**3. Isotherm models: theoretical knowledge**

Isotherm models are used to determine an adsorbent's maximal sorption capacity and is represented in terms of the amount of metal absorbed per unit mass of adsorbent.

#### **3.1 One parameter model**

The simple adsorption isotherm model has only one parameter to explain the adsorption mechanism. The most basic adsorption isotherm is one in which the amount of solute adsorbed is proportional to the equilibrium effluent concentration [16]. Eq. (3) is the model.

$$\mathbf{q}\_{\rm eq} = \mathbf{K} \mathbf{C}\_{\rm eq} \tag{3}$$

#### **3.2 Two parameter models**

This section focuses on gaining theoretical insight into models with two parameters that explain the adsorption mechanism. Henry's law with constant, Langmuir, Freundlich, Dubinin-Radushkevich, Temkin, Hill-de Boer, Fowler-Guggenheim, Flory-Huggins, Halsey, Harkin-Jura, Jovanovic, Elovich, and Kiselev are among the 13 models covered.

### *3.2.1 Henry's law with intercept isotherm model*

This model was created to address the contradiction highlighted by the one parameter model and to be applicable over a wide range of metal ion concentrations [16]. Eq. (4) describes the model.

$$\mathbf{q}\_{\rm eq} = \mathbf{K} \mathbf{C}\_{\rm eq} + \mathbf{C} \tag{4}$$

#### *3.2.2 Langmuir isotherm model*

The Langmuir model assumes a homogenous surface and explains how the adsorbate forms monolayer coverage on the adsorbent's outer surface [17, 18]. The rate of adsorption is proportional to the percentage of open adsorbent surface, and the rate of *Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

desorption is related to the fraction of covered adsorbent surface. The model is described in Eq. (5).

$$\mathbf{q\_{eq}} = \frac{\mathbf{q\_{max}} \mathbf{b\_L C\_{eq}}}{\mathbf{1} + \mathbf{b\_L C\_{eq}}} \tag{5}$$

bL is the Langmuir constant, which links the fluctuation of the appropriate area and porosity of the adsorbent with its adsorption capacity (mg/g). A dimensionless constant called the Langmuir separation factor RL, which is computed as Eq. (6), helps explain the key properties of the Langmuir isotherm.

$$\mathbf{R}\_{\rm L} = \frac{1}{\mathbf{1} + \mathbf{b}\_{\rm L} \mathbf{q}\_{\rm max}} \tag{6}$$

When RL > 1, adsorption is unfavourable, when RL = 1, linear when RL = 1, favourable when RL = 0, and irreversible when RL = 0.

#### *3.2.3 Freundlich isotherm model*

The Freundlich adsorption isotherm model depicts the adsorbent surface heterogeneity. The adsorptive sites are made up of tiny homogenous heterogeneous adsorption sites [19]. Eq. (7) is the model.

$$\mathbf{q\_{eq}} = \mathbf{a\_F C\_{eq}^{\rm tr}} \tag{7}$$

Freundlich adsorption capacity (L/mg) is denoted by aF, while adsorption intensity is denoted by nF. The higher the adsorption capacity, the larger the aF value. The magnitude of 1/nF, which varies from 0 to 1, indicates favourable adsorption and becomes more heterogeneous as it approaches zero [18–21].

### *3.2.4 Dubinin-Radushkevich isotherm model*

This empirical model implies that physical adsorption processes are multilayered and involve Van Der Waal's forces [22], it is frequently used to estimate the characteristic porosity. This model [23] gives insight into gas and vapour adsorption on micro porous sorbents.

The Dubinin-Radushkevich isotherm's temperature dependence is another distinguishing trait [24, 25]. Eq. (8) represents the Dubinin-Radushkevich isotherm.

$$\mathbf{q\_{eq}} = \mathbf{K\_{DR}} \exp\left[-\mathbf{B\_{DR}} \left(\mathbf{RTIn} \left(\mathbf{1} + \frac{\mathbf{1}}{\mathbf{C\_{eq}}}\right)\right)^2\right] \tag{8}$$

$$\mathbf{e} = \text{RTln}\left[\frac{\mathbf{1}}{\mathbf{C\_{eq}}}\right] \tag{9}$$

ε is known as the Polanyi potential, as seen in Eq. (9). The activation energy or mean free energy E (kJ/mol) of adsorption per molecule of adsorbate when it is transported from infinity in the solution to the surface of the solid may be computed using Eq. (10).

$$\mathbf{E} = \frac{1}{\sqrt{2\mathbf{K}\_{\rm DR}}}\tag{10}$$

The value of E is used to forecast whether an adsorption is physical or chemical. In nature, physisorption occurs when E = 8 KJ/mol, whereas chemisorption occurs when E=8–16 KJ/mol [23].

#### *3.2.5 Temkin isotherm model*

For forecasting the gas phase adsorption equilibrium, the Temkin adsorption isotherm model is quite useful [25, 26]. Eq. (11) illustrates the Temkin adsorption isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{RT}}{\mathbf{b}\_{\rm T}} \left( \ln \mathbf{A}\_{\rm T} \mathbf{C}\_{\rm eq} \right) \tag{11}$$

The heat of adsorption is represented by the equation RT/bT, and the equilibrium binding constant (L/mg) corresponding to the maximal binding energy is represented by AT.

#### *3.2.6 Hill-de Boer isotherm model*

This Hill-Deboer isotherm model accurately describes mobile adsorptions as well as lateral interactions between adsorbed molecules [27, 28]. Eq. (12) depicts the Hill-Deboer isotherm model.

$$\mathbf{K}\_{\mathbf{1}} \mathbf{C}\_{\mathbf{eq}} = \frac{\boldsymbol{\theta}}{\mathbf{1} - \boldsymbol{\theta}} \cdot \exp\left(\frac{\boldsymbol{\theta}}{\mathbf{1} - \boldsymbol{\theta}} - \frac{\mathbf{K}\_2 \boldsymbol{\theta}}{\mathbf{RT}}\right) \tag{12}$$

A positive value of K2 implies attraction between adsorbed species, whereas a negative value of K2 suggests repulsion. If K2 is equal to zero, there is no interaction between adsorbed molecules, and the Volmer equation [29] is used

#### *3.2.7 Fowler-Guggenheim isotherm model*

The Fowler-Guggenheim adsorption isotherm model describes how adsorbed molecules interact laterally. Eq. (13) represents the Fowler-Guggenheim adsorption isotherm.

$$\mathbf{K\_{FG}C\_{eq}} = \frac{\boldsymbol{\Theta}}{\mathbf{1} - \boldsymbol{\Theta}} \exp\left[\frac{2\boldsymbol{\Theta}\mathbf{W}}{\mathbf{RT}}\right] \tag{13}$$

The presence of a positive W indicates that the contact between the adsorbed molecules is attractive. In contrast, if W = 0, the contact between adsorbed molecules is repulsive, and the heat of adsorption decreases with loading, the Fowler-Guggenheim equation reduces to the Langmuir model. W = 0 when there is no contact between adsorbed molecules. Furthermore, this model is only viable when the surface coverage is less than 0.6 [30, 31].

*Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

#### *3.2.8 Flory-Huggins isotherm model*

The degree of surface coverage of the adsorbate onto the adsorbent is discussed by the Flory-Huggins isotherm [32–35]. Eq. (14) shows the Flory-Huggins adsorption isotherm.

$$\frac{\Theta}{\mathbf{C}\_{\rm in}} = \mathbf{K}\_{\rm FH} \left[ \mathbf{1} - \boldsymbol{\theta} \right]^{\rm nF} \tag{14}$$

KFH is used to calculate the spontaneity Gibbs free energy.

#### *3.2.9 Halsey isotherm model*

In multi layer adsorption, the hetero porous nature of the adsorbent is demonstrated by the fitting of experimental data to this equation [36]. Eq. (15) denotes the Halsey adsorption isotherm model.

$$\mathbf{q}\_{\rm eq} = \exp\left[\frac{\ln|\mathbf{K}\_{\rm Ha} - \ln|\mathbf{C}\_{\rm eq}|}{\mathbf{n}\_{\rm Ha}}\right] \tag{15}$$

#### *3.2.10 Harkin-Jura isotherm model*

The Hurkin-Jura adsorption isotherm is used to explain multilayer adsorption on the surface of absorbents with heterogeneous pore distribution [37]. Eq. (16) describes the Hurkin-Jura adsorption isotherm model.

$$\mathbf{q\_{eq}} = \sqrt{\frac{\mathbf{A\_{Hj}}}{\mathbf{B\_{Hl}} - \log \mathbf{C\_{eq}}}} \tag{16}$$

#### *3.2.11 Jovanovic isotherm model*

With the approximation of monolayer localised adsorption without lateral contacts, the Jovanovic adsorption isotherm model is analogous to the Langmuir model [38]. Eq. (17) shows the Jovanovic adsorption isotherm model.

$$\mathbf{q}\_{\rm eq} = \mathbf{q}\_{\rm max} \left[ \mathbf{1} - \mathbf{e}^{-\left(\mathbf{k}\_{\rm l} \mathbf{C}\_{\rm eq}\right)} \right] \tag{17}$$

It forms the Langmuir isotherm at large adsorbate concentrations but does not obey Henry's rule.

#### *3.2.12 Elovich isotherm model*

Adsorption sites expand exponentially with adsorption, demonstrating multilayer adsorption, according to the Elovich isotherm model [39]. The Elovich adsorption isotherm model is depicted in Eq. (18).

$$\frac{\mathbf{q\_{eq}}}{\mathbf{q\_{max}}} = \mathbf{K\_{E}C\_{eq}} \exp^{\frac{q\_{eq}}{q\_{max}}} \tag{18}$$

#### *3.2.13 Kiselev isotherm model*

The Kiselev adsorption isotherm model is also known as localised monomolecular layer model [40]. This model is valid only for surface coverage θ > 0.68. The Kiselev adsorption isotherm model is given in Eq. (19).

$$\mathbf{K\_{eqK}} \mathbf{C\_{eq}} = \frac{\boldsymbol{\theta}}{\left(\mathbf{1} - \boldsymbol{\theta}\right)\left(\mathbf{1} + \mathbf{K\_{nK}}\boldsymbol{\theta}\right)}\tag{19}$$

#### **3.3 Three parameter models**

Models containing three parameters to explain the mechanism of adsorption are discussed using 16 models viz., Hill, Redlich-Peterson, Sips, Langmuir-Freundlich, Fritz-Schlunder-III, Radke-Prausnits, Toth, Khan, Koble-Corrigan, Jossens, Jovanovic-Freundlich, Brouers-Sotolongo, Vieth-Sladek, Unilan, Holl-Krich and Langmuir-Jovanovic.

#### *3.3.1 Hill isotherm model*

To characterise the adhesion of diverse species to homogeneous substrates, the Hill isotherm model is developed [41]. Eq. (20) depicts the Hill adsorption isotherm model.

$$\mathbf{q\_{eq}} = \frac{\mathbf{q\_{max}} \mathbf{C\_{eq}^{n\_H}}}{\mathbf{K\_H} + \mathbf{C\_{eq}^{n\_H}}} \tag{20}$$

If nH is greater than 1, this isotherm indicates positive co-operativity in binding, nH is equal to 1, it indicates non-cooperative or hyperbolic binding and nH is less than 1, indicating negative co-operativity in binding.

#### *3.3.2 Redlich-Peterson isotherm model*

The Redlich-Peterson isotherm model is created by combining elements of the Langmuir and Freundlich isotherms [42]. Eq. (21) depicts the Redlich-Peterson isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{A}\_{\rm RP} \mathbf{C}\_{\rm eq}}{\mathbf{1} + \mathbf{B}\_{\rm RP} \mathbf{C}\_{\rm eq}^{\beta}} \tag{21}$$

When the liquid phase concentration is low, this model approaches Henrys Law. βRP, the exponent, is usually between 0 and 1. When βRP = 1, this model is similar to the Langmuir model, and when βRP = 0, this isotherm is similar to the Freundlich model.

#### *3.3.3 Sips isotherm model*

The Sips adsorption isotherm model [43] was designed to represent localised adsorption without adsorbate-adsorbate interactions [44] at high adsorbate concentrations. Eq. (22) is the Sips adsorption isotherm model.

*Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm m} \mathbf{K}\_{\rm s} \mathbf{C}\_{\rm eq}^{\theta\_{\rm s}}}{\mathbf{1} + \mathbf{K}\_{\rm s} \mathbf{C}\_{\rm eq}^{\theta\_{\rm s}}} \tag{22}$$

When β<sup>S</sup> equal to 1 this isotherm approaches Langmuir isotherm and β<sup>S</sup> equal to 0, this isotherm approaches Freundlich isotherm.

#### *3.3.4 Langmuir-Freundlich isotherm model*

Adsorption in heterogeneous surfaces is described by the Langmuir-Freundlich isotherm model [17, 18]. Eq. (23) depicts the Langmuir-Freundlich isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \left(\mathbf{K}\_{\rm LF} \mathbf{C}\_{\rm eq}\right)^{\rm m\_{\rm LF}}}{\mathbf{1} + \left(\mathbf{K}\_{\rm LF} \mathbf{C}\_{\rm eq}\right)^{\rm m\_{\rm LF}}} \tag{23}$$

mLF is a heterogeneous parameter with a value between 0 and 1. This value rises when the degree of surface heterogeneity decreases. For mLF is equal to 1, this model covert to Langmuir model.

#### *3.3.5 Fritz-Schlunder-III isotherm model*

Because of the large number of coefficients in their isotherm, the Fritz-Schunder three parameter isotherm model was constructed to suit a wide variety of experimental findings [45]. Eq. (24) has this expression.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \mathbf{K}\_{\rm FS3} \mathbf{C}\_{\rm eq}}{\mathbf{1} + \mathbf{q}\_{\rm max} \mathbf{C}\_{\rm eq}^{\rm m\_{FS3}}} \tag{24}$$

If mFS3 is equal to 1, the Fritz-Schlunder-III model becomes the Langmuir model but for high concentrations of adsorbate, the Fritz-Schlunder-III reduces to the Freundlich model.

#### *3.3.6 Radke-Prausnits isotherm model*

At low adsorbate concentrations, the Radke-Prausnits isotherm model has numerous essential qualities that make it preferable in most adsorption systems [46]. When the Radke-Prausnits model exponent mRaP3 is equal to zero. Eqs. (25)–(27) show Radke-Prausnits isotherm models.

$$\textbf{Model 1} : \textbf{q}\_{\text{eq}} = \frac{\textbf{q}\_{\text{max}} \textbf{K}\_{\text{RaP1}} \textbf{C}\_{\text{eq}}}{\left[\textbf{1} + \textbf{K}\_{\text{RP1}} \textbf{C}\_{\text{eq}}\right]^{\text{m}\_{\text{RaP1}}}} \tag{25}$$

$$\textbf{Model 2} : \textbf{q}\_{\text{eq}} = \frac{\textbf{q}\_{\text{max}} \textbf{K}\_{\text{RaP2}} \textbf{C}\_{\text{eq}}}{1 + \textbf{K}\_{\text{RP2}} \textbf{C}\_{\text{eq}}} \tag{26}$$

$$\textbf{Model 3} : \textbf{q}\_{\text{eq}} = \frac{\textbf{q}\_{\text{max}} \textbf{K}\_{\text{RaP3}} \textbf{C}\_{\text{eq}}^{\text{m}\_{\text{RaP3}}}}{\textbf{1} + \textbf{K}\_{\text{RP3}} \textbf{C}\_{\text{eq}}^{\text{m}\_{\text{RaP3}-1}}} \tag{27}$$

When both mRaP1 and mRaP2 are equal to 1, the Radke-Prausnitz 1, 2 models decrease to the Langmuir model; nevertheless, at low concentrations, the models become Henry's law; but, at high adsorbate concentrations, the Radke-Prausnitz 1 and 2 models become the Freundlich model. When the exponent mRaP3 is equal to 1, the Radke-Prausnitz-3 equation simplifies to Henry's law, and when the exponent mRaP3 is equal to 0, it becomes the Langmuir isotherm.

#### *3.3.7 Toth isotherm model*

The Toth adsorption isotherm model is used to explain heterogeneous adsorption systems that fulfil both the low and high end boundaries of adsorbate concentration [47]. Eq. (28) represents the Toth isotherm model.

$$\mathbf{q\_{eq}} = \frac{\mathbf{q\_{max}} \mathbf{C\_{eq}}}{\left(\frac{1}{\mathbf{K\_T}} + \mathbf{C\_{eq}^{n\_T}}\right)^{\frac{1}{m\_T}}} \tag{28}$$

When n = 1, this equation simplifies to the Langmuir isotherm equation, indicating that the process is approaching the homogenous surface. As a result, the value n describes the adsorption system's heterogeneity. The system is considered to be heterogeneous if it deviates farther from unity.

#### *3.3.8 Khan isotherm model*

For adsorption of bi-adsorbate from pure dilute equations solutions, the Kahn isotherm model is devised [48]. Eq. (29) presents the Kahn isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm m} \mathbf{b}\_{\rm K} \mathbf{C}\_{\rm eq}}{\left(\mathbf{1} + \mathbf{b}\_{\rm K} \mathbf{C}\_{\rm eq}\right)^{\rm a\rm c}} \tag{29}$$

When aK is equal to one, the Toth model approaches the Langmuir isotherm model, and when aK is more than one, the Toth model simplifies to the Freundlich isotherm model.

#### *3.3.9 Koble-Corrigan isotherm model*

The Sips isotherm model is similar to the Koble-Carrigan isotherm model. This model includes both the Langmuir and the Freundlich isotherms [49]. Eq. (30) depicts the Koble-Carrigan isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{A}\_{\rm KC} \mathbf{C}\_{\rm eq}^{\rm n\rm c}}{\mathbf{1} + \mathbf{B}\_{\rm KC} \mathbf{C}\_{\rm eq}^{\rm n\rm c}} \tag{30}$$

This model reduces to the Freundlich isotherm at large adsorbate concentrations. It is only acceptable when n is higher than or equal to 1. When n is lesser than one, it indicates that the model, despite a high concentration coefficient or a low error value, is incapable of describing the experimental data.

#### *3.3.10 Jossens isotherm model*

The Jossens isotherm model is based on the energy distribution of adsorbateadsorbent interactions at adsorption sites [50]. At low concentrations, this model is reduced to Henry's law. Eq. (29) depicts the Jossens isotherm model.

*Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{K}\_{\rm f} \mathbf{C}\_{\rm eq}}{\mathbf{1} + \mathbf{J} \begin{array}{c} \mathbf{C}\_{\rm eq} \\ \end{array}} \tag{31}$$

At low capacity, J equates to Henry's constant. bJ is the Jossens isotherm constant, which is constant regardless of temperature or the composition of the adsorbent.

#### *3.3.11 Jovanovic-Freundlich isotherm model*

To depict single-component adsorption equilibrium on heterogeneous surfaces, the Jovanovic-Freundlich isotherm model is developed [51]. Eq. (32) represents the Jovanovic-Freundlich isotherm model.

$$\mathbf{q}\_{\rm eq} = \mathbf{q}\_{\rm max} \left[ \mathbf{1} - \mathbf{e}^{-\left(\mathbf{K}\_{\rm IF} \ \mathbf{C}\_{\rm eq} \ \mathbf{"} \right)} \right] \tag{32}$$

#### *3.3.12 Brouers-Sotolongo isotherm model*

This isotherm is built in the form of a deformed exponential function for adsorption onto a heterogeneous surface [52]. Eq. (33) depicts the Brouers-Sotolongo model.

$$\mathbf{q}\_{\rm eq} = \mathbf{q}\_{\rm max} \left[ \mathbf{1} - \mathbf{e}^{\left( -\mathbb{K}\_{\rm BS} \mathbf{C}\_{\rm eq}^{a\_{\rm RS}} \right)} \right] \tag{33}$$

The parameter αBS is related with distribution of adsorption energy and the energy of heterogeneity of the adsorbent surfaces at the given temperature [53].

### *3.3.13 Vieth-Sladek isotherm model*

This model includes two independent parts for calculating transient adsorption diffusion rates in solid adsorbents [54]. Eq. (34) represents the Vieth-Sladek isotherm model.

$$\mathbf{q\_{eq}} = \mathbf{K\_{VS}}\mathbf{C\_{eq}} + \frac{\mathbf{q\_{max}}\beta\_{\rm VS}\mathbf{C\_{eq}}}{\mathbf{1} + \beta\_{\rm VS}\mathbf{C\_{eq}}} \tag{34}$$

#### *3.3.14 Unilan isotherm model*

The application of the local Langmuir isotherm and uniform energy distribution is assumed for the Unilan isotherm model [44]. Eq. (35) presents the Unilan isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max}}{2\beta\_{\rm U}} \ln \left[ \frac{\mathbf{1} + \mathbf{K}\_{\rm U} \mathbf{C}\_{\rm eq} \mathbf{e}^{\beta\_{\rm U}}}{\mathbf{1} + \mathbf{K}\_{\rm U} \mathbf{C}\_{\rm eq} \mathbf{e}^{-\beta\_{\rm U}}} \right] \tag{35}$$

The higher the model exponent βU, the system is more heterogeneous. If β<sup>U</sup> is equal to 0, the Unilan isotherm model becomes the classical Langmuir model as the range of energy distribution is zero in this limit [50, 55, 56].

#### *3.3.15 Holl-Krich isotherm model*

The Langmuir Isotherm [57] is a version of the Holl-Krich Isotherm Model. The Freundlich isotherm is formed when the concentration of the solvent is low [22]. The Holl-Krich Isotherm Model may be seen in Eq. (36).

$$\mathbf{q\_{eq}} = \frac{\mathbf{q\_{max}} \mathbf{K\_{HK}} \mathbf{C\_{eq}^{n\_{HK}}}}{\mathbf{1} + \mathbf{K\_{HK}} \mathbf{C\_{eq}^{n\_{HK}}}} \tag{36}$$

#### *3.3.16 Langmuir-Jovanovic isotherm model*

This empirical model is the combined form of both Langmuir and Jovanovic isotherm [58]. The Langmuir-Jovanovic model is given in Eq. (37).

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \mathbf{C}\_{\rm eq} \left[ \mathbf{1} - \mathbf{e}^{\left( \mathbf{K}\_{\rm l} \mathbf{C}\_{\rm eq}^{\rm NL} \right)} \right]}{\mathbf{1} + \mathbf{C}\_{\rm eq}} \tag{37}$$

#### **3.4 Four parameter models**

The four parameter models discussed in this study are Fritz-Schlunder-IV, Baudu, Weber-van Vliet and Marczewski-Jaroniec models.

#### *3.4.1 Fritz-Schlunder-IV isotherm model*

Fritz-Schlunder IV model is another model comprised of four-parameter with combine features of Langmuir-Freundlich isotherm [45]. The model is given in Eq. (38).

$$\mathbf{q\_{eq}} = \frac{\mathbf{A\_{FS\ $}C\_{eq}^{\mathbf{a\_{FS\$ }}}}}{\mathbf{1} + \mathbf{B\_{FS\ $}C\_{eq}^{\mathbf{0\_{FS\$ }}}}} \tag{38}$$

When the values of αFS5 and βFS5 are less than or equal to one, this isotherm is true. The Fritz-Schlunder-IV isotherm transforms into the Freundlich equation at high adsorbate concentrations. If both αFS5 and βFS5 are equal to one, the isotherm is reduced to the Langmuir isotherm. This isotherm model becomes the Freundlich at large concentrations of adsorbate in the liquid-phase.

#### *3.4.2 Baudu isotherm model*

The Baudu isotherm model was created in response to a disagreement in computing the Langmuir constant and coefficient from slope and tangent across a wide range of concentrations [59]. The Langmuir isotherm model has been modified into the Baudu isotherm model. Eq. (39) explains it.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \ \mathbf{b}\_{\rm o} \ \mathbf{C}\_{\rm eq}^{(1+\mathbf{x}+\mathbf{y})}}{\mathbf{1} + \mathbf{b}\_{\rm o} \mathbf{C}\_{\rm eq}^{(1+\mathbf{x})}} \tag{39}$$

*Removal of Divalent Nickel from Aqueous Solution Using Blue Green Marine Algae… DOI: http://dx.doi.org/10.5772/intechopen.103940*

This model is only applicable in the range of (1 + x + y) < 1 and (1 + x) < 1. For lower surface coverage, Baudu model reduces to the Freundlich equation [58], i.e.:

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm m0} \,\mathrm{b}\_{\rm o} \mathbf{C}\_{\rm eq}^{(\rm + \, x + y)}}{\mathbf{1} + \mathbf{b}\_0} \tag{40}$$

#### *3.4.3 Weber-van Vliet isotherm model*

With four parameters, the Weber and van Vliet isotherm model is used to represent equilibrium adsorption data [60–62]. Eq. (41) depicts the model.

$$\mathbf{C\_{eq}} = \mathbf{P\_1q\_{eq}}^{\left(\mathbf{p\_{2}q\_{eq}^{\mathbf{P\_3}} + p\_4}\right)} \tag{41}$$

Multiple nonlinear curve fitting approaches based on the reduction of the sum of squares of residuals can be used to define the isotherm parameters P1, P2, P3, and P4.

#### *3.4.4 Marczewski-Jaroniec isotherm model*

The Marczewski-Jaroniec isotherm model is analogous to the Langmuir isotherm model [62, 63]. Eq. (42) represents the Marczewski-Jaroniec isotherm model.

$$\mathbf{q}\_{\rm eq} = \mathbf{q}\_{\rm max} \left[ \frac{\left( \mathbf{K}\_{\rm MJ} \mathbf{C}\_{\rm eq} \right)^{\rm n\_{\rm MJ}}}{\mathbf{1} + \left( \mathbf{K}\_{\rm MJ} \mathbf{C}\_{\rm eq} \right)^{\rm n\_{\rm MJ}}} \right] \tag{42}$$

The spreading of distribution along the route of increasing adsorption energy is described by KMJ.

#### **3.5 Five parameter model**

Accounting for the high parameter models offers unmistakable information on the process of adsorption under equilibrium conditions. Only one five-parameter model, the Fritz-Schlunder-V isotherm model, is used in this section.

### *3.5.1 Fritz-Schlunder-V isotherm model*

The Fritz-Schlunder adsorption isotherm model was created with the goal of more precisely reproducing model modifications for applicability over a wide range of equilibrium data [45]. Eq. (43) represents the Fritz-Schlunder adsorption isotherm model.

$$\mathbf{q}\_{\rm eq} = \frac{\mathbf{q}\_{\rm max} \mathbf{K}\_{\rm IFSS} \mathbf{C}\_{\rm eq}^{\rm aggs}}{\mathbf{1} + \mathbf{K}\_{\rm zFSS} \mathbf{C}\_{\rm eq}^{\beta\_{\rm FS}}} \tag{43}$$
