**4. Mathematical modeling**

Adsorption isotherm is the mathematical representation of adsorption capacity (*Q*) versus equilibrium concentration of the solute (*Ce*). Modeling adsorption isotherm data is important for prediction/comparison among adsorption performances. Two, three and four-parameter isotherm models are suggested to model the sorption data. Some of the important sorption isotherm models used in the sorption studies include, the Langmuir, Freundlich, Toth and Sips models.

The Langmuir model [53] was fundamentally derived to define the sorption (gas-solid phase) of activated carbon. However, in later years, it was employed to assess and calculate the adsorption behavior of various adsorbents. In its formulation, binding to the surface was primarily by physical forces and implicit in its derivation was the assumption that all sites possess equal affinity for the sorbate. Its use was extended to empirically describe equilibrium relationships between a bulk liquid phase and a solid phase [53]. The model can be expressed as *<sup>Q</sup>* = \_

$$Q = \frac{Q\_{\text{max}} \, b\_L \, \mathbf{C}\_e}{\mathbf{1} + b\_L \, \mathbf{C}\_e} \tag{1}$$

where *Q* is the sorptional capacity (mg/g); *Ce* is the equilibrum concentration (mg/L); *Qmax* is the maximum uptake of toxin by the adsorbent (mg/g) and *bL* is the equilibrium coefficient of the Langmuir model (L/mg).

The Freundlich model [54] was empirically derived equation; however it can be applied to adsorption onto diverse surfaces or surfaces with sites of varied affinities. It is assumed that the stronger binding sites are occupied first and that the binding strength decreases with increasing degree of site occupation. It can expressed as,

$$\mathbf{Q} = \mathbf{K}\_F \mathbf{C}\_e^{1/n\_y} \tag{2}$$

where *nF* is the exponent of the Freundlich model and *KF* is the Freundlich model coefficient (L/g)1/*<sup>n</sup> F*,

The Sips model [55] is based on the assumption that binding sites on the adsorbent have varied strengths and each active binding site interact with one sorbate molecule. The constant *K*s represents sorptional uptake of the adsorbent, whereas aS denotes affinity of adsorbent toward metal ions. At high metal ion concentrations, the model ultimately takes the Langmuir form, whereas at low metal concentrations reduces to the Freundlich model [56]. The model can be expressed as *Qe* = *KS Ce* \_

$$\mathbf{Q}\_{\varepsilon} = \frac{K\_{\rm S} \mathbf{C}\_{\varepsilon}^{\rho\_{\rm S}}}{\mathbf{1} + a\_{\rm S} \mathbf{C}\_{\varepsilon}^{\rho\_{\rm S}}} \tag{3}$$

where *aS* is the Sips model coefficient (L/mg)*<sup>β</sup> <sup>S</sup>*, *βS* is the Sips model exponent and *KS* is the Sips model isotherm coefficient (L/g)*<sup>β</sup> S*.

The Toth model [57] is the other three parameter model frequently employed to describe metal-adsorent isotherms. The model assumes quasi-Gaussian energy distribution and is derived from the potential theory. The Toth model can be expressed as

Toth model:

## 10th model: 
$$Q = \frac{Q\_{\text{max}} \, b\_T C\_\epsilon}{\left[1 + (b\_T C\_\epsilon)^{1/n\_f}\right]^{n\_T}} \tag{4}$$

where *bT* is the Toth model constant (L/mg) and *nT* is the Toth model exponent.

For any practical applications, the process design, operation control and sorption kinetics are very important [13]. The sorption kinetics can be described using several models.

The most commonly used method to identify the contribution of intraparticle diffusion during adsorption is through fitting the kinetic data to an intraparticle diffusion plot, as presented by Weber and Morris [58] as below:

$$Q\_t = k\_i \, t^{1/2} \tag{5}$$

where *Q t* is the sorptional capacity at any time *t* (mg/g) and *ki* is the intraparticle diffusion constant. This involves plotting the sorptional capacity at a given time vs. the square root of that time. If the plot passes through the origin, then intraparticle diffusion is the rate determining step.

The pseudo-first-order model assumes that the rate of change of solute uptake with time is directly proportional to the difference in saturation concentration and the amount of solid uptake with time. The model can be expressed as,

$$Q\_t = Q\_t \left(1 - \exp\left(-k\_1 t\right)\right) \tag{6}$$

where *Qe* is the equilibrium uptake (mg/g) and *k1* is the pseudo-first-order constant (1/min). In an attempt to understand the Cd(II) adsorption mechanism of rice straw-derived biochar from aqueous solutions, Fan et al. [59] fitted Cd(II) kinetics data using pseudo-first-order and pseudo-second-order kinetic models. The results indicated that Cd(III) adsorption kinetics by rice straw biochar was better described by the pseudo-first-order kinetic model.

The pseudo-second-order kinetics is framed to predict adsorption capcity over entire experimental conditions (ranges) as the model based on the adsorption capacity of the solid phase. *e* \_

$$Q\_t = \frac{Q\_\epsilon^\circ K\_2 t}{1 + Q\_\epsilon K\_2 t} \tag{7}$$

**215**

**Author details**

Sultanate of Oman

Ramalingham Senthilkumar1

\* and Donipathi Mogili Reddy Prasad<sup>2</sup>

2 Faculty of Engineering, Universiti Teknologi Brunei, Gadong, Brunei Darussalam

© 2020 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,

type strongly influences the sorption capacity of biochar. In addition, the process operating parameters such as pH, temperature, initial solute concentration and biochar dosage strongly influences the extent of metal sorption by biochar. Despite the application of biochar as sorbents is increasing as indicated through published literatures, more knowledge needed especially in the area of column sorption and

1 Department of Engineering, College of Applied Sciences, Sohar,

\*Address all correspondence to: kumar.soh@cas.edu.om

provided the original work is properly cited.

*Sorption of Heavy Metals onto Biochar DOI: http://dx.doi.org/10.5772/intechopen.92346*

real effluent clean-up.

where *k2* is pseudo-second-order constant. Xiao et al. [60] prepared biochar using cow bone meal for adsorption of Cd(II), Pb(II) and Cu(II) ions. On analyzing the kinetics data, the authors identified that pseudo-second-order model fitted the kinetics data well compared to the pseudo-first-order model based on the correlation coefficients and calculated equilibrium uptake values.

### **5. Conclusions**

Biochar represents an effective class of sorbent for remediation of heavy metals from solutions. Several studies recognized superior adsorption potential of biochar compared to other established sorbents. The pyrolysis temperature and feedstock

### *Sorption of Heavy Metals onto Biochar DOI: http://dx.doi.org/10.5772/intechopen.92346*

*Applications of Biochar for Environmental Safety*

*<sup>Q</sup>* = \_\_\_\_\_\_\_\_\_\_\_\_\_ *Qmax bT Ce*

diffusion plot, as presented by Weber and Morris [58] as below:

*Qt* = *ki t*

diffusion is the rate determining step.

capacity of the solid phase.

[1 + (*bT Ce*)1/*nT*

For any practical applications, the process design, operation control and sorption kinetics are very important [13]. The sorption kinetics can be described using

The most commonly used method to identify the contribution of intraparticle diffusion during adsorption is through fitting the kinetic data to an intraparticle

where *Q t* is the sorptional capacity at any time *t* (mg/g) and *ki* is the intraparticle diffusion constant. This involves plotting the sorptional capacity at a given time vs. the square root of that time. If the plot passes through the origin, then intraparticle

The pseudo-first-order model assumes that the rate of change of solute uptake with time is directly proportional to the difference in saturation concentration and

where *Qe* is the equilibrium uptake (mg/g) and *k1* is the pseudo-first-order constant (1/min). In an attempt to understand the Cd(II) adsorption mechanism of rice straw-derived biochar from aqueous solutions, Fan et al. [59] fitted Cd(II) kinetics data using pseudo-first-order and pseudo-second-order kinetic models. The results indicated that Cd(III) adsorption kinetics by rice straw biochar was

The pseudo-second-order kinetics is framed to predict adsorption capcity over entire experimental conditions (ranges) as the model based on the adsorption

*e* \_ *K2 t* 1 + *Qe K2 t*

*Qt* = *<sup>Q</sup>* <sup>2</sup>

correlation coefficients and calculated equilibrium uptake values.

where *k2* is pseudo-second-order constant. Xiao et al. [60] prepared biochar using cow bone meal for adsorption of Cd(II), Pb(II) and Cu(II) ions. On analyzing the kinetics data, the authors identified that pseudo-second-order model fitted the kinetics data well compared to the pseudo-first-order model based on the

Biochar represents an effective class of sorbent for remediation of heavy metals from solutions. Several studies recognized superior adsorption potential of biochar compared to other established sorbents. The pyrolysis temperature and feedstock

the amount of solid uptake with time. The model can be expressed as,

better described by the pseudo-first-order kinetic model.

where *bT* is the Toth model constant (L/mg) and *nT* is the Toth model

] *nT*

(4)

1/2 (5)

(7)

*Qt* = *Qe*(1 − exp (− *k*<sup>1</sup> *t*)) (6)

Toth model:

exponent.

several models.

**214**

**5. Conclusions**

type strongly influences the sorption capacity of biochar. In addition, the process operating parameters such as pH, temperature, initial solute concentration and biochar dosage strongly influences the extent of metal sorption by biochar. Despite the application of biochar as sorbents is increasing as indicated through published literatures, more knowledge needed especially in the area of column sorption and real effluent clean-up.
