**2.3 Supporting theory**

In a typical adsorption process, species/materials in gaseous or liquid form (the adsorptive) become attached to a solid or liquid surface (the adsorbent) and form the adsorbate [Scheme 1], ( Christmann, 2010).

the subsequent layers. Often observed during Physisorption

Scheme 1. Presentation of the typical adsorption process (after Christmann, 2010)

following layers. Typical for Chemisorption case

Comparison of the Thermodynamic Parameters Estimation for

adsorbate per (g) of adsorbent;

adsorption process (Christmann, 2010).

*eql*

*KBET* is a parameter related to the binding intensity for all layers;

polarity increases get higher solubility because water is a polar solvent).

coverage.

(Christmann, 2010).

for the first layer [4]:

of adsorption decreases.

efficiency, respectively.

the Adsorption Process of the Metals from Liquid Phase on Activated Carbons 101

1

*eql*

max

*L eql*

(3)

(4)

*L eql*

*K q C <sup>q</sup> K C*

where *qmax* is the maximum adsorption capacity (monolayer coverage), i.e. mmol of the

*KL* is the constant of Langmuir isotherm if the enthalpy of adsorption is independent of

The constant *KL* depends on (i) the relative stabilities of the adsorbate and adsorptive species involved, (ii) on the temperature of the system, and (iii) on the initial concentration of the metal ions in the solution. Factors (ii) and (iii) exert opposite effects on the concentration of adsorbed species: the surface coverage may be increased by raising the initial metal concentration in the solution but will be reduced if the surface temperature is raised

If the desorption energy is equal to the energy of adsorption, then the first-order processes has been assumed both for the adsorption and the desorption reaction. Whether the deviation exists, the second-order processes should be considered, when adsorption/desorption reactions involving rate-limiting dissociation. From the initial slope of a log - log plot of a Langmuir adsorption isotherm the order of adsorption can be easily determined: if a slope is of 1, that is 1st order adsorption; if a slope is of 0.5, that is 2nd order

*BET (Brunauer, Emmett and Teller) Isotherm:* This is a more general, multi-layer model.

It assumes that a Langmuir isotherm applies to each layer and that no transmigration occurs between layers. It also assumes that there is equal energy of adsorption for each layer except

> max ( ) 1 ( 1) ( / ) *BET eql*

*init eql BET eql init*

where *Cinit* is saturation (solubility limit) concentration of the metal ions (in mmol/l) and

Two limiting cases can be distinguished: (i) when *Ceql* << *Cinit* and *KBET* >> 1 BET isotherm approaches Langmuir isotherm (*KL* = *KBET*/*Cinit*); (ii) when the constant *KBET* >> 1, the heat of adsorption of the very first monolayer is large compared to the condensation enthalpy; and adsorption into the second layer only occurs once the first layer is completely filled. Conversely, if *KBET* is small, then a multilayer adsorption already occurs while the first layer is still incomplete (Christmann, 2010). In general, as solubility of solute increases the extent

This is known as the "Lundelius' Rule". Solute-solid surface binding competes with solutesolvent attraction. Factors which affect solubility include molecular size (high MW- low solubility), ionization (solubility is minimum when compounds are uncharged), polarity (as

*Freundlich Isotherm*: For the special case of heterogeneous surface energies in which the energy term (*KF*) varies as a function of surface coverage the Freundlich model are used [5]:

where *K*F and 1/*n* are Freundlich constants related to adsorption capacity and adsorption

1/*n*

*eql F eql q KC* (5)

*K qC <sup>q</sup> C C K CC* 

Since the adsorptive and the adsorbent often undergo a chemical reactions, the chemical and physical properties of the adsorbate is not always just the sum of the individual properties of the adsorptive and the adsorbent, and often represents a phase with new properties (Christmann, 2010).

When the adsorbent and adsorptive are contacted long enough, the equilibrium is established between the amount of adsorptive adsorbed on the carbon surface (the adsorbate) and the amount of adsorptive in the solution. The equilibrium relationship is described by isotherms. Therefore, the adsorption isotherm for the metal adsorption is the relation between the specific amount adsorbed (*qeql,* expressed in (mmol) of the adsorbate per (g) of the solid adsorbent) and the equilibrium concentrations of the adsorptive in liquid phase (*Ceql*, in expressed in (mmol) of the adsorptive per (l) of the solution), when amount adsorbed is equals *qeql*.

Chemical equilibrium between adsorbate and adsorptive leads to a constant surface concentration (*Γ*) in [mmol/m2]. Constant (*Γ*) is maintained when the fluxes of adsorbing and desorbing particles are equal, thus the initial adsorptive concentration and temperature dependence of the liquid-solid phase equilibrium are considered (Christmann, 2010).

A common procedure is to equate the chemical potentials and their derivatives of the phases involved. Note: the chemical potential (*μ*) is the derivative of the Gibbs energy (*dG)* with respect to the mole number (*n*i) in question (Christmann, 2010), which is for the adsorption process from the liquid phase is the equilibrium concentrations of the adsorptive in liquid phase (*Ceql*), when amount adsorbed on the carbon surface is equals (*qeql*) [1]:

$$\mu\_i = \left\lfloor \frac{dG}{dn\_i} \right\rfloor\_{\mathbb{P}^{\mathbb{T}^r}}, \text{other mole numbers (C}\_{\text{eq}}) \tag{1}$$

The decisive quantities when studying the adsorption process are the heat of adsorption and its coverage dependence to lateral particle–particle interactions, as well as the kind and number of binding states (Christmann, 2010). The most relevant thermodynamic variable to describe the heat effects during the adsorption process is the differential isosteric heat of adsorption (*Hx*), kJ mol-1), that represents the energy difference between the state of the system before and after the adsorption of a differential amount of adsorbate on the adsorbent surface (Christmann, 2010). The physical basis is the Clausius-Clapeyron equation [2]:

$$\frac{1}{d(\mathcal{C}\_{eq})} \left[ \frac{d(\mathcal{C}\_{eq})}{dT} \right]\_{\mathbb{C}} = \left| \frac{d \ln(\mathcal{C}\_{eq})}{d(\frac{1}{T})} \right|\_{\mathbb{C}} = -\frac{\Delta H\_z}{R} \tag{2}$$

Knowledge of the heats of sorption is very important for the characterization and optimization of an adsorption process. The magnitude of (*ΔHx*) value gives information about the adsorption mechanism as chemical ion-exchange or physical sorption: for physical adsorption, (*ΔHx*) should be below 80 kJmol-1 and for chemical adsorption it ranges between 80 and 400 kJmol-1 (Saha & Chowdhury, 2011). It also gives some indication about the adsorbent surface heterogeneity.

*Langmuir Isotherm*: A model assumes monolayer coverage and constant binding energy between surface and adsorbate [3]:

Since the adsorptive and the adsorbent often undergo a chemical reactions, the chemical and physical properties of the adsorbate is not always just the sum of the individual properties of the adsorptive and the adsorbent, and often represents a phase with new properties

When the adsorbent and adsorptive are contacted long enough, the equilibrium is established between the amount of adsorptive adsorbed on the carbon surface (the adsorbate) and the amount of adsorptive in the solution. The equilibrium relationship is described by isotherms. Therefore, the adsorption isotherm for the metal adsorption is the relation between the specific amount adsorbed (*qeql,* expressed in (mmol) of the adsorbate per (g) of the solid adsorbent) and the equilibrium concentrations of the adsorptive in liquid phase (*Ceql*, in expressed in (mmol) of the adsorptive per (l) of the solution), when amount

Chemical equilibrium between adsorbate and adsorptive leads to a constant surface concentration (*Γ*) in [mmol/m2]. Constant (*Γ*) is maintained when the fluxes of adsorbing and desorbing particles are equal, thus the initial adsorptive concentration and temperature

A common procedure is to equate the chemical potentials and their derivatives of the phases involved. Note: the chemical potential (*μ*) is the derivative of the Gibbs energy (*dG)* with respect to the mole number (*n*i) in question (Christmann, 2010), which is for the adsorption process from the liquid phase is the equilibrium concentrations of the adsorptive in liquid

, , ( ) *<sup>i</sup> P T eql*

The decisive quantities when studying the adsorption process are the heat of adsorption and its coverage dependence to lateral particle–particle interactions, as well as the kind and number of binding states (Christmann, 2010). The most relevant thermodynamic variable to describe the heat effects during the adsorption process is the differential isosteric heat of

system before and after the adsorption of a differential amount of adsorbate on the adsorbent surface (Christmann, 2010). The physical basis is the Clausius-Clapeyron

> ( ) *eql eql x*

*dC d C H*

 

*C dT <sup>R</sup> <sup>d</sup> <sup>T</sup>*

Knowledge of the heats of sorption is very important for the characterization and optimization of an adsorption process. The magnitude of (*ΔHx*) value gives information about the adsorption mechanism as chemical ion-exchange or physical sorption: for physical adsorption, (*ΔHx*) should be below 80 kJmol-1 and for chemical adsorption it ranges between 80 and 400 kJmol-1 (Saha & Chowdhury, 2011). It also gives some indication about the

*Langmuir Isotherm*: A model assumes monolayer coverage and constant binding energy

1 ( ) ln( ) ( ) 1

*Hx*), kJ mol-1), that represents the energy difference between the state of the

*dG other mole numbers C*

(1)

(2)

dependence of the liquid-solid phase equilibrium are considered (Christmann, 2010).

phase (*Ceql*), when amount adsorbed on the carbon surface is equals (*qeql*) [1]:

*i*

*dn*

 

*eql*

(Christmann, 2010).

adsorbed is equals *qeql*.

adsorption (

equation [2]:

adsorbent surface heterogeneity.

between surface and adsorbate [3]:

$$q\_{eq} = \frac{K\_{\perp} \times q\_{max} \,\mathrm{C}\_{eq}}{1 + K\_{\perp} \,\mathrm{C}\_{eq}} \tag{3}$$

where *qmax* is the maximum adsorption capacity (monolayer coverage), i.e. mmol of the adsorbate per (g) of adsorbent;

*KL* is the constant of Langmuir isotherm if the enthalpy of adsorption is independent of coverage.

The constant *KL* depends on (i) the relative stabilities of the adsorbate and adsorptive species involved, (ii) on the temperature of the system, and (iii) on the initial concentration of the metal ions in the solution. Factors (ii) and (iii) exert opposite effects on the concentration of adsorbed species: the surface coverage may be increased by raising the initial metal concentration in the solution but will be reduced if the surface temperature is raised (Christmann, 2010).

If the desorption energy is equal to the energy of adsorption, then the first-order processes has been assumed both for the adsorption and the desorption reaction. Whether the deviation exists, the second-order processes should be considered, when adsorption/desorption reactions involving rate-limiting dissociation. From the initial slope of a log - log plot of a Langmuir adsorption isotherm the order of adsorption can be easily determined: if a slope is of 1, that is 1st order adsorption; if a slope is of 0.5, that is 2nd order adsorption process (Christmann, 2010).

*BET (Brunauer, Emmett and Teller) Isotherm:* This is a more general, multi-layer model.

It assumes that a Langmuir isotherm applies to each layer and that no transmigration occurs between layers. It also assumes that there is equal energy of adsorption for each layer except for the first layer [4]:

$$\eta\_{eql} = \frac{K\_{\text{BIT}} \times \eta\_{\text{max}} \mathbf{C}\_{eq}}{(\mathbf{C}\_{\text{init}} - \mathbf{C}\_{eq}) \times \left[1 + (K\_{\text{BIT}} - 1) \times (\mathbf{C}\_{eq} / \mathbf{C}\_{\text{init}})\right]} \tag{4}$$

where *Cinit* is saturation (solubility limit) concentration of the metal ions (in mmol/l) and *KBET* is a parameter related to the binding intensity for all layers;

Two limiting cases can be distinguished: (i) when *Ceql* << *Cinit* and *KBET* >> 1 BET isotherm approaches Langmuir isotherm (*KL* = *KBET*/*Cinit*); (ii) when the constant *KBET* >> 1, the heat of adsorption of the very first monolayer is large compared to the condensation enthalpy; and adsorption into the second layer only occurs once the first layer is completely filled. Conversely, if *KBET* is small, then a multilayer adsorption already occurs while the first layer is still incomplete (Christmann, 2010). In general, as solubility of solute increases the extent of adsorption decreases.

This is known as the "Lundelius' Rule". Solute-solid surface binding competes with solutesolvent attraction. Factors which affect solubility include molecular size (high MW- low solubility), ionization (solubility is minimum when compounds are uncharged), polarity (as polarity increases get higher solubility because water is a polar solvent).

*Freundlich Isotherm*: For the special case of heterogeneous surface energies in which the energy term (*KF*) varies as a function of surface coverage the Freundlich model are used [5]:

$$
\boldsymbol{\sigma}\_{eq} = \boldsymbol{\mathcal{K}}\_{\boldsymbol{\nu}} \times \boldsymbol{\mathsf{C}}\_{eq}^{1/n} \tag{5}
$$

where *K*F and 1/*n* are Freundlich constants related to adsorption capacity and adsorption efficiency, respectively.

Comparison of the Thermodynamic Parameters Estimation for

The BET model linearization equation [10] was used:

this case, a plot of ( )

**2.4.2 Thermodynamic parameters** 

chemical, physical adsorption, etc.

gas constant (8.314 Jmol-1K-1):

*H*0 and *S*0,

the values of

equations [14]

intercept of the plots of ln (*qeql/Ceql*) vs*. qeql* Then, the standard free energy change

were calculated from the Van't-Hoff equation [12].

obtain.

entropy

equals *qeql*; *eql eql*

*q*

( )

*C Cq C* 

Thermodynamic parameters such as change in Gibb's free energy

*S*0 were determined using the following equation [11]:

*eql eql init eql eql init C C vs*

the Adsorption Process of the Metals from Liquid Phase on Activated Carbons 103

For a successful determination of a BET model the limiting case of *KBET* >> 1 is required. In

intercept from which the constant (*KBET*) and the monolayer sorption capacity (*qmax*) can be

*eql*

*eql*

*d*

*<sup>q</sup> <sup>K</sup>*

where *Kd* is the apparent equilibrium constant, *qeql* (or [*Cr III*]*uptake*); is the amount of metal adsorbed on the unitary sorbent mass (mmol/g) at equilibrium and *Ceql* (or [*Cr III*]*eql*) equilibrium concentrations of metal ions in solution (mmol/l), when amount adsorbed is

*<sup>C</sup>* - relationship depends on the type of the adsorption that occurs, i.e. multi-layer,

The thermodynamic equilibrium constants (*Kd*) of the Cr III adsorption on studied activated carbons were calculated by the method suggested by (Khan and Singh, 1987) from the

where *Kd* is the apparent equilibrium constant; *T* is the temperature in Kelvin and R is the

The slope and intercept of the Van't-Hoff plot [13] of ln *Kd* vs. *1/T* were used to determine

0 0 <sup>1</sup> ln *<sup>d</sup> H S <sup>K</sup> RTR* 

Then, the influence of the temperature on the system entropy was evaluated using the

The thermodynamic parameters of the adsorption were also calculated by using the Langmuir constant *(KL*), Freundlich constants (*KF*) and the BET constant (*KBET*) for the

*G*0, enthalpy change

*G*0=–*RT* ln *Kd*, (12)

*G*0=*H*0–*T*

*eql BET eql init eql eql BET init BET*

*C C K C Cq Kq C Kq*

max max 1 1

(10)

yields a straight line with positive slope and

*C* (11)

*G*0, enthalpy

*H*0 and entropy change

(13)

*S*0 (14)

*S*<sup>0</sup>

*H*0 and

To determine which model (Scheme 2) to use to describe the adsorption isotherms for particular adsorbate/adsorbent systems, the experimental data were analyzed using model's linearization.

Scheme 2. Models presentation of the adsorption process (after Christmann 2010), where symbol (*θ*) is the fraction of the surface sites occupied.

#### **2.4 Theoretical calculations 2.4.1 Isotherms analysis**
