**3. Adsorption operation**

Adsorption from solution is usually conducted using either the column or the batch operation. It should be possible to characterize the solution - adsorbent system by both technique operations and arrive at the same result. This is due to the physical and/or chemical forces applicable in each case must be identical. Furthermore, the results obtained from the batch experiment should be somewhat more reliable. Among the most serious objections of the column experiments are: (1)the inherent difficulties associated to maintain a constant flow rate; (2) the difficulty of ensuring a constant temperature throughout the column; (3) the appreciable probability of presence the channels within the packed column; and (4) the relatively large expenditure both in time and manpower required for a column experiment.

### **3.1 Batch operation**

In a batch operation, fixed amount of adsorbent is mixed all at once with specific volume of adsorbate (with the range of initial concentration). Afterwards, the system kept in agitation for a convenient period of time. Separation of the resultant solution is accomplished by filtering, centrifuging, or decanting. The optimum pH, contact time, agitation speed and optimum temperature are fixed and used in this technique. For instance, the contact time study, the experiment are carried out at constant initial concentration, agitation speed, pH, and temperature. During the adsorption progress, the mixture container must be covered by

Thermodynamics Approach in the Adsorption of Heavy Metals 745

adsorbent materials is reached, and the rate of adsorption equals the rate of desorption. The theoretical adsorption capacity of an adsorbent can be calculated with an adsorption isotherm. There are basically two well established types of adsorption isotherm the Langmuir and the Freundlich adsorption isotherms. The significance of adsorption isotherms is that they show how the adsorbate molecules (metal ion in aqueous solution) are distributed between the solution and the adsorbent solids at equilibrium concentration on the loading capacity at different temperatures. That mean, the amount of sorbed solute

Langmuir is the simplest type of theoretical isotherms. Langmuir adsorption isotherm describes quantitatively the formation of a monolayer of adsorbate on the outer surface of the adsorbent, and after that no further adsorption takes place. Thereby, the Langmuir represents the equilibrium distribution of metal ions between the solid and liquid phases [29]. The Langmuir adsorption is based on the view that every adsorption site is identical and energically equivalent (thermodynamically, each site can hold one adsorbate molecule). The Langmuir isotherm assume that the ability of molecule to bind and adsorbed is independent of whether or not neighboring sites are occupied. This mean, there will be no interactions between adjacent molecules on the surface and immobile adsorption. Also mean, trans-migration of the adsorbate in the plane of the surface is precluded. In this case, the Langmuir isotherms is valid for the dynamic equilibrium adsorption desorption processes on completely homogeneous surfaces with negligible interaction between

*Q* and *b* are related to standard monolayer adsorption capacity and the Langmuir constant,

In summary, the Langmuir model represent one of the the first theoretical treatments of non-linear sorption and suggests that uptake occurs on a homogenous surface by monolyer sorption without interaction between adsorbed molecules. The Langmuir isotherm assumes that adsorption sites on the adsorbent surfaces are occupied by the adsorbate in the solution. Therefore the Langmuir constant (*b*) represents the degree of adsorption affinity the adsorbate. The maximum adsorption capacity (*Q*) associated with complete monolayer cover is typically expressed in (mg/g). High value of *b* indicates for much stronger affinity

The shape of the isotherm (assuming the (x) axis represents the concentration of adsorbing material in the contacting liquid) is a gradual positive curve that flattens to a constant value.

*qe* = (*Q*×*b*×*Ce*)/(1+*b*×*Ce*) (6)

*q*max = *Q*×*b* (7)

*Ce*/*qe* = 1/(*q*max×*b*) + (1/ *q*max) × *Ce* (8)

versus the amount of solute in solution at equilibrium.

**4.1 Langmuir adsorption isotherm** 

adsorbed molecules that exhibit the form:

Equation 6 could be re-written as:

of metal ion adsorption.

respectively.

*Ce* = The equilibrium concentration in solution *qe* = the amount adsorbed for unit mass of adsorbent

*q*max = is the constant related to overall solute adsorptivity (l/g).

alumina foil to avoid the evaporation. The samples are withdrawn at different time intervals, for example, every 5 minuets or every 15 minutes.

The uptake of heavy metal ions was calculated from the mass balance, which was stated as the amount of solute adsorbed onto the solid. It equal the amount of solute removed from the solution. Mathematically can be expressed in equation 1 [27]:

$$q\_c = \frac{\{\mathcal{C}\_i - \mathcal{C}\_e\}}{S} \tag{1}$$

*<sup>e</sup> q* : Heavy metal ions concentration adsorbed on adsorbent at equilibrium (mg of metal ion/g of adsorbent).

*Ci* : Initial concentration of metal ions in the solution (mg/l).

*Ce* : Equilibrium concentration or final concentration of metal ions in the solution (mg/l). *S* : Dosage (slurry) concentration and it is expressed by equation 2:

$$S = \frac{m}{v} \tag{2}$$

Where ν is the initial volume of metal ions solution used (L) and *m* is the mass of adsorbent. The percent adsorption (%) was also calculated using equation 3:

$$\% \text{ adsorption} = \frac{C\_i - C\_c}{C\_i} \times 100\% \tag{3}$$

#### **3.2 Column operation**

In a column operation, the solution of adsorbate such as heavy metals (with the range of initial concentration) is allowed to percolate through a column containing adsorbent (ion exchange resin, silica, carbon, etc.) usually held in a vertical position. For instance, column studies were carried out in a column made of Pyrex glass of 1.5 cm internal diameter and 15 cm length. The column was filled with 1 g of dried PCA by tapping so that the maximum amount of adsorbent was packed without gaps. The influent solution was allowed to pass through the bed at constant flow rate of 2 mL/min, in down flow manner with the help of a fine metering valve. The effluent solution was collected at different time intervals.

The breakthrough adsorption capacity of adsorbate (heavy metal ions) was obtained in column at different cycles using the equation 4 [28].

$$\mathbf{q}\_{\mathbf{c}} = \left[ \left( \mathbf{C}\_{i} - \mathbf{C}\_{e} \right) / m \right] \times \hbar \mathbf{v} \tag{4}$$

Where *Ci* and *Ce* denote the initial and equilibrium (at breakthrough) of heavy metal ions concentration (mg/L) respectively. bv was the breakthrough volume of the heavy metal ions solution in liters, and m was the mass of the adsorbent used (g). After the column was exhausted, the column was drained off the remaining aqueous solution by pumping air. The adsorption percent is given by equation 5.

$$\% \text{ Desorption} = (C\_t/C\_i) \times 100\tag{5}$$

### **4. Thermodynamic and adsorption isotherms**

Adsorption isotherms or known as equilibrium data are the fundamental requirements for the design of adsorption systems. The equilibrium is achieved when the capacity of the adsorbent materials is reached, and the rate of adsorption equals the rate of desorption. The theoretical adsorption capacity of an adsorbent can be calculated with an adsorption isotherm. There are basically two well established types of adsorption isotherm the Langmuir and the Freundlich adsorption isotherms. The significance of adsorption isotherms is that they show how the adsorbate molecules (metal ion in aqueous solution) are distributed between the solution and the adsorbent solids at equilibrium concentration on the loading capacity at different temperatures. That mean, the amount of sorbed solute versus the amount of solute in solution at equilibrium.

### **4.1 Langmuir adsorption isotherm**

744 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

alumina foil to avoid the evaporation. The samples are withdrawn at different time

The uptake of heavy metal ions was calculated from the mass balance, which was stated as the amount of solute adsorbed onto the solid. It equal the amount of solute removed from

> ( ) *i e <sup>e</sup> C C <sup>q</sup> <sup>S</sup>*

*<sup>e</sup> q* : Heavy metal ions concentration adsorbed on adsorbent at equilibrium (mg of metal

*Ce* : Equilibrium concentration or final concentration of metal ions in the solution (mg/l). *S* :

Where ν is the initial volume of metal ions solution used (L) and *m* is the mass of adsorbent.

% adsorption *i e* 100%

In a column operation, the solution of adsorbate such as heavy metals (with the range of initial concentration) is allowed to percolate through a column containing adsorbent (ion exchange resin, silica, carbon, etc.) usually held in a vertical position. For instance, column studies were carried out in a column made of Pyrex glass of 1.5 cm internal diameter and 15 cm length. The column was filled with 1 g of dried PCA by tapping so that the maximum amount of adsorbent was packed without gaps. The influent solution was allowed to pass through the bed at constant flow rate of 2 mL/min, in down flow manner with the help of a

The breakthrough adsorption capacity of adsorbate (heavy metal ions) was obtained in

 qe = [(*Ci Ce*)/*m*] × *bv* (4) Where *Ci* and *Ce* denote the initial and equilibrium (at breakthrough) of heavy metal ions concentration (mg/L) respectively. bv was the breakthrough volume of the heavy metal ions solution in liters, and m was the mass of the adsorbent used (g). After the column was exhausted, the column was drained off the remaining aqueous solution by pumping air. The

Adsorption isotherms or known as equilibrium data are the fundamental requirements for the design of adsorption systems. The equilibrium is achieved when the capacity of the

% Desorption = (*Ce*/*Ci*) × 100 (5)

fine metering valve. The effluent solution was collected at different time intervals.

*i C C C*

(1)

*<sup>m</sup> <sup>S</sup> <sup>v</sup>* (2)

(3)

intervals, for example, every 5 minuets or every 15 minutes.

ion/g of adsorbent).

**3.2 Column operation** 

the solution. Mathematically can be expressed in equation 1 [27]:

*Ci* : Initial concentration of metal ions in the solution (mg/l).

Dosage (slurry) concentration and it is expressed by equation 2:

The percent adsorption (%) was also calculated using equation 3:

column at different cycles using the equation 4 [28].

**4. Thermodynamic and adsorption isotherms** 

adsorption percent is given by equation 5.

Langmuir is the simplest type of theoretical isotherms. Langmuir adsorption isotherm describes quantitatively the formation of a monolayer of adsorbate on the outer surface of the adsorbent, and after that no further adsorption takes place. Thereby, the Langmuir represents the equilibrium distribution of metal ions between the solid and liquid phases [29]. The Langmuir adsorption is based on the view that every adsorption site is identical and energically equivalent (thermodynamically, each site can hold one adsorbate molecule). The Langmuir isotherm assume that the ability of molecule to bind and adsorbed is independent of whether or not neighboring sites are occupied. This mean, there will be no interactions between adjacent molecules on the surface and immobile adsorption. Also mean, trans-migration of the adsorbate in the plane of the surface is precluded. In this case, the Langmuir isotherms is valid for the dynamic equilibrium adsorption desorption processes on completely homogeneous surfaces with negligible interaction between adsorbed molecules that exhibit the form:

$$\mathfrak{l}\_{\mathfrak{l}} = (\mathbb{Q} \ltimes \mathfrak{b} \ltimes \mathbb{C}\_{\mathfrak{e}}) / (1 + \mathfrak{b} \ltimes \mathbb{C}\_{\mathfrak{e}}) \tag{6}$$

*Ce* = The equilibrium concentration in solution

*qe* = the amount adsorbed for unit mass of adsorbent

*Q* and *b* are related to standard monolayer adsorption capacity and the Langmuir constant, respectively.

$$
\omega\_{\text{max}} = Q \rtimes b \tag{7}
$$

*q*max = is the constant related to overall solute adsorptivity (l/g). Equation 6 could be re-written as:

*Ce*/*qe* = 1/(*q*max×*b*) + (1/ *q*max) × *Ce* (8)

In summary, the Langmuir model represent one of the the first theoretical treatments of non-linear sorption and suggests that uptake occurs on a homogenous surface by monolyer sorption without interaction between adsorbed molecules. The Langmuir isotherm assumes that adsorption sites on the adsorbent surfaces are occupied by the adsorbate in the solution. Therefore the Langmuir constant (*b*) represents the degree of adsorption affinity the adsorbate. The maximum adsorption capacity (*Q*) associated with complete monolayer cover is typically expressed in (mg/g). High value of *b* indicates for much stronger affinity of metal ion adsorption.

The shape of the isotherm (assuming the (x) axis represents the concentration of adsorbing material in the contacting liquid) is a gradual positive curve that flattens to a constant value.

Thermodynamics Approach in the Adsorption of Heavy Metals 747

isotherm curves in the opposite way of Langmuir isotherm and is exponential in form. The heat of adsorption, in many instances, decreases in magnitude with increasing extent of adsorption. This decline in heat is logarithmic implying that the adsorption sites are distributed exponentially with respect to adsorption energy. This isotherm does not indicate an adsorption limit when coverage is sufficient to fill a monolayer (*θ* = 1). The equation that

Kf = Freundlich constant related to maximum adsorption capacity (mg/g). It is a

*n* = Freundlich contestant related to surface heterogeneity (dimensionless). It gives an

The plotting qe versus Ce yield a non-regression line, which permits the determination of (1/*n*) and Kf values of (1/*n*) ranges from 0 to 1, where the closer value to zero means the more heterogeneous the adsorption surface. On linearization, these values can be obtained by plotting (*ln qe*) versus (*ln Ce*) as presented in equation 11. From the plot, the vales Kf and *n*

The DKR isotherm is reported to be more general than the Langmuir and Freundlich isotherms. It helps to determine the apparent energy of adsorption. The characteristic porosity of adsorbent toward the adsorbate and does not assume a homogenous surface or

<sup>2</sup> ln ln *q X e m*

where *Xm* is the maximum sorption capacity, *β* is the activity coefficient related to mean

The slope of the plot of ln *<sup>e</sup> q* versus *ε* <sup>2</sup> gives *β* (mol <sup>2</sup> */*J <sup>2</sup> ) and the intercept yields the sorption capacity, *X*m (mg*/*g) as shown in Fig. 6. The values of β and X*m*, as a function of temperature are listed in table 1 with their corresponding value of the correlation coefficient, *R* <sup>2</sup> . It can be observed that the values of β increase as temperature increases while the

> 1 2 *E* (2 )

*RT*

The values of the adsorption energy, *E*, was obtained from the relationship [33]

<sup>1</sup> ln(1 ) *e*

*C*

The Dubinin–Kaganer–Radushkevich (DKR) model has the linear form

sorption energy, and *ε* is the Polanyi potential, which is equal to

*qe* = Kf (*Ce*)1/*<sup>n</sup> n* > 1 (10)

*ln qe* = *ln Kf* + (1/n)*ln Ce* (11)

(13)

(12)

describes such isotherm is the Freundlich isotherm, given as [31]:

temperature-dependent constant.

where, the slop = (1/*n*), and the intercept = *ln Kf*

**4.3 Dubinin–Kaganer–Radushkevich (DKR)** 

where *R* is the gas constant (kJ/kmol- K) .

values of *Xm* decrease with increasing temperature.

constant sorption potential [32].

can be obtained.

indication of how favorable the adsorption processes. With *n* = 1, the equation reduces to the linear form: *qe* = k × *Ce*

A plot of *Ce*/*qe* versus *Ce* gives a straight line of slope 1/ *q*max and intercept 1/(*q*max×*b*), for example, as shown in Figure 7.

Fig. 7. The linearized Langmuir adsorption isotherms for Fe3+ ions adsorption by natural quartz (NQ) and bentonite (NB) at constant temperature 30 ºC. (initial concentration: 400 ppm, 300 rpm and contact time: 2.5 hours).

The effect of isotherm shape is discussed from the direction of the predicting whether and adsorption system is "favorable" or "unfavorable". Hall et al (1966) proposed a dimensionless separation factor or equilibrium parameter, *R*L, as an essential feature of the Langmuir Isotherm to predict if an adsorption system is "favourable" or "unfavourable", which is defined as [30]:

$$R\_L = \mathbf{1} / \langle \mathbf{1} + b\mathbf{C}\_i \rangle \tag{9}$$

*Ci* = reference fluid-phase concentration of adsorbate (mg/l) (initial Fe3+ ions concentration) *b* = Langmuir constant (ml mg−1)

Value of RL indicates the shape of the isotherm accordingly as shown in Table 1 below. For a single adsorption system, *Ci* is usually the highest fluid-phase concentration encountered.


Table 1. Type of isotherm according to value of *R*<sup>L</sup>

#### **4.2 Freundlich adsorption isotherms**

Freundlich isotherm is commonly used to describe the adsorption characteristics for the heterogeneous surface [31]. It represents an initial surface adsorption followed by a condensation effect resulting from strong adsorbate-adsorbate interaction. Freundlich

A plot of *Ce*/*qe* versus *Ce* gives a straight line of slope 1/ *q*max and intercept 1/(*q*max×*b*), for

R2 = 0.961

= 0.9385

Fe-NQ Fe-NB

R2

30 40 50 60 70 80 90 100 **Ce**

Fig. 7. The linearized Langmuir adsorption isotherms for Fe3+ ions adsorption by natural quartz (NQ) and bentonite (NB) at constant temperature 30 ºC. (initial concentration: 400

The effect of isotherm shape is discussed from the direction of the predicting whether and adsorption system is "favorable" or "unfavorable". Hall et al (1966) proposed a dimensionless separation factor or equilibrium parameter, *R*L, as an essential feature of the Langmuir Isotherm to predict if an adsorption system is "favourable" or "unfavourable",

 *R*L = 1/(1+*bCi*) (9) *Ci* = reference fluid-phase concentration of adsorbate (mg/l) (initial Fe3+ ions concentration)

Value of RL indicates the shape of the isotherm accordingly as shown in Table 1 below. For a single adsorption system, *Ci* is usually the highest fluid-phase concentration encountered.

Freundlich isotherm is commonly used to describe the adsorption characteristics for the heterogeneous surface [31]. It represents an initial surface adsorption followed by a condensation effect resulting from strong adsorbate-adsorbate interaction. Freundlich

Value of *R*<sup>L</sup> Type of Isotherm 0 < r < 1 Favorable r > 1 Unfavorable r = 1 Linear R = 0 Irreversible

example, as shown in Figure 7.

0

ppm, 300 rpm and contact time: 2.5 hours).

5

10

**Ce/qe**

which is defined as [30]:

*b* = Langmuir constant (ml mg−1)

Table 1. Type of isotherm according to value of *R*<sup>L</sup>

**4.2 Freundlich adsorption isotherms** 

15

20

25

isotherm curves in the opposite way of Langmuir isotherm and is exponential in form. The heat of adsorption, in many instances, decreases in magnitude with increasing extent of adsorption. This decline in heat is logarithmic implying that the adsorption sites are distributed exponentially with respect to adsorption energy. This isotherm does not indicate an adsorption limit when coverage is sufficient to fill a monolayer (*θ* = 1). The equation that describes such isotherm is the Freundlich isotherm, given as [31]:

$$q\_{\varepsilon} = \mathbb{K}\_{\mathbb{f}}\left(\mathbb{C}\_{\varepsilon}\right)^{1/n} \ n \geqslant 1 \tag{10}$$

Kf = Freundlich constant related to maximum adsorption capacity (mg/g). It is a temperature-dependent constant.

*n* = Freundlich contestant related to surface heterogeneity (dimensionless). It gives an indication of how favorable the adsorption processes.

With *n* = 1, the equation reduces to the linear form: *qe* = k × *Ce*

The plotting qe versus Ce yield a non-regression line, which permits the determination of (1/*n*) and Kf values of (1/*n*) ranges from 0 to 1, where the closer value to zero means the more heterogeneous the adsorption surface. On linearization, these values can be obtained by plotting (*ln qe*) versus (*ln Ce*) as presented in equation 11. From the plot, the vales Kf and *n* can be obtained.

$$
\ln q\_{\epsilon} = \ln K\_{\delta} + (1/\ln) \ln \,\mathrm{C}\_{\epsilon} \tag{11}
$$

where, the slop = (1/*n*), and the intercept = *ln Kf*

### **4.3 Dubinin–Kaganer–Radushkevich (DKR)**

The DKR isotherm is reported to be more general than the Langmuir and Freundlich isotherms. It helps to determine the apparent energy of adsorption. The characteristic porosity of adsorbent toward the adsorbate and does not assume a homogenous surface or constant sorption potential [32].

The Dubinin–Kaganer–Radushkevich (DKR) model has the linear form

$$
\ln q\_e = \ln X\_m - \beta \varepsilon^2 \tag{12}
$$

where *Xm* is the maximum sorption capacity, *β* is the activity coefficient related to mean sorption energy, and *ε* is the Polanyi potential, which is equal to

$$
\omega = RT \ln(1 + \frac{1}{C\_e}) \tag{13}
$$

where *R* is the gas constant (kJ/kmol- K) .

The slope of the plot of ln *<sup>e</sup> q* versus *ε* <sup>2</sup> gives *β* (mol <sup>2</sup> */*J <sup>2</sup> ) and the intercept yields the sorption capacity, *X*m (mg*/*g) as shown in Fig. 6. The values of β and X*m*, as a function of temperature are listed in table 1 with their corresponding value of the correlation coefficient, *R* <sup>2</sup> . It can be observed that the values of β increase as temperature increases while the values of *Xm* decrease with increasing temperature.

The values of the adsorption energy, *E*, was obtained from the relationship [33]

$$E = \left(-2\beta\right)^{-1/2}$$

Thermodynamics Approach in the Adsorption of Heavy Metals 749

water and can contribute to disease development in several ways. For instance, an excessive amounts of iron ions in specific tissues and cells (iron-loading) promote development of infection, neoplasia, cardiomyopathy, arthropathy, and various endocrine and possibly neurodegenerative disorders. Finally, the industrial consideration such as blocking the pipes and increasing of corrosion. In addition to that, iron oxides promote the growth of micro-

In response to the human body health, its environmental problems and the limitation water sources especially in Jordan [44], the high-levels of Fe3+ ions must be removed from the aqueous stream to the recommended limit 5.0 and 0.3 ppm for both inland surface and drinking water, respectively. These values are in agreement with the Jordanian standard parameters of water quality[45]. For tracing Fe3+ ions into recommended limit, many chemical and physical processes were used such as supercritical fluid extraction, bioremediation, oxidation with oxidizing agent [46]. These techniques were found not effective due to either extremely expensive or too inefficient to reduce such high levels of ions from the large volumes of water [47 - 48]. Therefore, the effective process must be low cost-effective technique and simple to operate [49 - 52]. It found that the adsorption process using natural adsorbents realize these prerequisites. In addition to that, the natural adsorbents are environmental friend, existent in a large quantities and has good adsorption properties. The binding of Iron(III) ion with the surface of the natural adsorbent could change their forms of existence in the environment. In general they may react with particular species, change oxidation states and precipitate [53]. In spite of the abundant reported researches in the adsorption for the removal of the dissolved heavy metals from the aqueous streams, however the iron(III) ions still has limited reported studies. Therefore our studies are concentrated in this field. From our previous work, the natural zeolite [19], quartz and Bentonite [20], olive cake [21], in addition to the chitin [24], activated carbon [54 - 55] and alumina [56] have been all utilized for this aspect at low levels. The adsorption isotherm models (Langmuir and Freundlich) are used in order to correlate the experimental

organism in water which inhibit many industrial processes in our country [43].

**5.1 Sorption Fe3+ ions using natural quartz (NQ) and bentonite (NB))** 

then achieving the maximum adsorption percentages [20].

It is known from the chemistry view that surface of NB and NQ is ending with (Si-O) negatively charged. These negative entities might bind metal ions *via* the coordination aspects especially at lower pH values as known in the literatures. Fe3+ ions are precipitated in the basic medium. Therefore, the 1 % HNO3 stock solution is used to soluble Fe3+ ions and

The binding of metal ions might be influenced on the surface of NB more than NQ. This is due to the expected of following ideas: (i) NQ have pure silica entity with homogeneous negatively charged, therefore the binding will be homogeneous. (ii) The natural bentonite has silica surface including an inner-layer of alumina and iron oxide, which cause a heterogeneous negatively charged. Therefore, the binding Fe3+ ions on the surface on NB

The adsorption thermodynamics modelsof Fe3+ ions on NQ and NB at 30 ºC are examined [20]. The calculated results of the Langmuir and Freundlich isotherm constants are given in Table 2. The high values of *R2* (*>*95%) indicates that the adsorption of Fe3+ ions onto both NQ and NB was well described by Freundlich isotherms.It can also be seen that the max *q* and

results.

might be complicated.

### **4.4 Thermodynamics parameters for the adsorption**

In order to fully understand the nature of adsorption the thermodynamic parameters such as free energy change (Gº) and enthalpy change (Hº) and entropy change (Sº) could be calculated. It was possible to estimate these thermodynamic parameters for the adsorption reaction by considering the equilibrium constants under the several experimental conditions. They can calculated using the following equations [34]:

$$
\Delta \mathbf{G} = -\mathbf{R} \ln \mathbf{K}\_0(\mathbf{T}) \tag{14}
$$

$$
\Delta \mathbf{K}\_{\rm d} = \Delta \mathbf{S} / \mathbf{R} - \Delta \mathbf{H} / \mathbf{R} \mathbf{T} \tag{15}
$$

$$
\Delta \mathbf{G} = \Lambda \mathbf{H} - \mathbf{T} \Lambda \mathbf{S} \tag{16}
$$

The Kd value is the adsorption coefficient obtained from Langmuir equation. It is equal to the ratio of the amount adsorbed (x/m in mg/g) to the adsorptive concentration (y/a in mg/dm3)

$$K\_d = (\mathbf{x}/\mathbf{m}).\,\mathrm{(y/a)}\tag{17}$$

These parameters are obtained from experiments at various temperatures using the previous equations. The values of Hº and Sº are determined from the slop and intercept of the linear plot of (*ln Kd*) vs. (1/T).

In general these parameters indicate that the adsorption process is spontaneous or not and exothermic or endothermic. The standard enthalpy change (Hº) for the adsorption process is: (i) positive value indicates that the process is endothermic in nature. (ii) negative value indicate that the process is exothermic in nature and a given amount of heat is evolved during the binding metal ion on the surface of adsorbent. This could be obtained from the plot of percent of adsorption (% Cads) *vs.* Temperature (T). The percent of adsorption increase with increase temperature, this indicates for the endothermic processes and the opposite is correct [35]. The positive value of (Sº) indicate an increase in the degree of freedom (or disorder) of the adsorbed species.
