Kinetic models applied to erbium adsorption on activated charcoal from aqueous solutions

The adsorption of erbium (Er) ions on activated charcoal (AC) is investigated at temperatures 10–40 °C from aqueous solutions to understand the kinetics behavior. The intra-particle diffusion, the pseudo-first order kinetic and pseudo-second order kinetic models were used to describe the kinetic data. Results shows that the adsorption of Er ions on AC occurs in two stages and the surface adsorption and diffusion phenomena are operative in the adsorption process. The result also reveals that intra-particle diffusion is not only the main rate determining step through out the adsorption process, but the boundary layer diffusion also play significant role in rate determination. Values of the intra-particle diffusion rate constant and the extent of the boundary layer diffusion were calculated. A comparison of the kinetics models on the overall adsorption rate indicates that the Er/AC system is best described by the pseudo-second order kinetic model than the pseudo-first order model, and the overall rate of the Er ions adsorption on AC appears to be controlled by more than one step, i.e., external mass transfer and diffusion mechanism.     

Molecular transport in nanopores: a theoretical perspective

Molecular transport in nanopores plays a central role in many emerging nanotechnologies for gas separation and storage, as well as in nanofluidics. Theories of the transport provide an understanding of the mechanisms that influence the transport and their interplay, and can lead to tractable models that can be used to advance these nanotechnologies through process analysis and optimisation. We review some of the most influential theories of fluid transport in small pores and confined spaces. Starting from the century old Knudsen formulation, the dusty gas model and several other related approaches that share a common point of departure in the Maxwell-Stefan diffusion equations are discussed. In particular, the conceptual basis of the models and the validity of the assumptions and simplifications necessary to obtain their final results are analysed. It is shown that the effect of adsorption is frequently either neglected, or treated on an ad hoc basis, such as through the division of the pore flux into gas-phase and surface diffusion contributions. Furthermore, while it is commonplace to assume that cross-sectional pressure is uniform, it is demonstrated that this violates the Gibbs-Duhem relation and that it is the chemical potential that essentially remains constant in the cross-section, as near-equilibrium density profiles are preserved even during transport. The Dusty Gas model and Maxwell-Stefan model for surface diffusion are analysed, and their strengths and weaknesses discussed, illustrating the use of conflicting choices of frames of reference in the former case, and the importance of assigning appropriate values for the binary diffusivity in the latter case. The oscillator model, developed in this laboratory, which is exact in the low density limit under diffuse reflection conditions, is shown to represent an advance on the classical Knudsen formula, although the latter frequently appears as a fundamental part of many transport models. The distributed friction model, also developed in this laboratory for the study of multi-component transport at any Knudsen number is discussed and compared with previous approaches. Finally, the outlook for theory and future research needs are discussed.     

Theoretical approach to non-constant uptake rates for tube-type diffusive samplers

A simple theoretical model was developed for evaluating the validity of the simplified uptake model of diffusive sampling. In the model based on the plate theory diffusion to the adsorbent surface, phase equilibrium of the adsorbate and mass transport in the adsorbent bed were considered. It was found that in the early stage of sampling, the rate of sampling is close to its theoretical value. As sampling progresses, the concentration increases and the mass transfer front gradually moves into the adsorbent layer. Above a certain threshold limit, the mass uptake becomes a steady state process in which the diffusion in the air gap and the mass transport in the adsorbent bed are balanced. As uptake is a cumulative process, sampling should continue long enough to render the effects of these initial changes negligible. That is why constant uptake rates can still be obtained above a critical exposure dose. This critical exposure dose should be exceeded both in the determination of uptake rates and outdoor measurements, to obtain consistent and reliable analytical data. Evaluation of the time and concentration dependence of uptake rate in laboratory experiments and the time dependence of uptake rate in filed test was performed to justify the model results. Since the determination of uptake rates always takes places in the laboratory, where the exposure time is much shorter and the concentration is much higher than in the environment, the uptake rates are thus overestimated by 10-30%. Therefore, the uptake rates should be determined in the field under ambient conditions by means of an independent reference method.     

Adsorption of short-chain thiols and disulfides onto gold under defined mass transport conditions: coverage, kinetics, and mechanism

The adsorption of water-soluble alkane thiols and their corresponding disulfides onto gold was followed in real time using highly sensitive surface conductivity measurements. Particular attention was paid to producing clean surfaces and to the purity of the adsorbates. The rate of mass transport to the surface was constant, controlled, and measured, over the whole time course of the experiment (1-10(4) s), by convective diffusion. An adsorption rate equation derived for coupled steady state convective-diffusion mass transport and Langmuir kinetics shows that systems limited by mass transport must also be slowed by Langmuir kinetics. Thiols and disulfides adsorbed at the same rate, limited mainly by mass transport. The distinct slowdown in adsorption rate for longer alkanethiols, attributed to conformational transitions (lying down → standing up), was less evident for the neutral thiols/disulfides. The slower rate of charged thiol adsorption is thought to stem from steric interactions of large, hydrated tail groups, although calcium as a counterion accelerated monolayer formation. The adsorption kinetics of a charged thiol were almost the same under screened (by extra added salt) or unscreened conditions. Therefore, long-range electrostatic interactions appear to be less important than short-range steric ones in limiting adsorption rates at surfaces.     

Modeling adsorption of colloids and proteins

Recent developments in the modeling of particle and protein adsorption kinetics on solid surfaces are discussed. Emphasis is focused on the coarse-grained methods, where protein molecules are treated as particles having a regular shape (spheres, spheroids) or a system of spherical beads of various sizes. Using such approaches hydrodynamic radii and diffusion coefficients of protein molecules are calculated in an exact way using the linear Stokes equation. Additionally, the surface blocking functions and jamming coverages for such molecule shapes are determined using the random sequential adsorption simulations. Theoretical results obtained in this way for various molecule shapes, including the bead models of fibrinogen are discussed. Knowing the jamming coverage and blocking functions one can formulate boundary conditions for bulk transport equations. Solutions of these equations for the convection and diffusion-controlled transport are presented. These theoretical predictions proved adequate for interpreting experimental data obtained for fibrinogen using AFM, ellipsometry and fluorescence methods. It is, therefore, concluded that these coarse grained approaches combined with solutions of the continuity equation can be efficiently used for quantitatively predicting protein adsorption kinetics for the time scale met under practical situations.     

The Effect of Diffusion Limitations on the Response of Amperometric Biosensors with Substrate Cyclic Conversion

A mathematical model of amperometric biosensors in which chemical amplification by cyclic substrate conversion takes place in a single enzyme membrane has been developed. The model involves three regions: the enzyme layer where enzyme reaction as well as mass transport by diffusion takes place, a diffusion limiting region where only the diffusion takes place, and a convective region where the analyte concentration is maintained constant. Using computer simulation the influence of the thicknesses of the enzyme layer and the diffusion region on the biosensor response was investigated. This paper deals with conditions when the mass transport in the exterior region may be neglected to simulate the biosensor response in a well-stirred solution. The digital simulation was carried out using the finite difference technique.     

Adsorption Calculations Using the Film Model Approximation for Intraparticle Mass Transfer

An approximate rate equation based on a film-model representation of diffusional mass transfer has been developed to describe the kinetics of multicomponent adsorption. The model describes mass transfer as a pseudo-steady state diffusion process through a flat film of thickness equal to one fifth of the particle radius. The flux relationships are integrated across the film yielding analytical expressions for the rate of mass transfer in a multicomponent adsorption system. The usefulness of the film model approximation is tested by carrying out calculations for three different practical adsorption systems: the adsorption of n-pentane and n-heptane mixtures on NaCaA zeolite discussed by Marutovsky and Bülow (1987); the adsorption of air in molecular sieve RS-10 discussed by Farooq et al. (1993); and the separation of air in a kinetically-controlled nitrogen PSA process discussed by Farooq and Ruthven (1990) and Sundaram and Yang (1998). In each case, the film model approximation predicts the expected trends accounting for the coupling of diffusion fluxes in the adsorbed phase.     

Theoretical modeling of water vapor transport in cellulose-based materials

The theory of mass transport in porous media is of fundamental importance for different applications such as food, paper packaging, textiles, and wood for building materials. In this study, a theoretical water vapor transport model has been developed for cellulose-based materials, such as paper and regenerated cellulose film. Pore diffusivities were determined from the dynamic moisture breakthrough experiments comprising a stack of paper sheets and regenerated cellulose films in a configuration similar to a packed adsorption column. Other mass transfer parameters were determined from transient moisture uptake rate measurements. The model incorporates pore and surface diffusion as a lump parameter into a variable effective diffusion coefficient. The mass transport, involving both pore and surface diffusions, is evaluated independently. The theoretical water vapor transmission rates (WVTRs) obtained from the model were compared with experimentally determined WVTRs measured under steady-state conditions. The theoretical model, based on intrinsic diffusion, stipulates higher WVTR values compared to the experimental results. However, the theoretical water vapor transfer rates agree well with the experimental results when external mass transfer resistance is incorporated in the model.     

Effect of Temperature on Interfacial Adsorption of Cr(VI) on Wollastonite

An extensive study on the effect of temperature on interfacial adsorption of Cr(VI) on wollastonite has been carried out. Adsorption on the wollastonite surface increased from 69.5 to 91.7% by increasing the temperature from 30 to 50 degrees C under optimum conditions. Kinetic modeling of the process of adsorption of Cr(VI) was done and various parameters were determined. The process follows a first-order kinetic equation and the rate of uptake was found to be 2.40x10(-2) min(-1) at 30 degrees C, 2.5 pH, 0.5x10(-4) M Cr(VI) concentration, and 0.01 M NaClO(4) ionic strength. Kinetic and equilibrium modeling of the process of adsorption was undertaken and the equilibrium parameters were determined. The process of adsorption follows pore diffusion and the value of the rate constant of pore diffusion was found to be 5.00x10(-3) mg g(-1) min(-1/2) at 30 degrees C and optimum conditions. The values of the coefficient of mass transfer, beta(L), were determined at different temperatures. Thermodynamic studies of the removal process were performed. The study suggests that the process is a typical example of endothermic adsorption. Copyright 2001 Academic Press.     

Effects of restricted diffusion at ultramicroelectrodes in brain tissue: The pool model: theory and experiment for chronoamperometry

The theory for chronoamperometry has been considered for the case where diffusion involves transport from media with lower diffusion coefficient and reduced volume fractions to a medium of unrestricted diffusion adjacent to the electrode. An analytical expression is developed in spherical coordinates for the special case of slow charge transfer kinetics and conditions where steady state behavior is approached, that is, for the experimental case of interest here. It is shown that the form of the resulting expression is similar to that obtained for slow charge transfer in homogeneous media. However, the expression obtained in restricted media is dependent on the radius of the media adjacent to the electrode and the volume fraction of the outer media. The chronoamperometric response of disk-shaped voltammetric electrodes in rat brain tissue is found to agree with the model predicted by these expressions. Evaluation of the data with the model enables an estimate of the size of the region of free diffusion next to the electrode to be made.     

Modeling adsorption rate of organic micropollutants present in landfill leachates onto granular activated carbon

The overall adsorption rate of single micropollutants present in landfill leachates such as phthalic acid (PA), bisphenol A (BPA), diphenolic acid (DPA), 2,4-dichlorophenoxy-acetic acid (2,4-D), and 4-chloro-2-methylphenoxyacetic acid (MCPA) on two commercial activated carbons was studied. The experimental data obtained were interpreted by using a diffusional model (PVSDM) that considers external mass transport, intraparticle diffusion, and adsorption on an active site. Furthermore, the concentration decay data were interpreted by using kinetics models. Results revealed that PVSDM model satisfactorily fitted the experimental data of adsorption rate on activated carbon. The tortuosity factor of the activated carbons used ranged from 2 to 4. The contribution of pore volume diffusion represented more than 92% of intraparticle diffusion confirming that pore volume diffusion is the controlling mechanism of the overall rate of adsorption and surface diffusion can be neglected. The experimental data were satisfactorily fitted the kinetic models. The second-order kinetic model was better fitted the experimental adsorption data than the first-order model.     

Modeling of adsorption dynamics at air-liquid interfaces using statistical rate theory (SRT)

A large number of natural and technological processes involve mass transfer at interfaces. Interfacial properties, e.g., adsorption, play a key role in such applications as wetting, foaming, coating, and stabilizing of liquid films. The mechanistic understanding of surface adsorption often assumes molecular diffusion in the bulk liquid and subsequent adsorption at the interface. Diffusion is well described by Fick's law, while adsorption kinetics is less understood and is commonly described using Langmuir-type empirical equations. In this study, a general theoretical model for adsorption kinetics/dynamics at the air-liquid interface is developed; in particular, a new kinetic equation based on the statistical rate theory (SRT) is derived. Similar to many reported kinetic equations, the new kinetic equation also involves a number of parameters, but all these parameters are theoretically obtainable. In the present model, the adsorption dynamics is governed by three dimensionless numbers: psi (ratio of adsorption thickness to diffusion length), lambda (ratio of square of the adsorption thickness to the ratio of adsorption to desorption rate constant), and Nk (ratio of the adsorption rate constant to the product of diffusion coefficient and bulk concentration). Numerical simulations for surface adsorption using the proposed model are carried out and verified. The difference in surface adsorption between the general and the diffusion controlled model is estimated and presented graphically as contours of deviation. Three different regions of adsorption dynamics are identified: diffusion controlled (deviation less than 10%), mixed diffusion and transfer controlled (deviation in the range of 10-90%), and transfer controlled (deviation more than 90%). These three different modes predominantly depend on the value of Nk. The corresponding ranges of Nk for the studied values of psi (10(-2)相似文献   

Heterogeneous catalysis on solids of gases diffusing through a liquid layer,studied by inverse gas chromatography

Physicochemical parameters for heterogeneous catalytic reactions when the catalytic bed was under a liquid phase have been determined, using a non-linear adsorption isotherm by the reversed-flow version of inverse gas chromatography (RF-GC). The mathematical analysis developed in heterogeneous catalysis, mass transfer across gas-liquid boundaries, and diffusion coefficients of gases in liquids was associated with a non-linear adsorption isotherm to find the relevant equations pertaining to the problem. These equations were then used to calculate the adsorption/desorption rate constant, the rate constant for the first-order catalytic reaction and the equilibrium constant for the non-linear adsorption isotherm. The diffusion coefficients of the reactant in the liquid and gaseous phases and the partition coefficients for the distribution of the reactant between the gaseous and liquid phase were also determined.     

Parallel pore and surface diffusion of levulinic acid in basic polymeric adsorbents

The equilibrium and kinetics of levulinic acid (LA) adsorption on two basic polymeric adsorbents, 335 (highly porous gel) and D315 (macroreticular), were investigated. Experimental adsorption rates in batch stirred vessels under a variety of operating conditions were described successfully by the parallel pore and surface diffusion model taking into account external mass transfer and nonlinear Toth isotherm. The film-pore diffusion model was matched with the rate data and the resulting apparent pore diffusivities were strongly concentration-dependent and approached to a constant value for 335 adsorbent. Thus, the constant value was taken as the accurate pore diffusivity, while the pore diffusivity in D315 was estimated from the particle porosity. The surface diffusivities decreased with increasing initial bulk concentration for both adsorbents. The inverse concentration dependence was correlated reasonably well to the change of isosteric heat of adsorption as amount adsorbed.     

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.     

Modeling of preparative chromatography processes with slow intraparticle mass transport kinetics

Mathematical modeling of the preparative chromatography process accompanied with complex intraparticle mass transport mechanism involving surface diffusion is discussed. As an experimental base for the analysis two steroid compounds, methyl esters of hydroxycholanic acids (bile acids), deoxycholic and cholic acids were selected. For these compounds surface diffusion kinetics were found to have a marked influence on the band broadening. The isocratic chromatography process was performed in a normal-phase preparative system with ternary mixture of solvents containing hexane, ethyl acetate and methanol as a modifier under different operating conditions, e.g., at various mobile phase compositions and inlet concentrations. The efficiency of the system was found to be dependent on the mass of sample injected as well as on the contents of the modifier. Such a phenomenon was suggested to originate from the contribution of the surface diffusion kinetics to the overall mass transport mechanism. For identifying the general trends and concentration dependencies of the surface diffusion coefficient the simplified approach was proposed. The set of chromatographic band profiles registered at different inlet concentration and mobile phase composition were used for determining the influence of the local solid-phase concentration on the mass transport mechanism. For the simulations the transport-dispersive model was used, in which all sources of mass transport resistances were lumped in the properly adjusted mass transport coefficient. The accuracy of this model was verified by comparing its predictions to the solutions of the general rate model.     

Analysis of diffusion models for protein adsorption to porous anion-exchange adsorbent

The ion-exchange adsorption kinetics of bovine serum albumin (BSA) and gamma-globulin to an anion exchanger, DEAE Spherodex M, has been studied by batch adsorption experiments. Various diffusion models, that is, pore diffusion, surface diffusion, homogeneous diffusion and parallel diffusion models, are analyzed for their suitabilities to depict the adsorption kinetics. Protein diffusivities are estimated by matching the models with the experimental data. The dependence of the diffusivities on initial protein concentration is observed and discussed. The adsorption isotherm of BSA is nearly rectangular, so there is little surface diffusion. As a result, the surface and homogeneous diffusion models do not fit to the kinetic data of BSA adsorption. The adsorption isotherm of gamma-globulin is less favorable, and the surface diffusion contributes greatly to the mass transport. Consequently, both the surface and homogeneous diffusion models fit to the kinetic data of gamma-globulin well. The adsorption kinetics of BSA and gamma-globulin can be very well fitted by parallel diffusion model, because the model reflects correctly the intraparticle mass transfer mechanism. In addition, for both the favorably bound proteins, the pore diffusion model fits the adsorption kinetics reasonably well. The results here indicate that the pore diffusion model can be used as a good approximate to depict protein adsorption kinetics for protein adsorption systems from rectangular to linear isotherms.