Open access peer-reviewed chapter - ONLINE FIRST

Written By

Andres Abin-Bazaine, Alfredo Campos Trujillo and Mario Olmos-Marquez

Submitted: January 27th, 2022Reviewed: March 4th, 2022Published: April 19th, 2022

DOI: 10.5772/intechopen.104260

From the Edited Volume

## Wastewater Treatment [Working Title]

Prof. Muharrem Ince and Dr. Olcay Kaplan Ince

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## Abstract

### Keywords

• isotherms
• Henry’s
• Langmuir
• Freundlich
• Temkin
• Redlich–Peterson
• sips
• Halsey
• Harkin–Jura
• Elovich
• Flory–Huggins
• fowler–Guggenheim
• Jovanovic and Kiselev

## 1. Introduction

### 2.1 Calculation of isotherms

Batch adsorption experiments are performed. A constant concentration of adsorbent is added and mixed with a constant volume of solution, progressively varying the concentration of the solute to be removed. Stir at a constant rate until equilibrium is reached. The percentage of adsorption is calculated using the following expression:

qe=CiCfm×VE2

where Ci and Cf are the initial and final concentrations, respectively, of the ion to be removed (mg L−1), qe is the adsorption capacity of the ion (adsorbate) by the adsorbent (mg g−1), m is the mass of the adsorbent (g), and V is the volume of the ion solution to be removed (L) [6, 7, 8, 9, 10, 11, 12, 15, 16, 17].

Once the values of mass removal capacity (qe) and equilibrium concentration (Cf) have been calculated using a Cartesian coordinate system to represent the two-dimensional distribution, the dispersion diagram provides a set of known points whose analysis allows a qualitative study of the relationship between the two variables. The next step is to determine the functional dependence between the two graphically represented variables that best fits the two-dimensional distribution. It is called linear regression if the function is linear.

The correlation coefficient is a technique used to study the two-dimensional distribution, indicating the intensity or degree of dependence between the variables X and Y. The correlation coefficient r is a number obtained using the following formula:

r=xix¯yiy¯xix¯2+yiy¯2E3

where r is correlation coefficient, xivalue of variable of x, x¯is the mean of the variable x, yi value of variable y, y¯is the mean value of variable y, xix¯2sum of the square of the deviation x, yiy¯2sum of the square of the deviation y. The values of the correlation coefficient vary in the range from −1 to +1. A value of −1 indicates a perfect inverse linear correlation while a value of +1 indicates a positive linear correlation.

This last procedure can easily be done with the help of software that uses spreadsheets.

### 2.2 Henry’s isotherm

This isothermal model adequately describes the adsorption process at low concentrations so that all adsorbate molecules are without interaction with the neighboring molecules [18]. The concentrations in the phases are associated with a linear expression. It is expressed as:

qe=KHECeE4

where qeis the adsorption capacity in equilibrium (mg g−1); KHE is the equilibrium constant of Henry’s; and Ceis the equilibrium concentration of the metal ions in the solution (mg L−1). From the straight line adjusted to graph Ceversus qe, the KHE coefficient was calculated, which is represented by the slope.

### 2.3 Langmuir isotherm model

The Langmuir equation assumes that the maximum adsorption corresponds to a monosaturated layer of adsorbate molecules on the surface of the adsorbent, the adsorption energy is constant, and there is no transmigration of the adsorbate on the surface of the adsorbent [8]. All adsorption sites are energetically identical, and the intermolecular forces decrease with increasing distance from the adsorption surface [19]. The Langmuir model is expressed by the following equation:

qe=qmKLCe1+KLCeE5

where qmis the maximum adsorption capacity (mg g−1) and KLis the Langmuir constant (L mg−1) [6, 7, 8, 10, 12, 16, 20, 21]. The linear form of the Langmuir model is:

Ceqe=1qmCe+1qmKLE6

The values of the constants qmand KLare determined by the slope and the intersection of the fitted line on the abscissa Ceand the ordinate Ce/qe, respectively. The characteristic of the Langmuir isotherm is that they can express a dimensionless constant called Balance Parameter or Separation Factor, which is expressed with the following equation:

RL=11+KL×CiE7

where RLis the equilibrium parameter (dimensionless), and Ciis the initial concentration (mg L−1). The RLvalues indicate what type of adsorption can be expected. RL > 1 is unfavorable, RL = 1 the adsorption is linear, RL = 0 is irreversible, and if 0 < RL < 1, adsorption is favorable [19, 22, 23].

### 2.4 Freundlich isotherm model

The Freundlich model takes into account the heterogeneity of the surface of the adsorbent and an exponential distribution of the active sites and their energies. The Freundlich model is expressed by the following equation:

qe=KF+CenE8

where KFis Freundlich’s constant in (mg g−1), and nis the Freundlich exponent related to the intensity of the adsorption; it is dimensionless. The equation of this model can be linearized as follows:

logqe=1nlogCe+logKFE9

The values of the constants KFand nare determined by the intercept and the slope of the graphical line on the ordinate ln qeand the abscissa ln Ce, respectively. The range of values of 1/nis between 0 and 1 showing the degree of nonlinearity between the concentration of the solution and the adsorption. If the value of 1/nis equal to 1, the adsorption is linear [6, 7, 8, 10, 12, 16, 21, 24]. High values of nindicate a relatively uniform surface, where low values mean high adsorption at low concentrations of solution. In addition, low values of nindicate the existence of a high proportion of high-energy active sites [25].

### 2.5 Temkin isotherm model

The Temkin isotherm model contains a factor that explicitly considers the adsorption interactions between the species and the adsorbate [14, 26]. The Temkin model is described with the following equation:

qe=BLnKTCeE10

where B = RT/b, bis Temkin’s constant, KTis the Bound Equilibrium Constant (L g−1), and Bis related to the heat of adsorption (J mol−1) [27]. The Temkin isotherm model is presented in linear form with the following equation:

qe=BLnCe+BLnKTE11

To obtain the constants Band KTdefined by the slope and intersection of the lines, respectively, plot qeon the ordinate and Ln Ceon the abscissa.

### 2.6 Dubinin–Radushkevich isotherm model (D–R)

The model of the Dubinin–Radushkevich isotherm (D–R) not assumes a homogeneous surface or a constant adsorption potential. The model is described with the following equation:

qe=qseKDRε2E12

where qsis the adsorption capacity (mol g−1), KDRis the Dubinin–Radushkevich constant (mol2/KJ2), and εis correlated with the following expression:

ε=RTLn1+1CeE13

where εis the potential of Polanyi, Ceis the concentration in equilibrium (mol L−1), qeis the equilibrium concentration in the adsorbent (mol g−1), Ris the universal gas constant (8.314 × 10−3 KJ mol−1 K−1), and Tis the absolute temperature °K. The linear form of this isotherm is expressed with the following equation:

qe=KDRε2+LnqsE14

where qsis the saturation capacity of the isotherm (mg g−1). The isotherm constants qsand KDR are obtained from the intercept and the slope, respectively, from plotting on the ordinate Ln qeand on the abscissa ε2. The constant KDR gives the mean of the free energy, E, is the adsorption per molecule of the sorbate when it is transferred to the surface of the solid from infinity in the solution and can be calculated using the following relationship:

E=12KDRE15

The magnitude of Eis used to estimate the type of adsorption process. The adsorption process is a chemical absorption if the magnitude of Eis greater than 16 KJ mol−1. In turn, when the magnitude of Eis less than 8 KJ mol−1, the type of absorption can be defined as a physical process [14, 27, 28]. There is talk of an ion exchange if 8 < E < 16 KJ mol−1 [29].

### 2.7 Redlich–Peterson isotherm model (R–P)

This model is a combination of the Langmuir and Freundlich isotherms. It is used to describe the adsorption on both homogeneous and heterogeneous surfaces. It is considered as a comparison between these two models [30]. This isotherm is expressed with the following expression:

qe=KRCe1+αRCeβE16

where KR is the constant of Redlich–Paterson (L g−1), Redlich–Paterson constant αR (L mg−1), βconstant of Redlich-Paterson (dimensionless). The values of βfluctuate between 0 and 1. At low concentrations, the Redlich–Paterson isotherm approximates Henry’s law [31]. When the constant βis very close to 1, it is the same as the Langmuir equation and in high concentrations its behavior approaches that of the Freundlich isotherm, since the βexponent tends to zero [32, 33]. From the transformation of the original equation, two linear forms are obtained. One of the linear forms of this isotherm is expressed with the following equation:

logKRCeqe1=βlogCe+logαRE17

The values of αRand βfor the above equation can be determined from the intercept and the slope, respectively, of the straight line of graphing logKRCeqe1versus log Ce[34]. Several values of the constants must be tested before obtaining the optimal line, in order to obtain the values of these. The range of values of these constants is very wide, ranging from 0.01 to several hundred, so it is not easy to obtain the correct values [31]. Another linear form of this equation is:

Ceqe=1KR+αRKRCeβE18

The constants of the Redlich–Paterson isotherm can be determined from the graph between Ceβand Ce/qe. However, its application is very complex since it includes three unknown parameters αR, KR, and β. Therefore, a minimization procedure is adopted to obtain the maximum value of the coefficient of determination R2, between the theoretical data for qeobtained from the linearized form of the isotherm equation of Redlich–Peterson and the experimental data [30]. By trial and error, values ​of βare adopted to obtain an optimal line. In the specific range, the values ​of bare limited, and it is easy to obtain the correct value [31].

### 2.8 Sips isotherm model

The Sips isotherm is a combined form of Langmuir and Freundlich isotherms applied for the prediction of heterogeneous adsorption systems. The Sips model avoids the inconveniences and limitations of the Langmuir or Freundlich models. At low concentrations, the adsorbate becomes Freundlich’s isotherm and, therefore, does not obey Henry’s law [35]. While in high concentrations, the isothermal formula of Langmuir is reduced [36]. The equation of the Sips isotherm is characterized by containing a dimensionless heterogeneity factor, βs. If βs = 1, the Sips equation is reduced to the Langmuir equation, which indicates that the adsorption process is homogeneous. The Sips isotherm constant (βs) confirms that the surface of the adsorbent is heterogeneous or not [33]. The Sips isotherm is expressed with the following equation:

qe=KSCeβs1+asCeβsE19

where KSis the equilibrium constant of the Sips isotherm (L mg−1), as is the maximum adsorption capacity (mg g−1), and βsis the model exponent (dimensionless) [35, 37, 38]. The linear form of the Sips isotherm is [39]:

Lnqe=βsLnCe+LnKSas×qeE20

The coefficients of the Sips isotherm are calculated from graphing ln(Ce) versus ln(qe), where βsis slope.

### 2.9 Halsey isotherm model

This model is used to evaluate multilayer adsorption in a system where metal ions are located relatively far from the surface of the adsorbent. The model is expressed with the following Equation [40]:

qe=explnKHlnCenHE21

The linear form from Halsey isotherm is [62, 63]:

Lnqe=1nHLn1Ce+1nHLnKHE22

where nHis the constant of the equation, and KHis Halsey’s equilibrium constant. The constants of the isotherm may be calculated graphing ln(1/Ce) versus ln(qe) and from the straight line obtained, the slope is nH, and the intercept represents KH[41].

### 2.10 Harkins–Jura isotherm model

The Harkins–Jura model describes a multilayer adsorption and the existence of a heterogeneous distribution of the pores of the adsorbent. The model is defined with the following expression:

1qe2=1AHJlogCe+BHJAHJE23

where BHJis a model constant, and AHJis another model constant. Graphing 1/qe2versus log(Ce), the model constants are calculated with slope AHJand intercept BHJ[41].

### 2.11 Elovich isotherm model

It assumes that the adsorption sites increase exponentially with adsorption, which implying a multilayer adsorption. This is expressed with [18, 41, 42, 43]:

qeqm=KECeexpqeqmE24

The linear form is expressed:

LnqeCe=1qmqe+LnKEqmE25

where qmis the maximum adsorption capacity of Elovich (mg g−1), and KEis the equilibrium constant of Elovich (L mg−1). KEand qmare calculated from the intercept and slope, respectively, of the straight line of ln(qe/Ce) versus qe[18, 41, 42, 43].

### 2.12 Flory-Huggins isotherm model

This model assumes the degree of coverage of the adsorbate on the adsorbent and expresses the degree of feasibility and spontaneity of the adsorption process. It included a parameter indicating the degree of coverage of the surface of the adsorbent, expressed as θ. The general form is given by the following Equation [43]:

θC0=KFH1θnE26

The linear form of it is expressed:

logθC0=nlog1θ+logKFHE27
θ=1CeC0E28

where KFHis the equilibrium constant of Flory–Huggins (L mg−1), nis the exponent of the model, and θis the coverage parameter of the adsorbent surface. For the calculation of the isotherm parameters, log θ/C0 versus log(1 − θ) should be plotted, where the slope and the intercept represent nFHand KFH, respectively. The constant KFHcan be used to calculate the spontaneity of Gibbs free energy [32, 43]. For its calculation is used the following Eq. (32):

ΔG°=RTlnKFHE29

where ΔG°Gibbs free energy change (KJ mol−1). The negative values of ΔG° indicate that the adsorption is thermodynamically spontaneous and feasible [44].

### 2.13 Fowler–Guggenheim isotherm model

It is one of the simplest equations that takes into account the lateral interaction of adsorbate molecules. Its general form is expressed below:

KFGCe=θ1θexp2θWRTE30

The linear form of it is expressed:

lnCe1θθ=2WRTθlnKFGE31

where KFGis the equilibrium constant of Fowler–Guggenheim (L mg−1), and Wis the energy of interaction between the molecules of the adsorbate (KJ mol−1) [37, 42]. The parameters of the equation are calculated by graphing ln[Ce(1 − θ)/θ] versus θ. From this straight line the intercept and the slope represent KFGand W, respectively. The charge and heat of adsorption vary linearly. When the values of Ware greater than zero, it indicates that the interaction between the adsorbate molecules is attractive, but if the values of Ware negative, the interaction is of the repulsion type, and if W = 0, there is no interaction [42].

### 2.14 Jovanovic isotherm model

This model assumes a superficial adsorption, it is an approximation of adsorption in a monolayer as expressed in the Langmuir model, but it assumes that there is no lateral interaction between the molecules. This model tolerates surface vibration of an adsorbed species [42] and allows some mechanical contact between the adsorbate and the adsorbent [18]. This model is expressed with the following expression [42]:

qe=qm1eKJCeE32

Its linear form is [18]:

qt=KJt12E33

where KJis the equilibrium constant of Jovanovic (L mg−1). When plotting ln qeversus Ce, the slope and the intercept are KJand qm, respectively.

### 2.15 Kiselev isotherm

This model is known as the model of the localized monomolecular layer and is only valid when θ ≥ 0.68. Its linear expression is [18]:

1Ce×1θ=Ki1θ+KiKnE34

where Kiis Kiselev’s constant (L mg−1), and Knis the equilibrium constant of complex formation between the molecules of the adsorbate. Constants are calculated by plotting 1/[Ce × (1 − θ)] versus 1/θwhere the slope and the intercept represent Kiand Ki × Kn, respectively [18].

### 2.16 Evaluating the suitability of the isothermal equations using experimental data

Statistical tools are used to evaluate the suitability of the values calculated with the selected isotherm equation in comparison with the experimentally determined values. To measure the differences between the experimentally observed values and the values calculated with the isothermal model, the so-called goodness of fit is used. Among the best known are the following:

#### 2.16.1 Average relative error (%ARE)

To evaluate the goodness of fit of the isothermal equations against the experimentally obtained data, the average relative error (% ARE) can be used. It is explained by the following equation:

%ARE=100ni=1nqi,calqi,expqi,expE35

where nis the data number, qi,calare the equilibrium values calculated with the mathematical isotherm expression (mg g−1), and qi,expare values obtained experimentally (mg g−1) [14, 28, 43]. An average relative error lower than or equal to 5% is considered adequate.

#### 2.16.2 Chi-square (χ2)

To determine the best-fitting isothermal model, the linear Chi-square values (χ2) were used along with the linear regressions (R2). The Chi-square test statistic is basically the sum of the squared errors of the differences between the experimental data and the data obtained by calculations with the models. Each square difference is divided by the corresponding data obtained by calculations using the models. If the values obtained using a model are similar to the experimental values, the value of χ2 is very small and close to zero. High values of χ2 imply a high discrepancy between the experiment and the model. Therefore, the analysis of the dataset of the Chi-square test may confirm the isotherm that best fits the adsorption system. The mathematical expression of the Chi-square test is explained below [29].

x2=i=1nqi,expqi,cal2qi,calE36

If values χ2 ≤ 0.05, the experimental data and the data obtained by calculations using the models have a statistically significant association.

## 3. Conclusions

The adsorption process of various compounds and elements is one of the most important environmental issues of the last decade due to its relevance, especially in emerging countries. To better understand how a solute can interact with the surface of a solid, it is useful to use the adsorption isotherm.

The effect of different variables on the adsorption process is of great importance for the development of treatment systems. The use of isotherms helps to determine these effects.

The analysis of equilibrium data is necessary to understand and interpret the adsorption mechanism and predict the removal of contaminants. They are necessary to know the mechanisms or transport phenomena that control the adsorption rate and thus be able to calculate, scale, and design an adsorption treatment system.

The obtained equilibrium data used to calculate isotherms can be used to scale up to a pilot plant experiment.

Among all the phenomena that determine the mobility of substances in porous aqueous media and in the aquatic environment, the transfer of substances from a mobile phase (liquid or gas) to a solid phase is the relevant phenomenon.

An isotherm is a curve describing the retention of a substance in a solid at different concentrations. It is an important tool for describing and predicting the mobility of this substance in the environment.

The adsorption process is a widely accepted treatment method because it is easy to apply and effective even at low concentrations. Adsorption has advantages over other treatment methods because it has low start-up and installation costs, is easy to operate, has low environmental risk, is resistant to toxic components, and offers significant potential for removing hazardous and unsafe contaminants.

The description of equilibrium data is valuable when using an equation that has a physical meaning, as it is then possible to correlate what is observed in one experiment with what is physically seen in a larger experiment.

## Conflict of interest

The authors declare no conflict of interest.

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Written By

Andres Abin-Bazaine, Alfredo Campos Trujillo and Mario Olmos-Marquez

Submitted: January 27th, 2022Reviewed: March 4th, 2022Published: April 19th, 2022