Adsorption kinetics at the solid/solution interface: statistical rate theory at initial times of adsorption and close to equilibrium

The kinetics of solute adsorption at the solid/solution interface has been studied by statistical rate theory (SRT) at two limiting conditions, one at initial times of adsorption and the other close to equilibrium. A new kinetic equation has been derived for initial times of adsorption on the basis of SRT. For the first time a theoretical interpretation based on SRT has been provided for the modified pseudo-first-order (MPFO) kinetic equation which was proposed empirically by Yang and Al-Duri. It has been shown that the MPFO kinetic equation can be derived from the SRT equation when the system is close to equilibrium. On the basis of numerically generated points ( t, q) by the SRT equation, it has been shown that we can apply the new equation for initial times of adsorption in a larger time range in comparison to the previous q vs radical t linear equation. Also by numerical analysis of the generated kinetic data points, it is shown that application of the MPFO equation for modeling of whole kinetic data causes a large error for the data at initial times of adsorption. The results of numerical analysis are in perfect agreement with our theoretical derivation of the MPFO kinetic equation from the SRT equation. Finally, the results of the present theoretical study were confirmed by analysis of an experimental system.     

Kinetic modeling of liquid-phase adsorption of reactive dyes on activated carbon

In this paper, adsorption equilibrium and kinetics of three reactive dyes from their single-component aqueous solutions onto activated carbon were studied in a batch reactor. Effects of the initial concentration and adsorbent particle size on adsorption rate were investigated Adsorption equilibrium data were then correlated with several well-known equilibrium isotherm models. The kinetic data were fitted using the pseudo-first-order equation, the pseudo-second-order equation, and the intraparticle diffusion model. The respective characteristic rate constants were presented. A new adsorption rate model based on the pseudo-first-order equation has been proposed to describe the experimental data over the whole adsorption process. The results show that the modified pseudo-first-order kinetic model generates the best agreement with the experimental data for the three single-component adsorption systems.     

On the applicability of the pseudo-second order equation to represent the kinetics of adsorption at solid/solution interfaces: a theoretical analysis based on the statistical rate theory

It is shown that the empirical pseudo-second order kinetic equation is a very efficient formula to correlate the kinetic data generated by applying theoretical expressions developed from the fundamental SRT (Statistical Rate Theory) approach to the interfacial transport. This is especially true when the most popular linear representation is used in which time/adsorbed amount is plotted vs. time. However, the commonly observed goodness of such linear plots does not necessarily speak for the applicability of the pseudo-second order kinetic equation. A reliable estimation, for instance, of the equilibrium adsorbed amount is possible only when a substantial part of a kinetic isotherms corresponds to the conditions close to equilibrium. Energetic surface heterogeneity increases the goodness of these linear regressions. Then, experimental errors have only little effect on the pseudo-second linear plots. This article is dedicated to Professor Mietek Jaroniec on the occasion of his 60^th birthday.     

Equilibrium and Kinetics of NO and CO Chemisorptions on Nonstoichiometric Nickel-Copper Manganites

The equilibrium and kinetics of adsorption of NO and CO on nonstoichiometric nickel-copper manganites have been investigated through volumetric measurements. The adsorption isotherms were satisfactorily fitted to the Freundlich equation. The equilibrium coverages at 298 K were found to depend closely on the chemical composition of the oxide; thus, a decrease in the coverage beyond a maximum copper extent was observed. The adsorption isotherms of NO at various temperatures in the range from 298 to 473 K showed that the equilibrium coverage decreases with increasing temperature. This behavior enabled us to follow the logarithmic decrease of the heat of adsorption of NO on such surfaces. The adsorptions of NO and CO on surfaces preadsorbed with CO and NO, respectively, were also studied. These experiments showed the ability of NO to displace CO preadsorbed molecules whereas the contrary did not hold, suggesting the existence of common adsorption sites as well as some specific CO adsorption sites. Finally, some kinetic data are reported showing that the experimental adsorption results fit the Elovich equation (with t(0) approximately 0), although two distinct rate processes could be identified. Copyright 2000 Academic Press.     

Application of a single model to study the adsorption equilibrium of prednisolone on six carbonaceous materials

The knowledge of sorption processes of nonelectrolytes in solution by solid adsorbents implies the study of kinetics, equilibrium, and thermodynamic functions. However, quite frequently the equilibrium isotherms are studied by comparing them with those corresponding to the Giles et al. classification (1); these isotherms are also analyzed by fitting them to equations based on thermodynamic or kinetic criteria, and even to empirical equations. Nevertheless, information obtained is more coherent and satisfactory if the adsorption isotherms are fitted by using an equation describing the equilibrium isotherms according to the kinetic laws. These mentioned laws would determine each one of the unitary processes (one or more) which condition the global process. In this paper, an adsorption process of prednisolone in solution by six carbonaceous materials is explained according to a previously proposed single model, which allows to establish a kinetic law which fits satisfactorily most of C vs t isotherms (2). According to the above-mentioned kinetic law, equations describing sorption equilibrium processes have been deducted, and experimental data points have been fitted to these equations; such a fitting yields to different values of adsorption capacity and kinetic equilibrium constants for the different processes at several temperatures. However, in spite of their practical interest, these constants have no thermodynamic signification. Thus, the thermodynamic equilibrium constant (K) has been calculated by using a modified expression of the Gaines et al. equation (3). Global average values of the thermodynamic functions have also been calculated from the K values. Information related to variations of DeltaH and DeltaS with the surface coverage fraction was obtained by using the corresponding Clausius-Clapeyron equations.     

Empirical Multicomponent Equilibrium and Film-Pore Model for the Sorption of Copper,Cadmium and Zinc onto Bone Char

The adsorption of three metal ions onto bone char has been studied in both equilibrium and kinetic systems. An empirical Langmuir-type equation has been proposed to correlate the experimental equilibrium data for multicomponent systems. The sorption equilibrium of three metal ions, namely, cadmium (II) ion, zinc (II) ion and copper (II) ion in the three binary and one ternary systems is well correlated by the Langmuir-type equation. For the batch kinetic studies, a multicomponent film-pore diffusion model was developed by incorporating this empirical Langmuir-type equation into a single component film-pore diffusion model and was used to correlate the multicomponent batch kinetic data. The multicomponent film-pore diffusion model shows some deviation from the experimental data for the sorption of cadmium ions in Cd-Cu, Cd-Zn and Cd-Cu-Zn systems. However, overall this model gives a good correlation of the experimental data for three binary and one ternary systems.     

时间参量法(Ⅳ)──对峙反应的热动力学研究

根据热动力学基本理论和对峙反应的通用微分动力学方程,将时间作为已知参量,建立了对峙反应热动力学时间参量法的数学模型,并利用该法分别研究了硝基乙烷与氨和三(羟甲基)氨基甲烷反应的热动力学.     

Proton-coupled electron transfer in soybean lipoxygenase

The proton-coupled electron transfer reaction catalyzed by soybean lipoxygenase-1 is studied with a multistate continuum theory that represents the transferring hydrogen nucleus as a quantum mechanical wave function. The inner-sphere reorganization energy of the iron cofactor is calculated with density functional theory, and the outer-sphere reorganization energy of the protein is calculated with the frequency-resolved cavity model for conformations obtained with docking simulations. Both classical and quantum mechanical treatments of the proton donor-acceptor vibrational motion are presented. The temperature dependence of the calculated rates and kinetic isotope effects is in agreement with the experimental data. The weak temperature dependence of the rates is due to the relatively small free energy barrier arising from a balance between the reorganization energy and the reaction free energy. The unusually high deuterium kinetic isotope effect of 81 is due to the small overlap of the reactant and product proton vibrational wave functions and the dominance of the lowest energy reactant and product vibronic states in the tunneling process. The temperature dependence of the kinetic isotope effect is strongly influenced by the proton donor-acceptor distance with the dominant contribution to the overall rate. This dominant proton donor-acceptor distance is significantly smaller than the equilibrium donor-acceptor distance and is determined by a balance between the larger coupling and the smaller Boltzmann probability as the distance decreases. Thus, the proton donor-acceptor vibrational motion plays a vital role in decreasing the dominant donor-acceptor distance relative to its equilibrium value to facilitate the proton-coupled electron transfer reaction.     

Electrostatic interaction between a charge-regulated particle and a solid surface in electrolyte solution: effect of cationic electrolytes

The effect of cations on the electrostatic interaction between a negative charge-regulated particle and a solid surface of constant negative potential in electrolyte solution is analyzed. Here, we assume that the rate of approach of a particle to a solid surface is faster than that of the dissociation of the ionogenic groups on the surface of particle. In other words, the effect of the time-dependent dissociation of ionogenic groups on the surface of a particle is taken into account. The result of the present study reveals that, although the solid surface is negatively charged, the presence of cations in the suspension medium has a negative effect on the rate of adhesion. The qualitative behaviors in the variation of the interaction force between a particle and a solid surface as a function of separation distance between them predicted by a kinetic model and the corresponding equilibrium model and constant charge density model are entirely different. The rate of approach of a particle to a solid surface is on the order (constant charge density model)>(kinetic model)>(equilibrium model).     

纳米γ-Al_2O_3吸附Ge(Ⅳ)的机理及性能

研究了纳米γ-Al2O3吸附剂对Ge(Ⅳ)的吸附行为,考察了吸附平衡时间、温度和溶液的pH值等因素对吸附过程的影响.结果表明,纳米Al2O3对Ge(Ⅳ)的吸附在2min时基本达到平衡,在pH=4～11范围内,Ge(Ⅳ)可以被纳米Al2O3定量富集,吸附率大于95%;吸附于纳米Al2O3上的Ge(Ⅳ)可以用0.3mol/LK3PO4和1mol/LH2SO4混合溶液洗脱,5min后基本达到解析平衡,解析率能达到97%;该吸附过程符合准二级反应动力学模型,计算了不同温度下的吸附速率常数,并求得纳米Al2O3对Ge(Ⅳ)的吸附活化能(Ea)为11.63kJ/mol;该体系的吸附过程符合Freundlich等温式,由D-R等温式求得常温下纳米Al2O3对Ge(Ⅳ)的平均吸附能为10.87kJ/mol.Ge(Ⅳ)吸附反应的ΔG0为负值,焓变ΔH0为正值,说明该吸附过程是自发的吸热反应.     

胡芦巴总皂苷在HPD 400A大孔树脂上的吸附性能研究

静态条件下,对胡芦巴中总皂苷在HPD 400A树脂的吸附平衡、吸附动力学、热力学进行了深入探讨,测定了不同温度下胡芦巴中总皂苷在该树脂上的吸附等温线,吸附动力学曲线以及热力学参数。结果表明,Freundlich方程可较好地描述胡芦巴中总皂苷在该树脂上的吸附平衡;吸附动力学规律可用二级速率方程表示,液膜扩散和颗粒内扩散是吸附的主要速率控制步骤,吸附活化能为8.314kJ/mol;热力学参数表明此吸附过程为吸热过程。     

Kinetic Aspects of the Adsorption on Aluminum Oxide Drying Agents Doped with Alkali Metal Ions

Fundamental dynamic (kinetic) aspects of the process in which water vapor interacts with the surface of drying agents that are synthesized on the basis of low-temperature modification of aluminum oxide produced from a pseudoboehmite-containing hydroxide and are modified (doped) with alkali atoms (K, Na). It is shown that the kinetics of adsorption on the samples under study, formed from the fine fraction (0.5–1.0 mm) of aluminum oxide adsorbents, can be described with the Glueckauf equation, which rather well describes the dynamics of water vapor absorption in the course of time. The equation parameters were determined: adsorption rate constants and the equilibrium adsorption capacities (a*). It was found that the alkaline modification of the surface of aluminum oxide adsorbents results in that a* increases (by ~40%) as compared with the unmodified drying agent. A correlation is observed between the equilibrium adsorption capacity of the samples under study and the acid-base properties of the surface.     

Application of a single model to study the adsorption kinetics of prednisolone on six carbonaceous materials

The knowledge of the adsorption processes of nonelectrolytes from liquid solution on solid materials involves the study of their kinetic and equilibrium aspects as well as the understanding of their thermodynamic functions. However, in most published papers adsorption isotherms are analyzed by using the Giles classification and other proposed equations which are either empirical or based on kinetic or thermodynamic criteria. Our opinion is that both the kinetic and the equilibrium studies must be complementary and that, in general, equations describing the adsorption isotherms come from the kinetic laws governing the different partial processes which determine the global process. These kinetic laws may be derived from single models. In this paper a single model is proposed, which makes it possible to establish a kinetic law satisfactorily fitting a great number of C (concentration) vs t (time) isotherms. This model has been applied to study the adsorption process of prednisolone by six carbonaceous materials from ethanol solution, the specific adsorption rate, and the activation thermodynamic functions being calculated. The results obtained have also been used to analyze the influence of the intraparticle diffusion on the kinetics of the process.     

A Method for Esterification Reaction Rate Prediction of Aliphatic Monocarboxylic Acids with Primary Alcohols in 1,4‐Dioxane Based on Two Parametrical Taft Equation

Esterification reaction rates of aliphatic monocarboxylic acids with primary alcohols in 1,4‐dioxane as inert solvent were investigated. Acids were esterified with 1‐propanol and alcohols with acetic acid as model reactants at a constant temperature of 60°C, at a fixed ionic strength and pH in a batch reactor with a constant volume. For evaluation of reaction rates, an exact kinetic equation for the equilibrium reaction was applied. Under these conditions and for low reactants, concentrations reaction rate depends only on the structure of reactants and, therefore, can be predicted by a correlation equation with two Taft coefficients (inductive and steric effects). From these equations, it is possible to estimate the esterification reaction rate constant for other acid‐alcohol pairs. This methodology may also be suitable for other kinetic systems measured under comparable experimental conditions.     

Thermal Dissociation in Dynamic Conditions by Modeling Thermogravimetric Curves Using The Logarithm of Conversion Degree

The dependence of conversion degree estimated from the (TG) curve of the mass loss on heating of temperature has been analyzed. It has been shown that dynamic TG curve can be modeled by an equation relating to the logarithm of conversion degree as a function of temperature. A coefficient in the equation developed provides information on the distance from the equilibrium, therefore, the coefficient a ₂=0–50 implies equilibrium, while a ₂>50 informs about some distance from the equilibrium. Further possibilities for the use of the models of lnα vs. 1/T in the analyses of thermodynamics and kinetics of thermal dissociation of solids has been shown. This revised version was published online in July 2006 with corrections to the Cover Date.     

Equilibrium and Kinetic Modelling of Adsorption of Phosphorus on Calcined Alunite

The adsorption of phosphorus onto calcined alunite has been studied. Its equilibrium isotherm has been measured. The isotherm was determined by shaking 1.0 g calcined alunite, particle size range 90–150 m, with 100 mL phosphorus solution of initial concentrations from 0.5 to 2.5 mmol/L. The water bath shaking a constant rate of 200-rpm was used and the temperature maintained at 298 ± 2 K. A contact time of 120 min was required to achieve equilibrium. The experimental isotherm data were analyzed using the Langmuir, Freundlich, Temkin and Dubinin-Radushkevich equations. The monolayer adsorption capacity is 1.355 mmol P per g calcined alunite. Three simplified kinetic models including a pseudo first-order equation, pseudo second-order equation and intraparticle diffusion equation were selected to follow the adsorption process. Kinetic parameters, rate constants, equilibrium sorption capacities and related correlation coefficients, for each kinetic model were calculated and discussed. It was shown that the adsorption of phosphorus could be described by the pseudo second-order equation.     

Kinetics and adsorption equilibrium in the system aqueous solution of copper ions—granulated activated carbon

The feasibility of using granulated activated carbon for adsorption removal of copper from aqueous solution was studied. The influence of pH, amount of the adsorbent, contact time, and copper concentration on adsorption of copper was investigated. The single-component equilibrium data on copper adsorption were analyzed using the Langmuir, Freundlich, Redlich—Peterson, Temkin, and Toth adsorption isotherms. The adsorption process was followed by two simplified kinetic models including pseudo-first- and pseudo-second-order equations. Kinetic parameters, rate constants, equilibrium sorption capacities, and the corresponding correlation coefficients were calculated and examined for each kinetic model. It was shown that copper adsorption can be described by the pseudo-second-order equation.     

Hydride transfer in liver alcohol dehydrogenase: quantum dynamics, kinetic isotope effects, and role of enzyme motion.

The quantum dynamics of the hydride transfer reaction catalyzed by liver alcohol dehydrogenase (LADH) are studied with real-time dynamical simulations including the motion of the entire solvated enzyme. The electronic quantum effects are incorporated with an empirical valence bond potential, and the nuclear quantum effects of the transferring hydrogen are incorporated with a mixed quantum/classical molecular dynamics method in which the transferring hydrogen nucleus is represented by a three-dimensional vibrational wave function. The equilibrium transition state theory rate constants are determined from the adiabatic quantum free energy profiles, which include the free energy of the zero point motion for the transferring nucleus. The nonequilibrium dynamical effects are determined by calculating the transmission coefficients with a reactive flux scheme based on real-time molecular dynamics with quantum transitions (MDQT) surface hopping trajectories. The values of nearly unity for these transmission coefficients imply that nonequilibrium dynamical effects such as barrier recrossings are not dominant for this reaction. The calculated deuterium and tritium kinetic isotope effects for the overall rate agree with experimental results. These simulations elucidate the fundamental nature of the nuclear quantum effects and provide evidence of hydrogen tunneling in the direction along the donor-acceptor axis. An analysis of the geometrical parameters during the equilibrium and nonequilibrium simulations provides insight into the relation between specific enzyme motions and enzyme activity. The donor-acceptor distance, the catalytic zinc-substrate oxygen distance, and the coenzyme (NAD(+)/NADH) ring angles are found to strongly impact the activation free energy barrier, while the donor-acceptor distance and one of the coenzyme ring angles are found to be correlated to the degree of barrier recrossing. The distance between VAL-203 and the reactive center is found to significantly impact the activation free energy but not the degree of barrier recrossing. This result indicates that the experimentally observed effect of mutating VAL-203 on the enzyme activity is due to the alteration of the equilibrium free energy difference between the transition state and the reactant rather than nonequilibrium dynamical factors. The promoting motion of VAL-203 is characterized in terms of steric interactions involving THR-178 and the coenzyme.