• Type II adsorption isotherm

**Figure 4** shows a typical adsorption isotherm curve of type II. This type of adsorption shows a large deviation from the Langmuir isotherm model and a flat region, which is corresponding to a monolayer formation.

• Type III adsorption isotherm

This type of isotherm indicates that there is no flat region as shown in **Figure 5**, and also there are formations of multilayer adsorption.

• Type IV adsorption isotherm

It can be depicted from **Figure 6** that there is a monolayer formation (intermediate region), which is followed by a multilayer formation at certain adsorbate concentration. At low concentration of adsorbate, the adsorption is mostly similar to type II adsorption isotherm.

• Type V adsorption isotherm

It is similar to type IV with a difference in the range of adsorbate's concentration where the monolayer and multilayer start the formation as shown in **Figure 7**.

The adsorption isotherms usually are being studied to understand the adsorption behavior modulation and to calculate the adsorption capacity for the adsorbents, so the data analysis is done using a linear/nonlinear least squares methods of adsorption isotherms, where they describe the relationship between the adsorbed amount of adsorbate and its equilibrium concentration in the solution.

The Freundlich, Langmuir, Temkin, Sips, and Redlich-Peterson models are the most common types of the adsorption isotherms to describe the metal ion bioadsorption from their single component solution.

The Freundlich isotherm (Eq. 1) is an empirical model where the adsorption occurs on heterogeneous adsorption sites on adsorbent surface, which is the general case in macroalgae bioadsorbents:

**157**

where

*Type IV adsorption isotherm.*

**Figure 6.**

**Figure 5.**

*Type III adsorption isotherm.*

*Marine Algae Bioadsorbents for Adsorptive Removal of Heavy Metals*

*DOI: http://dx.doi.org/10.5772/intechopen.80850*

*qe* = *Kf*C*<sup>e</sup>*

*Kf*: The relative adsorption capacity (mg1−1/n11/n/g). n: The unit less constants reflect the adsorption intensity.

to the well-known insensitivity of its linear form (ln-ln plot).

*qe*: The adsorption density at equilibrium (mg adsorbate/g of adsorbent). *Ce*: The residual adsorbate concentration in the solution (mg/L) at equilibrium.

A plot of ln*Ce* against ln*qe* will give a straight line with a slope 1/n and intercept LnK*f.* Smaller 1/n greater expected heterogeneity [35]. It is worthy here to note that usually the adsorption data have a good fit with the Freundlich isotherm model due

The Langmuir adsorption isotherms model is considered as the best known for describing a monolayer chemical adsorption process on homogenous adsorption

1/*<sup>n</sup>* (1)

**Figure 4.** *Type II adsorption isotherm.*

*Marine Algae Bioadsorbents for Adsorptive Removal of Heavy Metals DOI: http://dx.doi.org/10.5772/intechopen.80850*

*Advanced Sorption Process Applications*

• Type II adsorption isotherm

• Type III adsorption isotherm

• Type IV adsorption isotherm

to type II adsorption isotherm.

solution.

• Type V adsorption isotherm

sorption from their single component solution.

case in macroalgae bioadsorbents:

• Type I adsorption isotherm (shown in **Figure 2**)

region, which is corresponding to a monolayer formation.

and also there are formations of multilayer adsorption.

(ii) it might be explained using the Langmuir adsorption isotherm.

The main characteristics of this type are (i) there is a monolayer adsorption and

**Figure 4** shows a typical adsorption isotherm curve of type II. This type of adsorption shows a large deviation from the Langmuir isotherm model and a flat

This type of isotherm indicates that there is no flat region as shown in **Figure 5**,

It can be depicted from **Figure 6** that there is a monolayer formation (intermediate region), which is followed by a multilayer formation at certain adsorbate concentration. At low concentration of adsorbate, the adsorption is mostly similar

It is similar to type IV with a difference in the range of adsorbate's concentration

The Freundlich, Langmuir, Temkin, Sips, and Redlich-Peterson models are the most common types of the adsorption isotherms to describe the metal ion bioad-

The Freundlich isotherm (Eq. 1) is an empirical model where the adsorption occurs on heterogeneous adsorption sites on adsorbent surface, which is the general

where the monolayer and multilayer start the formation as shown in **Figure 7**.

The adsorption isotherms usually are being studied to understand the adsorption behavior modulation and to calculate the adsorption capacity for the adsorbents, so the data analysis is done using a linear/nonlinear least squares methods of adsorption isotherms, where they describe the relationship between the adsorbed amount of adsorbate and its equilibrium concentration in the

**156**

**Figure 4.**

*Type II adsorption isotherm.*

$$q\_e = K\_f C\_e^{1/n} \tag{1}$$

where

*qe*: The adsorption density at equilibrium (mg adsorbate/g of adsorbent).

*Ce*: The residual adsorbate concentration in the solution (mg/L) at equilibrium.

*Kf*: The relative adsorption capacity (mg1−1/n11/n/g).

n: The unit less constants reflect the adsorption intensity.

A plot of ln*Ce* against ln*qe* will give a straight line with a slope 1/n and intercept LnK*f.* Smaller 1/n greater expected heterogeneity [35]. It is worthy here to note that usually the adsorption data have a good fit with the Freundlich isotherm model due to the well-known insensitivity of its linear form (ln-ln plot).

The Langmuir adsorption isotherms model is considered as the best known for describing a monolayer chemical adsorption process on homogenous adsorption

**Figure 7.** *Type V adsorption isotherm.*

sites on adsorbent surfaces. It partially considers the thermodynamic in the adsorption process. It is expressed in Eq. (2):

$$q\_{\epsilon} = \frac{q\_{\max} \, b \, \mathcal{C}\_{\epsilon}}{1 + b \, \mathcal{C}\_{\epsilon}} \tag{2}$$

where

*qe*: The adsorption capacity at equilibrium (mg of adsorbate/g of adsorbent).

*Ce*: The residual adsorbate concentration at equilibrium in solution (mg/L). *qmax*: The maximum adsorption capacity corresponding to monolayer coverage (mg of analyte adsorbed/g of adsorbent).

*b*: The Langmuir constant correlated to the adsorption energy (1/mg adsorbate).

The essential features of the Langmuir isotherm may be expressed in terms of equilibrium parameter *RL* (Eq. 3), which is a dimensionless constant referred to as separation factor or equilibrium parameter [36]:

$$R\_L = \frac{1}{1 + (1 + K\_L C\_e)}\tag{3}$$

The most used linear form of the Langmuir model is the following form (Eq. 4), which is also called reciprocal Langmuir plot:

$$\frac{C\_{\epsilon}}{q\_{\epsilon}} = \frac{1}{q\_{\max}b} + \frac{C\_{\epsilon}}{q\_{\max}} \tag{4}$$

**159**

*Marine Algae Bioadsorbents for Adsorptive Removal of Heavy Metals*

The energy of adsorption can be described using the Temkin isotherm (Eq. 5). However, this isotherm is valid only for an intermediate range of adsorbate concen-

*<sup>b</sup>* ln (*ATCe*) (5)

*<sup>b</sup> ln* (*Ce*) (6)

*<sup>b</sup>* (7)

(1 <sup>+</sup> (*bCe*)*ns*) (8)

*nRP*)) (9)

*RT*

*<sup>b</sup>* ln (*AT*) <sup>+</sup> \_\_\_

Plotting *qe* versus ln(*Ce*) gives a linear regression where the slope for that plot gives the Temkin isotherm constant (*b)* and the intercept gives the Temkin isotherm equilibrium binding constant (AT) (L/g), where R is the universal gas constant (8.314 J/mol K), T is the temperature in Kelvin (K), and B in Eq. (7) is a constant

*RT*

The Sips isotherm model for mono-component system is a combination between the Freundlich and Langmuir isotherm models. Eq. (8) expresses the Sips model:

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

*qe*: The adsorption capacity at equilibrium (mg of adsorbate/g of adsorbent). *Ce*: The residual adsorbate concentration at equilibrium in solution (mg/L). *qmax*: The maximum adsorption capacity corresponding to monolayer coverage

*b*: The Langmuir constant correlated to the adsorption energy (1/mg adsorbate).

As an extension for the Langmuir isotherm, a model with three parameters was

(1 + (*bRPCe*

where *Ce* (mg/L) is the residual adsorbate concentration at equilibrium in the solution and *qe* (mg/g) is the adsorption capacity at equilibrium. However, *aRP* (1/g) and *bRP* (1/mg)nRP do not have physical or chemical meaning. The third parameter *nRP* is dimensionless that gives an idea about the heterogeneity of adsorption sites on

Studying the uptake rate of heavy metals is achieved by the adsorption kinetics where the metal ion uptake rate clearly controls residence time of these compounds at the solid-liquid interface, so and in sequence the mechanism of heavy metal adsorption on the biomass materials will be evaluated using the most common

The simplest one which expresses on the proportionality between the metal adsorption and the number of vacant adsorption sites on the surface of adsorbents

*ns*: The Sips constant for the heterogeneity of binding surface.

established expressed in Eq. (9). That is Redlich-Peterson isotherm:

*qe* <sup>=</sup> *aRP* (*Ce*) \_\_\_\_\_\_\_\_\_\_\_\_

*RT*

*RT*

*DOI: http://dx.doi.org/10.5772/intechopen.80850*

*qe* = \_\_\_

Rearranging Eq. (4) results in Eq. (6):

*qe* = \_\_\_

related to heat of adsorption (J/mol):

*B* = \_\_\_

*qe* <sup>=</sup> *qmax* (*b*C*e*)*ns*

(mg of analyte adsorbed/g of adsorbent).

the surface of adsorbents [39].

**2.2 Kinetic models**

kinetic models.

trations [38]:

where

Plotting *Ce/qe* versus C*e* from the experimental data gives a linear regression where the slope for that plot gives the experimental maximum adsorption capacity *qmax*, and the intercept gives the Langmuir constant *b*.

There are another three linear transformation forms of the Langmuir isotherm models: (1) the distribution coefficient or Scatchard plot, (2) Eadie-Hofstee plot, and (3) double reciprocal Lineweaver-Burk plot. Every one of these four linear transformation forms gives a greater weighing to low adsorption values than to high adsorption values, which leads to changing in the error distribution [37].

*Marine Algae Bioadsorbents for Adsorptive Removal of Heavy Metals DOI: http://dx.doi.org/10.5772/intechopen.80850*

The energy of adsorption can be described using the Temkin isotherm (Eq. 5). However, this isotherm is valid only for an intermediate range of adsorbate concentrations [38]:

$$q\_{\epsilon} = \frac{RT}{b} \ln \left( A\_T C\_{\epsilon} \right) \tag{5}$$

Rearranging Eq. (4) results in Eq. (6):

$$q\_{\epsilon} = \frac{RT}{b} \ln \left( A\_T \right) + \quad \frac{RT}{b} \ln \left( C\_{\epsilon} \right) \tag{6}$$

Plotting *qe* versus ln(*Ce*) gives a linear regression where the slope for that plot gives the Temkin isotherm constant (*b)* and the intercept gives the Temkin isotherm equilibrium binding constant (AT) (L/g), where R is the universal gas constant (8.314 J/mol K), T is the temperature in Kelvin (K), and B in Eq. (7) is a constant related to heat of adsorption (J/mol):

$$B\_{\perp} = \frac{RT}{b} \tag{7}$$

The Sips isotherm model for mono-component system is a combination between the Freundlich and Langmuir isotherm models. Eq. (8) expresses the Sips model:

$$q\_e = \frac{q\_{max} \left(b \, \mathrm{C}\_e\right)^{us}}{\left\{1 + \left(b \, \mathrm{C}\_e\right)^{us}\right\}} \tag{8}$$

where

*Advanced Sorption Process Applications*

tion process. It is expressed in Eq. (2):

(mg of analyte adsorbed/g of adsorbent).

separation factor or equilibrium parameter [36]:

which is also called reciprocal Langmuir plot:

\_\_\_

*RL* = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>1</sup>

*qmax*, and the intercept gives the Langmuir constant *b*.

*Ce*

adsorption values, which leads to changing in the error distribution [37].

*qe* <sup>=</sup> \_\_\_\_\_ <sup>1</sup> *qmax b*

Plotting *Ce/qe* versus C*e* from the experimental data gives a linear regression where the slope for that plot gives the experimental maximum adsorption capacity

There are another three linear transformation forms of the Langmuir isotherm models: (1) the distribution coefficient or Scatchard plot, (2) Eadie-Hofstee plot, and (3) double reciprocal Lineweaver-Burk plot. Every one of these four linear transformation forms gives a greater weighing to low adsorption values than to high

where

**Figure 7.**

*Type V adsorption isotherm.*

*qe* = \_\_\_\_\_\_\_

sites on adsorbent surfaces. It partially considers the thermodynamic in the adsorp-

*qe*: The adsorption capacity at equilibrium (mg of adsorbate/g of adsorbent). *Ce*: The residual adsorbate concentration at equilibrium in solution (mg/L). *qmax*: The maximum adsorption capacity corresponding to monolayer coverage

*b*: The Langmuir constant correlated to the adsorption energy (1/mg adsorbate). The essential features of the Langmuir isotherm may be expressed in terms of equilibrium parameter *RL* (Eq. 3), which is a dimensionless constant referred to as

The most used linear form of the Langmuir model is the following form (Eq. 4),

+ \_\_\_\_ *Ce*

<sup>1</sup> <sup>+</sup> (1 <sup>+</sup> *KL Ce*) (3)

*qmax* (4)

*qmax bCe* 1 + *bCe*

(2)

**158**

*qe*: The adsorption capacity at equilibrium (mg of adsorbate/g of adsorbent).

*Ce*: The residual adsorbate concentration at equilibrium in solution (mg/L).

*qmax*: The maximum adsorption capacity corresponding to monolayer coverage (mg of analyte adsorbed/g of adsorbent).

*b*: The Langmuir constant correlated to the adsorption energy (1/mg adsorbate). *ns*: The Sips constant for the heterogeneity of binding surface.

As an extension for the Langmuir isotherm, a model with three parameters was established expressed in Eq. (9). That is Redlich-Peterson isotherm:

$$q\_{\epsilon} = \frac{a\_{RP} \left(\mathbf{C}\_{\epsilon}\right)}{\left(1 + \left(b\_{RP} \mathbf{C}\_{\epsilon}^{u\_{RP}}\right)\right)}\tag{9}$$

where *Ce* (mg/L) is the residual adsorbate concentration at equilibrium in the solution and *qe* (mg/g) is the adsorption capacity at equilibrium. However, *aRP* (1/g) and *bRP* (1/mg)nRP do not have physical or chemical meaning. The third parameter *nRP* is dimensionless that gives an idea about the heterogeneity of adsorption sites on the surface of adsorbents [39].

#### **2.2 Kinetic models**

Studying the uptake rate of heavy metals is achieved by the adsorption kinetics where the metal ion uptake rate clearly controls residence time of these compounds at the solid-liquid interface, so and in sequence the mechanism of heavy metal adsorption on the biomass materials will be evaluated using the most common kinetic models.

The simplest one which expresses on the proportionality between the metal adsorption and the number of vacant adsorption sites on the surface of adsorbents is Lagergren model (pseudo-first-order). The nonlinear and linear forms of the model are represented in Eqs. (10) and (11), respectively [40]:

$$q\_t = q\_\epsilon \{ \mathbf{1} + e^{-k\mu} \} \tag{10}$$

$$
\ln \langle q\_\varepsilon - q\_t \rangle = \ln \langle q\_\varepsilon \rangle - k\_1 t \tag{11}
$$

where *qt* and *qe* (mg/g), respectively, are the adsorption capacity at any time (*t*) and at equilibrium. *k1* (1/min) is the pseudo-first-order rate constant.

The kinetic model that has the correlation between the adsorption of metal ions and the square of active vacant adsorption sites on the surface of adsorbents is called pseudo-second-order rate model (Eq. 12) [38]:

$$q\_{\epsilon} = \frac{q\_{\epsilon}^{2}(k\_{2}t)}{\left(\mathbf{1} + \left(k\_{2}q\_{\epsilon}t\right)\right)}\tag{12}$$

Eq. (8) can be rearranged to be in the following linear form (Eq. 13):

$$\frac{t}{q\_t} = \frac{1}{q\_\epsilon^2 k\_2} + \frac{t}{q\_\epsilon} \tag{13}$$

where *qt* and *qe* (mg/g), respectively, are the adsorption capacity at any time (*t*) and at equilibrium. *k2* (g/mg min) is the pseudo-second-order rate constant.

By plotting ln*(qe−qt)* versus *t* and *t/qt* versus *t* in the previous equations (Eqs. (11) and (13)), all the adsorption kinetic parameters can be determined from the slope and the intercept.

The influence of mass transfer resistance on binding metal ions on adsorbents was tested using the intra-particle diffusion model (Weber and Morris model) represented in Eq. (14) [41]:

$$q\_t = \text{ kJ}^{0.5} + \text{C} \tag{14}$$

**161**

**Author details**

Mazen K. Nazal

provided the original work is properly cited.

*Marine Algae Bioadsorbents for Adsorptive Removal of Heavy Metals*

The author declares that there are no conflicts of interest.

The support of the Center for Environment and Water in the research institute

of King Fahd University of Petroleum and Minerals King Fahd University of

*DOI: http://dx.doi.org/10.5772/intechopen.80850*

Petroleum and Minerals is highly acknowledged.

**Acknowledgements**

**Conflict of interest**

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Center for Environment and Water (CEW), Research Institute (RI) at King Fahd

University of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia

\*Address all correspondence to: mazennazal@kfupm.edu.sa

where *qt* (mg/g) is the adsorption capacity at any time (*t*), *kid* (mg/g min0.5) is the intra-particle diffusion rate constant, and C (mg/g) is a constant related to the thickness of the boundary layer. From plotting of *qt* versus the square root of *t,* the diffusion constant kid can be calculated. If this plot passes through the origin, then intra-particle diffusion is the only rate-controlling step.

#### **3. Conclusion**

Removal of heavy metals from wastewater would provide an exceptional alternative water resource. Algae biomass adsorbents, which utilized for adsorptive removal of heavy metal pollutants from wastewater, show a promising alternative. Different empirical isotherm models for single analyte have been discussed (i.e., Freundlich, Langmuir, Temkin, Sips, and Redlich-Peterson). In a large number of studies, the Freundlich and Langmuir models are the most commonly and widely used isotherm models. The two kinetic models, which are still in a wide use for studying the rate uptake of heavy metals and their bioadsorption from aqueous solutions, are pseudofirst- and pseudo-second-order kinetic models. In chemisorption process, the pseudosecond-order kinetic model is superior to pseudo-first-order model as it takes into account the interaction of adsorbent-adsorbate through their valency forces.

*Marine Algae Bioadsorbents for Adsorptive Removal of Heavy Metals DOI: http://dx.doi.org/10.5772/intechopen.80850*
