Langmuir Kinetic Equation Research Articles

Role of the Number of Adsorption Sites and Adsorption Dynamics of Diffusing Particles in a Confined Liquid with Langmuir Kinetics

In this work, we investigate the effect of the number of available adsorption sites for diffusing particles in a liquid confined between walls where the adsorption (desorption) phenomena occur. We formulate and numerically solve a model for particles governed by Fickian’s law of diffusion, where the dynamics at the surfaces obey the Langmuir kinetic equation. The ratio between the available number of adsorption sites and the number of total particles are used as a control parameter. The investigation is carried out in terms of characteristic times of the system for different initial configurations, as well as the cases of identical or non-identical surfaces. We calculate the bulk and surface densities dynamics, as well as the variance of the system, and demonstrate that the number of sites affects the bulk, surface distributions, and diffusive regimes.

Self-Assembled Monolayer of Monomercaptoundecahydro-closo-dodecaborate on a Polycrystalline Gold Surface.

In this work, we present an electrochemical study of the boron cage monomercaptoundecahydro-closo-dodecaborate [B12H11SH]2− in solution and in a self-assembled monolayer over a polycrystalline gold electrode. Cyclic voltammetry of the anion [B12H11SH]2− in solution showed a shift in the peak potentials related to the redox processes of gold hydroxides, which evidences the interaction between the boron cage and the gold surface. For an Au electrode modified with the anion [B12H11SH]2−, cyclic voltammetry response of the probe Fe(CN)63−/Fe(CN)64− showed a ΔEp value typical for a surface modification. Electrochemical impedance spectroscopy presented Rtc and Cdl values related to the formation of a self-assembled monolayer (SAM). A comparison of electrochemical responses of a modified electrode with thioglycolic acid (TGA) reveals that the boron cage [B12H11SH]2− diminishes the actives sites over the Au surface due to the steric effects. Differential capacitance measurements for bare gold electrode and those modified with [B12H11SH]2− and (TGA), indicate that bulky thiols enhance charge accumulation at the electrode–solution interface. In addition to electrochemical experiments, DFT calculations and surface plasmon resonance measurements (SPR) were carried out to obtain quantum chemical descriptors and to evaluate the molecular length and the dielectric constant of the Boron cage. From SPR experiments, the adsorption kinetics of [B12H11SH]2− were studied. The data fit for a Langmuir kinetic equation, typical for the formation of a monolayer.

Removal of fluoride from coke wastewater by aluminum doped chelating ion-exchange resins: a tertiary treatment

Coke wastewater is one of the most problematic industrial wastewaters, due to its large volume and complex pollutant load. In this study, ion exchange technology was investigated with the objective of reducing the fluoride content of the effluent from a coke wastewater treatment plant (26.7 mg F-/L). Two Al-doped exchange resins with chelating aminomethyl-phosphonic acid and iminodiacetic groups were assessed: Al-doped TP260 and TP207 resins, respectively. The effect of resin dosage, varying from 5 to 25 g/L, was evaluated. F- removal was within the range 57.8–89.3% and 72.0–92.1% for Al-doped TP260 and TP207, respectively. A kinetic study based on a generalized integrated Langmuir kinetic equation fitted the experimental data (R2 > 0.98). The parameters of the said kinetics met the optimal conditions for the ion exchange process, which seemed to be more favorable with Al-doped TP260 resin than with Al-doped TP207 resin, using the same resin dosage. Furthermore, the experimental data were well described (R2 > 0.98) by Langmuir and Freundlich isotherm models, in agreement with the findings of the kinetic study: the maximum sorption capacity was obtained for the Al-doped TP260 resin.

Langmuir Adsorption Kinetics in Liquid Media: Interface Reaction Model.

Adsorption kinetic equation has been derived assuming that the process follows the behavior of a heterogeneous chemical reaction at the solid–liquid interface. This equation is converted into the Langmuir isotherm at equilibrium and describes well the unsteady-state adsorption process. Based on that, a working equation has been developed, which gives adsorption-rate-constant independent of operating parameters including concentration. Also, a kinetic model expressed as a sum of first- and second-order systems available in the literature has been applied (modified with the interface reaction concept) to determine the adsorption rate constant. Both methods gave similar results. Three dimensionless numbers have been developed to determine and distinguish pseudo-first-order and pseudo-second-order kinetics justified from the viewpoint of chemical kinetics. It is shown that curve-fitting with a high correlation coefficient could validate an empirical kinetic model, but the fitted model parameters could not automatically be related to chemical kinetic parameters if the model itself is not grounded on well-defined chemical kinetics. Finally, it is concluded that the currently applied empirical approach could not provide reliable data for comparison among similar systems, while the Langmuir kinetic equation developed based on the concept of heterogeneous reaction would be a good basis for standardization of the method for adsorption system characterization.

A fractal Langmuir kinetic equation and its solution structure

The Langmuir kinetic equation is analyzed by the variational iteration method, its solution property is revealed analytically. The effects of desorption time and adsorption coefficient on the solution properties are also discussed, and a fractal modification of Langmuir kinetic equation is suggested.

A Pseudo-Second Order Kinetic Equation for Sorption Processes

The problem of substantiating a pseudo-second order equation in the kinetics of sorption processes is considered. A simple way is presented of transforming the Langmuir kinetic equation written for a process within a limited volume into a polynomial relation in the form of the sum of difference terms of first- and second-order kinetic equations. It is shown that such a relation is reduced to a good approximation of a pseudo-second order equation over a much wider range of conditions determined by the equilibrium and kinetic characteristics of the sorbent and the experimental parameters than was predicted earlier. It is established that the errors of such a theoretical approximation are either negligible or do not exceed the corresponding errors at the initial stage of kinetic experiments within a confined space. It is concluded that the applicability of the pseudo-second order model is independent of the mechanisms determining the rate of sorption and does not require concepts of chemisorption or special equations of kinetics controlled via chemical reactions or diffusion.

Generalized Langmuir kinetic equation for ions adsorption model applied to electrical double layer capacitor

The electric response of an electrical double layer capacitor is analyzed by means of the impedance spectroscopy technique. From the spectrum of the imaginary part of the electric impedance of the cell, in the low frequency region, it follows that the electrodes are blocking. To interpret the experimental data we assume that the electrodes are adsorbing. The kinetic equation describing the adsorption phenomenon used in our analysis is a generalization of Langmuir isotherm, where the effective adsorption coefficient and desorption time are frequency dependent. Their frequency dependence is deduced considering the adsorption as a linear phenomenon relating the bulk density of particles to the surface density of particles, characterized by a distribution of relaxation times. The agreement between the experimental data and the theoretical predictions is very good within explored frequency range.

Frequency dependence of the phenomenological parameters describing adsorption processes in supercapacitors

It is shown that the frequency dependence of the phenomenological parameters entering Langmuir kinetic equation describing the surface adsorption could be related to a random distribution of relaxation times. A theoretical model based on a generalization of Langmuir isotherm, taking into account a homogeneous distribution of relaxation times is proposed for supercapacitor modeling. Its theoretical predictions agree well with experimental data derived by the analysis of Lissajous figures describing the response of an electrochemical cell excited with a well defined frequency. According to our data, the effective desorption time is a rapidly decreasing function of the frequency of the applied signal, whereas the effective adsorption coefficient is an increasing function of it. The effective relaxation time is independent of the ionic concentration, whereas the effective adsorption coefficient strongly depends on this quantity. A large increase in the adsorption coefficient is reported, for the electrochemical cells with high surface area of the electrodes.

Predicting equilibrium time by adsorption kinetic equations and modifying Langmuir isotherm by fractal-like approach

In this work, the pseudo-first-order (PFO), pseudo-second-order (PSO) and Vermeulen equations were modified by introducing an adjustable parameter i.e. the fractional surface coverage θ. The Langmuir kinetic equation was modified by the fractal-like approach and the separation factor RH was proposed to express essential characteristics of the fractal-like Langmuir isotherm. The kinetic and isotherm data obtained from nitrate adsorption on the PAN/AC composite were used to evaluate the validity of these models. Results indicated that the modified PSO equation could accurately predict the equilibrium time at different initial nitrate concentrations (R2 > 0.998) and that the fractal-like Langmuir isotherm could better describe the equilibrium data at different temperatures (R2 > 0.994). The magnitude of parameter b reflected the affinity of the adsorbent for the adsorbates. The innovation and significance of the present study was that the modified kinetic and isotherm models could predict the equilibrium time and describe the heterogeneous surfaces, respectively. Therefore, this work is expected to extend the application scope of the PFO, PSO and Vermeulen equations and Langmuir isotherm.

Kinetic analysis on precursors for iturin A production from Bacillus amyloliquefaciens BPD1

In this study, the precursor effect for iturin A production was quantitatively analyzed. A strain identified as Bacillus amyloliquefaciens BPD1 (Ba-BPD1) was selected due to its ability to produce iturin A. The enhancement of iturin A production in a submerged culture was tested using various additives, including palmitic acid, oils, and complex amino acids. Among these, complex amino acids triggered the highest yield at 559mg/L. The respective amino acids that contribute to the structure of iturin A were used as precursors. In fact, it was found that the addition of l-proline, l-glutamine, l-asparagine and l-serine could improve iturin A yield in the defined medium. However, during the kinetic analysis, all the amino acids exhibited a lower saturation level than l-serine, which exhibited a high saturation level at 1.2% resulting in an iturin A yield of 914mg/L. In contrast, a negative effect was observed following the addition of l-tyrosine. To analyze the kinetic behavior of l-serine, three kinetic models were adopted: the kinetic order equation, the Langmuir kinetic equation, and a modified logistic equation. The regression results showed that the modified logistic model was the best fit for the kinetic behavior of l-serine as the major precursor, which could be further referred to the biosynthesis pathway of iturin A. Among the proposed processes for iturin A production, this study achieved the highest iturin A levels as a result of the addition of precursors.

Kinetic modeling of CO2 adsorption on an amine-functionalized solid sorbent

A kinetic model of CO2 adsorption on an amine-functionalized silica sorbent, 0.37EB-PEI, was proposed by using experimental data from a miniature isothermal fixed bed reactor system called Autochem. Breakthrough curve tests were conducted at different CO2 concentrations and different temperatures to acquire experimental data under adsorption-dominant conditions. Additionally, temperature programmed desorption (TPD) experiments were carried out for different CO2 concentrations with different starting temperatures to investigate the desorption kinetics. A modified Langmuir kinetic equation was found to provide a satisfactory representation of the adsorption-dominant experimental data, including the long tails. A separate kinetic model was proposed to express the desorption-dominant TPD data. The dynamic effects of the empty volume in the Autochem device on the experimental measurements were rigorously handled through separate empty-volume dynamic tests.

Kinetic models for adsorption on mineral particles comparison between Langmuir kinetics and mass transfer

Abstract This paper presents a parametric study of kinetic adsorption model, for treating industrial wastewater contaminated with heavy metals by mineral particles, using Langmuir kinetics. It also compares the model with a mass transfer model, as a mathematical limiting case. Theoretical adsorption curves are presented in a simple form of fractional removal of pollutant as a function of time. They are obtained from the mathematical solution of ordinary differential equations, derived from differential mass balance and Langmuir kinetic rate equation. The effects of dimensionless process parameters on the fraction removal are presented. These include: equilibrium constant, particle dosage, ratio of the initial mass of pollutant to the maximum adsorbed mass, and Langmuir rate coefficient. The model provides a good fit to experimental results for adsorption of Cr(III) on both marl stone and stone cutting particles. A simplified fitting approach is presented.

Further examination of the relationship between the Langmuir kinetics and the Lagergren and the second-order rate models of batch adsorption

This study presents a further examination of the relationship between the Langmuir kinetics and the Lagergren (first-order) and the second-order rate models of batch adsorption. The earlier work of Liu and Shen on the transformation of the Langmuir kinetic equation to a linear combination of the Lagergren rate expression and the second-order rate expression was given a rational treatment and solutions of the transformed equation were obtained. Based on the solutions, correct relationships between the Langmuir equation and the first-order and second-order equations were established and the significance of the relationships discussed.

A generalized Langmuirian approach in adsorption kinetic modeling

Liquid-phase adsorption kinetic models used in the literature include a limited number of equations, the most frequently used being those of the pseudo first-and pseudo-second order. The applicability of these two equations is discussed by researchers, which can be deduced from the Langmuir kinetic equation under specific experimental conditions. This conclusion is confirmed in this work for the pseudo-first-order model. The assumptions under which the equation is valid, i.e. single-site adsorption, are determined, whereas the pseudo-second-order-kinetic equation is deduced from the Langmuir dual-site adsorption model and the respective equation. A calculation procedure based on those models is proposed for the kinetic modeling of adsorption. The main characteristic is that the modeling procedure makes use of the Langmuir theory of adsorption and the corresponding equilibrium and kinetic equations. The only restriction is that the adsorption mechanism of the adsorbent–adsorbate system under consideration must follow the Langmuir theoretical model.

Adsorption and desorption kinetics of benzene derivatives on mesoporous carbons

Adsorption and desorption of benzoic and salicylic acids and phenol from a series of synthesized mesoporous carbons is measured and analyzed. Equilibrium adsorption isotherms are best described by the Langmuir–Freundlich isotherm. Intraparticle diffusion and McKay’s pore diffusion models, as well as mixed 1,2-order (MOE), integrated Langmuir kinetic equation (IKL), Langmuir–Freundlich kinetic equation and recently derived fractal-like MOE (f-MOE) and IKL models were compared and used to analyze adsorption kinetic data. New generalization of Langmuir kinetics (gIKL), MOE and f-MOE were used to describe desorption kinetics. Analysis of adsorption and desorption half-times shows simple relation to the size of carbon pores.

A fundamental approach in liquid phase adsorption kinetics

Liquid–solid adsorption kinetic data are mostly interpreted by a limited number of rate equations, (pseudo-first, pseudo-second order and Elovich). Besides, a fundamental tool in heterogeneous reactions, the Langmuir theory of adsorption, is used almost exclusively to model equilibrium isotherms but, surprisingly, not kinetics. Considering the limited applicability of the mentioned kinetic models, especially in estimating kinetic constants and predicting adsorption mechanism, it seems more reasonable to apply the Langmuir kinetic equation in modeling the liquid phase adsorption. In the diffusion-free regime, by constructing the experimental equilibrium isotherm of a system, one can easily check the validity of the Langmuir model. Afterwards, kinetic constants can be readily calculated and used to predict adsorption performance under different operating conditions. The proposed procedure was tested in the experimental results of cyanide adsorption in activated carbon to verify the applicability of the Langmuir adsorption mechanism and estimate the equilibrium and kinetic constants.

Electrode potential and selective ionic adsorption

A simple description of the electrode potential based on the selective ionic adsorption is proposed. It is shown that if the adsorption–desorption coefficients entering in the Langmuir kinetic equation for the adsorption at the limiting surfaces are not identical, a difference of potential between the electrode and the bulk of the solution exists. In the case where the thickness of the sample is large with respect to the length of Debye, this difference of potential depends only on the adsorption–desorption coefficients and on the length of Debye of the ionic solution.

Influence of surface acidic functional groups on activated carbons for adsorption and desorption kinetics of phenol from aqueous solutions

Adsorption and desorption kinetics of phenol onto activated carbons with and without surface acidic oxygen functional groups were examined in aqueous solutions. Aniline was also used to compare with phenol. The surface oxygen groups was introduced by nitric acid oxidation after de-ashing, and completely removed by out-gassing, respectively. Langmuir isotherms could be successfully employed for both activated carbons. Application of Langmuir kinetic equation was limited to the oxidized carbons, whereas it was not fitted very well for the out-gassed carbons. For desorption experiments from the pre-adsorbed out-gassed carbons, first rapid desorption and consecutive slow re-adsorption was clearly observed for phenol and aniline, while only rapid desorption was measured from the oxidized carbons. Based on the results, for the oxidized carbons the adsorption was progressed by collision control kinetics between phenol and adsorption sites on the external carbon surface and meso- and macro-pore on the one hand, not only collision control but also diffusion control kinetics was included for the out-gassed carbons on the other hand. From the results of the temperature dependence of adsorption capacity and the aniline adsorption kinetics, coupling reaction accompanied by dehydration would be taken place on the out-gassed carbons simultaneously with the adsorption process.

Surface heterogeneity and surface area from linear inverse gas chromatography: Application to glass fibers

A new method of measuring surface heterogeneity of non-porous adsorbents by inverse gas chromatography (IGC) is developed. In contrast with the methods described in the literature which are based on nonlinear chromatography (thermodynamics of adsorption) this method is based on linear chromatography (kinetics of adsorption). The mass balance equation of linear chromatography with axial diffusion term and the Langmuir kinetic equation on an open (non-porous) heterogeneous surface are solved by the method of Laplace transforms. The expressions for four semi-invariants of an elution profile are obtained. These are linear combinations of the moments of the distribution of sites in residence time of adsorbed molecule with coefficients determined from elution profile of non-adsorbing gas. Four semi-invariants of elution profiles of benzene and methane on a column packed with E-glass fiber were determined experimentally at temperatures around 100 °C. The mean, the standard deviation of the normal distribution in adsorption energy as well as the sticking coefficient and pre-exponential factor of the Frenkel equation are determined for benzene on E-glass. A method of the specific surface determination from linear IGC is proposed.

Derivation of a general kinetic equation for transfer-controlled adsorption at liquid interfaces using Statistical Rate Theory

Recently, Joos proposed a new kinetic model for transfer-controlled surfactant adsorption at liquid interfaces. His model is general, in the sense that the change in interfacial tension is incorporated into adsorption rate descriptions, and its governing kinetic equation can be reduced to various specific expressions for individual situations, such as the Langmuir kinetic equation. Joos developed his kinetic model by simply drawing analogies between adsorption and electrode kinetics, where the rate expression was obtained by using conventional chemical reaction theory. In this paper, we present a quantum statistical formulation for surfactant adsorption kinetics. With the assumption of excited states for molecular adsorption, we derive a rate equation by using Statistical Rate Theory (SRT), which gives the expression for transition probabilities from a first-order perturbation analysis of the Schrödinger equation. This new derivation avoids using the principle of mass action and the law of Arrhenius–Eyring, and hence many empirical assumptions in conventional chemical reaction theory are not needed for describing adsorption kinetics. The resulting rate expression can be reduced to Joos's kinetic equation and hence serves as a confirmation of it. Simplifications of the general rate expression are also presented for specific interfacial situations of near equilibrium and far from equilibrium. A combination of these rate expressions with a surface tension–adsorption relation, e.g. the Frumkin equation, can result in equations that are experimentally accessible. Future work will be concentrated on experimental investigations that may provide further validation of the theory and estimates for the parameters used in the rate expressions, especially the symmetry factor α.