*2.3.1. Adsorption isotherms*

To describe the concentration-dependent equilibrium between the pollutant amount adsorbed on the cells (*a*) and the pollutant concentration dissolved in aqueous solution *(Ce* ) at equilibrium conditions and constant temperature, which is referred to as the adsorption isotherm. Langmuir, Freundlich, Langmuir-Freundlich, Redlich-Peterson, Brunauer-Emmett-Teller (BET), and Radke-Prausnitz models are the most frequently cited literature in the literature [23–26].

When sorption equilibrium is reached, the adsorption capacity can be calculated from mass balance in a batch sorption system consisting of a discrete volume of water and adsorbent:

$$a = \frac{V}{m} (\mathbf{C}\_0 - \mathbf{C}\_\epsilon) \tag{1}$$

*a* = *k* . *C*<sup>1</sup>/*<sup>n</sup>* (3)

where *k* (L.kg−1) is a Freundlich constant referring to biosorbent capacity and *n* (dimensionless) is a Freundlich constant indicating the intensity of biosorption. Freundlich isotherm does

Tempkin isotherm [25] assumes that biosorption energy decreases linearly with increasing saturation of biosorption sites, rather than decreasing exponentially, as Freundlich isotherma

where *aTe* is the Tempkin isotherm constant, *bTe* is the Tempkin constant referring to the biosorption energy, *R* is the universal gas constant (8.314 J.mol−1.K−1), *T* is the thermodynamic

BET (Brunauer, Emmett, and Teller) isotherm is described by the following equation [26]:

is the equilibrium pollutant concentration in solution.

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (*Ce* <sup>−</sup> *<sup>C</sup>*) . (<sup>1</sup> <sup>+</sup> (*<sup>d</sup>* <sup>−</sup> <sup>1</sup>) . *<sup>C</sup>*/*Ce*

is the equilibrium concentration of adsorbate (kg.m−3), *d* is the constant expressing

A pseudo-first order model [27] and the pseudo-second order kinetic model [28] can be

The Lagergren pseudo-first order model suggests that the rate of sorption is proportional to the number of sites unoccupied by the solutes. The pseudo-first order model can be written

*at* = *ae*(1 − *exp*(−k<sup>1</sup> . *t*)) (6)

1 + *ae*

is the amount of pollutant biosorbed at equilibrium (mg.g−1), *at*

The pseudo-second order kinetic model can be written in linearized form as follows:

. *ln*(*aTe* . *Ce*) (4)

) (5)

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is the amount of pol-

is the first order rate constant (min−1).

. *<sup>k</sup>*<sup>1</sup> . *<sup>t</sup>*)) (7)

*bTe*

not take into account the saturation of biosorbents.

suggests. Tempkin isotherm is given as follows:

*<sup>a</sup>* <sup>=</sup> *amax* . *<sup>d</sup>* . *<sup>C</sup>*

the energy of sorbate interaction with the sorbent surface.

applied to fit the experimental data and evaluate the adsorption kinetics.

*a* = \_\_\_\_\_\_\_\_\_\_\_\_\_ *<sup>R</sup>* . *<sup>T</sup>*

temperature (K), and *Ce*

*2.3.2. Kinetics of adsorption*

in linearized form as follows:

lutant biosorbed (mg.g−1) at any time *t,* and k<sup>1</sup>

*at* <sup>=</sup> *ae*((<sup>1</sup> <sup>−</sup> \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ <sup>1</sup>

is the second order rate constant (g.mg−1.min−1).

The pseudo-second order model does not identify the diffusion mechanism.

where *Ce*

where *ae*

where *k*<sup>2</sup>

where *a* is the sorption capacity (kg.kg−1), *V* is the volume of water/wastewater (m<sup>3</sup> ) treated in a single sorption step, *m* is the mass of the adsorbent (kg), *C0* and *Ce* are the initial and equilibrium aqueous adsorbate concentration (kg.m−3), respectively.

Most often, pollutant distribution is concentration-dependent and in such case non-linear forms of adsorption isotherms are used to describe experimental data resulted from batch adsorption measurements. Langmuir isotherm is defined assuming that adsorption takes place at specific homogeneous sites at the surface of the adsorbent. This means that once the molecule of the adsorbed substance occupies a sorption site, no further adsorption can occur at this site. The Langmuir adsorption isotherm [23] has the form:

$$a = \begin{array}{c} a\_{mn} \cdot b \cdot \text{C}\_{\text{e}} \\ \hline 1+b \cdot \text{C}\_{\text{e}} \end{array} \tag{2}$$

Where *amax* (kg.kg−1) is the maximum biosorbent capacity of the adsorbent in the formation of a saturated monomolecular adsorption layer and *b* (L.kg−1) is Langmuir's empirical constant associated with the free energy of biosorption.

Freundlich's isotherm [24] was postulated for adsorption at heterogeneous surfaces and it takes the form:

$$a = \mathbb{k} \quad \text{C}^{1/n} \tag{3}$$

where *k* (L.kg−1) is a Freundlich constant referring to biosorbent capacity and *n* (dimensionless) is a Freundlich constant indicating the intensity of biosorption. Freundlich isotherm does not take into account the saturation of biosorbents.

Tempkin isotherm [25] assumes that biosorption energy decreases linearly with increasing saturation of biosorption sites, rather than decreasing exponentially, as Freundlich isotherma suggests. Tempkin isotherm is given as follows:

$$a = \frac{R \cdot T}{b\_{1\epsilon}} \; . \; \ln \langle a\_{1\epsilon} \; . \; \text{C}\_{\epsilon} \rangle \tag{4}$$

where *aTe* is the Tempkin isotherm constant, *bTe* is the Tempkin constant referring to the biosorption energy, *R* is the universal gas constant (8.314 J.mol−1.K−1), *T* is the thermodynamic temperature (K), and *Ce* is the equilibrium pollutant concentration in solution.

BET (Brunauer, Emmett, and Teller) isotherm is described by the following equation [26]:

\*\*E 1\*\* (Brùnáuer, Emmert, and \*\*euleer) istémers is aescríbea by me rolowong equation (\$\omega\$):

$$a = \frac{a\_{nn} \cdot d \cdot C}{\text{(C}\_r - \text{C}) \cdot \begin{array}{l} \text{\$a\_{nn}\$ } \cdot d \cdot \text{\$C|} \\ \text{(\$\bf C\$)} \end{array}} \tag{5}$$

where *Ce* is the equilibrium concentration of adsorbate (kg.m−3), *d* is the constant expressing the energy of sorbate interaction with the sorbent surface.

#### *2.3.2. Kinetics of adsorption*

and complexation [22]. For more detailed information we refer the reader to the work Fomina

The equilibrium distribution of the sorbed pollutant (sorbate) between the sorbent and the aqueous phase is required to determine the maximum sorbent's uptake capacity for a sorbate

Besides sorbate distribution at equilibrium, the sorption kinetics provides additional important information about the sorption mechanism, especially the rate of pollutant removal. When applied in water treatment technology, information on sorption kinetics is important for setting an optimum residence time of the wastewater at the biosolid phase interface.

To describe the concentration-dependent equilibrium between the pollutant amount adsorbed

rium conditions and constant temperature, which is referred to as the adsorption isotherm. Langmuir, Freundlich, Langmuir-Freundlich, Redlich-Peterson, Brunauer-Emmett-Teller (BET), and Radke-Prausnitz models are the most frequently cited literature in the literature [23–26].

When sorption equilibrium is reached, the adsorption capacity can be calculated from mass balance in a batch sorption system consisting of a discrete volume of water and adsorbent:

Most often, pollutant distribution is concentration-dependent and in such case non-linear forms of adsorption isotherms are used to describe experimental data resulted from batch adsorption measurements. Langmuir isotherm is defined assuming that adsorption takes place at specific homogeneous sites at the surface of the adsorbent. This means that once the molecule of the adsorbed substance occupies a sorption site, no further adsorption can occur

1 + *b* . *Ce*

Where *amax* (kg.kg−1) is the maximum biosorbent capacity of the adsorbent in the formation of a saturated monomolecular adsorption layer and *b* (L.kg−1) is Langmuir's empirical constant

Freundlich's isotherm [24] was postulated for adsorption at heterogeneous surfaces and it

) at equilib-

) treated

(2)

are the initial and

*<sup>m</sup>*(*C*<sup>0</sup> − *Ce*) (1)

and *Ce*

on the cells (*a*) and the pollutant concentration dissolved in aqueous solution *(Ce*

where *a* is the sorption capacity (kg.kg−1), *V* is the volume of water/wastewater (m<sup>3</sup>

in a single sorption step, *m* is the mass of the adsorbent (kg), *C0*

at this site. The Langmuir adsorption isotherm [23] has the form:

*<sup>a</sup>* <sup>=</sup> *amax* . *<sup>b</sup>* . *<sup>C</sup>* \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_*<sup>e</sup>*

associated with the free energy of biosorption.

takes the form:

equilibrium aqueous adsorbate concentration (kg.m−3), respectively.

and Gadd [21].

4 Biosorption

**2.3. Modeling of biosorption**

*2.3.1. Adsorption isotherms*

and to understand the sorption mechanism.

*a* = \_\_*<sup>V</sup>*

A pseudo-first order model [27] and the pseudo-second order kinetic model [28] can be applied to fit the experimental data and evaluate the adsorption kinetics.

The Lagergren pseudo-first order model suggests that the rate of sorption is proportional to the number of sites unoccupied by the solutes. The pseudo-first order model can be written in linearized form as follows:

$$a\_i = a\_i(1 - \exp\{-\mathbf{k}\_1 \dots \ t\})\tag{6}$$

where *ae* is the amount of pollutant biosorbed at equilibrium (mg.g−1), *at* is the amount of pollutant biosorbed (mg.g−1) at any time *t,* and k<sup>1</sup> is the first order rate constant (min−1).

The pseudo-second order kinetic model can be written in linearized form as follows:

$$a\_t = a\_i \left( \left( 1 - \frac{1}{1 + a\_i \cdot k\_i \cdot t} \right) \right) \tag{7}$$

where *k*<sup>2</sup> is the second order rate constant (g.mg−1.min−1).

The pseudo-second order model does not identify the diffusion mechanism.

From the majority of biosorption-related work, it follows that the pseudo-first order equation does not describe well-meaning values throughout the contact time. Generally, this equation is only applicable in the initial phase of the adsorption process. This is due to the fact that, using the linearized form of Eq. (6) it is necessary to know the value of the equilibrium adsorption capacity, which can be approximated by the extrapolation of experimental data for infinite time, i.e., the trial and error method. On the other hand, it is not necessary to know this value for the use of the linearized form of the kinetic equation of the pseudo-second-order.

waste; in fact, they are the source for the production of agar, alginate and, carrageenan. This means that the choice of algae for biosorption purposes needs to be given the utmost attention. Scientists work mainly with brown algae using one of the best metal sorbents seaweed, Sargassum seaweed. They focus on the study of sorption properties and biosorption mechanisms. Biosorbents using algae, bacteria, fibrous fungi, and yeasts are also used for analytical techniques, specifically for solid phase extraction to determine metals present in trace

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Microbial biomass (bacteria, fungi, and micorrhagia) shows better results of biosorption of dyes than macroscopic materials (seaweed, squirrel crabs, etc.). The reason is the difference in cell wall and functional groups involved in dye binding. Many bacteria, fungi, and microor-

The results of the study by Simionato et al. [9] show that the use of chitosan obtained from silkworm chrysalis is a viable alternative for the removal of blue remazol and black remazol five dyes from the wastewater of the textile industry. Potential biosorbents belonging to the class of bacteria include *Bacillus, Geobacillus, Lactobacillus, Pseudomonas, Streptomyces,* 

Several studies have recently been carried out to develop cheap sorbents from industrial and agricultural waste. Partial attention was paid in particular to crab shells, activated sludge, rice husks, egg shells, mosses, and lichens. The results showed that, in particular, crab shells have

A preferred biosorbent material is activated sludge. There are a large number of binding sites on the cell walls of microorganisms, which are predominantly composed of polysaccharides, proteins, and lipids. This is due to the high biosorption capacity of activated sludge. The amount of excess sludge produced mostly outweighs the possibilities of its use and represents one related

Authors [30–32] disclose the advantages of using aerobic and anaerobic deactivated sludge to remove dyestuffs and hazardous effluent from wastewater. Qiu et al. [33] presented the results of research into the use of active aerobic and anaerobic sludge for sewage treatment. The extent of biosorption depends on the type of biomass [34]. In the past, biosorbent phenomena have often been found to bioaccumulate highly hydrophobic organic substances directly depending on the lipid content of biomass. However, non-polar substances have been found to accumulate in organisms according to the distribution equilibrium between the medium and the lipid content of the organism [35]. Other authors found the opposite phenomenon to track DDT [Dichloro-Diphenyl-Trichloroethane or 1,1,1-Trichloro-2,2-bis(p-chlorophenyl)-ethane] adsorption by different soil fractions [36]. Some soil fractions were first extracted with ether and ethanol to remove lipid-like substances. Absence of lipid-like materials did not decrease, on the contrary, increased DDT adsorption with soil, indicating that other substances other than lipids may also play a role in biosorption. A similar finding was obtained by monitoring the adsorption of chlorites with microbial biomass [37]. Bacterial biomass with the highest lipid content among the observed samples had the lowest biosorption capacity. Further, it has been found that in different samples of fibrous fungi biomass, despite the similar lipid content in the

problem of wastewater treatment. Thus, this biosorbent is reely available and low-cost.

excellent sorption abilities in relation to arsenic, chromium, cobalt, and nickel.

amounts in different aqueous matrices [29].

ganisms bind different types of dyes.

*Staphylococcus, Streptococcus,* and others.

In this context, it should be emphasized that using a non-linear method of determining the values of parameters of non-linear equations in general it is possible to avoid such errors in the modeling of process kinetics.
