General Isotherm Equation Research Articles

General Cluster Sorption Isotherm

Abstract Adsorption isotherms are an essential tool in chemical physics of surfaces. However, several approaches based on a different theoretical basis exist and for some isotherms existing approaches can fail. Here, the sorbate-sorbate interaction is not limited to an orientation vertical to the adsorbent surface. Instead, a formal orientation-independent clustering, also comprising the possibility of multilayers as a special case, is underlain. The flexible clusters are described by classical thermodynamics including a free energy relationship depending on the cluster size. In this paper a rigorous unified treatment of the adsorbate-adsorbate-adsorbent interaction is shown to result in a general isotherm equation which is applied to literature data both concerning type IV isotherms of argon and nitrogen in ordered mesoporous silica, and type II isotherms of disordered macroporous silica. The new isotherm covers the full range of partial pressure (10−6 - 0.7). The determination of surface areas is not possible by this isotherm because the cross-sectional area of a cluster is unknown. Based on the full description of type IV isotherms, most known isotherms including BET are accessible by respective simplifications. The presented model is an extension of the classical derivation of the ζ-isotherm which was shown to describe type IV isotherms restricted to adsorbates with very strong tendency of clustering such as water on hydrophobic surface (Phys. Chem. Chem. Phys. 21 (2019) 5614).

Adsorption of Sudan-IV contained in oily wastewater on lipophilic activated carbons: kinetic and isotherm modelling.

Up to nine kinetic and fourteen isotherm adsorption models are employed to model the adsorption of Sudan IV, a lipophilic model pollutant present in a biphasic mixture of cyclohexane-water system to simulate oily wastewater. Six different modified activated carbons were used as adsorbents. The highest amount adsorbed of Sudan IV was found in the material prepared by successive treatments of the parent commercial activated carbon Norit ROX 0.8 with nitric acid and urea, followed by thermal treatment at 800°C under continuous flow of nitrogen. Kinetic and isotherm adsorption models can be employed to simulate the process, since the effect of the presence of water in the adsorption of Sudan IV from the cyclohexane phase was found to be negligible, owing to the high lipophilic character of both adsorbent and adsorbate. All kinetic and isotherm coefficients, coupling with statistical parameters (r2, adjusted r2 and sum of squared errors), are determined by non-linear regression fitting and compared to literature data. The model of Avrami is found to be the most appropriate model to represent the adsorption of the pollutant in any of the six modified carbons tested, the highest value of the kinetic constant being 0.055min-1. The isotherm adsorption is well-modelled by using the general isotherm equation of Tóth and the multilayer Jovanović expression for the adsorption of Sudan-IV on that material, resulting in a high monolayer uptake capacity (qm = 193.6mgg-1).

Investigation of thermodynamic parameters of cetyl pyridinium bromide sorption onto ZSM-5 and natural clinoptilolite

A study was undertaken to determine the influence of temperature and surfactant concentration on the adsorption of cetyl pyridinium bromide on two ZSM-5 zeolites, and on natural clinoptilolite. The effect of temperature on adsorption and thermodynamic properties was investigated by making measurements at (297, 303, 307, 313) K. The results show that the calculated curves from the general isotherm equation can represent the experimental data very well up to T=313 K. Based on the calculation of the thermodynamics of adsorption, it is evident that the principle contribution to the Δ G ∘ hm of negative value is the large positive value of Δ S ∘ hm (36 to 52 J · mol −1), whereas Δ H ∘ hm is positive and in the range of 1 to 7 kJ · mol −1. Therefore, similar to the micellisation in bulk solution, the second step of surfactant adsorption is an entropy-driven process.

Transport of Zinc and Cadmium in Soils: Experimental Evidence and Modeling Approaches

AbstractThis study was conducted to assess the suitability of several isotherm equations in describing single solute and binary ion exchange of Zn and Cd in two soils (Windsor and Olivier). Isotherms were determined using batch methods for equilibration times of 1 and 14 d. Ion exchange batch experiments (based on NH4OAc extraction) were carried out at different heavy metal/Ca ratios. Several isotherm equations were capable of describing 1‐ and 14‐d sorption isotherms. However, the general Langmuir‐Freundlich was most suitable for describing single solute (Zn and Cd) isotherms, whereas the Rothmund‐Kornfeld ion exchange equation well described binary data sets (Zn‐Ca and Cd‐Ca). Miscible displacement experiments were carried out to study Zn and Cd transport in uniformly packed soil columns (10 cm in length) for different flow velocities. For Windsor soil, the general isotherm equation predicted Zn transport results adequately but not Cd results. Moreover, under conditions of variable ionic strength, the use of constant selectivity (equal affinity) for ion exchange provided consistently superior predictions of Zn and Cd transport than the Rothmund‐Kornfeld (variable affinity) approach. Attempts to predict Cd and Zn transport in Olivier soil, using several isotherms and ion exchange approaches, were not successful. Local equilibrium appeared dominant for Windsor soil, whereas nonequilibrium sorption behavior may be rate limiting for Zn and Cd transport in Olivier soil.

Statistical mechanics approach to the general isotherm equation for adsorption of surfactant at the solid/liquid interface

The general isotherm equation for adsorption of surfactant at the solid/liquid interface has been derived using a statistical method. Its application range and the corresponding meaning of the parameter n for the equation are discussed.

Surfactant adsorption at solid-liquid interfaces

The purpose of this review is to introduce systematically a new adsorption theory which is particularly applicable to surfactant adsorption, but might also be useful in other cases. The adsorption of surfactant from solution onto solids generally shows two-step character. In the first step the surface-active species are adsorbed through the interactions (i.e. electrostatic interaction and/or van der Waals interaction) between surface-active species and the solid interface. In the second step the surfactants are adsorbed through the interaction (i.e. hydrophobic interaction if the adsorption is from aqueous solution, and hydrogen bonding or polar interaction if the adsorption is from non-polar solvents) between the adsorbed surfactants. Based on the two-step adsorption model and the mass action treatment, a general adsorption isotherm equation has been derived. The equation shows that the adsorbed amount Γ depends on the equilibrium concentration C, saturated adsorption Γ∞, the surface aggregation number (of surface micelle) n, and the equilibrium constants k 1 and k 2 for the first and second step, respectively. The equation has been applied to various types of surfactant adsorption isotherms on various adsorbents from both aqueous and non-aqueous solutions (including Langmuir-, S-, and LS-types) successfully. From the results of the thermodynamics of adsorption, one can conclude that when dealing with adsorption from aqueous solution, the second step of surfactant adsorption (i.e. surface micellization) is an entropy-driven process similar to the micellization in bulk aqueous solution. However, as expected, the reverse surface micelle formation from non-polar solvent is not an entropy-driven process. The isotherm equations for individual adsorptions of surfactant monomers, surface micelles and unoccupied sites at the surface have also been derived. In addition, by combining the general isotherm equation and the Gibbs adsorption equation, the interfacial pressure-concentration relationships have been derived. Therefore, one can expect that knowledge of two-dimensional states and transitional phases of adsorbed surfactant films at solid/liquid interfaces could be provided by this kind of investigation.

Interfacial film of surfactant adsorbed at a solid/liquid interface

Based on the two-step adsorption model, the isotherm equations for individual adsorptions of surfactant monomers and hemi-micelles and for the unoccupied sites at a solid/liquid interface have been derived. In this way the compositions of the adsorption films were determined. By combining the general isotherm equation for adsorption of surfactant at a solid/liquid interface (B-Y. Zhu and T. Gu, J. Chem. Soc., Faraday Trans. 1, 1989, 85, 3813) with the Gibbs adsorption equation, solid/solution interfacial pressures at various concentrations may be calculated. The π–C and π–A curves for real systems have been provided. It is revealed that there are different states, including a liquid expanded film, a liquid condensed film and a transitional phase, in the interfacial film of surfactant adsorbed on a solid.

General isotherm equation for adsorption of surfactants at solid/liquid interfaces. Part 2. Applications

This work describes a number of results of surfactant adsorption (including non-ionic, cationic and anionic surfactants on various adsorbents) illustrating the applications of the general isotherm equation proposed in part 1. The results show that the calculated curves from the general isotherm equation can represent the experiment data very well. Based on the calculation of the thermodynamics of adsorption, it is evident that the principal contribution to the ΔG°hm of negative value is the large negative value is the large positive value of ΔS°hm, whereas ΔH°hm is positive. Therefore, similar to the micellization in bulk solution, the second step of surfactant adsorption (see part 1) is an entropy-driven process. For S-type and LS-type isotherms, two approaches and corresponding equations for calculation of h.m.c. have been proposed respectively, and the calculated h.m.c.s. by both approaches agree with the empirical value very well.

General isotherm equation for adsorption of surfactants at solid/liquid interfaces. Part 1. Theoretical

Based on the two-step adsorption mechanism (first step : the surface-active species are adsorbed through the interactions between the surface-active species and the solid surface; second step: through the hydrophobic interaction between the adsorbed surface-active species) and the mass-action treatment, the following general adsorption isotherm equation is derived: Γ=Γ∞k1C([graphic omitted] +k2Cn–1)//1 +k1C(1 +k2Cn–1) where Γ is the amount of surfactant adsorbed at concentration C, Γ∞ is the limiting adsorption at high concentrations, k1 and k2 are the equilibrium constants involved in the first and second step, respectively, and n is the aggregation number of surface hydrophobic aggregates or hemimicelles. The equation can interpret various types of surfactant adsorption isotherms (including L-, and LS-types) published in the literature.

Ideal adsorption from binary liquid mixtures on heterogeneous solid surfaces: Equations for excess isotherms and heats of immersion

Previous theoretical approaches to the problem of surface heterogeneity in adsorption from binary liquid mixtures on actual solid surfaces are reexamined. A general isotherm equation is developed, correctly yielding the transition from the well-known Dubinin-Radushkevich, and Freundlich isotherm equations to Henry's law. The forms are generalized by taking into account the different cross-sectional areas of adsorbed molecules. The related equation for heat of immersion is also developed and proved by analyzing some available data on excess isotherms and heats of immersion. Numerous model calculations are reported to better illustrate the complicated nature of actual solid/solution interface systems.

Low-pressure adsorption of single gases on heterogeneous solid surfaces

A general isotherm equation is developed, to describe the low-pressure adsorption of single gases on the heterogeneous surfaces of real solids. This is a new approach, which is close to the Condensation Approximation, but yields correctly Henry's Law at very low adsorbate pressure. Special attention is given to the adsorption systems exhibiting the skew-gaussian distribution of the adsorption energy, which is responsible for the Dubinin-Raduskevich behaviour of the low pressure adsorption isotherms. The theory is illustrated by suitable model calculations, which explain all the deviations observed experimentally, from the traditional DR plot.

A uniform interpretation of gas/solid adsorption

The uniform interpretation of gas/solid physical adsorption means that the different types of isotherms can be described by means of a single isotherm equation derived from the dynamical character of adsorption. The equation applicable to all types of isotherms (the general isotherm equation or GIE) takes into account the surface heterogeneity and the lateral and vertical interaction energies of the molecules adsorbed. Furthermore it can be demonstrated that in special cases the GIE transforms into already well-known or quite new isotherm equations applicable to single isotherms (for types I and III).

Adsorption and Desorption of Water Vapor by Cereal Grains and Their Products Part II: Development of the General Isotherm Equation

Adsorption and Desorption of Water Vapor by Cereal Grains and Their Products Part II: Development of the General Isotherm Equation