File Name: freundlich and langmuir adsorption isotherms .zip
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The need to design low-cost adsorbents for the detoxification of industrial effluents has been a growing concern for most environmental researchers.
So modelling of experimental data from adsorption processes is a very important means of predicting the mechanisms of various adsorption systems. Therefore, this paper presents an overall review of the applications of adsorption isotherms, the use of linear regression analysis, nonlinear regression analysis, and error functions for optimum adsorption data analysis.
The migration of pollutant s in aqueous media and subsequent development of containment measures have resulted in the use of adsorption among other techniques [ 1 , 2 ]. Adsorption equilibrium information is the most important piece of information needed for a proper understanding of an adsorption process.
A proper understanding and interpretation of adsorption isotherms is critical for the overall improvement of adsorption mechanism pathways and effective design of adsorption system [ 3 ]. In recent times, linear regression analysis has been one of the most applied tools for defining the best fitting adsorption models because it quantifies the distribution of adsorbates, analyzes the adsorption system, and verifies the consistency of theoretical assumptions of adsorption isotherm model [ 4 ].
Because of the inherent bias created by linearization, several error functions have been used to address this shortfall. Concomitant with the evolution of computer technology, the use of nonlinear isotherm modelling has been extensively used. This is the simplest adsorption isotherm in which the amount of surface adsorbate is proportional to the partial pressure of the adsorptive gas [ 4 ].
This isotherm model describes an appropriate fit to the adsorption of adsorbate at relatively low concentrations such that all adsorbate molecules are secluded from their nearest neighbours [ 5 ]. The Hill-Deboer isotherm model describes a case where there is mobile adsorption as well as lateral interaction among adsorbed molecules [ 6 , 7 ]. Equilibrium data from adsorption experiments can be analyzed by plotting versus [ 8 — 10 ]. Fowler-Guggenheim proposed this isotherm equation which takes into consideration the lateral interaction of the adsorbed molecules [ 11 ].
The linear form of this isotherm model is as follows [ 8 ]: where is Fowler-Guggenheim equilibrium constant , is fractional coverage, is universal gas constant , is temperature k , and is interaction energy between adsorbed molecules.
This isotherm model is predicated on the fact that the heat of adsorption varies linearly with loading. Therefore, if the interaction between adsorbed molecules is attractive, then the heat of adsorption will increase with loading because of increased interaction between adsorbed molecules as loading increases i.
However, if the interaction among adsorbed molecules is repulsive, then the heat of adsorption decreases with loading i. But when then there is no interaction between adsorbed molecules, and the Fowler-Guggenheim isotherm reduces to the Langmuir equation. A plot of versus is used to obtain the values for and. It is important to note that this model is only applicable when surface coverage is less than 0.
Kumara et al. Langmuir adsorption which was primarily designed to describe gas-solid phase adsorption is also used to quantify and contrast the adsorptive capacity of various adsorbents [ 12 ]. Langmuir isotherm accounts for the surface coverage by balancing the relative rates of adsorption and desorption dynamic equilibrium.
Adsorption is proportional to the fraction of the surface of the adsorbent that is open while desorption is proportional to the fraction of the adsorbent surface that is covered [ 13 ]. The essential characteristics of the Langmuir isotherm can be expressed by a dimensionless constant called the separation factor [ 15 ].
Freundlich isotherm is applicable to adsorption processes that occur on heterogonous surfaces [ 15 ]. This isotherm gives an expression which defines the surface heterogeneity and the exponential distribution of active sites and their energies [ 16 ]. The linear form of the Freundlich isotherm is as follows [ 17 ]: where is adsorption capacity and is adsorption intensity; it also indicates the relative distribution of the energy and the heterogeneity of the adsorbate sites.
Boparai et al. Although several isotherm models were applied, the equilibrium data was best represented by Freundlich and Flory-Huggins isotherms due to high correlation coefficients [ 18 ]. Dubinin-Radushkevich isotherm model [ 19 ] is an empirical adsorption model that is generally applied to express adsorption mechanism with Gaussian energy distribution onto heterogeneous surfaces [ 20 ]. The model is a semiempirical equation in which adsorption follows a pore filling mechanism [ 22 ].
It is usually applied to differentiate between physical and chemical adsorption of metal ions [ 22 ]. A distinguishing feature of the Dubinin-Radushkevich isotherm is the fact that it is temperature dependent; hence when adsorption data at different temperatures are plotted as a function of logarithm of amount adsorbed versus the square of potential energy, all suitable data can be obtained [ 13 ].
Ayawei et al. The Temkin isotherm is valid only for an intermediate range of ion concentrations [ 25 ]. The linear form of Temkin isotherm model is given by the following [ 22 ]: where is Temkin constant which is related to the heat of sorption and is Temkin isotherm constant [ 26 ]. Hutson and Yang applied Temkin isotherm model to confirm that the adsorption of cadmium ion onto nanozero-valent iron particles follows a chemisorption process.
Similarly, Elmorsi et al. Flory-Huggins isotherm describes the degree of surface coverage characteristics of the adsorbate on the adsorbent [ 27 ]. The linear form of the Flory-Huggins equation is expressed as where is degree of surface coverage, is number of adsorbates occupying adsorption sites, and is Flory-Huggins equilibrium constant. The equilibrium constant is used to calculate spontaneity Gibbs free energy as shown in the following expression [ 28 ]: where is standard free energy change, is universal gas constant 8.
Hamdaoui and Naffrechoux used the Flory-Huggins isotherm model in their study of the biosorption of Zinc from aqueous solution using coconut coir dust [ 29 ]. The Hill isotherm equation describes the binding of different species onto homogeneous substrates. This model assumes that adsorption is a cooperative phenomenon with adsorbates at one site of the adsorbent influencing different binding sites on the same adsorbent [ 30 ].
The linear form of this isotherm is expressed as follows [ 29 ]: where , , and are constants. Hamdaoui and Naffrechoux investigated the equilibrium adsorption of aniline, benzaldehyde, and benzoic acid on granular activated carbon GAC using the Hill isotherm model; according to their report, the Hill model was very good in comparison with previous models with for all adsorbates [ 29 ].
The Halsey isotherm is used to evaluate multilayer adsorption at a relatively large distance from the surface [ 16 ]. The adsorption isotherm can be given as follows [ 31 ]: where and are Halsey isotherm constant and they can be obtained from the slope and intercept of the plot of versus. Fowler and Guggenheim reported the use of Halsey isotherm in their equilibrium studies of methyl orange sorption by pinecone derived activated carbon. The fitting of their experimental data to the Halsey isotherm model attests to the heteroporous nature of the adsorbent [ 31 ].
Similarly, Song et al. The Halsey isotherm fits the experimental data well due to high correlation coefficient , which may be attributed to the heterogeneous distribution of activate sites and multilayer adsorption on coconut shell carbons [ 16 ]. Harkin-Jura isotherm model assumes the possibility of multilayer adsorption on the surface of absorbents having heterogeneous pore distribution [ 32 ].
This model is expressed as follows: where and are Harkin-Jura constants that can be obtained from plotting versus. Foo and Hameed reported that the Harkin-Jura isotherm model showed a better fit to the adsorption data than Freundlich, Halsey, and Temkin isotherm models in their investigation of the adsorptive removal of reactive black 5 from wastewater using Bentonite clay [ 32 ].
The Jovanovic model is predicated on the assumptions contained in the Langmuir model, but in addition the possibility of some mechanical contacts between the adsorbate and adsorbent [ 33 ]. Kiseler reported the use of Jovanovic isotherm model while determining adsorption isotherms for L-Lysine imprinted polymer. Their report showed that the best prediction of retention capacity was obtained by applying the Jovanovic isotherm model [ 33 ]. The equation that defines this model is based on a kinetic principle which assumes that adsorption sites increase exponentially with adsorption; this implies a multilayer adsorption [ 35 ].
The equation was first developed to describe the kinetics of chemisorption of gas onto solids [ 36 ]. The linear forms of the Elovich model are expressed as follows [ 37 ]: but the linear form is expressed as follows [ 8 ]: Elovich maximum adsorption capacity and Elovich constant can be calculated from the slope and intercept of the plot of versus.
Rania et al. Equilibrium data from adsorption processes can be modelled by plotting versus [ 8 , 38 — 40 ]. The Redlich-Peterson isotherm is a mix of the Langmuir and Freundlich isotherms.
The numerator is from the Langmuir isotherm and has the benefit of approaching the Henry region at infinite dilution [ 41 ]. This isotherm model is an empirical isotherm incorporating three parameters. It combines elements from both Langmuir and Freundlich equations; therefore the mechanism of adsorption is a mix and does not follow ideal monolayer adsorption [ 42 ]. At high liquid-phase concentrations of the adsorbate, 16 reduces to the Freundlich equation: where and of the Freundlich isotherm model.
The linear form of the Redlich-Peterson isotherm can be expressed as follows [ 34 ]: A plot of versus enables the determination of Redlich-Peterson constants, where is slope and is intercept [ 30 , 42 — 45 ]. This isotherm model has a linear dependence on concentration in the numerator and an exponential function in the denomination which altogether represent adsorption equilibrium over a wide range of concentration of adsorbate which is applicable in either homogenous or heterogeneous systems because of its versatility [ 46 , 47 ].
The linearized form is given as follows [ 12 ]: This model is suitable for predicting adsorption on heterogeneous surfaces, thereby avoiding the limitation of increased adsorbate concentration normally associated with the Freundlich model [ 19 ].
Therefore at low adsorbate concentration this model reduces to the Freundlich model, but at high concentration of adsorbate, it predicts the Langmuir model monolayer adsorption. The parameters of the Sips isotherm model are , temperature, and concentration dependent [ 12 , 49 ] and isotherm constants differ by linearization and nonlinear regression [ 50 ]. The Toth isotherm is another empirical modification of the Langmuir equation with the aim of reducing the error between experimental data and predicted value of equilibrium data [ 51 ].
This model is most useful in describing heterogeneous adsorption systems which satisfy both low and high end boundary of adsorbate concentration [ 52 ]. It is clear that when , this equation reduces to Langmuir isotherm equation.
Therefore the parameter characterizes the heterogeneity of the adsorption system [ 51 ] and if it deviates further away from unity 1 , then the system is said to be heterogeneous. The Toth isotherm may be rearranged to give a linear form as follows: The values of parameters of the Toth model can be evaluated by nonlinear curve fitting method using sigma plot software [ 53 ].
This isotherm model has been applied for the modelling of several multilayer and heterogeneous adsorption systems [ 53 , 54 ]. Koble-Carrigan isotherm model is a three-parameter equation which incorporates both Langmuir and Freundlich isotherms for representing equilibrium adsorption data [ 55 ].
All three Koble-Carrigan isotherm constants can be evaluated with the use of a solver add-in function of the Microsoft Excel [ 56 ]. At high adsorbate concentrations, this model reduces to Freundlich isotherm. The Kahn isotherm model is a general model for adsorption of biadsorbate from pure dilute equations solutions [ 57 ].
Nonlinear methods have been applied by several researchers to obtain the Khan isotherm model parameters [ 59 , 60 ]. The Radke-Prausnitz isotherm model has several important properties which makes it more preferred in most adsorption systems at low adsorbate concentration [ 61 ].
At low adsorbate concentration, this isotherm model reduces to a linear isotherm, while at high adsorbate concentration it becomes the Freundlich isotherm and when , it becomes the Langmuir isotherm. Another important characteristic of this isotherm is that it gives a good fit over a wide range of adsorbate concentration. The Radke-Prausnitz model parameters are obtained by nonlinear statistical fit of experimental data [ 61 , 62 ].
Langmuir-Freundlich isotherm includes the knowledge of adsorption heterogeneous surfaces. It describes the distribution of adsorption energy onto heterogeneous surface of the adsorbent [ 54 ]. At low adsorbate concentration this model becomes the Freundlich isotherm model, while at high adsorbate concentration it becomes the Langmuir isotherm.
These parameters can be obtained by using the nonlinear regression techniques [ 63 ]. The Jossens isotherm model predicts a simple equation based on the energy distribution of adsorbate-adsorbent interactions at adsorption sites [ 64 ]. This model assumes that the adsorbent has heterogeneous surface with respect to the interactions it has with the adsorbate.
However, upon rearranging 29 [ 65 ], The values of and can be obtained from either a plot of versus or using a least square fitting procedure. A good representation of equilibrium data using this equation was reported for phenolic compounds on activated carbon [ 66 ] and on amberlite XAD-4 and XAD-7 macroreticular resins [ 67 ].
Fritz and Schlunder derived an empirical equation which can fit a wide range of experimental results because of the large number of coefficients in the isotherm [ 68 ].
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Santa Fe. This paper studies the thermodynamic aspects of the processes of adsorption of phenol from dilute aqueous solutions on different commercial carbons, evaluating how to optimize the removal of this persistent contaminant. Two powdered activated carbons from two different companies were used: Tetrahedron Carbon Andes Chemistry Lab. Both specific surface areas were measured by means of the BET method. Experimental isotherms were determined at K, K and K.
The need to design low-cost adsorbents for the detoxification of industrial effluents has been a growing concern for most environmental researchers. So modelling of experimental data from adsorption processes is a very important means of predicting the mechanisms of various adsorption systems. Therefore, this paper presents an overall review of the applications of adsorption isotherms, the use of linear regression analysis, nonlinear regression analysis, and error functions for optimum adsorption data analysis. The migration of pollutant s in aqueous media and subsequent development of containment measures have resulted in the use of adsorption among other techniques [ 1 , 2 ]. Adsorption equilibrium information is the most important piece of information needed for a proper understanding of an adsorption process.
[us97redmondbend.org]. Fig. In contrast to the theoretically justified the Langmuir isotherm, the Freundlich isotherm is of a purely.
The data used to support the findings of this study are available from the corresponding author upon request. To understand the nature of sorption process, linear and nonlinear forms of five sorption isotherms including Freundlich and Langmuir models were employed. Feasibility and viability of sorption process were evaluated by calculating kinetics and thermodynamics of the process, while error analysis suggested best fitted sorption model on sorption data.
The Freundlich equation or Freundlich adsorption isotherm , an adsorption isotherm , is an empirical relationship between the quantity of a gas adsorbed into a solid surface and the gas pressure. The same relationship is also applicable for the concentration of a solute adsorbed onto the surface of a solid and the concentration of the solute in the liquid phase. In , Herbert Freundlich gave an expression representing the isothermal variation of adsorption of a quantity of gas adsorbed by unit mass of solid adsorbent with gas pressure. As this relationship is entirely empirical, in the case where adsorption behavior can be properly fit by isotherms with a theoretical basis, it is usually appropriate to use such isotherms instead see for example the Langmuir and BET adsorption theories.
The applicability of Freundlich and Langmuir adsorption isotherms to guide bleaching of vegetable oils was examined using rubber [ Hevea brasiliensis Willd. Juss Muell.
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The applicability of Freundlich and Langmuir adsorption isotherms to guide bleaching of vegetable oils was examined using rubber [ Hevea brasiliensis Willd. Juss Muell. The degree of bleaching was monitored spectrophotometrically. The results obtained show good agreement with Freundlich and Langmuir isotherms, indicating that the adsorption of the coloring matter from the oils proceeds by monolayer formation on the surface of the adsorbent. This is a preview of subscription content, access via your institution.
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