for the saturation vapor pressure of halide derivatives of hydrocarbons calculated from tabular reference data agree satisfactorily in practice.     

Isothermal phase equilibria and structural phase transition in the carbon dioxide + cyclopropane mixed-gas hydrate system

The isothermal phase equilibria of the carbon dioxide + cyclopropane mixed-gas hydrate system were investigated by means of static temperature measurement and Raman spectroscopic analysis. Raman spectra indicated that the crystal structure of the carbon dioxide + cyclopropane mixed-gas hydrate changes from structure-I to structure-II and back to structure-I with an increase of the equilibrium carbon dioxide composition at 279.15 K, while each simple gas hydrate belongs to structure-I at the temperature. Whereas, unlike 279.15 K, no structural phase transition occurs along the isothermal stability boundary at 284.15 K.     

An activity coefficient model for predicting salt effects on vapor–liquid equilibria of mixed solvent systems

In the present study, an activity coefficient model, based on the concept of local volume fractions and the Gibbs–Helmholtz relation, has been developed. Some modifications were made from Tan–Wilson model (1987) and TK–Wilson model (1975) to represent activity coefficients in mixed solvent–electrolyte systems. The proposed model contains two groups of binary interaction parameters. One group for solvent–solvent interaction parameters corresponds to that given by the TK–Wilson model (1975) in salt-free systems. The other group of salt–solvent interaction parameters can be calculated either from vapor pressure or bubble temperature data in binary salt–solvent systems. It is shown that the present model can also be used to describe liquid–liquid equilibria. No ternary parameter is required to predict the salt effects on the vapor–liquid equilibria (VLE) of mixed solvent systems. By examining 643 sets of VLE data, the calculated results show that the prediction by the present model is as good as that by the Tan–Wilson model (1987), with an overall mean deviation of vapor phase composition of 1.76% and that of the bubble temperature of 0.74 K.     

Theoretical consideration of the use of a Langmuir adsorption isotherm to describe the effect of light intensity on electron transfer in photosystem II

Electron transport through photosystem II (PSII), measured as oxygen evolution, was investigated in isolated PSII particles and thylakoid membranes irradiated with white light of intensities (I) of 20 to about 4000 micromol of photons/(m2.s). In steady-state conditions, the evolution of oxygen varies with I according to the hyperbolic expression OEth = OEth(max)I/(L1/2 + I) (eq i) where OEth is the theoretical oxygen evolution, OEth(max) is the maximum oxygen evolution, and L1/2 is the light intensity giving OEth(max)/2. In this work, the mathematical derivation of this relationship was performed by using the Langmuir adsorption isotherm and assuming that the photon interaction with the chlorophyll (Chl) in the PSII reaction center is a heterogeneous reaction in which the light is represented as a stream of particles instead of an electromagnetic wave (see discussion in Turro, N. J. Modern Molecular Photochemistry; University Science Books: Mill Valley, CA, 1991). In accordance with this approximation, the Chl molecules (P680) were taken as the adsorption surfaces (or heterogeneous catalysts), and the incident (or exciting) photons as the substrate, or the reagent. Using these notions, we demonstrated that eq i (Langmuir equation) is a reliable interpretation of the photon-P680 interaction and the subsequent electron transfer from the excited state P680, i.e., P680*, to the oxidized pheophytin (Phe), then from Phe- to the primary quinone QA. First, eq i contains specific functional and structural information that is apparent in the definition of OEth(max) as a measure of the maximal number of PSII reaction centers open for photochemistry, and L1/2 as the equilibrium between the electron transfer from Phe- to QA and the formation of reduced Phe in the PSII reaction center by electrons in provenance from P680*. Second, a physiological control mechanism in eq i is proved by the observation that the magnitudes of OEth(max) and L1/2 are affected differently by exogenous PSII stimulators of oxygen evolution (Fragata, M.; Dudekula, S. J. Phys. Chem. B 2005, 109, 14707). Finally, an unexpected new concept, implicit in eq i, is the consideration of the photon as the substrate in the photochemical reactions taking place in the PSII reaction center. We conclude that the Langmuir equation (eq i) is a novel mathematical formulation of energy and electron transfer in photosystem II.     

Effects of molecular siting and adsorbent heterogeneity on the ideality of adsorption equilibria

The ideal adsorbed solution (IAS) theory is the benchmark for the prediction of mixed-gas adsorption equilibria from pure-component isotherms. In this work, we use atomistic grand canonical Monte Carlo simulations to test the effects of molecular siting and adsorbent energetic heterogeneity on the applicability of the IAS theory. Pure-component isotherms generated by atomistic simulation are used to predict binary isobaric isotherms using the IAS theory. These predicted isotherms are compared with those obtained by a full atomistic simulation of the binary mixture. Binary mixtures of argon, methane, and CF4 in silicalite are found to obey IAS theory, while benzene/methane and cyclohexane/methane in silicalite are nonideal. The mixture of argon and CF4 is ideal despite the large difference in the sizes of the two species. This contradicts previous hypotheses in the literature, which state that mixtures of species of unequal size do not adsorb ideally. The nonideal behavior of the benzene/methane and cyclohexane/methane systems occurs because of adsorbent heterogeneity in these systems, which depends on both sorbent and sorbate. In addition, we use a lattice gas model with parameters derived from atomistic simulation to demonstrate analytically that a sufficiently energetically heterogeneous adsorbent will result in the breakdown of IAS theory even in the absence of interactions between sorbates.