Analysis of kinetic Langmuir model. Part I: Integrated kinetic Langmuir equation (IKL): a new complete analytical solution of the Langmuir rate equation

Comparison of the natural and surfactant-modified zeolites in the adsorption efficiency of sunset yellow food dye from aqueous solutions

The removal of Sunset Yellow (E110) on natural zeolite and zeolite modified with the cationic surfactant cetyl pyridinium chloride (CPC) was studied using the adsorption method. The structural characteristics of the surfactant-modified zeolite (SMZ-CPC) were investigated using X-ray diffraction(XRD), Fourier transform infrared spectroscopy (FTIR), Brunauer-Emmett-Teller (BET) analysis, and the scanning electron microscopy (SEM) images. The effect of different parameters on the adsorption process, such as equilibration time and amount of adsorbent at 298 K, were determined using UV-Vis spectroscopy. The maximum dye removal percentage on SMZ-CPC was obtained with 0.08 g of adsorbent in 30 min. The results show that the zeolite modified with CPC surfactant (SMZ) has a higher adsorption capacity for Sunset Yellow than the unmodified zeolite (natural form). The experimental adsorption data were nonlinearly analyzed using isotherm equations such as Langmuir, Freundlich, Temkin, Langmuir-Freundlich (or Sips), and Extended Langmuir (EL). The experimental data were better fitted with the Sips isotherm. The adsorption kinetic data were analyzed using eight models in nonlinear forms: the pseudo-first-order (PFO), pseudo-second-order (PSO), integrated kinetic Langmuir (IKL), Elovich, intraparticle diffusion (IPD), mixed order rate equation (MOE), fractal-like pseudo-first-order (FL-PFO) and fractal-like pseudo-second-order (FL-PSO). According to the results of the coefficient of determination (R2), Elovich kinetic model well expressed E110 adsorption onto SMZ-CPC. Most of the dye removal takes place in less than 5 min and the maximum adsorption capacity is 5.06 mg/g. The results of this study show that zeolite modified with cationic surfactant is an effective adsorbent for removal anionic dyes from an aqueous solution.

Preparation and Selective Adsorption Performance of the Carboxymethyl Salix psammophila Wood Powder-Imprinted Membrane for Tetracycline

Carboxymethyl Salix psammophila wood powder-imprinted membranes (CMSM-MIPs) were prepared by using wet spinning technology and molecular-imprinting technology for the selective removal of tetracycline from wastewater. Scanning electron microscopy, X-ray diffraction, thermogravimetry, and X-ray photoelectron spectroscopy characterizations demonstrate that CMSM-MIPs retain the membranous structure of Carboxymethyl Salix psammophila wood powder membranes, successfully encapsulate thin layers of imprinted polymers on the membrane surface, and exhibit excellent thermal stability. The adsorption results showed that CMSM-MIPs had the highest selective adsorption capacity for tetracycline, which was 253.8 mg/g. In addition, the adsorption capacities for oxytetracycline and chlortetracycline were 208.8 and 188 mg/g, respectively. It can be observed that CMSM-MIPs not only exhibit a high adsorption capacity for tetracycline but also demonstrate good adsorption capacities for oxytetracycline and chlortetracycline. The experimental results showed that CMSM-MIPs were best fitted with pseudo-second-order kinetics and most consistent with Freundlich fitting. The regeneration experiment showed that CMSM-MIPs still had good regeneration performance after 5 regeneration cycles. In conclusion, the CMSM-MIPs can not only have the natural adsorption performance of Salix psammophila wood powder but also give it higher selectivity through molecular imprinting, so as to achieve efficient removal of target organic pollutants in water.

Equilibrium and Kinetic Studies on Adsorption of Neutral and Ionic Species of Organic Adsorbates from Aqueous Solutions on Activated Carbon

This work presents comprehensive studies of the adsorption of neutral and ionic forms of organic adsorbates from aqueous solutions on activated carbon. The influence of pH on the equilibrium and kinetics of the adsorption of methylene blue (MB) and organic acids, benzoic (BA), 2-nitrobenzoic (2-NBA), 3-nitrobenzoic (3-NBA), and 4-nitrobenzoic (4-NBA) acid, was investigated. Experimental adsorption isotherms were analyzed using the generalized Langmuir isotherm equation (R2 = 0.932-0.995). Adsorption rate data were studied using multiple adsorption kinetics equations, of which the multi-exponential equation gave the best fit quality (R2 - 1 = (6.3 × 10-6)-(2.1 × 10-3)). The half-time was also used to represent the effect of pH on adsorption kinetics. Strong dependences of the adsorption efficiency on the solution pH were demonstrated. In the case of organic acid adsorption, the amount and rate of this process increased with a decrease in pH. Moreover, larger adsorbed amounts of methylene blue were recorded in an alkaline environment in a relatively short time. The maximum absorbed amounts were 11.59 mmol/g, 6.57 mmol/g, 9.38 mmol/g, 2.70 mmol/g, and 0.24 mmol/g for BA, 2NBA, 3-NBA, 4-NBA, and MB. The pure activated carbon and the selected samples after adsorption were investigated using thermal analysis and X-ray photoelectron spectroscopy.

An HPLC-SEC-based rapid quantification method for vesicular stomatitis virus particles to facilitate process development

Virus particle (VP) quantification plays a pivotal role in the development of production processes of VPs for virus-based therapies. The yield based on total VP count serves as a process performance indicator for evaluating process efficiency and consistency. Here, a label-free particle quantification method for enveloped VPs was developed, with potential applications in oncolytic virotherapy, vaccine development, and gene therapy. The method comprises size-exclusion chromatography (SEC) separation using high-performance liquid chromatography (HPLC) instruments. Ultraviolet (UV) was used for particle quantification and multi-angle light scattering (MALS) for particle characterization. Consistent recoveries of over 97% in the SEC were achieved upon mobile phase screenings and addition of bovine serum albumin (BSA) as sample stabilizer. A calibration curve was generated, and the method's performance and applicability to in-process samples were characterized. The assay's repeatability variation was <1% and its intermediate precision variation was <3%. The linear range of the method spans from 7.08 × 108 to 1.72 × 1011 VP/mL, with a limit of detection (LOD) of 7.72 × 107 VP/mL and a lower limit of quantification (LLOQ) of 4.20 × 108 VP/mL. The method, characterized by its high precision, requires minimal hands-on time and provides same-day results, making it efficient for process development.

Comprehensive Studies of Adsorption Equilibrium and Kinetics for Selected Aromatic Organic Compounds on Activated Carbon

This work presents a comprehensive analysis of the adsorption of selected aromatic organic compounds on activated carbons. Both the equilibrium and kinetics of adsorption were studied using UV-Vis spectrophotometry. The influence of a number of factors: pH, contact time, presence of an accompanying substance, adsorbate concentration, as well as the mass and size of adsorbent grains, on the adsorption process from aqueous solutions was investigated. Phenol, 2-nitrophenol, 3-nitrophenol, 4-nitrophenol and methylene blue (as an accompanying substance) were selected as adsorbates. GAC 1240W and RIAA activated carbons were used as adsorbents. The equilibrium data were analyzed using the generalized Langmuir isotherm equation (R2 = 0.912-0.996). Adsorption rate data were fitted using a multi-exponential kinetic equation (1 - R2 = (1.0 × 10-6)-(8.2 × 10-4)). As an additional parameter, the half-time was also used to present the influence of selected factors on the adsorption kinetics. An increase in the amount of adsorption was demonstrated with increasing contact time as well as with decreasing solution pH and adsorbent grain size. For selected systems, an increase in the adsorption rate was observed with increasing adsorbate concentration, adsorbent mass and at lower pH values. In some cases, the presence of an accompanying substance also resulted in an increase in adsorption kinetics. In the tested experimental systems, optimal conditions for adsorption were established (T = 298 K, pH = 2, contact time: 7 days, grain diameter: >0.5 mm and the ratio of the mass of the adsorbent to the volume of the adsorbate solution: 1 g/L). Additionally, the acid-base properties (potentiometric titration), morphology (SEM) and structure (TEM) of the used adsorbents were also examined.

Metformin and lamotrigine sorption on a digestate amended soil in presence of trace metal contamination

The antidiabetic drug metformin and antiepileptic drug lamotrigine are contaminants of emerging concern that have been detected in biowaste-derived amendments and in the environment, and their fate must be carefully studied. This work aimed to evaluate their sorption behaviour on soil upon digestate application. Experiments were conducted on soil and digestate-amended soil as a function of time to study kinetic processes, and at equilibrium also regarding the influence of trace metals (Pb, Ni, Cr, Co, Cu, Zn) at ratio pharmaceutical/metal 1/1, 1/10, and 1/100. Pharmaceutical desorption experiments were also conducted to assess their potential mobility to groundwater. Results revealed that digestate amendment increased metformin and lamotrigine adsorbed amounts by 210% and 240%, respectively, increasing organic matter content. Metformin adsorption kinetics were best described by Langmuir model and those of lamotrigine by Elovich and intraparticle diffusion models. Trace metals did not significantly affect the adsorption of metformin in amended soil while significantly decreased that of lamotrigine by 12-39%, with exception for Cu2+ that increased both pharmaceuticals adsorbed amounts by 5 - 8%. This study highlighted the influence of digestate amendment on pharmaceutical adsorption and fate in soil, which must be considered in the circular economy scenario of waste-to-resource.

Experimental and Numerical Study of Radioiodine Sorption and Transport in Hanford Sediments

Radioiodine (129I) poses a potential risk to human health and the environment at several U.S. Department of Energy sites, including the Hanford Site, located in southeastern Washington State. Experimental studies and numerical modeling were performed to provide a technical basis for field-scale modeling of iodine sorption and transport behavior. The experiments were carried out using six columns of repacked contaminated sediments from the Hanford Site. Although iodate has been determined to be the dominant iodine species at the Hanford Site, the sorption and transport behaviors of different iodine species were investigated in a series of column experiments by first leaching sediments with artificial groundwater (AGW) followed by AGW containing iodate (IO3-), iodide (I-), or organo-iodine (2-iodo-5-methoxyphenol, C7H7IO2). Ferrihydrite amendments were added to the sediments for three of the columns to evaluate the impact of ferrihydrite on 129I attenuation. The results showed that ferrihydrite enhanced the iodate sorption capacity of the sediment and retarded the transport but had little effect on iodide or organo-I, providing a technical basis for developing a ferrihydrite-based remedial strategy for iodate under oxidizing conditions. Data from the column transport experiments were modeled using the linear equilibrium Freundlich isotherm model, the kinetic Langmuir adsorption model, and a distributed rate model. Comparisons of the experimental data and modeling results indicated that sorption was best represented with the distributed rate model with rates and maximum sorption extents varying by iodine species and ferrihydrite treatment. However, the linear Freundlich isotherm (Kd) model was also found to fit the laboratory experimental data relatively well, suggesting that the Kd model could also be used to represent iodine transport at the field scale.

The Synthesis and Evaluation of Porous Carbon Material from Corozo Fruit (Bactris guineensis) for Efficient Propranolol Hydrochloride Adsorption

This study explores the potential of the corozo fruit (Bactris guineensis) palm tree in the Colombian Caribbean as a source for porous carbon material. Its specific surface area, pore volume, and average pore size were obtained using N₂ adsorption/desorption isotherms. The images of the precursor and adsorbent surface were obtained using scanning electron microscopy (SEM). Fourier transform infrared (FTIR) spectra were obtained to detect the main functional groups present and an X-ray diffraction analysis (XRD) was performed in order to analyze the structural organization of the materials. By carbonizing the fruit stone with zinc chloride, a porous carbon material was achieved with a substantial specific surface area (1125 m² g⁻¹) and pore volume (3.241 × 10-¹ cm³ g⁻¹). The material was tested for its adsorption capabilities of the drug propranolol. The optimal adsorption occurred under basic conditions and at a dosage of 0.7 g L⁻¹. The Langmuir homogeneous surface model effectively described the equilibrium data and, as the temperature increased, the adsorption capacity improved, reaching a maximum of 134.7 mg g⁻¹ at 328.15 K. The model constant was favorable to the temperature increase, increasing from 1.556 × 10^-1 to 2.299 × 10^-1 L mg^-1. Thermodynamically, the adsorption of propranolol was found to be spontaneous and benefited from higher temperatures, indicating an endothermic nature (12.39 kJ mol⁻¹). The negative ΔG⁰ values decreased from -26.28 to -29.99 kJ mol^-1, with the more negative value occurring at 328 K. The adsorbent material exhibited rapid kinetics, with equilibrium times ranging from 30 to 120 min, depending on the initial concentration. The kinetics data were well-represented by the general order and linear driving force models. The rate constant of the general order model diminished from 1.124 × 10^-3 to 9.458 × 10^-14 with an increasing concentration. In summary, the leftover stone from the Bactris guineensis plant can be utilized to develop activated carbon, particularly when activated using zinc chloride. This material shows promise for efficiently adsorbing propranolol and potentially other emerging pollutants.

Synergistic effects of binary surfactant mixtures in the adsorption of diclofenac sodium drug from aqueous solution by modified zeolite

In this paper, the surfactant-modified clinoptilolite zeolite (with two methods) were used to remove diclofenac sodium (DFS) as a widely used drug in an aqueous solution. Clinoptilolite was modified by using pure cationic surfactant (cetyltrimethylammonium chloride, CTAC) and the mixed surfactants of CTAC + Triton-X100 (TX100). In the new approach, the synergistic effects between CTAC and TX100 were determined by surface tension measurements in different mole fractions and the optimum ratio (y₁ ≈ 0.8) was identified with the maximum synergism. According to the mole fraction of this composition, the surface of clinoptilolite was modified by mixed surfactants (MSMZ) for the adsorption of DFS and then results compared with modified zeolite with pure cationic surfactant (SMZ). The raw and modified (SMZ and MSMZ) zeolites were characterized by Fourier transform infrared spectroscopy (FT-IR), BET analysis, the scanning electron microscopy (SEM) images, Zeta potential and X-ray. The experimental data of adsorption in equilibrium conditions were also analyzed using different adsorption isotherm models (Langmuir, Freundlich, Hill, Khan, Sips, Redlich-Peterson and Toth) in non-linear forms, and finally, the best model consistent with experimental data is determined (SMZ:Sips and MSMZ:Toth). According to the best isotherm model, the amount of absorption capacity in MSMZ was obtained almost 57% higher than SMZ. In addition, the kinetic adsorption data were correlated with eight various models in order to selection the best model for these systems. The kinetic adsorption data were well described by fractal-like pseudo-first-order (FL-PFO) and IKL models for SMZ and MSMZ adsorbents, respectively. Eight error functions were used to estimate the best fitted isotherm and kinetic models.

Adsorption of Non-Steroidal Anti-Inflammatory Drugs on Alginate-Carbon Composites-Equilibrium and Kinetics

In this work, alginate-carbon composites with different active carbon content were synthesized and studied by various techniques. The obtained materials can be used as adsorbents in the processes of removing organic pollutants from water. In this study, the effect of the immobilization of activated carbon in calcium alginate was investigated. Textural properties were determined by measuring low-temperature nitrogen adsorption/desorption isotherms. The largest specific surface area was recorded for ALG_C8 and amounted to 995 m2/g. The morphology of alginate materials was determined on the basis of scanning electron microscopy. The adsorption properties were estimated based on the measurements of equilibrium and adsorption kinetics. The highest sorption capacities were 0.381 and 0.873 mmol/g for ibuprofen and diclofenac, respectively. The generalized Langmuir isotherm was used to analyze the equilibrium data. A number of equations and kinetic models were used to describe the adsorption rate data, including first (FOE) and second (SOE) order kinetic equations, 1,2-mixed-order kinetic equation (MOE), fractal-like MOE equation (f-MOE), multi-exponential equation (m-exp), in addition to diffusion models: intraparticle diffusion model (IDM) and pore diffusion model (PDM). Thermal stability was determined on the basis of data from thermal analysis in an atmosphere of synthetic air.

Purification of acidic lignocellulose hydrolysate using anion-exchange resin: Multicomponent adsorption, kinetic and thermodynamic study

Acid hydrolysis of lignocellulosic biomass to produce high value-added products presents a breathtaking industrial application foreground. However, the hydrolysate under harsh conditions contains extremely complex degradations, resulting in many restrictions or lethal toxicity on the following utilization and bioconversion. In this study, the anion-exchange resin 335 was exploited to separate and purify main degradations from the acidic corncob-hydrolysate. A comprehensive investigation was explored on equilibrium isotherms, adsorption kinetics, and thermodynamic parameters of the representative substances in the hydrolysate. The results indicated that the removal of acetic acid, furfural, and lignin reached 90.13%, 92.58%, and 94.85% respectively, while the loss rate of xylose was well controlled within 20%. Based on these studies, various models and parameters were evaluated to uncover the mechanisms. In conclusion, this work offered a theoretical basis for the application in the separation and purification of acidic lignocellulose-hydrolysate and further bioconversion.

Modelling the adsorption of phospholipid vesicles to a silicon dioxide surface using Langmuir kinetics

Supported Lipid Bilayers (SLBs) are model biological membranes that have been developed to study the interactions between biomolecules in a cell membrane. Though forming SLBs is relatively easy, their formation mechanism remains a topic of debate. When buffered solutions containing phosphatidylcholine vesicles are flowed over a silicon dioxide (SiO₂) surface they adsorb intact to the surface to form a Supported Vesicle Layer (SVL) if the pH of the buffer is above 9. We have run experiments with buffers with a pH at or above 9 to study the kinetics of the adsorption of 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) vesicles to an SiO₂ surface, which is the first step in the formation of an SLB. We used a quartz crystal microbalance (QCM) to monitor the real-time changes in the mass of the SVL as it formed from solutions with different lipid concentrations. Increases in the maximum frequency change with increasing lipid concentration indicated that both adsorption and desorption of DOPC vesicles were occurring, and that an equilibrium was established between the DOPC vesicles in the SVL and in the bulk solution. From the data acquired we were able to determine that the equilibrium constant for the adsorption and desorption of DOPC vesicles was 18 ± 1. The data was fitted to a Langmuir adsorption model from which the rate constants for the adsorption and desorption of DOPC vesicles were determined to be k_a = (0.0107 ± 0.0004) mL mg^-1 s^-1 and k_d = (5.8 ± 0.3) × 10^-4 s^-1. The best fit to the experimental data was achieved if a parameter (α = (0.035 ± 0.003) s^-1) was used to account for the time taken for the lipid concentration to reach its steady state value in the flow cell used in the experiments.

Lipid Anchoring Improves Lubrication and Wear Resistance of the Collagen I Matrix

Osteoarthritis is a prevalent degenerative joint disease characterized by progressive articular cartilage loss and destruction. The resultant increase in friction causes severe pain. The collagen I matrix (COL I) has been used clinically for cartilage repair; however, how COL I acts at cartilage surfaces is unclear. Here, we studied adsorption and lubrication of synovial fluid components, albumin, γ-globulin, and the phospholipid DPPC, on COL I under physiological conditions using surface plasmon resonance and an in situ sensing surface force apparatus. Our results revealed COL I had poor lubrication ability, a fairly high coefficient of friction (COF, μ = 0.651 ± 0.013), and surface damage under a 7 mN load. DPPC formed an improved lubricating layer on COL I (μ = 0.072 ± 0.016). In sharp contrast, albumin and γ-globulin exhibited poor lubrication with an order of magnitude higher COF but still provided benefits by protecting COL I from wear. Hence, DPPC on COL I may help optimize COL I implantation design.

Investigation of carbon dioxide photoreduction process in a laboratory-scale photoreactor by computational fluid dynamic and reaction kinetic modeling

The production of solar fuels via the photoreduction of carbon dioxide to methane by titanium oxide is a promising process to control greenhouse gas emissions and provide alternative renewable fuels. Although several reaction mechanisms have been proposed, the detailed steps are still ambiguous, and the limiting factors are not well defined. To improve our understanding of the mechanisms of carbon dioxide photoreduction, a multi-physics model was developed using COMSOL. The novelty of this work is the computational fluid dynamic model combined with the novel carbon dioxide photoreduction intrinsic reaction kinetic model, which was built based on three-steps, namely gas adsorption, surface reactions and desorption, while the ultraviolet light intensity distribution was simulated by the Gaussian distribution model and Beer-Lambert model. The carbon dioxide photoreduction process conducted in a laboratory-scale reactor under different carbon dioxide and water moisture partial pressures was then modeled based on the intrinsic kinetic model. It was found that the simulation results for methane, carbon monoxide and hydrogen yield match the experiments in the concentration range of 10−4 mol·m−3 at the low carbon dioxide and water moisture partial pressure. Finally, the factors of adsorption site concentration, adsorption equilibrium constant, ultraviolet light intensity and temperature were evaluated.

Removal Performance and Mechanism of Benzo(b)Fluorathene Using MnO2 Nanoflower/Graphene Oxide Composites

High-ring polycyclic aromatic hydrocarbons (PAHs, Benzo[b]fluorathene (BbFA), etc.) are difficult to biodegrade in the water environment. To address this issue, an innovative method for the preparation of MnO₂ nanoflower/graphene oxide composite (MnO₂ NF/GO) was proposed for adsorption removal of BbFA. The physicochemical properties of MnO₂ NF/GO were characterized by SEM, TEM, XRD, and N₂ adsorption/desorption and XPS techniques. Results show that the MnO₂ NF/GO had well-developed specific surface area and functional groups. Batch adsorption experiment results showed that adsorption capacity for BbFA was 74.07 mg/g. The pseudo-second-order kinetic model and Freundlich isotherm model are fitted well to the adsorption data. These show electron-donor-acceptor interaction; especially π-π interaction and π complexation played vital roles in BbFA removal onto MnO₂ NF/GO. The study highlights the promising potential adsorbent for removal of PAHs.

A critical overview of adsorption kinetics for cooling and refrigeration systems

The dynamic uptake of adsorbate onto the porous adsorbent plays a crucial role in determining the performance of the adsorption-based cooling system. Therefore, it is imperative to know the kinetics parameters of an adsorbate - adsorbent pair to design a system to be operated at variable working conditions. The kinetics models of adsorption, used to simulate the adsorption rate of different pairs, are derived and presented in this paper. Besides, the limitations and advantages of the models are also mentioned. Moreover, the dynamic performance of different adsorption pairs is analyzed, and the values of kinetics parameters, determined through experimental procedures and fitting of kinetics models, are also summarized. It is opined that during the initial unsaturated condition of adsorption, the semi-infinite model can be preferred to determine the diffusion time constant. The modification of different models, e.g., Langmuir and linear driving force models, can significantly overcome the drawbacks of the models, as shown by several researchers. However, research may be carried out to investigate different models' fitting errors from a statistical perspective. Furthermore, to evaluate the dynamic performance of different adsorbates, a lot of research needs to be done, specifically, on the adsorption of the newly developed environment-friendly refrigerants, onto the promising composite adsorbents possessing high thermal conductivity and significantly improved adsorption uptakes.

Nonspecific Binding-Fundamental Concepts and Consequences for Biosensing Applications

Nature achieves differentiation of specific and nonspecific binding in molecular interactions through precise control of biomolecules in space and time. Artificial systems such as biosensors that rely on distinguishing specific molecular binding events in a sea of nonspecific interactions have struggled to overcome this issue. Despite the numerous technological advancements in biosensor technologies, nonspecific binding has remained a critical bottleneck due to the lack of a fundamental understanding of the phenomenon. To date, the identity, cause, and influence of nonspecific binding remain topics of debate within the scientific community. In this review, we discuss the evolution of the concept of nonspecific binding over the past five decades based upon the thermodynamic, intermolecular, and structural perspectives to provide classification frameworks for biomolecular interactions. Further, we introduce various theoretical models that predict the expected behavior of biosensors in physiologically relevant environments to calculate the theoretical detection limit and to optimize sensor performance. We conclude by discussing existing practical approaches to tackle the nonspecific binding challenge in vitro for biosensing platforms and how we can both address and harness nonspecific interactions for in vivo systems.

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.

Mesoporous Carbons of Well-Organized Structure in the Removal of Dyes from Aqueous Solutions

Mesoporous carbons with differentiated properties were synthesized by using the method of impregnation of mesoporous well-organized silicas. The obtained carbonaceous materials and microporous activated carbon were investigated by applying different methods in order to determine their structural, surface and adsorption properties towards selected dyes from aqueous solutions. In order to verify applicability of adsorbents for removing dyes the equilibrium and kinetic experimental data were measured and analyzed by applying various equations and models. The structural and acid-base properties of the investigated carbons were evaluated by Small-Angle X-ray Scattering (SAXS) technique, adsorption/desorption of nitrogen, potentiometric titration, and Transmission Electron Microscopy (TEM). The results of these techniques are complementary, indicating the type of porosity and structural ordering, e.g., the pore sizes determined from the SAXS data are in good agreement with those obtained from nitrogen sorption data. The SAXS and TEM data confirm the regularity of mesoporous carbon structure. The adsorption experiment, especially kinetic measurements, reveals the utility of mesoporous carbons in dye removing, taking into account not only the adsorption uptake but also the adsorption rate.

A Revised Pseudo-Second-Order Kinetic Model for Adsorption, Sensitive to Changes in Adsorbate and Adsorbent Concentrations

The development of new adsorbent materials for the removal of toxic contaminants from drinking water is crucial toward achieving the United Nations Sustainable Development Goal 6 (clean water and sanitation). The characterization of these materials includes fitting models of adsorption kinetics to experimental data, most commonly the pseudo-second-order (PSO) model. The PSO model, however, is not sensitive to parameters such as adsorbate and adsorbent concentrations (C₀ and C_s) and consequently is not able to predict changes in performance as a function of operating conditions. Furthermore, the experimental conditionality of the PSO rate constant, k₂, can lead to erroneous conclusions when comparing literature results. In this study, we analyze 103 kinetic experiments from 47 literature sources to develop a relatively simple modification of the PSO rate equation, yielding dqtdt=k'Ct(1-qtqe)2. Unlike the original PSO model, this revised rate equation (rPSO) provides the first-order and zero-order dependencies upon C₀ and C_s that we observe empirically. Our new model reduces the residual sum of squares by 66% when using a single rate constant to model multiple adsorption experiments with varying initial conditions. Furthermore, we demonstrate how the rPSO rate constant k' is more appropriate for comparing literature studies, highlighting faster kinetics in the adsorption of arsenic onto alumina versus iron oxides. This revised rate equation should find applications in engineering studies, especially since the rPSO rate constant k' does not show a counter-intuitive inverse relationship with increasing reaction rates when C₀ is increased, unlike the PSO rate constant k₂.

Investigation of Fractal-like Characteristics According to New Kinetic Equation of Desorption

Most of the adsorbents have porous structures and a suitable kinetic model is essential for studying these systems. The kinetic Langmuir model is one of the first theoretical models, which can be used for desorption studies. In the present research, the fractal-like concept was added to the kinetic Langmuir model of desorption. A new integrated kinetic Langmuir equation was provided to investigate the rate of desorption from a solid surface. The preferred characteristic of the provided rate equation is the application of the fractal concept for the kinetic study of the desorption process from porous surfaces. The derivation of a new equation was confirmed using the generated data. The fractal-like concept for some experimental desorption studies was obtained. This parameter can show how the porous structure of an adsorbent can affect the desorption kinetics.

Prediction of half-life for adsorption kinetics in a batch reactor

Besides adsorption rate constant, the half-life was also a basic factor that described the characteristics of adsorption kinetics. However, the direct prediction of the half-life was still a problem to be addressed urgently. In this work, the parameter τ was introduced into the pseudo-first-order (PFO), pseudo-second-order (PSO), pseudo-nth-order (PNO), and the corresponding fractal-like kinetic models (fractal-like PFO, fractal-like PSO, and fractal-like PNO) to directly predict the half-life by changing the boundary condition, i.e., the replacement of q_t = 0, t = 0 by q_t = q_e/2, t = τ. The fitting performance of these kinetic models after modification was evaluated by nitrate adsorption on polyaniline-modified activated carbon (PAN/AC) and phosphate adsorption on zirconium-loaded Ca-montmorillonite. The results indicated that this type of model modifications did not influence the fitting performance and that the half-life was easily obtained only by the curve fitting. The practical significance of this work was to simultaneously predict the adsorption rate constant and half-life using the modified kinetic models.

Activated Carbon from Agricultural Wastes for Adsorption of Organic Pollutants

Agricultural waste materials (strawberry seeds and pistachio shells) were used for preparation of activated carbons by two various methods. Chemical activation using acetic acid and physical activation with gaseous agents (carbon dioxide and water vapor) were chosen as mild and environmentally friendly methods. The effect of type of raw material, temperature, and activation agent on the porous structure characteristics of the materials was discussed applying various methods of analysis. The best obtained activated carbons were characterized by high values of specific surface area (555-685 m²/g). The Guinier analysis of small-angle X-ray scattering (SAXS) curves showed that a time of activation affects pore size. The samples activated using carbon dioxide were characterized mostly by the spherical morphology of pores. Adsorbents were utilized for removal of the model organic pollutants from the single- and multicomponent systems. The adsorption capacities for the 4-chloro-2-methyphenoxyacetic acid (MCPA) removal were equal to 1.43-1.56 mmol/g; however, for adsorbent from strawberry seeds it was much lower. Slight effect of crystal violet presence on the MCPA adsorption and inversely was noticed as a result of adsorption in different types of pores. For similar herbicides strong competition in capacity and adsorption rate was observed. For analysis of kinetic data various equations were used.

Depletion-Mediated Uniform Deposition of Nanorods with Patterned, Multiplexed Assembly

Device-scale, uniform, and controllable deposition of nanoparticles on various substrates is fundamentally important not only for the fabrication of thin-film devices but also for the large sample statistics of single-particle performances. However, it is challenging to obtain such predefined depositions using a simple and efficient method. Here, we present a novel strategy for obtaining the uniform and particle density/spacing-tunable deposition of nanorods on a linker-free substrate. The deposition is driven by the tailored particle-substrate depletion attraction owing to the size-matched design of the substrate roughness and the nanorod diameter. Both gold nanorods and upconversion nanorods were applied to demonstrate the generality of the method. The high particle density of more than 21 per μm² and correspondingly the small particle spacing of fewer than 0.3 μm were achieved on a scalable substrate template. On this basis, orientational ordering and pattern-selective deposition of nanorods were realized by controlling the liquid flow rate and employing the substrate with patterned roughness areas, respectively. With the roughness-directed density-tunable depositions of nanorods integrated onto a single platform, multiplexed gold nanorod assembly and programmable surface-enhanced Raman mapping were achieved, with a promising prospect in information encoding by using the Raman signals as the translation units. The thermal stability and related transition temperature of about 160 °C of gold nanorods were also revealed as an application of single-particle statistics. This practical method could be extended to wide ranges of potential applications in plasmonic coupling devices, cryptography, or single-particle performance statistics with the feature of the high-throughput, low-cost, and scalable fabrication.

Mesocellular Silica Foams (MCFs) with Tunable Pore Size as a Support for Lysozyme Immobilization: Adsorption Equilibrium and Kinetics, Biocomposite Properties

The effect of the porous structure of mesocellular silica foams (MCFs) on the lysozyme (LYS) adsorption capacity, as well as the rate, was studied to design the effective sorbent for potential applications as the carriers of biomolecules. The structural (N2 adsorption/desorption isotherms), textural (SEM, TEM), acid-base (potentiometric titration), adsorption properties, and thermal characteristics of the obtained lysozyme/silica composites were studied. The protein adsorption equilibrium and kinetics showed significant dependence on silica pore size. For instance, LYS adsorption uptake on MCF-6.4 support (pore diameter 6.4 nm) was about 0.29 g/g. The equilibrium loading amount of LYS on MCF-14.5 material (pore size 14.5 nm) increased to 0.55 g/g. However, when the pore diameter was larger than 14.5 nm, the LYS adsorption value systematically decreased with increasing pore size (e.g., for MCF-30.1 was only 0.27 g/g). The electrostatic attractive interactions between the positively charged lysozyme (at pH = 7.4) and the negatively charged silica played a significant role in the immobilization process. The differences in protein adsorption and surface morphology for the biocomposites of various pore sizes were found. The thermal behavior of the studied bio/systems was conducted by TG/DSC/FTIR/MS coupled method. It was found that the thermal degradation of lysozyme/silica composites was a double-stage process in the temperature range 165-420-830 °C.

Chitosan Deposited onto Fumed Silica Surface as Sustainable Hybrid Biosorbent for Acid Orange 8 Dye Capture: Effect of Temperature in Adsorption Equilibrium and Kinetics

Chitosan was deposited on fumed silica without the addition of cross-linkers or activating agents. The chitosan surface layer has a high affinity toward organic molecules, e.g., Acid Orange 8 (AO8) dye, robust to a broad range of simulated conditions (variance with respect to temperature, time, and concentration of solute). Experimental equilibrium data were analyzed by the generalized Langmuir equation taking into consideration the energetic heterogeneity of the adsorption system. The effect of temperature on dye uptake and adsorption rate was studied. According to the calculated thermodynamic functions ΔG°, ΔH°, and ΔS° from the equilibrium data at different temperatures, the adsorption of AO8 onto chitosan-fumed silica composite is exothermic and spontaneous. The studies of temperature effect on adsorption equilibrium show that the maximum adsorption capacity (determined from the Langmuir-Freundlich equation) of synthesized composite toward AO8 is about one-third higher in the case of an isotherm measured at 5 °C than this value obtained for the isotherm measured at 45 °C. The quantitative binding of dye molecules to chitosan coating on the surface of silica was proved by ¹H MAS NMR. The deep kinetics study through the application of various theoretical models-the first-order equation, pseudo-first-order equation, second-order equation, pseudo-second-order equation, mixed first, second-order equation, and multiexponential equation-was applied for getting inside the mechanism of AO8 binding to the chitosan coating. Structural characteristics of chitosan-coated silica were obtained from the low-temperature adsorption/desorption isotherms of nitrogen and imaging by scanning electron microscopy. The effects of a synthetic route for polymer coating on thermal stability and the ability to degrade were studied by differential scanning calorimetry.

Adsorption kinetic models: Physical meanings, applications, and solving methods

Adsorption technology has been widely applied in water and wastewater treatment, due to its low cost and high efficiency. The adsorption kinetic models have been used to evaluate the performance of the adsorbent and to investigate the adsorption mass transfer mechanisms. However, the physical meanings and the solving methods of the kinetic models have not been well established. The proper interpretation of the physical meanings and the standard solving methods for the adsorption kinetic models are very important for the applications of the kinetic models. This paper mainly focused on the physical meanings, applications, as well as the solving methods of 16 adsorption kinetic models. Firstly, the mathematical derivations, physical meanings and applications of the adsorption reaction models, the empirical models, the diffusion models, and the models for adsorption onto active sites were analyzed and discussed in detail. Secondly, the model validity evaluation equations were summarized based on literature. Thirdly, a convenient user interface (UI) for solving the kinetic models was developed based on Excel software and provided in supplementary information, which is helpful for readers to simulate the adsorption kinetic process.

Extended Rate Constants Distribution (RCD) Model for Sorption in Heterogeneous Systems: 2. Importance of Diffusion Limitations for Sorption Kinetics on Cryogels in Batch

Here we address the problem of what we can expect from investigations of sorption kinetics on cryogel beads in batch. Does macroporosity of beads indeed help eliminate diffusion limitations under static sorption conditions? Are sorption rate constants calculated using phenomenological kinetic models helpful for predicting sorption properties under dynamic conditions? Applying the rate constants distribution (RCD) model to kinetic curves of Cu(II) ions sorption on polyethyleneimine (PEI) cryogel and gel beads and fines, we have shown that diffusion limitations in highly swollen beads are very important and result in at least ten-fold underestimation of the sorption rate constants. To account for intraparticle diffusion, we have developed the RCD-diffusion model, which yields "intrinsic" kinetic parameters for the sorbents, even if diffusion limitations were important in kinetic experiments. We have shown that introduction of a new variable-characteristic diffusion time-to the RCD model significantly improved the reliability of sorption kinetic parameters and allowed prediction of the minimal residence time in column required for efficient uptake of the adsorbate under dynamic conditions. The minimal residence time determined from kinetic curves simulated using the RCD-diffusion model was in good agreement with experimental data on breakthrough curves of Cu(II) ion sorption on monolith PEI cryogel at different flow rates.

Phenoxyacid pesticide adsorption on activated carbon - Equilibrium and kinetics

The adsorption of herbicides belonging to the group of halogenated phenoxyacids on the activated carbon was studied. They are differentiated in terms of quantity and type of functional groups (such as chloride, bromide, fluoride) and their position on an aromatic ring. The experimental equilibrium data were analyzed using adsorption isotherm equations taking into account energetic heterogeneity of the adsorption systems. The calculated concentration profiles from the kinetic data were discussed applying two diffusion models, MOE, f-MOE and multi-exponential equations. The dependences between the properties of adsorbates, adsorption uptake and rate were analyzed. The adsorption affinity of pesticides was correlated with adsorbate hydrophobicity, character of functional group, molecular structure. The applicability of kinetic models and equations was investigated; the assumptions of the models were analyzed with regard to consistency with adsorption mechanism. Similarity of adsorption mechanism was found for all adsorbates confirmed by similarity of kinetic curves and corresponding distributions of rate coefficients. The differences in kinetic profiles were attributed to differentiation of herbicide's molecules - number and type of functional groups and their positions on aromatic ring.

Chitosan-Silica Hybrid Composites for Removal of Sulfonated Azo Dyes from Aqueous Solutions

In this study, the influence of the chitosan immobilization method on the properties of final hybrid materials was performed. Chitosan was immobilized on the surface of mesoporous (ChS2) and fumed silica (ChS3) by physical adsorption and the sol-gel method (ChS1). It was found that physical immobilization of chitosan allows to obtain hybrid composites (ChS) with a homogeneous distribution of polymer on the surface, relatively wide pores, and specific surface area of about 170 m²/g, pH_PZC = 5.7 for ChS3 and 356 m²/g and pH_PZC = 6.0 for ChS2. The microporous chitosan-silica material with a specific surface area of 600 m²/g and a more negatively charged surface (pH_PZC = 4.2) was obtained by the sol-gel reaction. The mechanisms of azo dye adsorption were studied, and the correlation with the composite structure was distinguished. The generalized Langmuir equation and its special cases, that is, Langmuir-Freundlich and Langmuir equations, were applied for the analysis of adsorption isotherm data. The adsorption study showed that physically adsorbed chitosan (ChS1 and ChS2) on a silica surface has a higher sorption capacity, for example, 0.48 mmol/g for the acid red 88 (AR88) dye (ChS2) and 0.23 mmol/g for the acid orange 8 (AO8) dye (ChS1), compared to the composite obtained by the sol-gel method [ChS1, 0.05 mmol/g for the AO8 dye]. For a deeper understanding of the behavior of immobilized chitosan in the adsorption processes, various kinetic equations were applied: first-order, second-order, mixed 1,2-order (MOE), multiexponential, and fractal-like MOE as well as intraparticle and pore diffusion model equations. In the case of AO8 dye, the adsorption rates were differentiated for three composites: for ChS3, 50% of the dye was removed from the solution after merely 5 min and almost 90% after 80 min. The slowest adsorption process controlled by the diffusion rate of dye molecules into the internal space of the pore structure was found for ChS1 (225 min halftime). In the case of ChS2, the rates for various dyes change in the following order: acid orange (AO7) > orange G (OG) > acid red 1 (AR1) > AR88 > AO8 (halftimes: 10.5 < 15.7 < 23.7 < 34.9 < 42.9 min).

Kinetics of Ion-Capturing/Ion-Releasing Processes in Liquid Crystal Devices Utilizing Contaminated Nanoparticles and Alignment Films

Various types of nanomaterials and alignment layers are considered major components of the next generation of advanced liquid crystal devices. While the steady-state properties of ion-capturing/ion-releasing processes in liquid crystals doped with nanoparticles and sandwiched between alignment films are relatively well understood, the kinetics of these phenomena remains practically unexplored. In this paper, the time dependence of ion-capturing/ion-releasing processes in liquid crystal cells utilizing contaminated nanoparticles and alignment layers is analyzed. The ionic contamination of both nanodopants and alignment films governs the switching between ion-capturing and ion-releasing regimes. The time dependence (both monotonous and non-monotonous) of these processes is characterized by time constants originated from the presence of nanoparticles and films, respectively. These time constants depend on the ion adsorption/ion desorption parameters and can be tuned by changing the concentration of nanoparticles, their size, and the cell thickness.