**2. Adsorption isotherms**

### **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:

$$\text{\%Adsorption} = \frac{\text{C}\_{\text{i}} - \text{C}\_{\text{f}}}{\text{C}\_{\text{i}}} \times 100\tag{1}$$

$$q\_e = \frac{\mathbf{C}\_i - \mathbf{C}\_f}{\mathbf{m}} \times \mathbf{V} \tag{2}$$

where Ci and Cf are the initial and final concentrations, respectively, of the ion to be removed (mg L�<sup>1</sup> ), qe is the adsorption capacity of the ion (adsorbate) by the adsorbent (mg g�<sup>1</sup> ), m is the mass of the adsorbent (g), and V is the volume of the ion solution to be removed (L) [6–12, 15–17].

Once the values of mass removal capacity (qe) and equilibrium concentration (Cf) have been calculated using a Cartesian coordinate system to represent the twodimensional 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: *Adsorption Isotherms: Enlightenment of the Phenomenon of Adsorption DOI: http://dx.doi.org/10.5772/intechopen.104260*

$$r = \frac{\sum \left[ \left( \mathbf{x}\_i - \overline{\mathbf{x}} \right) \left( \mathbf{y}\_i - \overline{\mathbf{y}} \right) \right]}{\sqrt{\sum \left( \mathbf{x}\_i - \overline{\mathbf{x}} \right)^2 + \sum \left( \mathbf{y}\_i - \overline{\mathbf{y}} \right)^2}} \tag{3}$$

where r is correlation coefficient, *xi* value 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*, <sup>P</sup>ð Þ *xi* � *<sup>x</sup>* <sup>2</sup> sum of the square of the deviation x, <sup>P</sup> *yi* � *<sup>y</sup>* � �<sup>2</sup> sum of the square of the deviation *<sup>y</sup>*. 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:

$$q\_{\varepsilon} = \mathcal{K}\_{HE} \mathcal{C}\_{\varepsilon} \tag{4}$$

where *qe* is the adsorption capacity in equilibrium (mg g�<sup>1</sup> ); *K*HE is the equilibrium constant of Henry's; and *Ce* is the equilibrium concentration of the metal ions in the solution (mg L�<sup>1</sup> ). From the straight line adjusted to graph *Ce* versus *qe*, the *K*HE 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:

$$q\_{\epsilon} = \frac{q\_{m}K\_{L}C\_{\epsilon}}{[1 + K\_{L}C\_{\epsilon}]} \tag{5}$$

where *qm* is the maximum adsorption capacity (mg g�<sup>1</sup> ) and *KL* is the Langmuir constant (L mg�<sup>1</sup> ) [6–8, 10, 12, 16, 20, 21]. The linear form of the Langmuir model is:

$$\frac{C\_{\epsilon}}{q\_{\epsilon}} = \frac{1}{q\_{m}} C\_{\epsilon} + \frac{1}{q\_{m} K\_{L}} \tag{6}$$

The values of the constants *qm* and *KL* are determined by the slope and the intersection of the fitted line on the abscissa *Ce* and 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:

$$R\_L = \frac{1}{\left(1 + (K\_L \times C\_i)\right)}\tag{7}$$

where *RL* is the equilibrium parameter (dimensionless), and *Ci* is the initial concentration (mg L�<sup>1</sup> ). The *RL* values 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:

$$q\_e = K\_F + C\_e^\* \tag{8}$$

where *KF* is Freundlich's constant in (mg g�<sup>1</sup> ), and *n* is the Freundlich exponent related to the intensity of the adsorption; it is dimensionless. The equation of this model can be linearized as follows:

$$\log q\_{\epsilon} = \frac{1}{n} \log \mathcal{C}\_{\epsilon} + \log \mathcal{K}\_{F} \tag{9}$$

The values of the constants *KF* and *n* are determined by the intercept and the slope of the graphical line on the ordinate ln *qe* and the abscissa ln *Ce*, respectively. The range of values of 1/*n* is between 0 and 1 showing the degree of nonlinearity between the concentration of the solution and the adsorption. If the value of 1/*n* is equal to 1, the adsorption is linear [6–8, 10, 12, 16, 21, 24]. High values of *n* indicate a relatively uniform surface, where low values mean high adsorption at low concentrations of solution. In addition, low values of *n* indicate 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:

$$q\_{\epsilon} = BLn(K\_T C\_{\epsilon}) \tag{10}$$

where *B* = *RT*/*b*, *b* is Temkin's constant, *KT* is the Bound Equilibrium Constant (L g�<sup>1</sup> ), and *B* is related to the heat of adsorption (J mol�<sup>1</sup> ) [27]. The Temkin isotherm model is presented in linear form with the following equation:

$$q\_{\epsilon} = BLnC\_{\epsilon} + BLnK\_{T} \tag{11}$$

To obtain the constants *B* and *KT* defined by the slope and intersection of the lines, respectively, plot *qe* on the ordinate and Ln *Ce* on 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:

$$\mathfrak{q}\_{\mathfrak{e}} = \mathfrak{q}\_{\mathfrak{e}} \mathfrak{e}^{-\mathbb{K}\_{\text{DR}} \mathfrak{e}^2} \tag{12}$$

where *qs* is the adsorption capacity (mol g�<sup>1</sup> ), *KDR* is the Dubinin–Radushkevich constant (mol<sup>2</sup> /KJ<sup>2</sup> ), and *ε* is correlated with the following expression:

$$
\varepsilon = RT\text{Ln}\left(\mathbf{1} + \frac{\mathbf{1}}{\mathbf{C}\_{\epsilon}}\right) \tag{13}
$$

where *ε* is the potential of Polanyi, *Ce* is the concentration in equilibrium (mol L�<sup>1</sup> ), *qe* is the equilibrium concentration in the adsorbent (mol g�<sup>1</sup> ), *R* is the universal gas constant (8.314 � <sup>10</sup>�<sup>3</sup> KJ mol�<sup>1</sup> <sup>K</sup>�<sup>1</sup> ), and *T* is the absolute temperature °K. The linear form of this isotherm is expressed with the following equation:

$$q\_e = -K\_{\rm DR} e^2 + L n q\_s \tag{14}$$

where *qs* is the saturation capacity of the isotherm (mg g�<sup>1</sup> ). The isotherm constants *qs* and *K*DR are obtained from the intercept and the slope, respectively, from plotting on the ordinate Ln *qe* and on the abscissa *ε* 2 . The constant *K*DR 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:

$$E = \frac{1}{\sqrt{-\text{2}K\_{\text{DR}}}} \tag{15}$$

The magnitude of *E* is used to estimate the type of adsorption process. The adsorption process is a chemical absorption if the magnitude of *E* is greater than 16 KJ mol�<sup>1</sup> . In turn, when the magnitude of *E* is less than 8 KJ mol�<sup>1</sup> , 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�<sup>1</sup> [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:

$$q\_{\epsilon} = \frac{K\_{\mathbb{R}} C\_{\epsilon}}{\left(1 + a \mathcal{R} C\_{\epsilon}^{\beta}\right)}\tag{16}$$

where *K*<sup>R</sup> is the constant of Redlich–Paterson (L g�<sup>1</sup> ), Redlich–Paterson constant αR (L mg�<sup>1</sup> ), *β* constant of Redlich-Paterson (dimensionless). The values of *β* fluctuate between 0 and 1. At low concentrations, the Redlich–Paterson isotherm

approximates Henry's law [31]. When the constant *β* is 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 *β* exponent 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:

$$\log\left[\left(K\_R \frac{\mathbf{C}\_\epsilon}{q\_\epsilon}\right) - \mathbf{1}\right] = \beta \log \mathbf{C}\_\epsilon + \log\left(a\mathbf{R}\right) \tag{17}$$

The values of *αR* and *β* for the above equation can be determined from the intercept and the slope, respectively, of the straight line of graphing *log KR Ce qe* � � � <sup>1</sup> h i versus 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:

$$\frac{C\_{\epsilon}}{q\_{\epsilon}} = \frac{1}{K\_{R}} + \frac{aR}{K\_{R}} C\_{\epsilon}^{\beta} \tag{18}$$

The constants of the Redlich–Paterson isotherm can be determined from the graph between *C<sup>β</sup> <sup>e</sup>* and *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 *R*<sup>2</sup> , between the theoretical data for *qe* obtained from the linearized form of the isotherm equation of Redlich–Peterson and the experimental data [30]. By trial and error, values of *β* are adopted to obtain an optimal line. In the specific range, the values of *b* are 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:

$$q\_{\epsilon} = \frac{K\_S C\_{\epsilon}^{\beta\_s}}{1 + asC\_{\epsilon}^{\beta\_s}} \tag{19}$$

where *KS* is the equilibrium constant of the Sips isotherm (L mg�<sup>1</sup> ), as is the maximum adsorption capacity (mg g�<sup>1</sup> ), and *β<sup>s</sup>* is the model exponent (dimensionless) [35, 37, 38]. The linear form of the Sips isotherm is [39]:

*Adsorption Isotherms: Enlightenment of the Phenomenon of Adsorption DOI: http://dx.doi.org/10.5772/intechopen.104260*

$$Ln\left(q\_{\epsilon}\right) = \beta\_{\iota}Ln\left(\mathbf{C}\_{\epsilon}\right) + Ln\left(\mathbf{K}\_{\mathcal{S}} - \mathbf{a}\_{\iota} \times q\_{\epsilon}\right) \tag{20}$$

The coefficients of the Sips isotherm are calculated from graphing ln(*Ce*) versus ln (*qe*), where *β<sup>s</sup>* is 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]:

$$q\_e = \exp\left(\frac{\ln K\_H - \ln C\_e}{n\_H}\right) \tag{21}$$

The linear form from Halsey isotherm is [41, 43]:

$$Ln(q\_{\epsilon}) = -\frac{1}{n\_H}Ln\frac{1}{C\_{\epsilon}} + \left[\left(\frac{1}{n\_H}\right)Ln(K\_H)\right] \tag{22}$$

where *nH* is the constant of the equation, and *KH* is 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:

$$\frac{1}{q\_{\epsilon}^{2}} = -\frac{1}{A\_{H\!\!\!\!}} \log \left( \text{Ce} \right) + \frac{B\_{H\!\!\!\!\!\!}}{A\_{H\!\!\!\!\!\!}} \tag{23}$$

where *BHJ* is a model constant, and *AHJ* is another model constant. Graphing 1*=q*<sup>2</sup> *<sup>e</sup>* versus log(*Ce*), the model constants are calculated with slope *AHJ* and 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, 41–43]:

$$\frac{q\_\epsilon}{q\_m} = K\_E \mathbf{C}\_\epsilon \exp\left(-\frac{q\_\epsilon}{q\_m}\right) \tag{24}$$

The linear form is expressed:

$$Ln\frac{q\_{\epsilon}}{C\_{\epsilon}} = -\frac{1}{q\_{m}}q\_{\epsilon} + LnK\_{E}q\_{m} \tag{25}$$

where *qm* is the maximum adsorption capacity of Elovich (mg g�<sup>1</sup> ), and *KE* is the equilibrium constant of Elovich (L mg�<sup>1</sup> ). *KE* and *qm* are calculated from the intercept and slope, respectively, of the straight line of ln(*qe*/*Ce*) versus *qe* [18, 41–43].

### **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]:

$$\frac{\theta}{C\_0} = K\_{FH} (1 - \theta)^n \tag{26}$$

The linear form of it is expressed:

$$\log\frac{\theta}{C\_0} = n\log(1-\theta) + \log K\_{FH} \tag{27}$$

$$\theta = \left(\mathbf{1} - \frac{\mathbf{C}\_{\epsilon}}{\mathbf{C}\_{0}}\right) \tag{28}$$

where *KFH* is the equilibrium constant of Flory–Huggins (L mg�<sup>1</sup> ), *n* is the exponent of the model, and *θ* is the coverage parameter of the adsorbent surface. For the calculation of the isotherm parameters, log *θ*/*C*<sup>0</sup> versus log(1 � *θ*) should be plotted, where the slope and the intercept represent *nFH* and *KFH*, respectively. The constant *KFH* can be used to calculate the spontaneity of Gibbs free energy [32, 43]. For its calculation is used the following Eq. (32):

$$
\Delta G^\circ = RT \ln K\_{FH} \tag{29}
$$

where Δ*G*° Gibbs free energy change (KJ mol�<sup>1</sup> ). The negative values of Δ*G*° 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:

$$K\_{FG} \mathcal{C}\_{\varepsilon} = \frac{\theta}{1 - \theta} \exp\left(\frac{2\theta W}{RT}\right) \tag{30}$$

The linear form of it is expressed:

$$
\ln \left[ \frac{\mathbf{C}\_{\epsilon} (1 - \theta)}{\theta} \right] = \frac{2W}{RT} \theta - \ln K\_{\text{FG}} \tag{31}
$$

where *KFG* is the equilibrium constant of Fowler–Guggenheim (L mg�<sup>1</sup> ), and *W* is the energy of interaction between the molecules of the adsorbate (KJ mol�<sup>1</sup> ) [37, 42]. The parameters of the equation are calculated by graphing ln[*Ce*(1 � *θ*)/*θ*] versus *θ*.

*Adsorption Isotherms: Enlightenment of the Phenomenon of Adsorption DOI: http://dx.doi.org/10.5772/intechopen.104260*

From this straight line the intercept and the slope represent *KFG* and *W*, respectively. The charge and heat of adsorption vary linearly. When the values of *W* are greater than zero, it indicates that the interaction between the adsorbate molecules is attractive, but if the values of *W* are 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]:

$$q\_{\epsilon} = q\_m \left( \mathbf{1} - e^{K\_f G\_r} \right) \tag{32}$$

Its linear form is [18]:

$$q\_t = K\_l t^{\frac{1}{2}} \tag{33}$$

where *KJ* is the equilibrium constant of Jovanovic (L mg�<sup>1</sup> ). When plotting ln *qe* versus *Ce*, the slope and the intercept are *KJ* and *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]:

$$\frac{1}{C\_{\epsilon} \times (1 - \theta)} = K\_i \frac{1}{\theta} + K\_i K\_n \tag{34}$$

where *Ki* is Kiselev's constant (L mg�<sup>1</sup> ), and *Kn* is the equilibrium constant of complex formation between the molecules of the adsorbate. Constants are calculated by plotting 1/[*Ce* � (1 � *θ*)] versus 1/*θ* where the slope and the intercept represent *Ki* and *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:

$$\%ARE = \frac{100}{n} \sum\_{i=1}^{n} \left\| \frac{q\_{i,cal} - q\_{i,exp}}{q\_{i,exp}} \right\| \tag{35}$$

where *n* is the data number, *qi,cal* are the equilibrium values calculated with the mathematical isotherm expression (mg g�<sup>1</sup> ), and *qi,exp* are values obtained experimentally (mg g�<sup>1</sup> ) [14, 28, 43]. An average relative error lower than or equal to 5% is considered adequate.

#### *2.16.2 Chi-square (*χ*<sup>2</sup> )*

To determine the best-fitting isothermal model, the linear Chi-square values (*χ*<sup>2</sup> ) were used along with the linear regressions (*R*<sup>2</sup> ). 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 *χ* <sup>2</sup> is very small and close to zero. High values of *χ* <sup>2</sup> 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].

$$\infty^2 = \sum\_{i=1}^n \frac{\left(q\_{i,cap} - q\_{i,cal}\right)^2}{q\_{i,cal}} \tag{36}$$

If values *χ* <sup>2</sup> ≤ 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.

*Adsorption Isotherms: Enlightenment of the Phenomenon of Adsorption DOI: http://dx.doi.org/10.5772/intechopen.104260*

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.
