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Adsorption heat transformation (AHT) systems can play a major role in protecting our environment by decreasing the usage of fossil fuels and utilizing natural and alternative working fluids. The adsorption isotherm is the most important feature in characterizing an AHT system. There are eight types of International Union of Pure and Applied Chemistry (IUPAC) classified adsorption isotherms for different “adsorbent-adsorbate” pairs with numerous empirical or semi-empirical mathematical models to fit them. Researchers face difficulties in choosing the best isotherm model to describe their experimental findings as there are several models for a single type of adsorption isotherm. This study presents the optimal models for all eight types of isotherms employing several useful statistical approaches such as average error; confidence interval (CI), information criterion (ICs), and proportion tests using bootstrap sampling. Isotherm data of 13 working pairs (which include all eight types of IUPAC isotherms) for AHT applications are extracted from literature and fitted with appropriate models using two error functions. It was found that modified Brunauer–Emmet–Teller (BET) for Type-I(a) and Type-II; Tóth for Type-I(b); GAB for Type-III; Ng et al. model for Type-IV(a) and Type-IV(b); Sun and Chakraborty model for Type-V; and Yahia et al. model for Type-VI are the most appropriate as they ensure less information loss compared to other models. Moreover; the findings are affirmed using selection probability; overall; and pairwise proportion tests. The present findings are important in the rigorous analysis of isotherm data.

In the present study, recovery of lithium as lithium aluminate from Urmia Lake was investigated. A coprecipitation method was utilized by using an aluminum salt AlCl 3 . 6 H 2 O . Lithium ions are adsorbed on aluminum hydroxide, which is prepared by adding NaOH and AlCl 3 . 6 H 2 O to the brines at Al 3 + Li + molar ratio ≈5. The results showed that the maximum Li + ion adsorption was adsorbed at pH ~ 7. Also, by increasing the temperature from 30 °C to 40 °C, lithium ions adsorption was decreased. The maximum adsorption amount of Al OH 3 was at 30 °C, pH = 7 and density 1.31. The obtained results from adsorption of Li + of Urmia Lake were compared with four isotherm models, Langmuir, Dubinin–Radushkevich, Freundlich and Temkin isotherms. In addition, sulfuric acid was used for Li + desorption from aluminum hydroxide.

The recently developed thermodynamic Langmuir isotherm model is used to estimate the isosteric heat of adsorption for pure component adsorption. Specifically, pure component adsorption isotherms at different temperatures are first correlated with the thermodynamic Langmuir isotherm model. Then the pure component isosteric heat of adsorption is calculated with the Clausius–Clapeyron equation. We show the thermodynamic Langmuir isotherm model correctly estimates the isosteric heat of adsorption as a function of surface loading for pure component adsorption.

Adsorption is a major basic and applied phenomenon in many scientific disciplines. In particular, the design of adsorbents to remove pollutants from wastewater requires advanced knowledge of adsorption isotherms, which are useful tools to identify physicochemical factors that control adsorption performance. Recent modeling of adsorption isotherms has focused on the use of statistical physics. Here, we review the major adsorption isotherm models based on statistical physics. We discuss steric, energetic, and thermodynamic parameters.

Sorption isotherms describe the relation between the equilibrium moisture content of a material and the ambient relative humidity. Most materials exhibits sorption hysteresis, that is, desorption give higher equilibrium moisture contents than absorption at equal ambient climate conditions. Sorption hysteresis is commonly evaluated by determination of an absorption isotherm followed by desorption starting from the highest relative humidity used in the absorption measurement (typically 95%). The latter is often interpreted as the desorption isotherm but is in fact a scanning isotherm, i.e. an isotherm obtained from neither dry nor water-saturated state. In the present study, we investigated the difference between desorption isotherms and scanning isotherms determined by desorption from different high relative humidity levels reached by absorption and how this difference influenced the evaluation of sorption hysteresis. The measurements were performed on Norway spruce ( Picea abies (L.) Karst.) using automated sorption balances. Hysteresis evaluated from desorption isotherms gave linear absolute sorption hysteresis for the studied relative humidity range (0–96%), whereas hysteresis evaluated from scanning isotherms gave non-linear curves with a peak between 50 and 80% relative humidity. The position of this peak depended on the relative humidity from which desorption was initiated. Consequently, understanding and evaluation of sorption hysteresis might be challenging if scanning isotherms are used instead of desorption isotherms, hereby increasing the risk of misinterpreting the results. Graphical Abstract

Simulation and design of adsorptive separation units demand accurate estimation of thermodynamic properties. Isosteric heat of adsorption as calculated from generalized Langmuir (gL) isotherm coupled with Clausius–Clapeyron expression for pure component and mixed-gas adsorption equilibria is presented in this work. The estimated isosteric heat of adsorption as functions of surface loading and composition is validated against the experimental data for various adsorption systems. Furthermore, the gL results are compared against classical Langmuir (cL) and Toth isotherm for pure components and with Ideal Adsorbed Solution Theory (IAST) for mixed-gas adsorption equilibria. The comparison highlights that gL outperforms cL and Toth for pure component adsorption and IAST for mixed-gas adsorption, and gL reliably captures the loading dependence and the composition dependence for isosteric heat of adsorption.

The present experimental study reports the performance of tea waste (TW) derived adsorbent for the adsorption of sodium diclofenac (SD) from aqueous solution (SD concentration = 10–50 mg/L). The waste-derived activated carbon was prepared by chemical activation process of raw waste using H 2 SO 4 , KOH, ZnCl 2 , and K 2 CO 3 as activating agents (TW: activating agent = 1:1 by weight). Subsequently, the oven-dried material was carbonized at 600-°C temperature for 2 h. The synthesized adsorbents were porous and their Brunauer-Emmett-Teller (BET) surface area was ranged 115–865 m 2 /g. Among all synthesized adsorbents, the adsorbent activated by ZnCl 2 exhibited the highest adsorption capacity (= 62 mg/g), though it was much lower compared to 91 mg/g obtained with commercial activated carbon (CAC) (SD concentration = 30 mg/L, adsorbent dose = 300 mg/L and initial wastewater pH = 6.47). SD equilibrium data could be described by Langmuir isotherm adequately, while pseudo-second-order rate model showed better fit to the time based adsorption data. Low activation energy of the adsorption process suggests the reaction to be temperature independent. Thermodynamic parameters showed the spontaneous and endothermic nature of adsorption process conducted in the presence of waste derived adsorbent.

Adsorption is an important phenomenon widely used for the removal of contaminants. Several drinking water contaminants such as arsenic and fluoride, vanadium and chromium, nickel, cadmium and cobalt  are found to coexist in nature as multi-component mixtures in water. Hence, the modeling of multi-component adsorption isotherms for designing water treatment systems has gained importance recently. However, review studies of multi-component adsorption and competitive adsorption modeling are limited. The current review paper summarizes twenty-six multi-component adsorption isotherm models. Also, case studies of several common multi-component adsorption systems and the mechanisms of multi-component adsorption are discussed. Furthermore, a modeling analysis of four multi-component isotherms models for three commonly found two-component adsorption systems, i.e., cadmium-nickel, nickel–cobalt, and cadmium-cobalt, is reported. The Extended Langmuir isotherm, Competitive Langmuir isotherm, Extended Langmuir–Freundlich isotherm and Extended Freundlich isotherm models were applied in the modeling study for the competitive adsorption of Cd, Ni, and Co. The goodness of fit parameters and adsorption isotherm constants were estimated for these models. The factors influencing competitive adsorption, mechanisms of adsorption, various single and multi-component isotherm models, their significance, and limitations are also discussed in this review article. Highlights • Twenty-six multi-component and ten single-component isotherm models are compiled • Factors affecting multi-component competitive adsorption isotherms are discussed in this review paper • The applications of four multi-component isotherm models for three binary contaminant systems are presented. Graphical Abstract

Lead (Pb) is one of the greatly toxic heavy metals that can enter into the biological system through food chain and induce serious health impairments. Lead ion (Pb (II)) can be released in the environment through natural and anthropogenic sources. Among different types of remediation techniques, adsorption is one of the most well-accepted techniques due to its economic feasibility and versatility as well. Herein, the removal of Pb (II) was examined using chitosan-activated carbon-polyvinyl alcohol (CS-AC-PVA) composite beads. Results demonstrated that CS-AC-PVA composite beads have more potential to adsorb Pb (II) from the aqueous solution than lone chitosan (CS), activated carbon (AC), and chitosan-activated carbon composite beads. Moreover, CS-AC-PVA composite beads showed 44% more adsorbent capacity than CA-AC beads. The adsorption capacity of CS-AC-PVA composite beads for Pb (II) showed highest adsorption at 25 °C which is increased with increasing concentrations and decrease with increasing temperature. The equilibrium data is well described by Freundlich isotherm labeling chemical interaction via multiple adsorption sites. Thus, this hybrid CS-AC-PVA composite bead is proven as potential adsorbent for environmental remediation for Pb (II) from the aqueous solution.

Nitrogen adsorption isotherms, along with the BET model for interpretation, are recommended for estimating biochar surface area. The frequently measured small surface areas of biochars contrast with their high sorption and cation exchange capacities. We hypothesised that water adsorption provides a better tool for estimating the surface area of biochars. Although adsorption energy also appears to be a valuable surface characteristic, there is a lack of studies on this subject. We studied the surface areas and adsorption energies of three waste deposits – peat, willow dust and biochar prepared from these materials at different temperatures – using nitrogen and water vapour adsorption isotherms. The BET model accurately described all water vapour adsorption isotherms but failed for some nitrogen isotherms. Alternative methods for estimating surface areas and adsorption energies were proposed in cases where the BET model did not apply. Nitrogen adsorption was typically much lower than water vapour adsorption, and the estimated surface areas reflected this. However, nitrogen adsorption energies were significantly higher. Nitrogen surface areas increased with pyrolysis temperature, while water vapour surface areas decreased. The surface area estimated from nitrogen adsorption was generally much lower than needed to accommodate the surface-charged groups responsible for the cation exchange capacity of biochars.