Langmuir-type Model Research Articles

The inhibition of crystal growth of mirabilite in aqueous solutions in the presence of phosphonates

The formation of sodium sulfate decahydrate (Mirabilite) has been known to cause serious damages to structural materials both of modern and of historical buildings. Methods which can retard or completely suppress the development of mirabilte crystals are urgently needed especially as remedies or preventive measures for the preservation of the built cultural heritage. In the present work we present results on the effect of the presence of phosphonate compounds on the kinetics of crystal growth from aqueous supersaturated solutions at 18°C using the seeded growth technique. The phosphonate compounds tested differed with respect to the number of ionizable phosphonate groups and with respect to the number of amino groups in the respective molecules. The crystal growth process was monitored by the temperature changes during the exothermic crystallization of mirabilite in the stirred supersaturated solutions. The crystal growth of mirabilite in the presence of: (1-hydroxyethylidene)-1, 1-diphosphonic acid (HEDP), amino tri (methylene phosphonic acid) (ATMP), hexamethylenediaminetetra (methylene)phosphonic acid (HTDMP), and diethylene triamine penta(methylene phosphonic acid)(DETPMP) over a range of concentrations between 0.1–5% w/w resulted in significant decrease of the rates of mirabilite crystal growth. All phosphonic compounds tested reduced the crystallization rates up to 60% in comparison with additive-free solutions. The presence of the test compounds did not cause changes of the mechanism of crystal growth which was surface diffusion controlled, as shown by the second order dependence of the rates of mirabilite crystal growth on the relative supersaturation. The excellent fit of the measured rates to a kinetic Langmuir-type model suggested that the activity of the tested inhibitors could be attributed to the adsorption and subsequent reduction of the active crystal growth sites of the seed crystals. In all cases, the inhibitory activity was reduced with increasing solution supersaturation, while the presence of DETPMP, which showed the best inhibition activity, showed the least reduction of inhibition with increasing supersaturation.

High-Pressure/High-Temperature Methane-Sorption Measurements on Carbonaceous Shales by the Manometric Method: Experimental and Data-Evaluation Considerations for Improved Accuracy

Summary In exploration for shale gas, experimental methane-sorption measurements represent a valuable source of information for resource estimates and for reservoir-modeling studies. Here, the main difficulty is the relatively low adsorption capacity of shales (typically 10% of the sorption capacity of coals), as well as the fact that the measurements need to be performed over a wide range of pressures and temperatures characteristic of past or present geological conditions. In this work, we demonstrate the capabilities of an adapted manometric apparatus to reliably measure excess sorption isotherms at pressures of up to 30 MPa and temperatures up to 423 K on carbonaceous shales. This is accomplished with an experimental design comprising separate heating zones for the sample cell and for the rest of the apparatus. An experimental and mass-balance approach is presented to quantify the temperature gradient existing between the two heating zones, as well as the thermal expansion of the sample cell, and to account for these in the calculation of the excess sorption. We demonstrate that the analysis of the helium-void-volume data over a large temperature range can be interpreted with respect to the thermal expansion of the sample and, in some cases, changes in pore-volume accessibility to helium. We propose to perform blank-expansion tests with non-adsorbing specimens (e.g., steel cylinders) as a quality check to eliminate device-specific artifacts resulting from unknown measurement uncertainties or from uncertainty in the equation of state. Two evaluation procedures are presented to quantitatively account for the blank tests in the final result of sorption measurements on shale samples. As an example, methane-sorption isotherms for carbonaceous shale at 311, 338, 373, and 423 K are presented. By use of a Monte Carlo algorithm to simulate the propagation of the experimental uncertainties, the final estimated uncertainty in excess sorption resulting from systematic errors was found to be ± 0.007 mmol/g at 25 MPa. The consideration of the blank-expansion tests in the mass balance further reduces the systematic error, at least to a point at which an excellent intralaboratory consistency is obtained. The estimated uncertainty resulting from random errors was found to significantly overestimate the actual precision of the experimental setup, and an explanation is provided with respect to experimental design. A data-reduction approach using an excess-sorption function based on a Langmuir-type absolute-sorption model was found to provide an excellent representation of the measured sorption data. By means of simplified model calculations we demonstrate that the excess-sorption formalism is a sufficient, simple, and adequate approach to applications in shale-gas-resource estimation. The uncertainties pertaining to representativeness of experimental sorption data of in-situ reservoir conditions are briefly discussed.

Salt-free dyeing of ramie fabric with an amino-terminated hyperbranched polymer

A new amino-terminated hyperbranched polymer (at-HBP) was synthesized, and its salt-free dyeing property on ramie fabric was studied. The structure and molecular weight of at-HBP were established by Fourier transform infrared spectrometer, H nuclear magnetic resonance and gel permeation chromatography. The untreated ramie fabric and modified ramie fabrics were characterized by an X-ray diffraction (XRD) and field emission scanning electron micrograph (FE-SEM). XRD results showed a transformation of the crystalline structure from ramie cellulose I to cellulose II allomorph during mercerization and epichlorohydrin modification, and the crystalline structure of cellulose II was maintained with an obvious crystallinity index increase after at-HBP modification. FE-SEM results confirmed that at-HBP was successfully grafted onto the fabric surface. Dyed with reactive dye C.I. reactive Blue 4, the color strength of the at-HBP-modified fabric was enhanced, even when dyeing was carried out without the electrolyte. The washing and rubbing fastness of the salt-free dyeing of fabrics was also good compared with those obtained by conventional dyeing. The adsorption isotherm of C.I. reactive Blue 4 on modified fabric was examined and found to follow a Langmuir-type adsorption model. The at-HBP modification mechanism of ramie fabric and dyeing mechanism with reactive dye were suggested.

Discrimination of fresh fruit juices by a fluorescent sensor array for carboxylic acids based on molecularly imprinted titania

Design of chemical sensor arrays that can discriminate real-world samples has been highly attractive in recent years. Herein a fluorescent indicator-displacement sensor array for discrimination of fresh fruit juices was developed. By coupling the unique high affinity of titania to electron-donating anions and the cross-reactivity of molecularly imprinted materials to structurally similar species, a small array was fabricated using only one rhodamine-based fluorescent dye and three synthesized materials. Citric, malic, succinic and tartaric acids were chosen as indices. The recognition mechanism was investigated by spectrofluorimetric titration using a non-linear Langmuir-type adsorption model. The proposed method was applied to discriminate thirteen fruit juices through their carboxylic acid contents. Principal component analysis of the data clearly grouped the thirteen juices with the first principal component owning 98.2% of the total variation. The comparison of the sensor array with HPLC determination of the carboxylic acids was finally made.

Ultrasonic degradation of acetaminophen in water: Effect of sonochemical parameters and water matrix

This paper deals about the sonochemical water treatment of acetaminophen (ACP, N-acetyl-p-aminophenol or paracetamol), one of the most popular pharmaceutical compounds found in natural and drinking waters. Effect of ultrasonic power (20–60W), initial ACP concentration (33–1323μmolL−1) and pH (3–12) were evaluated. High ultrasonic powers and, low and natural acidic pH values favored the efficiency of the treatment. Effect of initial substrate concentration showed that the Langmuir-type kinetic model fit well the ACP sonochemical degradation. The influence of organic compounds in the water matrix, at concentrations 10-fold higher than ACP, was also evaluated. The results indicated that only organic compounds having a higher value of the Henry’s law constant than the substrate decrease the efficiency of the treatment. On the other hand, ACP degradation in mineral natural water showed to be strongly dependent of the initial substrate concentration. A positive matrix effect was observed at low ACP concentrations (1.65μmolL−1), which was attributed to the presence of bicarbonate ion in solution. However, at relative high ACP concentrations a detrimental effect of matrix components was noticed. Finally, the results indicated that ultrasonic action is able to transform ACP in aliphatic organic compounds that could be subsequently eliminated in a biological system.

Experimental and modelling studies of ion exchange equilibria between zeolite NaY and an electrolytic solution of iron

In this work, ion exchange equilibrium in NaY-packed columns was examined and equilibrium data was generated for Na+↔Fe2+ in zeolite. In this context, different mathematical models were evaluated to describe the equilibrium: the through adsorption approach (Langmuir type model, Jain & Snoeyink model and Langmuir–Freundlich model), the ion exchange approach (the ideal and non-ideal mass action law) and the thermodynamic approach (IAST—ideal adsorbed solution theory and RAST—real adsorbed solution theory). Experiments were carried out at 30°C using solutions of 1, 2 and 3meqL−1 at pH 4.5 and a column flow rate of 8mLmin−1. Fe(II) was retained through ion exchange followed by adsorption. The non-ideal mass action law best fit the experimental data as it predicted the exchange process of electrolytes and detected the adsorption multilayer.

Application of Fickian and non-Fickian diffusion models to study moisture diffusion in asphalt mastics

The objective of this study was to investigate certain aspects of asphalt mastic moisture diffusion characteristics in order to better understand the moisture damage phenomenon in asphalt mixtures. Moisture sorption experiments were conducted on four asphalt mastics using an environmental chamber capable of automatically controlling both relative humidity (85 %) and temperature (23 °C). The four mastics tested were identical in terms of bitumen type (40/60 pen), bitumen amount (25 % by of wt% total mix), mineral filler amount (25 % by wt%) and fine aggregate amount (50 % by wt%). The materials differed in terms of mineral filler type (granite or limestone) and fine aggregate type (granite or limestone). Preliminary data obtained during the early part of the study showed certain anomalous behavior of the materials including geometry (thickness)-dependent diffusion coefficient. It was therefore decided to investigate some aspects related to moisture diffusion in mastics by applying the Fickian and two non-Fickian (anomalous) diffusion models to the moisture sorption data. The two non-Fickian models included a two-phase Langmuir-type model and a two-parameter time-variable model. All three models predicted moisture diffusion in mastics extremely well (R 2 > 0.95). The observed variation of diffusion coefficient with thickness was attributed in part to microstructural changes (settlement of the denser fine aggregates near the bottom of the material) during the rather long-duration diffusion testing. This assertion was supported by X-ray computed tomography imaging of the mastic that showed significant accumulation of aggregate particles near the bottom of the sample with time. The results from the Langmuir-type model support a two-phase (free and bound) model for moisture absorbed by asphalt mastic and suggests about 80 % of absorbed water in the free phase remain bound within the mastic. The results also suggest that moisture diffusion in asphalt mastic may be time-dependent with diffusion decreasing by about four times during a typical diffusion test lasting up to 500 h. The study concludes that both geometry and time-dependent physical characteristics of mastic are important factors to consider with respect to moisture diffusion in asphalt mastics.

Kinetic study on hot-wire-assisted atomic layer deposition of nickel thin films

High-purity Ni films were deposited using hot-wire-assisted atomic layer deposition (HW-ALD) at deposition temperatures of 175, 250, and 350 °C. Negligible amount of nitrogen or carbon contamination was detected, even though the authors used NH2 radical as the reducing agent and nickelocene as the precursor. NH2 radicals were generated by the thermal decomposition of NH3 with the assist of HW and used to reduce the adsorbed metal growth precursors. To understand and improve the deposition process, the kinetics of HW-ALD were analyzed using a Langmuir-type model. Unlike remote-plasma-enhanced atomic layer deposition, HW-ALD does not lead to plasma-induced damage. This is a significant advantage, because the authors can supply sufficient NH2 radicals to deposit high-purity metallic films by adjusting the distance between the hot wire and the substrate. NH2 radicals have a short lifetime, and it was important to use a short distance between the radical generation site and substrate. Furthermore, the impurity content of the nickel films was independent of the deposition temperature, which is evidence of the temperature-independent nature of the NH2 radical flux and the reactivity of the NH2 radicals.

Phosphorus atomic layer doping in SiGe using reduced pressure chemical vapor deposition

Phosphorus (P) atomic layer doping in SiGe is investigated at temperatures between 100°C to 600°C using a single wafer reduced pressure chemical vapor deposition system. SiGe(100) surface is exposed to PH3 at different PH3 partial pressures by interrupting SiGe growth. The impact of the SiGe buffer/cap growth condition (total pressure/SiGe deposition precursors) on P adsorption, incorporation, and segregation are investigated. In the case of SiH4-GeH4-H2 gas system, steeper P spikes due to lower segregation are observed by SiGe cap deposition at atmospheric (ATM) pressure compared with reduced pressure (RP). The steepness of P spike of ~5.7nm/dec is obtained for ATM pressure without reducing deposition temperature. This result may be due to the shift of equilibrium of P adsorption/desorption to desorption direction by higher H2 pressure. Using Si2H6-GeH4-H2 gas system for SiGe cap deposition in RP, lowering the SiGe growth temperature is possible, resulting in higher P incorporation and steeper P profile due to reduced desorption and segregation. In the case of Si2H6-GeH4-H2 gas system, the P dose could be simulated assuming a Langmuir-type kinetics model. Incorporated P shows high electrical activity, indicating P is adsorbed mostly in lattice position.

Modification of nano-TiO2 by doping with nitrogen and fluorine and study acetaldehyde removal under visible light irradiation

P25-TiO2 nanoparticles were doped with fluo- rine, nitrogen, and their combination. Samples of N-doped, F-doped, and N-F-codoped TiO2 were prepared by physi- cal and chemical treatments. The products were charac- terized by X-ray diffraction, Fourier transform infrared, Brunauer-Emmett-Teller technique, and ultraviolet-visi- ble diffuse reflectance spectroscopy. It was revealed that absorption spectra of N-doped, F-doped, and F-N-codoped TiO2 were extended to the visible region wavelengths, and the photocatalytic experiments showed enhancement of acetaldehyde removal under visible light irradiation. The photocatalytic activity of the powders was evaluated through the process of acetaldehyde degradation under visible light scattering in a continuous stirred tank reactor. F-N-codoped nano-TiO2 calcinated at 500 C possessed the highest photocatalytic activity. The photocatalytic kinetic consumption of acetaldehyde was studied on N-F- TiO2 powders under 80 W Hg lamp irradiation, and a Langmuir-type kinetic model was developed for the reac- tion with appropriate kinetic parameters.

Reactivity of Azole Anions with CO2 from the DFT Perspective

Azole anions are key components in CO₂ capture materials that include ionic liquids and porous solids. Herein, we use density functional theory (DFT) and a Langmuir-type adsorption model to study azole anion-CO₂ interactions. Linear CO₂ has to be bent by approximately 45° to form an N-C bond within the azole ring. The energy cost of bending renders CO₂ absorption much more difficult compared to SO₂ absorption. For different azole anions, the number of nitrogen atoms in the ring and the natural bond orbital energy of the reacting nitrogen lone pair, both linearly correlate with the calculated reaction enthalpy and are useful handles for new sorbent designs. Unlike for SO₂, the azole parent architecture (unsubstituted) does not allow successive CO₂ absorption under mild conditions (<0.12 MPa and at room temperature). Experimental CO₂ and SO₂ absorption isotherms are reproduced by using the Langmuir model parameterized with the calibrated DFT reaction enthalpies. This study provides insight for designing azole-based CO₂-capture materials.

Synergistic effect in the simultaneous removal of binary cobalt–nickel heavy metals from effluents by a novel e-waste-derived material

Simultaneous removal of heavy metals coupled with high adsorption capacity of an adsorbent material is an ultimate target in industry for the purification of wastewater. A novel adsorbent prepared from an electronic waste-based material has been used for this purpose. The results exhibit very high removal capacities of the adsorbent for both cobalt and nickel in the single-component systems. These capacities have been compared with those of three widely-used industrial adsorbents and it has been shown that the removal capability of this novel material is well above those of the industrial ones. Moreover, the effect of pH has also been explored and it has been shown that the pH value has a drastic effect on the metal removal capacity at pH values lower than 3. Furthermore, a synergistic effect of two metals has been observed and studied in the binary metal system. The results indicate that not only the simultaneous presence of the two metals does not reduce the adsorption capacity, but also enhances their removal from the effluent.Moreover, the adsorption equilibrium modeling for the single and binary systems have been presented and it has been inferred that the Langmuir-type models can fit the single component experimental data, while the binary system can be fitted only by the modified extended Freundlich model.

Sacrificial Hydrogen Generation from Formaldehyde with Pt/TiO2 Photocatalyst in Solar Radiation

In this work, solar TiO2 photocatalysis was used for sacrificial hydrogen generation from formaldehyde. Platinum was loaded onto a TiO2 photocatalyst by a solar photodeposition method to suppress the electron/hole recombination process. The photocatalyst inside the reactor was irradiated from the top with a solar simulator. Photocatalytic hydrogen generation from formaldehyde was influenced by the solution pH, platinum loading (wt %) on TiO2, catalyst concentration, light intensity, and initial formaldehyde concentration. A Langmuir-type model fitted well with the experimental data, and the values of surface reaction rate constant, k, and the adsorption equilibrium constant, K, are 2.3598 × 10–6 mol min–1 and 17.73 M–1, respectively. Apparent quantum yield (QY) was higher for the UV light-driven hydrogen generation (10.91%) compared to the solar light-driven hydrogen generation (1.24%).

Phosphorus atomic layer doping in Ge using RPCVD

Phosphorus atomic layer doping (P-ALD) in Ge is investigated at temperatures between 100°C and 400°C using a single wafer reduced pressure chemical vapor deposition (RPCVD) system. Hydrogen-terminated and hydrogen-free Ge (100) surfaces are exposed to PH3 at different PH3 partial pressures after interrupting Ge growth. The adsorption and reaction of PH3 proceed on a hydrogen-free Ge surface. For all temperatures and PH3 partial pressures used for the P-ALD, the P dose increased with increasing PH3 exposure time and saturated. The saturation value of the incorporated P dose at 300°C is ∼1.5×1014cm−3, which is close to a quarter of a monolayer of the Ge (100) surface. The P dose could be simulated assuming a Langmuir-type kinetics model with a saturation value of Nt=1.55×1014cm−2 (a quarter of a monolayer), reaction rate constant kr=77s−1 and thermal equilibrium constant K=3.0×10−2Pa−1. An electrically active P concentration of 5–6×1019cm−3, which is a 5–6 times higher thermal solubility of P in Ge, is obtained by multiple P spike fabrication using the P-ALD process.

Photoelectrocatalytic Performance of Benzoic Acid onTiO2Nanotube Array Electrodes

The photoelectrocatalytic performance of benzoic acid on TiO2nanotube array electrodes was investigated. A thin-cell was used to discuss the effect of the bias voltage, illumination intensity, and electrolyte concentration on the photoelectrocatalytic degradation efficiency of benzoic acid. The photogenerated current-time (I-t) profiles were found to be related to the adsorption and the degradation process. The relationship between the initial concentration and the photocurrent peaks (I0ph) fits the Langmuir-type adsorption model, thus confirming that the adsorption of benzoic acid on TiO2nanotube arrays (TNAs) was single monolayer adsorption. At low concentrations, theI-tprofiles simply decay after the initial transient peak due to the sufficient holes on the TNAs which would oxidize the benzoic acid quickly. However, theI-tprofiles varied with increasing benzoic acid concentrations because the rate of diffusion in the bulk solution and the degradation of the intermediate products affect the photoelectrocatalysis on the electrode surface.

Effects of perforation size on the response of perforated GFRP composites aged in acidic media

The response of non-perforated and perforated GFRP specimens, aged in a combined acidic and hot environment, under an externally applied load (stress corrosion), and no external load conditions, is investigated. The applicability of Fickian and Langmuir type models for predicting the mass absorption response of the composite is also examined, and the degradation in the flexural properties of the composite bearing different perforation sizes is also established. The results indicate that the perforation size, immersion time, and the presence of external load (during aging); all directly influence the flexural properties of the composite.

Adsorption Kinetic and Thermodynamic Studies of Phosphate onto Tantalum Hydroxide

Tantalum hydroxide exhibits the ability for the removal of phosphate from aqueous solution. The kinetic study, adsorption isotherm, thermodynamic study, desorption, and foreign anions effect were examined in batch experiments. The kinetic process was very well described by a pseudo-second-order rate model. The adsorption isotherms showed that phosphate uptake fitted with a Langmuir-type model very well, with an increase of PO4(3-) adsorption capacity from 78.5 to 97.0 mg/g when the temperature increased from 298 to 338 K. The negative values of deltaG(0) and the positive values of deltaH(0) indicated that the phosphate adsorption process was spontaneous and endothermic naturally. While the deltaS(0) values obtained were positive, indicating an increase in randomness at the solid-liquid interface during the adsorption. Foreign anions tests showed that the presence of competitive ions cause minimal interference with the adsorption of phosphate on tantalum hydroxide.

Reversible and Oriented Immobilization of Ferrocene-Modified Proteins

Adopting supramolecular chemistry for immobilization of proteins is an attractive strategy that entails reversibility and responsiveness to stimuli. The reversible and oriented immobilization and micropatterning of ferrocene-tagged yellow fluorescent proteins (Fc-YFPs) onto β-cyclodextrin (βCD) molecular printboards was characterized using surface plasmon resonance (SPR) spectroscopy and fluorescence microscopy in combination with electrochemistry. The proteins were assembled on the surface through the specific supramolecular host-guest interaction between βCD and ferrocene. Application of a dynamic covalent disulfide lock between two YFP proteins resulted in a switch from monovalent to divalent ferrocene interactions with the βCD surface, yielding a more stable protein immobilization. The SPR titration data for the protein immobilization were fitted to a 1:1 Langmuir-type model, yielding K(LM) = 2.5 × 10(5) M(-1) and K(i,s) = 1.2 × 10(3) M(-1), which compares favorably to the intrinsic binding constant presented in the literature for the monovalent interaction of ferrocene with βCD self-assembled monolayers. In addition, the SPR binding experiments were qualitatively simulated, confirming the binding of Fc-YFP in both divalent and monovalent fashion to the βCD monolayers. The Fc-YFPs could be patterned on βCD surfaces in uniform monolayers, as revealed using fluorescence microscopy and atomic force microscopy measurements. Both fluorescence microscopy imaging and SPR measurements were carried out with the in situ capability to perform cyclic voltammetry and chronoamperometry. These studies emphasize the repetitive desorption and adsorption of the ferrocene-tagged proteins from the βCD surface upon electrochemical oxidation and reduction, respectively.

Extended Tonks–Langmuir-type model with non-Boltzmann-distributed electrons and cold ion sources

AbstractA general formalism for calculating the potential distribution Φ(z) in the quasineutral region of a new class of plane Tonks–Langmuir (TL)-type bounded-plasma-system (BPS) models differing from the well-known ‘classical’ TL model (Tonks, L. and Langmuir, I. 1929 A general theory of the plasma of an arc. Phys. Rev. 34, 876) by allowing for arbitrary (but still cold) ion sources and arbitrary electron distributions is developed. With individual particles usually undergoing microscopic collision/sink/source (CSS) events, extensive use is made here of the basic kinetic-theory concept of ‘CSS-free trajectories’ (i.e., the characteristics of the kinetic equation). Two types of electron populations, occupying the ‘type-t’ and ‘type-p’ domains of electron phase space, are distinguished. By definition, the type-t and type-p domains are made up of phase points lying on type-t (‘trapped’) CSS-free trajectories (not intersecting the walls and closing on themselves) and type-p (‘passing’) ones (starting at one of the walls and ending at the other). This work being the first step, it is assumed that ε ≡ λD/l → 0+ (where λD and l are a typical Debye length and a typical ionization length respectively) so that the system exhibits a finite quasineutral ‘plasma’ region and two infinitesimally thin ‘sheath’ regions associated with the ‘sheath-edge singularities’ | dΦ/dz|z→±zs → ∞. The potential in the plasma region is required to satisfy a plasma equation (quasineutrality condition) of the form ni {Φ} =ne (Φ), where the electron density ne (Φ) is given and the ion density ni {Φ} is expressed in terms of trajectory integrals of the ion kinetic equation, with the ions produced by electron-impact ionization of cold neutrals. While previous TL-type models were characterized by electrons diffusing under the influence of frequent collisions with the neutral background particles and approximated by Maxwellian (Riemann, K.-U. 2006 Plasma-sheath transition in the kinetic Tonks–Langmuir model. Phys. Plasmas 13, 063508) or bi-Maxwellian (Godyak, V. A. et al. 1995 Tonks–Langmuir problem for a bi-Maxwellian plasma. IEEE Trans. Plasma Sci. 23, 728) electron velocity distribution functions (VDFs), which satisfy the zero-CSS-term (Vlasov) kinetic equation and imply zero electron currents, we here propose a more general class of electron VDFs allowing, in an approximate manner, for non-zero CSS terms and finite electron currents inside the plasma region. The sheath-edge and floating-wall potentials are calculated by balancing the ion and electron current densities at sheath-edge singularities. In a first detailed application, the type-t and type-p electron VDFs are assumed to be ‘inner’ and ‘outer’ cut-off Maxwellians respectively, with different amplitudes and ‘formal’ temperatures, implying the perfectly CSS-free limit. For the special case of equal type-t and type-p electron VDF amplitudes and formal temperatures, the classical Boltzmann distribution for electrons is formally retrieved. Special cases with other amplitude and formal-temperature ratios show significant deviations from the classical case.

Modeling of lead and cadmium uptake by plants in the presence of surfactants

The mechanism of simultaneous uptake of Cd(II) and Pb(II) by Indian mustard (Brassica juncea) in presence of Triton X-100 and Tween 80 was investigated. The metal uptake data were analyzed according to the linear as well as the nonlinear Langmuir- and Freundlich-type models. The modeling efficiency (EF) and the root mean square error (RMSE) were used to evaluate the models prediction. Compared to the linear and the Langmuir-type models, the Freundlich-type model marginally exhibits a better fit for the biosorption of solubilized Cd(II) by surfactants as reflected by higher EF and lower RMSE values. The values of observed Pb(II) uptake were in close agreement with the predictions of the Freundlich-type model than for the Langmuir-type model.