Kinetic and Equilibrium Models of Adsorption

Abstract

In adsorption study, isotherms and kinetics of adsorption process provide pieces of information underlying the mechanisms and dynamics of the process. Several equilibrium and kinetic models are usually employed for performing the experimental design of an adsorption system. In this chapter, the Langmuir, Freundlich, Sips, Liu, Redlich–Peterson nonlinear equations, as well as other unusual isotherm models (Hill, Khan, Radke–Prausnitz, Toth) are discussed. For the kinetics of adsorption, the pseudo-first-order, pseudo-second-order, general-order, Avrami fractionary order, and Elovich chemisorption models are explained. The importance of statistical parameters such as coefficient of determination ($R^{2}$), adjusted coefficient of determination ($R_{\text{adj}}^{2}$), and standard deviation (root of mean square error) are highlighted. The usage of linearized and nonlinearized equations are illustrated and explained. Some common mistakes commonly committed in the literature using linearized equilibrium and kinetic adsorption models as well as other polemic points in adsorption research are pointed out. Analytical techniques together with thermodynamical data of enthalpy and entropy changes are needed to ascertain if an adsorption process is a chemical or a physical process.

3.1 Introduction

At a constant temperature, an adsorption isotherm describes the relationship between the amount of adsorbate adsorbed by the adsorbent (q _e) and the adsorbate concentration remaining in solution after equilibrium is reached (C _e). The parameters from the adsorption equilibrium models provide useful pieces of information on the surface properties, adsorption mechanism and interaction between the adsorbent and adsorbate. There are numerous equations for describing the adsorption equilibrium of an adsorbate on an adsorbent. The most employed and discussed in the literature is the Langmuir equation [1]. Other isotherm models such as Freundlich isotherm [2], Sips isotherm [3], Liu isotherm [4], Redlich–Peterson isotherm [5] are also well discussed in the literature.

Adsorption kinetic studies are important in treatment of aqueous effluents because they provide valuable information on the mechanism of the adsorption process. Many kinetic models were developed in order to find intrinsic kinetic adsorption constants. In this chapter we will discuss the adsorption kinetic models based on the chemical reaction (pseudo-first order equation [6], pseudo-second-order equation [7], general-order equation [8]), and the empiric models (Avrami fractionary model [9], and Elovich chemisorption model [9]).

3.2 Equilibrium Isotherm Models

3.2.1 Langmuir Isotherm Model

The Langmuir [1] isotherm is based on the following assumptions:

• adsorbates are chemically adsorbed at a fixed number of well-defined sites;

• a monolayer of the adsorbate is formed over the surface of the adsorbent when it gets saturated;

• each site can hold only one adsorbate species;

• all sites are energetically equivalent;

• interactions between the adsorbate species do not exist.

The Langmuir isotherm equation is depicted by Eq. 3.1:

$$q_{\text{e}} = \frac{{Q_{\hbox{max} } \cdot K_{\text{L}} \cdot C_{\text{e}} }}{{1 + K_{\text{L}} \cdot C_{\text{e}} }}$$

(3.1)

where q _e is the amount of adsorbate adsorbed at the equilibrium (mg g⁻¹), C _e is the supernatant adsorbate concentration at the equilibrium (mg L⁻¹), K _L is the Langmuir equilibrium constant (L mg⁻¹), and Q _max is the maximum adsorption capacity of the adsorbent (mg g⁻¹) assuming a monolayer of adsorbate uptake by the adsorbent.

It is pertinent to use an example to describe the Langmuir model. Figure 3.1 presents the data of adsorption of Direct Blue 53 (DB-53) onto multi-walled carbon nanotubes (MWCNT) adsorbent [10]. In this section of the chapter, only nonlinear equations will be considered. An explanation for this will be given at the subsequent sections in the chapter, which will show the main problems that occur with the linearization of equilibrium and kinetic models of adsorption. Figure 3.1, therefore, shows the nonlinearized Langmuir isotherm plot for the adsorption of DB-53 dye using MWCNT as adsorbent [10].

Fig. 3.1

Langmuir adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

The $R_{\text{adj}}^{2}$ for nonlinear Langmuir isotherm was 0.9608, the Q _max was 332.4 mg g⁻¹ and the Langmuir equilibrium constant was 3.980 L mg⁻¹. The total standard deviation (SD-root of mean square error) of the fitting was 17.89 mg g⁻¹.

3.2.2 Freundlich Isotherm Model

Freundlich [2] isotherm model is an exponential equation, and assumes that the concentration of adsorbate on the adsorbent surface increases as the adsorbate concentration increases. Theoretically, using this expression, an infinite amount of adsorption will occur. Similarly, the model assumes that the adsorption could occur via multiple layers instead of a single layer. The equation has a wide application in heterogeneous systems. Equation 3.2 shows the Freundlich isotherm model;

$$q_{\text{e}} = K_{\text{F}} \cdot C_{\text{e}}^{{{1 \mathord{\left/ {\vphantom {1 {n_{\text{F}} }}} \right. \kern-0pt} {n_{\text{F}} }}}}$$

(3.2)

where K _F is the Freundlich equilibrium constant (mg g⁻¹(mg L⁻¹)^−1/n), and n _F is the Freundlich exponent (dimensionless). Figure 3.2 shows the nonlinearized Freundlich isotherm plot for the adsorption of DB-53 dye using MWCNT as adsorbent [10].

Fig. 3.2

Freundlich adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

The $R_{\text{adj}}^{2}$ for nonlinear Freundlich isotherm was 0.9946 while the Freundlich equilibrium constant was $249.1\,({\text{mg}}\;{\text{g}}^{ - 1} ({\text{mg}}\;{\text{L}}^{ - 1} )^{{ - 1/n_{F} }} )$. The total SD of the fitting was 13.21 mg g⁻¹. When compared with Langmuir isotherm, the Freundlich isotherm model with $R_{\text{adj}}^{2}$ is closer to unity and a lower SD presented a better fit.

3.2.3 Sips Isotherm Model

Sip model, an empirical model, consists of the combination of the Langmuir and Freundlich isotherm models. The Sips [3] model takes the following form:

$$q_{\text{e}} = \frac{{Q_{\hbox{max} } \cdot K_{\text{S}} \cdot C_{\text{e}}^{{1/n_{\text{S}} }} }}{{1 + K_{\text{S}} \cdot C_{\text{e}}^{{1/n_{\text{S}} }} }}\quad {\text{where }}0 < 1/n_{\text{s}} \le 1$$

(3.3)

In Eq. 3.3, K _S is the Sips equilibrium constant (mg L⁻¹)^−1/n, Q _max is the Sips maximum adsorption capacity (mg g⁻¹), and n _S is the exponent. It is assumed that the 1/n _S should be ≤1 for integration purpose [3].

Although several works in the literature using Sips isotherm really do not take into account this consideration. The nonlinearized Sips isotherm curve for the adsorption of DB-53 dye using MWCNT as adsorbent is shown in Fig. 3.3 [10].

Fig. 3.3

Sips adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

At low adsorbate concentrations, Sips equation relatively reduces to the Freundlich isotherm, but it predicts a monolayer adsorption capacity characteristic of the Langmuir isotherm at high adsorbate concentrations.

It was observed that the Sips exponent was restricted to 1 [3], in this manner, this isotherm model has the same parameters as the Langmuir isotherm, however, the $R_{\text{adj}}^{2}$ and the SD of the Sips isotherm were worse than those of the Langmuir isotherm. In that case, when n _s = 1, the Langmuir expression is preferred because the Sips isotherm has three parameters while Langmuir isotherm has just two. This parametric difference worsens the values of $R_{\text{adj}}^{2}$ and SD of the Sips isotherm. More details of the statistical analysis (reduced chi-squared; SD; coefficient of determination—$R^{2}$ and Adjusted coefficient of determination—$R_{\text{adj}}^{2}$) will be provided later in this chapter after the kinetic models.

3.2.4 Liu Isotherm Model

The Liu isotherm model [4] is a combination of the Langmuir and Freundlich isotherm models, but the monolayer assumption of Langmuir model and the infinite adsorption assumption that originates from the Freundlich model are discarded. The Liu model predicts that the active sites of the adsorbent cannot possess the same energy.

Therefore, the adsorbent may present active sites preferred by the adsorbate molecules for occupation [4], however, saturation of the active sites should occur unlike in the Freundlich isotherm model. Equation 3.4 defines the Liu isotherm model.

$$q_{\text{e}} = \frac{{Q_{\hbox{max} } \cdot \left( {K_{g} \cdot C_{\text{e}} } \right)^{{n_{\text{L}} }} }}{{1 + \left( {K_{g} \cdot C_{\text{e}} } \right)^{{n_{\text{L}} }} }}$$

(3.4)

where K _g is the Liu equilibrium constant (L mg⁻¹); n _L is dimensionless exponent of the Liu equation, and Q _max is the maximum adsorption capacity of the adsorbent (mg g⁻¹). Contrary to the Sips isotherm, n _L could assume any positive value.

Figure 3.4 shows the nonlinearized Liu isotherm plot for the adsorption of DB-53 dye using MWCNT as adsorbent [10].

Fig. 3.4

Liu adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

From the fit, the $R_{\text{adj}}^{2}$ obtained was 0.9998, which is very good for a nonlinear isotherm. Similarly, the SD for Liu isotherm model was only 1.143 mg g⁻¹. This value was 15.65, 11.56, and 16.29 times lower than the SD values of Langmuir, Freundlich, and Sips isotherms models, respectively, indicating that this isotherm model was a better fit to the experimental equilibrium data [10]. The advantages of Liu isotherm model (a 3-parameter isotherm) over the Sips isotherm model is that the exponent of Liu isotherm could admit any positive value unlike the exponent of Sips that is limited to 1/n ≤ 1.

3.2.5 Redlich–Peterson Isotherm Model

This is an empirical equation that describes an equilibrium isotherm as shown in Eq. 3.5 [5].

$$q_{\text{e}} = \frac{{K_{\text{RP}} \cdot C_{\text{e}} }}{{1 + a_{\text{RP}} \cdot C_{\text{e}}^{g} }}\quad {\text{where }}\,\, 0 < {g} \le 1$$

(3.5)

where K _RP and a _RP are Redlich–Peterson constants with the respective units of L g⁻¹ and (mg L⁻¹)^−g, and g is the Redlich–Peterson exponent (dimensionless) whose value should be ≤1. This equation becomes linear at a low surface coverage (g = 0) and reduces to a Langmuir isotherm when g = 1. Figure 3.5 presents the nonlinearized Redlich–Peterson isotherm curve for the adsorption of DB-53 dye using MWCNT as adsorbent [10].

Fig. 3.5

Redlich–Peterson adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

The $R_{\text{adj}}^{2}$ and the SD obtained with the Redlich–Peterson model were better than those of Langmuir, Freundlich, and Sips isotherm models, being just a little bit worse than the Liu isotherm model. This model has three parameters, and the exponent is limited to ≤1.

3.2.6 Other Unusual Isotherm Models

Apart from equilibrium models commonly described in the literature, there are other models that can be used to describe how an adsorbate is adsorbed onto an adsorbent [11, 12]. Mathematical expressions of some of these adsorption equilibrium models are given in Table 3.1 [11, 12].

Table 3.1 Unusual equilibrium adsorption isotherms

Figures 3.6, 3.7, 3.8, and 3.9 shows the nonlinearized isotherms of Hill, Khan, Radke–Prausnitz, and Toth, respectively, for the adsorption of DB-53 dye using MWCNT as adsorbent [10].

Fig. 3.6

Hill adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

Fig. 3.7

Khan adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

Fig. 3.8

Radke–Prausnitz adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

Fig. 3.9

Toth adsorption equilibrium isotherm of DB-53 dye using MWCNT at 50 °C and pH 2.0

3.2.7 General Comments About the Isotherm Models

Examining all the set of isotherm parameters, the $R_{\text{adj}}^{2}$ as well as the SD, it is observed that Liu and Hill isotherm models are identical, except for their equilibrium constants (K _L ≠ K _H). The Liu isotherm is actually derived from Hill isotherm model [4]. The $R_{\text{adj}}^{2}$, SD and the isotherm parameters for Khan isotherm model and Radke–Prausnitz are also similar. Therefore, there is no significant difference in the Khan and Radke–Prausnitz isotherm models. The only difference between these two equations (see Table 3.1) is the exponent (n _K = 1/n _RP). This observation confirms that several isotherms that are reported in the literature are exactly the same isotherm model with slightly differences. The Toth isotherm model has the lowest value of SD of all the nine models described in this chapter (Langmuir, Freudlich, Sips, Liu, Redlich–Peterson, Hill, Khan, Radke–Prausnitz, and Toth). Mathematically, Toth is the best fit isotherm model. On the other hand, the value of the equilibrium constant and the value of maximum amount adsorbed may not correspond to the correct values. An examination of just one temperature value may not provide a realistic conclusion. It is necessary to use different isotherm models with different temperature values. The model with the best values of $R_{\text{adj}}^{2}$ and the lowest values of SD for the majorities of temperature values should be taken as the best isotherm model to describe the adsorption behavior of an adsorbate on a given adsorbent. The Khan and Radke–Prausnitz isotherm models are three-parameter isotherm models, whose parameter values are completely different from other isotherm models; however, they exhibited good values of $R_{\text{adj}}^{2}$ and SD. Therefore, according to the authors’ experience, the statistical analysis is to guide the adsorption researcher to employ a good isotherm model; however, it is necessary to check if the obtained parameters have physical meaning, and also verify the behavior of each isotherm model at different temperature values. Based on the nine values of the parameters, and the experience of the authors, the best results are the Liu and the Hill isotherm values, where the maximum sorption capacity of the adsorbent was 409.4 mg g⁻¹ for adsorption of DB-53 dye onto MWCNT [10]. The complete analysis of different isotherms for the same adsorbent and adsorbate [10] intends to attract the attention of the reader to use different isotherm models (Liu, Sips, Redlich–Peterson, besides Langmuir and Freundlich) for explaining the equilibrium data obtained in the laboratory during adsorption experiment. Another critical issue is that some isotherm models present equilibrium constant values and maximum sorption capacities that do not sometimes correspond to the actual values of these parameters.

3.3 Kinetic Adsorption Models

3.3.1 Kinetic Models Based on the Order of Reaction

The study of adsorption kinetic is important in the treatment of aqueous effluents using nanomaterials because it provides valuable pieces of information on the reaction pathways and the mechanism of adsorption reactions.

Many kinetic models were developed to evaluate the intrinsic kinetic adsorption constants. Traditionally, the kinetics of adsorption of an adsorbate is described using the expressions originally given by Lagergren [6]. A simple kinetic analysis of adsorption is the pseudo-first-order equation in the form of Eq. 3.6:

$$\frac{\text{d}q}{\text{d}t} = k_{\text{f}} \cdot \left( {q_{\text{e}} - q_{t} } \right)$$

(3.6)

where q _t is the amount of adsorbate adsorbed at time t (mg g⁻¹), q _e is the equilibrium adsorption capacity (mg g⁻¹), k _f is the pseudo-first-order rate constant (min⁻¹), and t is the contact time (min). The integration of Eq. 3.6 with initial conditions, q _t  = 0 at t = 0, and q _t  = q _t at t = t leads to Eq. 3.7:

$${ \ln }\left( {q_{\text{e}} - q_{t} } \right) = { \ln }\left( {q_{\text{e}} } \right) - k_{\text{f}} \cdot t$$

(3.7)

A nonlinear rearrangement of Eq. 3.7 gives Eq. 3.8:

$$q_{t} = q_{\text{e}} \cdot \left[ {1 - \exp \left( { - k_{\text{f}} \cdot t} \right)} \right]$$

(3.8)

Equation 3.8 is known as pseudo-first-order kinetic adsorption model.

In addition, a pseudo-second-order equation [7] based on equilibrium adsorption capacity is shown in Eq. 3.9:

$$\frac{\text{d}q_{t}}{\text{d}t} = k_{\text{s}} \cdot \left( {q_{\text{e}} - q_{t} } \right)^{2}$$

(3.9)

where, k _s is the pseudo-second-order rate constant (g mg⁻¹ min⁻¹). The integration of Eq. 3.9 with initial conditions, q _t  = 0 at t = 0, and q _t  = q _t at t = t leads to Eq. 3.10:

$$q_{t} = \frac{{k_{\text{s}} \cdot q_{\text{e}}^{2} \cdot t}}{{1 + q_{\text{e}} \cdot k_{\text{s}} \cdot t}}$$

(3.10)

Equation 3.10 is known as pseudo-second-order kinetic adsorption model.

The initial sorption rate (h _o, expressed in mg g⁻¹ min⁻¹) can be obtained when t approaches zero [13] as shown in Eq. 3.11:

$$h_{\text{o}} = k_{\text{s}} \cdot q_{\text{e}}^{2}$$

(3.11)

The pseudo-first-order and pseudo-second-order are the most commonly employed kinetic models for describing adsorption process based on chemical reactions kinetic. Another approach to this theme is described below.

The exponents of rate laws of chemical reactions are usually independent of the coefficients of chemical equations, but are sometimes related. This assertion implies that the order of a chemical reaction depends solely on the experimental data. Adsorption process, which is considered to be the rate determining step, helps in establishing the general rate law equation of adsorption process [12, 14]. Attention is now focused on the change in the effective number of active sites at the surface of adsorbent during adsorption instead of concentration of adsorbate in the bulk solution. Applying reaction rate law to Eq. 3.12 gives adsorption rate expression.

$$\frac{{{\text{d}}q}}{{{\text{d}}t}} = k_{\text{N}} \cdot \left( {q_{\text{e}} - q_{\text{t}} } \right)^{n}$$

(3.12)

where k _N is the rate constant, q _e is the amount of adsorbate adsorbed by adsorbent at equilibrium, q _t is the amount of adsorbate adsorbed by adsorbent at a given time, t, and n is the order of adsorption with respect to the effective concentration of the adsorption active sites present on the surface of adsorbent [8]. Application of universal rate law to adsorption process led to Eq. 3.12, which can be used without assumptions. Theoretically, the exponent n in Eq. 3.12 can be an integer or noninteger rational number [8, 12, 14].

Equation 3.13 describes the number of the active sites (θ _t) available on the surface of adsorbent for adsorption [12, 14].

$$\theta_{t} = 1 - \frac{{q_{\text{e}} }}{{q_{t} }}$$

(3.13)

Equation 3.14 describes the relationship between the variable (θ _t ) and rates of adsorption.

$$\frac{\text{d}\theta_{\text{t}}}{\text{d}t} = - k\theta_{\text{t}}^{n}$$

(3.14)

where $k = k_{\text{N}} (q_{\text{e}} )^{n - 1} .\, \theta_{t} = 1$ if an adsorbent has not adsorbed. The value of θ _t decreases during adsorption process. θ _t approaches a fixed value when adsorption process reaches equilibrium. θ _t = 0 for a saturated adsorbent [8]. Equation 3.14 gives Eq. 3.15 after integration:

$$\int\limits_{1}^{\theta } {\frac{\text{d}\theta_{t}}{{\theta_{\text{t}}^{n} }}} = - k \int\limits_{0}^{t} {\text{d}t}$$

(3.15)

Equation 3.15 also gives Eq. 3.16:

$$\frac{1}{1 - n} \cdot \left[ {\theta_{t}^{1 - n} - 1} \right] = - kt$$

(3.16)

Rearrangement of Eq. 3.16 gives Eq. 3.17.

$$\theta_{t} = \left[ {1 - k\left( {1 - n} \right) \cdot t} \right]^{1/1 - n}$$

(3.17)

Substituting Eq. 3.13 into Eq. 3.17, and put $k = k_{\text{N}} (q_{\text{e}} )^{n - 1}$ yields Eq. 3.18.

$$q_{t} = q_{\text{e}} - \frac{{q_{\text{e}} }}{{\left[ {k_{\text{N}} \left( {q_{\text{e}} } \right)^{n - 1} \cdot t \cdot \left( {n - 1} \right) + 1} \right]^{1/1 - n} }}$$

(3.18)

Equation 3.18 is regarded as the general-order kinetic equation of adsorption process, which is valid for n ≠ 1 [8]. A special case of Eq. 3.14 is the pseudo-first-order kinetic model (n = 1) [12, 14].

$$\frac{\text{d}\theta_{\text{t} }}{\text{d}t} = - k\theta_{\text{t}}^{1}$$

(3.19)

Integration of Eq. 3.19 gives Eq. 3.20.

$$\theta_{t} = \exp \left( { - k \cdot t} \right)$$

(3.20)

Substitution of Eq. 3.13 into Eq. 3.20, and put k = k ₁ gives pseudo-first-order kinetic model as shown in Eq. 3.21.

$$q_{t} = q_{\text{e}} \left[ {1 - \exp \left( { - k_{1} \cdot t} \right)} \right]$$

(3.21)

Pseudo-first-order kinetic equation is a special case of general kinetic model of adsorption. It must be noted that Eq. 3.21 is the same as Eq. 3.8 using the adsorption rate expression. When n = 2, the pseudo-second-order kinetic model is a special case of Eq. 3.18 [8].

$$q_{t} = q_{\text{e}} - \frac{{q_{\text{e}} }}{{\left[ {k_{2} \left( {q_{\text{e}} } \right) \cdot t + 1} \right]}}$$

(3.22)

Equation 3.22 on rearrangement gives Eq. 3.23.

$$q_{t} = \frac{{q_{\text{e}}^{2} k_{2} t}}{{\left[ {k_{2} \left( {q_{\text{e}} } \right) \cdot t + 1} \right]}}$$

(3.23)

Equation 3.23 is the pseudo-second-order kinetic adsorption model, which is exactly the same as Eq. 3.10. Therefore, the general-order adsorption Eq. 3.18 could give rise to pseudo-second-order when n = 2, while the pseudo-first order is obtained from Eq. 3.12 (adsorption rate expression).

3.3.2 Empiric Models

The Elovich equation is generally applied to chemisorption kinetics [9]. The equation has been used satisfactorily for some chemisorption processes [15] and has been found to cover a wide range of slow adsorption rates. The same equation is often valid for systems in which the adsorbing surface is heterogeneous. The Elovich equation is given in Eq. 3.24.

$$\frac{\text{d}q_{\text{t} }}{\text{d}t} = {\alpha} {\exp} \left( { - \beta q_{t} } \right)$$

(3.24)

Integrating Eq. 3.24 using boundary conditions; q _t  = 0 at t = 0 and q _t  = q _t at t = t gives Eq. 3.25:

$$q_{t} = \frac{1}{\beta }\ln \left( {t + t_{\text{o}} } \right) - \frac{1}{\beta }\ln \left( {t_{\text{o}} } \right)$$

(3.25)

where α is the initial adsorption rate (mg g⁻¹ min⁻¹) and β is related to the extent of surface coverage and the activation energy involved in chemisorption (g mg⁻¹) and t _o = 1/αβ.

If t is much larger than t _o, the kinetic equation can be simplified as Eq. 3.26.

$$q_{t} = \frac{1}{\beta }\ln \left( {\alpha \cdot \beta } \right) + \frac{1}{\beta }\ln \left( t \right)$$

(3.26)

Equation 3.26 is known as Elovich–chemisorption kinetic adsorption model.

The determination of some kinetic parameters, possible changes of the adsorption rates as function of the initial concentration and the adsorption time as well as the determination of fractional kinetic orders, still lacks in the kinetic adsorption models [16]. In this way, an alternative Avrami kinetic equation to find a correlation between good experimental and calculated data was early proposed [16]. The adsorption should now be visualized using Avrami’s exponential function, which is an adaptation of kinetic thermal decomposition modeling [16].

$$\alpha = 1 - \exp \left[ {\left( { - k_{\text{AV}} \cdot t} \right)} \right]^{{n_{\text{AV}} }}$$

(3.27)

where α is adsorption fraction (q _t /q _e) at time t, k _AV is the Avrami kinetic constant (min⁻¹), and n _AV is a fractional adsorption order, which is related to the adsorption mechanism [16]. By inputting α in Eq. 3.27, the Avrami kinetic equation could be written as Eq. 3.28.

$$q_{t} = q_{\text{e}} \cdot \left\{ {1 - \exp \left[ { - \left( {k_{\text{AV}} \cdot t} \right)} \right]^{n} } \right\}$$

(3.28)

After describing the pseudo-first order, pseudo-second-order, general-order, Elovich chemisorption, and Avrami fractionary order equations, we will illustrate these models using nonlinear fit of the data. Figure 3.10 shows the pseudo-first-order kinetic curve of the Reactive Blue 4 dye (RB-4) using SWCNT as adsorbent at pH 2.0 and 25 °C [17].

Fig. 3.10

Pseudo-first-order kinetic adsorption model of RB-4 dye using SWCNT. Initial pH, 2.0; temperature, 25 °C; initial concentration of the adsorbate, 800.0 mg L⁻¹

Figure 3.11 presents the pseudo-second-order kinetic plot of the RB-4 dye using SWCNT as adsorbent at pH 2.0 and 25 °C [17].

Fig. 3.11

Pseudo-second-order kinetic adsorption model of RB-4 dye using SWCNT. Initial pH, 2.0; temperature, 25 °C; initial concentration of the adsorbate, 800.0 mg L⁻¹

The general-order kinetic curve of the RB-4 dye using SWCNT as adsorbent at pH 2.0 and 25 °C is presented in Fig. 3.12 [17].

Fig. 3.12

General-order kinetic adsorption model of RB-4 dye using SWCNT. Initial pH, 2.0; temperature, 25 °C; initial concentration of the adsorbate, 800.0 mg L⁻¹

Figure 3.13 presents the Elovich Chemisorption kinetic plot of the RB-4 dye using SWCNT as adsorbent at pH 2.0 and 25 °C [17].

Fig. 3.13

Elovich chemisorption kinetic adsorption model of RB-4 dye using SWCNT. Initial pH, 2.0; temperature, 25 °C; initial concentration of the adsorbate, 800.0 mg L⁻¹

Figure 3.14 shows the Avrami-fractional kinetic plot of the RB-4 dye using SWCNT as adsorbent at pH 2.0 and 25 °C [17].

Fig. 3.14

Avrami fractionary kinetic adsorption model of RB-4 dye using SWCNT. Initial pH, 2.0; temperature, 25 °C; initial concentration of the adsorbate, 800.0 mg L⁻¹

Figures 3.10, 3.11, 3.12, 3.13, and 3.14 present kinetic curves of the adsorption of RB-4 dye onto SWCNT adsorbent [17] for pseudo-first-order, pseudo-second-order, general-order, Elovich chemisorption, and Avrami fractionary models, respectively. The best fit model is the one with the lowest value of SD and the one in which the value of $R_{\text{adj}}^{2}$ is closer to unity. Equations 3.29–3.32 depict the expressions of reduced chi-square, SD, R ², and $R_{\text{adj}}^{2}$, respectively.

$${\text{Reduced Chi-squared}} = \sum\limits_{i}^{n} {\frac{{\left( {q_{{i},\exp } - q_{{i},{\text{model}}}} \right)^{2} }}{{n_{\text{p}} - p}}}$$

(3.29)

$${\text{SD}} = \sqrt {\left( {\frac{1}{{n_{\text{p}} - p}}} \right) \cdot \sum\limits_{i}^{n} {\left( {q_{{i},\exp } - q_{{i},{\text{model}}}} \right)}^{2} }$$

(3.30)

$$R^{2} = \left[ {\frac{{\sum\nolimits_{i}^{\text{np}} {\left( {q_{{i},\exp} - \bar{q}_{\exp} } \right)^{2} } - \sum\nolimits_{i}^{\text{np}} {\left( {q_{{i},\exp} - q_{{i},{\text{model}}}} \right)^{2} } }}{{\sum\nolimits_{i}^{n} {\left( {q_{{i},\exp} - \bar{q}_{\exp }} \right)^{2} } }}} \right]$$

(3.31)

$$R_{\text{adj}}^{2} = 1 - \left( {1 - R^{2} } \right) \cdot \left( {\frac{{n_{\text{p}} - 1}}{{n_{\text{p}} - p - 1}}} \right)$$

(3.32)

where q _i , model is each value of q predicted by the fitted model, q _{i, exp} is each value of q measured experimentally, $\bar{q}_{\exp }$ is the average of q experimentally measured, n _p is the number of experiments performed, and p is the number of parameters of the fitted model.

The reduced chi-square is the residual sum of squares divided by the degree of freedom (DOF) (np-p) (Eq. 3.29). The SD is the square root of reduced chi squared (Eq. 3.30). Both Eqs. 3.29 and 3.30 are very useful for evaluation of point to point of a given kinetic or equilibrium adsorption model, this is because for each experimental point there is a point in the model that corresponds exactly to the point on the curve (model). The lower the reduced chi-square and the SD, the lower the difference between the values of experimental q and theoretical q; therefore, the best fit is expected. However, it should be taken into account that it is not possible to compare the values of reduced chi-squared and SD among several kinetic and equilibrium isotherms that present other different concentrations or other conditions since the values of SD and chi-square tend to increase as the concentration increases. On the contrary, for the same set of experimental data, the values of reduced chi-square and SD are useful to ascertain the best model since R ² and $R_{\text{adj}}^{2}$ are of low sensitivity (their values are limited to unity) [18].

The R ² and $R_{\text{adj}}^{2}$ in Eqs. 3.31 and 3.32, respectively, are very useful parameters to evaluate kinetic and equilibrium adsorption fits. Limitedly, their values are between 0 and 1. Values of R ² and $R_{\text{adj}}^{2}$ that are closer to 1 means that the model has a better fit. It is important to note that ${\bar{q}}_{ \exp}$ of Eq. 3.31 is the average of all experimental data (q). If the range of q values is too large, ${{\bar{q}}}_{{{\text{i}, \exp}}}$ could distort the interpretation of the fit. If there is equidistant values of q _i,exp from the average value, the values of R ² tend to 1. Therefore, the analysis of a kinetic and equilibrium isotherm could not only be based on the values of R ² [18]. In the same way, comparison of two-parameter models (pseudo-first order, pseudo-second-order, Elovich chemisorption) with three-parameter models (general order, Avrami fractionary order) is not possible because the equations with higher number of parameters have the tendency to exhibit R ² values closer to 1. In these cases, it is recommended to use $R_{\text{adj}}^{2}$ [18]. This statistical parameter is used to penalize the models with more parameters in order to really know if the best fitting ($R_{\text{adj}}^{2}$) is due to the advantage of presenting more terms in the equation (mathematical advantage), or alternatively, the equation is physically closer to the reality of the system. $R_{\text{adj}}^{2}$ is, therefore, a very good parameter for evaluating a given kinetic and equilibrium of adsorption [18].

Based on the explanation above, we really recommend evaluation of SD and $R_{\text{adj}}^{2}$ to decide the best fitting model. In addition, it should be stressed that it is necessary to interpret the values of the obtained parameters to really know if the model is a suitably fitting model, as it was done in the adsorption equilibrium studies described above. For the graphs depicted in Figs. 3.10, 3.11, 3.12, 3.13, and 3.14, the general-order kinetic adsorption model (the best fitting model) showed the lowest values of SD, and the highest values of $R_{\text{adj}}^{2}$. Sequentially, Avrami fractionary kinetic model was the second best kinetic model followed by pseudo-first order (third best fitting model), pseudo-second-order (4th), and then the Elovich chemisorption model (5th). Taking into consideration that the general-order kinetic of adsorption model was the best fitting model, its parameters can now be discussed. The capacity of adsorption at equilibrium (q _e) is 317.5 mg g⁻¹, which is closer to the values of saturation attained in the Fig. 3.13. The kinetic adsorption rate constant is 8.087.10⁻³ (min⁻¹(g mg⁻¹)^0.337) and the order of kinetic model (n) is 1.337. What could be the reason for n not being 1 (pseudo-first order) or 2 (pseudo-second-order)? The order should be determined experimentally and not as previously stipulated as described above. The fractional number could be a change in the order of kinetic adsorption that took place during the adsorption process. Similarly, it was observed that pseudo-first-order kinetic model gave a better fit than pseudo-second-order. This phenomenon could be analyzed in the following way: 1.337 is closer to 1 (0.337) than 2 (0.663). Therefore, an n value of 1.337 could also be attributed to a change in the kinetics of adsorption, changing from pseudo-first-order to pseudo-second-order during the contact of the adsorbate with the adsorbent. Therefore, the general-order kinetic model represents a more detailed mechanism of adsorption between the adsorbent and adsorbate, when a change in the order of kinetic process is possible to occur during the adsorption process. This assertion explains why fractional numbers are obtained as the orders of adsorption processes.

3.3.3 Diffusive Mechanisms

The process of sorption falls into four basic stages: (1) transport of the adsorbate from the bulk solution to the film of solvent around the particles of the adsorbent, usually this stage is too fast for agitated systems, (2) diffusive mass transfer of the adsorbate through the film, (3) intraparticle diffusion of the adsorbate through the pores of the adsorbent, (4) binding of the adsorbate to the active sites in the pores of the adsorbent [19], this stage is usually faster than the diffusion of the adsorbate by the film and intraparticle diffusion; therefore, it is assumed that this stage does not limit mass transfer.

The possibility of intraparticle diffusion resistance affecting adsorption process could be explored using the intraparticle diffusion model as given in Eq. 3.33 [20]:

$$q_{t} = k_{\text{id}} \cdot \sqrt t + C$$

(3.33)

where q _t is the amount of adsorbate adsorbed by adsorbent at a given time, t (min), k _id is the intraparticle diffusion rate constant (mg g⁻¹ min^−0.5), and C is a constant related to the thickness of boundary layer (mg g⁻¹) [20]. In order to illustrate the intraparticle diffusion, Fig. 3.15 presents the Weber-Morris intraparticle kinetic plot of the RB-4 dye using SWCNT as adsorbent at pH 2.0 and 25 °C [17].

Fig. 3.15

Weber-Morris intraparticle kinetic adsorption curve of RB-4 dye using SWCNT. Initial pH, 2.0; temperature, 25 °C; initial concentration of the adsorbate, 800.0 mg L⁻¹

The intraparticle diffusion constant, k _id (mg g⁻¹ min^−0.5), can be obtained from the slope of the plot of q _t versus the square root of time. Figure 3.15 shows the plot of q _t versus t ^1/2, with multilinearity for RB-4 dye using SWCNT adsorbent. These results imply that the adsorption process involves more than a single kinetic stage (or adsorption rate) [17]. The adsorption process showed three zones which could be attributed to each linear portion as shown in Fig. 3.15. The first linear portion (1st zone) was assigned to the diffusional process of the dye on the adsorbent surface [17]; hence, it was the fastest sorption stage. The second portion (2nd zone) was ascribed to intraparticle diffusion, a delayed process [17]. The third stage (3rd zone) could be regarded as the diffusion through smaller pores, which is followed by the establishment of equilibrium [17]. From the slope of the linear portion of the second zone it was possible to determine k _id which is 9.583 mg g⁻¹ min^−0.5.

3.4 Problems with the Linearization of Equilibrium and Kinetic Equations Usually Employed in the Adsorption Studies

The misuse of linearization is probably the most common error in data analysis of adsorption studies. It was largely used some decades ago when computers and statistical software were not available. Presently, linearization in data analysis is on the increase because the authors usually employ the simplest tool to analyze their experimental data. One of the reasons for using linear equations is the possibility of performing less experimental points to define a line (that is the major problem of using nonlinear equations, which require more experimental points to define a curve) or alternatively, discard some points to increase the R ² values. The main problem is that some points are discarded to increase the R ² values, and the authors are not concerned about the values of the parameters of the equilibrium and kinetic of adsorption models.

There are problems associated with linearizing an inherently nonlinear equation using various transformations. The main concern when transforming data to obtain a linearized equation is the knowledge of the error structure of the data and how this structure is affected by the mathematical manipulation of the data [21]. Linearization is based on the fact that the variance of all q values (Y variable of the graph; dependent variable; amount of adsorbate adsorbed by the adsorbent) is equal for all range of the data. By assuming homoscedasticity of the data (assuming equal variance for all q values throughout the range of the data) when it is actually heteroscedastic (the variance of all q values are not equal in the full range of the data), may result in overestimating the goodness of fit as measured by the correlation coefficient (R), which translates into error in the coefficient of determination (R ²) and also in the adjusted coefficient of determination [21]. Therefore, higher R ² values do not necessarily mean better fit of the experimental data.

In practice, authors have to delete some points from their curves to justify the linearization procedure. This adjustment decreases the DOF of the fitting model, which consequently decreases the number of independent ways by which a dynamic system can move without violating any constraint imposed on it. The DOF can be defined as the minimum number of independent coordinates that can specify the position of a system completely. In order to make it clear to the readers, some case studies will be discussed below.

3.4.1 Equilibrium Isotherms

We start with the equilibrium isotherms. Taking into account that the majority of the published articles used the Langmuir model, this equilibrium model will be given extensive study in this chapter.

Let us consider the adsorption of RB-4 dye on SWCNT at 308 K [17]. According to Kumar 2007 [22], there are four linearized equations of the Langmuir as presented in Table 3.2.

Table 3.2 Langmuir equilibrium isotherms

Let us also consider the following experimental data [17] as depicted on Table 3.3.

Table 3.3 Experimental data of the adsorption of RB-4 on SWCNT at 308 K [17]

It should be noted that in manipulating the parameters the parametric units changed, and this change is not considered by many authors. The nonlinear isotherm directly uses experimental data: concentrations of the RB-4 left in the supernatant after adsorption process (abscissa axis) and the amount adsorbed (ordinate axis), which were calculated according to Eq. 3.34:

$$q = \frac{{\left( {C_{\text{o}} - C_{\text{f}} } \right)}}{m} \cdot V$$

(3.34)

where q is the amount of adsorbate adsorbed by the adsorbent (mg g⁻¹), C _o is the initial adsorbate concentration in contact with the adsorbent (mg L⁻¹), C _f is the final adsorbate concentration after adsorption process (mg L⁻¹), m is the mass of adsorbent (g), and V is the volume of adsorbate solution (L).

The nonlinear, Langmuir-1, Langmuir-2, Langmuir-3, and Langmuir-4 linearized isotherms are presented in Fig. 3.16.

Fig. 3.16

a Nonlinearized Langmuir equation; b linearized Langmuir-1 equation; c linearized Langmuir-2 equation; d linearized Langmuir-2 equation after deleting some points; e linearized Langmuir-3 equation; f linearized Langmuir-3 equation after deleting some points; g linearized Langmuir-4 equation; h linearized Langmuir-4 equation after deleting some points. Arrows indicate the set of point that need to be deleted to improve the values of R ²

Table 3.4 presents the statistical analysis of all isotherms presented in Fig. 3.16. As can be observed, Langmuir-1 linearized equation was the only model that showed the best values of R ² and $R_{\text{adj}}^{2}$, smaller SD, and reduced Chi-square and residual sum of squares, and with this model, it was not necessary to delete any manipulated point. However, for linearized Langmuir-2, linearized Langmuir-3, and linearized Langmuir-4, it was necessary to delete some data points to improve the statistical analysis. Usually, the authors are only worried about the value of R ² but not concerned about the other statistical parameters. When the points on an isotherm are deleted, the slope and intercept of the plot will be modified, which will invariably affect the values of Q _max, and K _L, apart from decreasing the values of DOF. It should also be stressed that reduced Chi-square and SD altered the units of dependent variable (Y-axis). Only linearized Langmuir-3 equation presents the same units for the statistical parameters compared with nonlinear Langmuir equation. Therefore, it is not possible to infer that linearized Langmuir-1 equation provides more accurate results than nonlinear equation since all the statistical parameters (Chi-squared, SD, residual sum of squares) presented in Table 3.3 reflect different situations (Y-axis of each graph) for nonlinear and linearized Langmuir-1 equations. Considering the fact that the nonlinear fit is the best fitting method of equilibrium isotherm, the values of Q _max and K _L obtained by nonlinear and linearized Langmuir-1, Langmuir-2, Langmuir-3, and Langmuir-4 were compared. Table 3.5 presents the relative differences of the parameters of the isotherm plots depicted on Fig. 3.16. As observed, the Langmuir-1 linearized equation produced the worst values of K _L when compared with nonlinear Langmuir equation. Another important observation is that in all cases when points of isotherms were deleted, although the R ² values improved in all cases for linearized Langmuir-2, Langmuir-3, and Langmuir-4 equations, but the relative values of Q _max and K _L became worse compared with the nonlinear reference values. The analyses of plots in Fig. 3.16 as shown in Tables 3.4 and 3.5 clearly indicate that R ² values close to unity did not indicate that the model was properly fitted, and, in addition, did not prove that the obtained values of Q _max and K _L reflect the reality of the system.

Table 3.4 Statistical analysis of data presented in Fig. 3.16

Table 3.5 Relative differences of Q _max and K _L isotherm parameters of linearized Langmuir-1, Langmuir-2, Langmuir-3 and Langmuir-4 in relation to nonlinear Langmuir equation

Considering that K _L values are very important equilibrium parameters for estimating equilibrium constants (determination of thermodynamic parameters) [10, 23–26], the use of Langmuir-1 linearized isotherms could lead to erroneous estimation of Gibb’s free energy change (ΔG°), enthalpy change (ΔH°), and entropy change (ΔS°) of adsorption.

3.4.2 Kinetics of Adsorption

The mostly employed pseudo-first-order and pseudo-second-order kinetic equations are given in Table 3.6 in nonlinear and linearized forms [26, 27]. Let us consider the kinetics of adsorption of DB-53 dye on MWCNT [10]. The experimental kinetic data of adsorption of DB-53 on MWCNT are presented in Table 3.7. The nonlinear pseudo-first-order and linearized pseudo-first-order plots are shown in Fig. 3.17, while nonlinear pseudo-second-order, and linearized pseudo-second-order-1, pseudo-second-order-2, pseudo-second-order-3 and pseudo-second-order-4 plots are presented in Fig. 3.18. It is observed from Table 3.7 that the manipulation of the data alters the units of the variables. The analysis of pseudo-first-order kinetics necessitated deletion of five experimental points to achieve a good value of R ²; however, the DOF decreases from 18 to 13.

Table 3.6 Kinetics of adsorption

Table 3.7 Experimental data of the adsorption of DB-53 onto MWCNT at 298 K [10]

Fig. 3.17

a Nonlinearized pseudo-first-order equation; b linearized pseudo-first order with all experimental points; c linearized pseudo-first-order equation after deleting some points. Arrows indicate the region of points that were deleted just to improve the values of R ²

Fig. 3.18

a Nonlinearized pseudo-second-order equation; b linearized pseudo-second order-1; c linearized pseudo-second order-2; d linearized pseudo-second-order-2 after deleting some experimental points; e linearized pseudo-second-order-3; f linearized pseudo-second-order-3 after deleting some experimental points; g linearized pseudo-second-order-4; h linearized pseudo-second-order-4 after deleting some experimental points. The arrows indicate region of experimental points that were deleted just to improve the values of R ²

The analysis of pseudo-second-order linearized models indicates that the linearized pseudo-second-order-1 equation apparently provides the best values of R ² for the four linearized models. However, it is necessary to highlight that this analysis did not put into consideration the units of SD and reduced Chi-square, which are not the same for nonlinearized model. Therefore, different incidents are being compared; however, some authors use the value of R ² to infer that a model is a good fit.

Only linearized pseudo-second-order-3 presents the same units as nonlinearized kinetic equation (Table 3.7), however, the statistical analysis of this equation (SD and reduced Chi squared) is worse than the nonlinearized equation, even after deleting 10 experimental points.

Considering that the kinetics of adsorption is nonlinear equation, the parameter values (q _e and k ₁ or k ₂) of this model were taken as a reference for computing the relative differences of the linearized models in relation to the nonlinearized equations; these values are depicted in Table 3.9. It is observed that the strategy of deleting some points to improve the values of R ² causes a serious error in the values of q _e and k ₁ or k ₂. The activation energy is calculated based on the values of kinetic rate constants of the Arrhenius equation (Eq. 3.35), [23]:

$$\ln k_{\text{kinetic}} = \ln A - \frac{{E_{\text{a}} }}{\text{RT}}$$

(3.35)

where k _kinetic is the kinetic rate constant of adsorption, A is the Arrhenius constant, E _a is the activation energy (kJ mol⁻¹), R is the universal gas constant (8.314 J K⁻¹ mol⁻¹), and T is the absolute temperature (K). If the values of k _kinetic are biased, the values of calculated activation energy will also be compromised.

We have demonstrated using the given examples (Figs. 3.16, 3.17, and 3.18; Tables 3.4, 3.5 3.8, and 3.9) the existing problems in using linear equations for Langmuir isotherm, pseudo-first-order, and pseudo-second-order kinetics of adsorption. Remarkably, although linearized Langmuir-1 and linearized pseudo-second-order-1 equations provide values of R ² close to unity, an assumption of a good fit of any model but the values of the parameters (equilibrium and kinetics) differ from those of the nonlinear equations. Similarly, it should be stressed that with the manipulation of data to convert a nonlinear equation to linear formats, the units of the Y and X axes changed remarkably. Therefore, the reduced Chi-square and SD should be taken into consideration while evaluating the fit of a model. Only the linearized model that presents the same Y-axis (q _e and q _t for equilibrium and kinetic equations, respectively) could be directly compared with the nonlinear equations. The linearized equations exhibited different units, although those values could be glancingly better when compared with nonlinear equations, but the numerical values do not correspond to the values of reduced Chi-square and SD of nonlinear equations, and therefore, they are incomparable. We do hope that the perusal of the figures and tables presented in this section will make it clearer to the readers that the nonlinear usage of equilibrium and kinetics of adsorption models give values that are reliable and statistical relevant to the modeling of an isotherm and kinetic of adsorption curve. We do not recommend the usage of linearized equilibrium and kinetics of adsorption models, even in the cases of linearized Langmuir-1 and linearized pseudo-second-order-1 equations.

Table 3.8 Statistical analysis of data presented in Fig. 3.17

Table 3.9 Statistical analysis of data presented in Fig. 3.18

Table 3.10 shows the relative difference on the kinetic parameters of pseudo-first order and pseudo-second order. In some cases, where experimental data points were deleted to increase the values of R ² (see Table 3.10), the relative difference of the kinetic parameters became worse compared to parameters before experimental data points were deleted. Therefore, we reiterated that linearization of equilibrium and kinetic of adsorption models could make the parameter values of the models meaningless.

Table 3.10 Relative differences of q _e and k ₁ or k ₂ kinetic parameters of linearized pseudo-first order, linearized pseudo-second order-1, pseudo-second order-2, pseudo-second order-3, and pseudo-second order-4 in relation to nonlinear pseudo-first order and nonlinear pseudo-second order equations

Other point that was not taken into account in this analysis is that for linearization, almost all manuscripts described that linearized pseudo-second-order-1 model is followed. Perusing through Table 3.9, this observation can be erroneously concluded. However, the linearized pseudo-first-order model is not always a good fit. If one compares $R_{\text{adj}}^{2}$ and SD values for nonlinearized kinetic models, in the example given in this chapter (Tables 3.8 and 3.9), one will observe that the kinetic data fit better to nonlinear pseudo-first-order ($R_{\text{adj}}^{2}$ 0.9995; SD 0.8386) than the nonlinear pseudo-second-order ($R_{\text{adj}}^{2}$ 0.9862; SD 4.244). It is important to stress that the SD of nonlinearized pseudo-second order is 5.1 times higher than the nonlinearized pseudo-first order. Therefore, it is evident that the kinetic should follow pseudo-first order in preference. However, using the distorted linearized equations, the opposite is obtained. This should be one of the reasons majorities of the papers reported in the literature used linearized equations, almost all, suggest that the kinetic should follow pseudo-second order in disadvantage of pseudo-first order. However, if the nonlinearized equations were employed, the opposite could be obtained. This is one of the reasons that the authors of this chapter do not recommend the use of linearized equations for the equilibrium and kinetic of adsorption, and we also reiterate that there are so many errors committed in the published literatures of adsorption research.

3.4.3 Other Common Mistakes in Adsorption Works

It is very common in the adsorption literature that the authors attribute that a mechanism of adsorption is chemisorption based only on the fitting parameters of kinetic data of the linearized pseudo-second-order-1 equation. As visibly seen in the above example, this is the mathematical model that provides the best fitting result (higher R ², $R_{\text{adj}}^{2}$, lower SD, among others). In this case the authors committed two errors: first, using linearized equations; only linearized pseudo-second-order-1 equation fits with good values of R ², even better than nonlinearized equation; however, the authors did not take into account that the values of k ₂ and q _e could be completely compromised (see Table 3.10). Second, using nonlinearized equations; other kinetic models could give better fits than the linearized pseudo-second-order-1 model (see Tables 3.8 and 3.9). Given that the pseudo-second-order kinetic model is the best fitting model (using nonlinear equations) does not mean that adsorption process is a chemical adsorption. The early works of pseudo-second-order model [7, 28] used the sorption of metallic ions with adsorbents that could form complexes with the metallic ions. In this case, the sorption process should be the formation of covalent bond between the adsorbate and the adsorbent. However, this specific case of a pseudo-second-order kinetic model being followed does not imply that the sorption process is a chemisorption. To establish if an adsorption process is chemical or physical, it is necessary to prove the formation of some chemical bonds using some analytical techniques (FTIR, Raman spectroscopy, TGA, and so on) combined with thermodynamical data of changes in enthalpy (ΔH) and changes in entropy (ΔS). It is expected that large organic molecules would be adsorbed by physical interactions and small inorganic ions could be complexed with nitrogen or oxygen atoms of amine, amide, phenols, alcohols, and so on, being an interaction with adsorbent, which is a chemical adsorption process. This chapter will not discuss the errors committed in the literature, where authors attribute adsorption process inadvertently to chemical sorption based just on the kinetic data as explained above.

Another problem that is always commented in almost all adsorption researches is the calculation of the thermodynamic parameters (Enthalpy and Entropy changes) using the equilibrium data. Majority of authors use the following equations for the determination of the thermodynamic parameters:

$$K_{\text{D}} = \frac{{\left( {C_{\text{o}} - C_{\text{e}} } \right)}}{{C_{\text{e}} }}$$

(3.36)

$$\Delta G^{0} =\Delta H^{0} - T\Delta S^{0}$$

(3.37)

$$\Delta G^{0} = {\text{RT}}\, {\text{In}} \left( {K_{\text{D}} } \right)$$

(3.38)

Equation 3.39 is obtained from Eqs. 3.37 and 3.38.

$${\text{In}} \left( {K_{\text{D}} } \right) = \frac{{\Delta S^{0} }}{R} - \frac{{\Delta H^{0} }}{R}\cdot\frac{1}{T}$$

(3.39)

where R is the universal gas constant (8.314 J K⁻¹ mol⁻¹); T is the absolute temperature (Kelvin); and K _D is the distribution equilibrium constant (dimensionless).

By plotting a graph of ln(K _D) versus 1/T, a straight line is obtained. From the intercept and slope of the plot, it is possible to calculate the changes in Entropy ΔS° and changes in enthalpy (ΔH°), respectively. Until this point, everything seems to be regular. However, some authors usually do not mention how initial concentrations (C _o) and equilibrium concentrations (C _e) they had employed in the calculation of K _D were arrived at. In addition, most authors did not report in their papers that K _D decreases exponentially as C _e and C _o increase, as shown in Fig. 3.19.

Fig. 3.19

Distribution constant K _D as function of equilibrium concentrations C _e. a 298 K; b 303 K; c 308 K; d 313 K; e 318 K; f 323 K. These data were culled from the adsorption of Reactive Orange-16 (RO-16) dye onto coffee wastes carbon composite [12]

As seen in Fig. 3.19, K _D has no fixed value at a given temperature. The most correct K _D value should therefore be calculated from infinite dilution (low values of C _o). By fitting the experimental data to an exponential decay equation, it is possible to evaluate the value of K _D for C _o = 0 (the value of K _D at infinite dilution). Using these considerations, it is possible to compute the values of K _D at each temperature value; subsequently, it is possible to calculate the thermodynamic parameters by employing Eq. 3.39. Figure 3.20 shows this graph.

Fig. 3.20

ln (K _D) versus 1/T

By observing Fig. 3.20, it is remarkable that ln(K _D) versus 1/T is not a linear relation ($R_{\text{adj}}^{2}$ 0.2802). Therefore, if the authors calculate K _D at infinite dilution, it is quite impossible to determine correctly the values of ΔH° and ΔS° of adsorption process.

On the contrary, instead of calculating the equilibrium constant (K _D), it is possible to calculate the value of adsorption equilibrium constant from the value of the nonlinear isotherm equilibrium model that gives the best fit to the experimental data. Therefore, K is the equilibrium adsorption constants of the isotherm fits (for this specific case (K = K _g ), K _g —Liu equilibrium constant, which must be converted to SI units using the molecular mass of the dye). Making a plot of ln(K) versus 1/T, a linear graph should be obtained. It is possible to evaluate ΔS° and ΔH° from the respective intercept and slope of the plot [24].

From Fig. 3.21, it is observed that the equilibrium constants calculated from the Liu isotherm (from 298 to 323 K) [12] showed a linear relationship with the reciprocal of temperature. From this figure, ΔS° (177.3 J K⁻¹ mol⁻¹) and ΔH° (30.4 kJ mol⁻¹) were calculated [24]. These values of adsorption enthalpy are in agreement with the electrostatic attraction of negatively charged dye reactive red 120 with the positively charged adsorbent at pH 2.0 as reported in the literature [24].

Fig. 3.21

ln K versus 1/T

3.5 Concluding Remarks

In this chapter, the commonly employed and unpopular adsorption equilibrium isotherm models such as Langmuir, Freundlich, Sips, Liu, Redlich–Peterson, Hill, Khan, Radke–Prausnitz and Toth were discussed with suitable illustrations. Adsorption kinetic models such as pseudo-first-order, pseudo-second-order, general-order and empiric models (Avrami fractionary and Elovich chemisorption) were adequately described. The usage of different isotherm models at different temperature values was suggested in this chapter. The importance and consideration of statistical parameters such as $R_{\text{adj}}^{2}$ and SD were emphasized. The statistical analysis is a guidance for adsorption researchers to select the best adsorption isotherm and kinetic model for their work, which is coupled with obtainability of useful constants or parameters that are meaningful. The model with the best value of $R_{\text{adj}}^{2}$ and the lowest value of SD should be considered as the best isotherm and kinetic model to describe the adsorption process. Based on the detailed analysis of linearized and nonlinearized models in this chapter, we do not recommend the use of linearized equations for the analysis of adsorption equilibrium and kinetic data. Analytical techniques together with thermodynamical data of enthalpy and entropy changes are needed to affirm if an adsorption process is a chemical or a physical process. It is advisable to evaluate the value of adsorption equilibrium constant (K) from the value of the best fit nonlinear isotherm equilibrium model instead of using calculated equilibrium constant (K _D) from initial and equilibrium concentrations of an adsorbate. Overall, this chapter proffered solutions to the common errors arising from usage of linearized equilibrium and kinetic adsorption models.