**3.5 Kinetic study**

The adsorption kinetic studies were carried out by contacting Zn3Al-Cl-LDH (Cm = 0.8 g/L) with NO3 � solutions (500 ml) of the initial concentration of 0.4 mg/L, respectively. The adsorption process was agitated at 25°C and a pH of 7.0 for several periods ranging from 5 to 60 min under inert atmosphere (N2). LDH obtained after adsorption was filtered and then washed several times. The concentration of the nitrate ion in the filtrate was determined by spectrophotometer at 415 nm. The nitrate amount *qe* (mg/g) loaded on adsorbents after adsorption experiments and the percentage removal (removal %) of NO3 � ions from solutions were calculated using the following equations:

$$q\_{\epsilon} = \frac{(C - Ce) \times V}{m} \tag{1}$$

and the removal

$$\mathbf{u}(\%) = \frac{\mathbf{C}\_0 - \mathbf{C}\_\varepsilon}{\mathbf{C}\_0} \times \mathbf{100} \tag{2}$$

where Ce (mg/L) is the equilibrium of nitrate ion concentration in solution, C0 (mg/L) is the initial of nitrate ion concentration in solution, m (g) is the mass of adsorbent, and V (L) is the volume of the solutions.

The equilibrium is reached after 30 min, with a maximum of approximately 59.12% adsorption capacity corresponding to a 295.62 mg/g of an affinity of the adsorbate for the active sites of the adsorbent [23]. From **Figure 3** it is quite clear that the percentage of nitrate ion adsorption calculated by kinetic study and

**Figure 4.** *Kinetic study evaluation of NO3* � *removal by Zn3AlCl-(NO3* �*)ads complex.*

efficiency adsorption calculated by impedance spectroscopy are less than 10% which shows that the values are close and both techniques are okay and best correlated. From **Figure 4** adsorption of nitrate ions by this system noted as a function of adsorption time is quite rich on information. The adsorption phenomena are due to the active sites in LDH interlayer with different electron donor sites (active adsorption sites) on the ions (N&O) and relative humidity [24–26].

#### **3.6 Analysis of adsorption kinetics**

The different kinetic models including pseudo-first-order, pseudo-secondorder, and intraparticle diffusion are employed to investigate the mechanism of adsorption and potential rate controlling steps such as chemical reaction mass transport and diffusion control processes [27]. The pseudo-first-order and pseudo-second-order are generally expressed as Eqs. (3) and (4), respectively (**Figure 5**):

$$\text{Log } \left( \mathbf{q}\_e - \mathbf{q}\_\prime \right) \, = \text{ Log } \left( \mathbf{q}\_e \right) - \frac{K\_{\text{l,adj}}}{2,303} \times \mathbf{t} \tag{3}$$

$$\frac{t}{q\_t} = \frac{1}{K\_{2,ads} \times q\_e^2} + \frac{t}{q\_e} \tag{4}$$

confirms the rapid process noted during the kinetic study. The maximum amount obtained by applying the pseudo-second-order model is very close to that deter-

*by Zn3-Al-Cl-LDH.*

*by (Zn3Al-Cl-LDH).*

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered…*

ment data of kinetics follows the pseudo-second-order model (R<sup>2</sup> ≈ 0.99).

The values of the correlation factors obtained (**Table 2**) show that the measure-

Intraparticle diffusion equation suggests that intraparticle diffusion is the ratelimiting step in adsorption. The diffusion process may affect the adsorption of

mined by the kinetic study (296 mg/g).

*Pseudo-second-order model of removal NO3*

**Figure 5.**

**Figure 6.**

**119**

*Pseudo-first-order model removal NO3*

*DOI: http://dx.doi.org/10.5772/intechopen.90883*

**3.7 Constant diffusion rate determination**

where qe (mg/g) and qt (mg/g) are the adsorption of NO3 � ions on to adsorbents at equilibrium and at time t (min), respectively; K1,ads (min�<sup>1</sup> ) and K2,ads (g/(mg min)) are the constants of the pseudo-first-order and pseudo-second-order adsorption, respectively. Additionally h (mg/(g min)) is the initial adsorption rate of pseudo-second-order which can be calculated using h = K2qe2.

The adsorption rate constants k1,ads and k2,ads of nitrate ions by (Zn3Al-Cl-LDH) are deducted, respectively, from the curve log(qe-q) = f(t) and (t/qt) = f(t/qe) (**Figures 6** and **7**).

The regression by the pseudo-second-order model agrees well to study the adsorption of the nitrate ions by Zn3Al-Cl-LDH. The constant of adsorption rate *Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered… DOI: http://dx.doi.org/10.5772/intechopen.90883*

**Figure 5.** *Pseudo-first-order model removal NO3 by (Zn3Al-Cl-LDH).*

**Figure 6.** *Pseudo-second-order model of removal NO3 by Zn3-Al-Cl-LDH.*

confirms the rapid process noted during the kinetic study. The maximum amount obtained by applying the pseudo-second-order model is very close to that determined by the kinetic study (296 mg/g).

The values of the correlation factors obtained (**Table 2**) show that the measurement data of kinetics follows the pseudo-second-order model (R<sup>2</sup> ≈ 0.99).

## **3.7 Constant diffusion rate determination**

Intraparticle diffusion equation suggests that intraparticle diffusion is the ratelimiting step in adsorption. The diffusion process may affect the adsorption of

**Figure 7.** *Intraparticle diffusion model for Zn3Al-Cl-(NO3* �*)ads-LDH complex.*


**Table 2.**

*Kinetic parameters for Zn3Al-Cl-(NO3* �*)ads-LDH complex.*

nitrate ions on Zn3-Al-Cl-LDH due to the porous structure of the adsorbent and the attractive effect of nitrate ions. Therefore the intraparticle diffusion is used to explore the behavior of intraparticle diffusion is obeys Eq. (5) [27].

$$\mathbf{q}\_{\mathbf{t}} = \mathbf{k} \mathbf{p}. \mathbf{t}^{1/2} + \mathbf{C} \tag{5}$$

recovered after adsorption at various times (**Figure 6**) show the increase of the

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered…*

It could be seen that the plots were multilinear over the whole time range

LDH. The first linear plot was the instantaneous adsorption or external surface adsorption attributing to the rapid consumption of the available adsorption sites on the adsorbent surface. The second stage was the gradual adsorption stage where the intraparticle was the rate-limiting step, and the second portion was attributed to the final equilibrium for which the intraparticle diffusion starts to slow down due to the extremely few adsorption sites left on adsorbent which will be clearly in impedance spectroscopy using the Nyquist diagram analysis by means of fit and

Using impedance spectroscopy one can distinguish between intrinsic (grain) and extrinsic (grain boundaries). The Nyquist and Bode plots were used to interpret the electrical relaxation processes associated with adsorption phenomenon and used to

In order to determine the mechanisms responsible for the conductivity, in general, the different variations of complex impedance spectra (Nyquist plot) consist of two semicircle arcs corresponding to the grain interior and grain boundary. The

The impedance analysis allows one to determine the contributions of various processes such as bulk effects and the grain boundaries. **Figure 8** shows the complex impedance plane plots (Nyquist plot) of the nitrate ion removal by the system

The analysis (**Figure 8a** and **b**) of the data by the Nyquist diagram allowed us to determine the resistance values for the two regions in the time interval of 0, 5, and 10 min (Region 1) and 20, 30, and 60 min (Region 2). The separation into two time intervals is justified by **Figure 7** of the kinetic study and **Figures 9** and **10** of the

*Nyquist plots for samples in Region 1 (a) and Region 2 (b), respectively, during adsorption phenomenon by*

arc at a high frequency usually represents the grain response, and the low-

suggesting that two steps were operational in the adsorption of NO3

extrapolation of experimental data for both adsorption regions.

at 1381 cm<sup>1</sup> in a function of contact

by Zn3-Al-Cl-

intensity of the characteristic band of NO3

*DOI: http://dx.doi.org/10.5772/intechopen.90883*

**4. Impedance spectroscopy analysis**

Zn3-Al-Cl-LDH (0 min ≤ t ≤ 60 min).

impedance spectroscopy study.

**Figure 8.**

**121**

*Zn3Al-Cl-(NO3*

*)ads-LDH.*

search for the electrical analogue of the adsorption.

frequency arc corresponds to the grain boundary [28].

time of ions.

where qt is the quantity retained at time t and kip are the diffusion rate constants.

The results obtained (**Figure 6**) show that there are two stages. *Region 1* is attributed to the most readily available site on the surface of the adsorbent. *Region 2* can be explained by a very slow diffusion of adsorption in the inner pores. Thus the nitrate ion adsorption by Zn3-Al-Cl-LDH may be governed by the intraparticle model [1]. The values of kp1 and kp2 diffusion rate constants for Region 1 and Region 2, respectively, obtained by using the regression linear are shown in **Table 2**.

These values are in good agreement with the kinetic study. Indeed, in Region 1 the value of the slope (86.04) is greater than that of the Region 2 whose value is of the order of 1.68. This can be explained by the availability of sites in Zn3-Al-Cl-LDH at the beginning of adsorption. The release rate constants intraparticle using the kinetics study was according the values respectively K1P = 86.04 with the R2 = 0.98 and K2P = 1.68 with the R<sup>2</sup> = 0.99. The adsorption of nitrate ions by Zn3-Al-Cl-LDH is confirmed by FT-IR spectroscopy. In fact the infrared spectra of the materials

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered… DOI: http://dx.doi.org/10.5772/intechopen.90883*

recovered after adsorption at various times (**Figure 6**) show the increase of the intensity of the characteristic band of NO3 at 1381 cm<sup>1</sup> in a function of contact time of ions.

It could be seen that the plots were multilinear over the whole time range suggesting that two steps were operational in the adsorption of NO3 by Zn3-Al-Cl-LDH. The first linear plot was the instantaneous adsorption or external surface adsorption attributing to the rapid consumption of the available adsorption sites on the adsorbent surface. The second stage was the gradual adsorption stage where the intraparticle was the rate-limiting step, and the second portion was attributed to the final equilibrium for which the intraparticle diffusion starts to slow down due to the extremely few adsorption sites left on adsorbent which will be clearly in impedance spectroscopy using the Nyquist diagram analysis by means of fit and extrapolation of experimental data for both adsorption regions.

## **4. Impedance spectroscopy analysis**

Using impedance spectroscopy one can distinguish between intrinsic (grain) and extrinsic (grain boundaries). The Nyquist and Bode plots were used to interpret the electrical relaxation processes associated with adsorption phenomenon and used to search for the electrical analogue of the adsorption.

In order to determine the mechanisms responsible for the conductivity, in general, the different variations of complex impedance spectra (Nyquist plot) consist of two semicircle arcs corresponding to the grain interior and grain boundary. The arc at a high frequency usually represents the grain response, and the lowfrequency arc corresponds to the grain boundary [28].

The impedance analysis allows one to determine the contributions of various processes such as bulk effects and the grain boundaries. **Figure 8** shows the complex impedance plane plots (Nyquist plot) of the nitrate ion removal by the system Zn3-Al-Cl-LDH (0 min ≤ t ≤ 60 min).

The analysis (**Figure 8a** and **b**) of the data by the Nyquist diagram allowed us to determine the resistance values for the two regions in the time interval of 0, 5, and 10 min (Region 1) and 20, 30, and 60 min (Region 2). The separation into two time intervals is justified by **Figure 7** of the kinetic study and **Figures 9** and **10** of the impedance spectroscopy study.

**Figure 8.**

*Nyquist plots for samples in Region 1 (a) and Region 2 (b), respectively, during adsorption phenomenon by Zn3Al-Cl-(NO3 )ads-LDH.*

**Figure 9.** *Variation of the ratio (Rbg/Rbgmax-Rbg) according to the square root of the adsorption time.*

**Figure 10.** *Variation of σ'ac conductivity for (a) Region 1 and (b) Region 2 as a function of frequency.*

**Figure 11** shown is intended to show that there are only two capacitive loops in the extrapolation technique of experimental data in a frequency range of 10<sup>2</sup> Hz to 1 MHz: one for the contribution of grain region and the other for grain boundary contribution.

The estimated value of R is the difference between the high intercept and low intercept values. If the data reflects a parallel R-C element with a depression angle of zero, the estimated R will be the same as the diameter of the semicircle. Estimated C is calculated using the relationship ωmax = 1/(RC) and depends on the accuracy of ωmax.

The Nyquist (**Figures 12** and **13**) plot studies discovered the presence of grain and grain boundary which then become dipoles when they are subjected to the action of an electric field. We thus observe two phenomena of relaxations, no more of which are observed as the maximum in the Nyquist plot (Cole-Cole). We also find that we have two different regions that are two constants of the time τ<sup>g</sup> and τbg using extrapolation by a corresponding equivalent circuit of data, which leads us to say this kinetic is mixed and it is what we confirmed by the chemical kinetic study.

After analyzing and evaluating the spectra using ZView 2.2 software, we extracted the parameters mentioned in **Table 3**. From the values shown in **Table 3**, it is observed that the adsorption efficiency increases when showing the fixing of nitrate ions on the surface of Zn3-Al-Cl-LDH. We used other quantities extracted from the equivalent circuit. The quantities are the dispersion coefficient for the grain and the grain boundary, on the one hand, and the grain and grain seal capacity, on the other hand, as a function of the adsorption time in order to follow

*Superposition of experimental data done by the equivalent electrical circuit and extrapolation of different*

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered…*

*DOI: http://dx.doi.org/10.5772/intechopen.90883*

**Figure 11.**

**123**

*adsorption times.*

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered… DOI: http://dx.doi.org/10.5772/intechopen.90883*

**Figure 11.** *Superposition of experimental data done by the equivalent electrical circuit and extrapolation of different adsorption times.*

After analyzing and evaluating the spectra using ZView 2.2 software, we extracted the parameters mentioned in **Table 3**. From the values shown in **Table 3**, it is observed that the adsorption efficiency increases when showing the fixing of nitrate ions on the surface of Zn3-Al-Cl-LDH. We used other quantities extracted from the equivalent circuit. The quantities are the dispersion coefficient for the grain and the grain boundary, on the one hand, and the grain and grain seal capacity, on the other hand, as a function of the adsorption time in order to follow

**Figure 12.** *Fitting and extrapolation of data experimental using equivalent circuit at all times of adsorption at room temperature.*

plotted as a function of square root of time (t1/2). The rate constant for intraparticle diffusion was obtained using the Weber-Morris equation given as follows [35]:

*t*

<sup>p</sup> <sup>þ</sup> *<sup>c</sup>* (6)

**) Tbg (Ω**�**1.s<sup>α</sup>**

**) X2**

*q t*ðÞ¼ *kp:* ffiffi

*Extrapolation of tan (d) measurement data of Zn3-Al-Cl-LDH for t = 5 min (a) and (b) t = 60 min.*

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered…*

*DOI: http://dx.doi.org/10.5772/intechopen.90883*

0 11,513 0 61,702 0 0.71 0.43 2.05 1.42 0.003 5 36,160 2.14 157,410 1.55 0.73 0.63 1.90 0.34 0.001 10 40,275 2.49 196,670 2.19 0.74 0.62 1.64 0.23 0.001 20 41,753 2.62 302,160 3.90 0.75 0.63 1.49 0.28 0.001 30 42,584 2.69 298,590 3.84 0.76 0.64 1.43 0.24 0.0008 60 43,504 2.77 319,590 4.18 0.77 0.66 1.39 0.14 0.0001

**Sample Rg (Ω) (E%)g Rbg (Ω) (E%)bg pg pbg Tg (Ω**�**1.s<sup>α</sup>**

*The values of the fitted corresponding equivalent circuit parameters.*

**Figure 13.**

**Table 3.**

**125**

where q is the amount of nitrate adsorbed in mg/g of adsorbent, kp is the intraparticle diffusion rate constant, and "t" is the agitation time in minutes. Due to stirring, there is a possibility of transport of nitrate species from the bulk into the

the phenomenon of adsorption of the nitrate ions to the available pores of Zn3-Al-Cl-LDH.

#### **4.1 Intraparticle diffusion rate constant**

In order to test the existence of intraparticle diffusion in the adsorption process, the amount of nitrate adsorbed per unit mass of adsorbents q at any time t was

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered… DOI: http://dx.doi.org/10.5772/intechopen.90883*

**Figure 13.** *Extrapolation of tan (d) measurement data of Zn3-Al-Cl-LDH for t = 5 min (a) and (b) t = 60 min.*


**Table 3.**

*The values of the fitted corresponding equivalent circuit parameters.*

plotted as a function of square root of time (t1/2). The rate constant for intraparticle diffusion was obtained using the Weber-Morris equation given as follows [35]:

$$q(t) = k\_p \sqrt{t} + c \tag{6}$$

where q is the amount of nitrate adsorbed in mg/g of adsorbent, kp is the intraparticle diffusion rate constant, and "t" is the agitation time in minutes. Due to stirring, there is a possibility of transport of nitrate species from the bulk into the

pores of the LDH as well as adsorption at an outer surface of the LDH. The ratelimiting step may be either adsorption or intraparticle diffusion.

Different regions of a system sample are characterized using a resistance and a constant phase element (CPE) placed usually in parallel, where subindexes "g" and "gb" refer to grain and grain boundary, respectively (Eqs. 7 and 8):

$$
\tau\_{\mathfrak{g}} = \left(R\_{\mathfrak{g}}, T\_{\mathfrak{g}}\right)^{\frac{\mathsf{L}}{p\_{\mathfrak{k}}}} \quad \text{and} \quad \tau\_{\mathfrak{j}\mathfrak{g}} = \left(R\_{\mathfrak{j}\mathfrak{g}}, T\_{\mathfrak{j}\mathfrak{g}}\right)^{\frac{\mathsf{L}}{p\_{\mathfrak{k}}}} \tag{7}
$$

$$\mathbf{C\_{\mathcal{g}}} = R^{\frac{1}{\mathbb{R}} - 1} . T^{\frac{1}{\mathbb{R}}} \quad \text{and} \quad \mathbf{C\_{\dot{\mathcal{g}}}} = R^{\frac{1}{\mathbb{R}\mathbb{R}} - 1} . T^{\frac{1}{\mathbb{R}\mathbb{R}}} \tag{8}$$

The values of the individual Rg.Cg and Rbg.Cbg components may then be quantified. Let us now see some practical examples of data and interpretation. A common type of impedance spectrum for Zn3-Al-Cl-(NO3 ─)ads LDH shows the presence of two distinct features attributable to intergrain or bulk and intergrain or grain boundary regions, using Eqs. (7) and (8) for obtained correspondent values listed in **Table 4**.

### **4.2 AC conductivity analysis**

The ionic conductivities extracted from the data using the equivalent circuit of **Figure 14** depicted in **Figure 15a** and **b** show the variation of AC conductivity with frequency at various times of adsorption for nitrate ions in the surface of the ionic clay. The log–log curves are flat in the low-frequency region as the conductivity values approach those of the DC conductivity. As frequency increases, the curves become dispersive. In the high-frequency range, weak time dependence may be noted, and it is evident that the shapes of the curves are similar. In most materials AC conductivity due to localized states may be described using the equation of double power law of Jonscher [29]. In the electrical conductivity at different times of adsorption, it is clear from the plot that above a certain point, the conductivity increases linearly with frequency. From **Figure 8**, it is also evident that the DC contribution is important at low frequencies and the high time of adsorption, whereas the frequency-dependent term dominates at high frequencies [30–34].

It can be observed (**Figure 15a** and **b**) that increased with increasing frequency. This can be explained in terms of conductivity of grains separated by highly resistive grain boundaries. According to this model, the AC conductivity at low frequencies exhibited the grain boundary behavior, while the dispersion at high frequency is attributed to the conductivity of grains. This variation corresponding to the interpretation of LDH materials has two types of charge carrier, which are responsible for the dielectric relaxation [35]. As reported in our earlier article [36], the proton of the polarized clusters of water is the first carrier, and the nitrate ions


of the adsorption surface of LDH region (**Figure 2b**) is the second one. The proton

water clusters in the presence of the applied electric field due to proton hopping (inter-cluster hopping at low-frequency region and intra-cluster hopping at high frequency) (**Figure 2c**). In this condition, nitrate ions also transfer from their

*Variation of the resistor (a) grain boundaries and (b) grains as a function of square root of time.*

*Separation behavior of the conductivity (a) of grains and grain boundaries. (b) Nyquist diagram showing the behavior of the grain and grain boundary for the adsorption time 5 min. (e) and (e) in the representation of the imaginary part of the conductivity. (c) Variation of the grain boundary pulsation for the adsorption*

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered…*

*DOI: http://dx.doi.org/10.5772/intechopen.90883*

equilibrium positions to serve as an additional charge carrier.

<sup>3</sup> ions (hydrogen bonding) at every path end of

transfers to produce *OH*� and *HO*<sup>þ</sup>

**Figure 14.**

*time 5 min.*

**Figure 15.**

**127**

**Table 4.** *Results obtained by the corresponding equivalent electrical circuit.* *Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered… DOI: http://dx.doi.org/10.5772/intechopen.90883*

#### **Figure 14.**

*Separation behavior of the conductivity (a) of grains and grain boundaries. (b) Nyquist diagram showing the behavior of the grain and grain boundary for the adsorption time 5 min. (e) and (e) in the representation of the imaginary part of the conductivity. (c) Variation of the grain boundary pulsation for the adsorption time 5 min.*

**Figure 15.** *Variation of the resistor (a) grain boundaries and (b) grains as a function of square root of time.*

of the adsorption surface of LDH region (**Figure 2b**) is the second one. The proton transfers to produce *OH*� and *HO*<sup>þ</sup> <sup>3</sup> ions (hydrogen bonding) at every path end of water clusters in the presence of the applied electric field due to proton hopping (inter-cluster hopping at low-frequency region and intra-cluster hopping at high frequency) (**Figure 2c**). In this condition, nitrate ions also transfer from their equilibrium positions to serve as an additional charge carrier.

**Figures 13** and **15** show that σ'ac (ω) becomes almost independent of frequency below a certain value when it decreases with decreasing frequency. The ionic σ'ac (ω) conductivity will be obtained using the technical extrapolation of this part of spectra toward lower frequency.

The conductivity σ'ac frequency dependence can be described in the majority of ionic conductors by the simple power law Jonscher according to [37] descript be one term dispersion although our system provides else dispersion term depicted in **Figures 7** and **8**.

$$
\sigma\_{ac}'(\alpha) = \sigma\_{\rm DC} + A\_{\rm g} \alpha^{p\_{\rm g}} + A\_{\rm bg} \alpha^{p\_{\rm bg}} \tag{9}
$$

material. The circuit consists of a resistor in parallel with a constant phase element (CPE) characterized by a pseudocapacitance and Tp (scattering coefficient).

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered…*

*S* � *y*<sup>0</sup>

Using the simulation of Eq. 11, we have obtained the equation of the admittance

� *<sup>ω</sup>pg*

In the term of conductivity, plots of the lower frequency dispersion correspond to the presence in grain boundary; this region can be approximately modeled by the

By similar calculations, the expression of the grain seal conductivity follows the

*bg* � cos *pbg<sup>π</sup>*

� � � �

� ð Þ *ωτ pgb* cos

*<sup>ω</sup>hbg* � �*pbg* � �

*bg* cos

� � � �

2

*pbgπ* 2

is the pulsation of hopping

� *<sup>ω</sup>pbg* ,

*pbgπ* 2 � �

1 þ ð Þ *ωτ <sup>g</sup>*

Finally the conductivity of grain is shown in the following equation:

*pgπ* 2 � �

<sup>1</sup> <sup>þ</sup> ð Þ *ωτ <sup>p</sup> cos <sup>p</sup><sup>π</sup>*

*pg* cos

*k R* 2

*pgπ* 2 � � � � (12)

� ð Þ *ωτ pg*

*<sup>g</sup>* cos

*pgπ* 2 � �

(13)

(14)

� � � � (11)

ð Þ *ω* (10)

*ac*ð Þ¼ *<sup>ω</sup> <sup>e</sup>*

The conductivity of this circuit is of the following form [40]:

*σ*0

ð Þ¼ *<sup>ω</sup>* <sup>1</sup> *R*

*<sup>g</sup>* ð Þ¼ *<sup>ω</sup>* <sup>1</sup> *R*

*<sup>g</sup>* cos *pg<sup>π</sup>* 2 h i � � <sup>¼</sup> *<sup>σ</sup>dcg* <sup>þ</sup>

� �

� � � � �<sup>1</sup> the pulsation of hopping

� <sup>1</sup> <sup>þ</sup> ð Þ *ωτ pbg*

bg <sup>¼</sup> ð Þ *<sup>σ</sup>dc* bg � <sup>1</sup> <sup>þ</sup> ð Þ*<sup>τ</sup> pbg*

bg <sup>¼</sup> ð Þ *<sup>σ</sup>dc* bg � <sup>1</sup> <sup>þ</sup> *<sup>ω</sup>*

*Corresponding equivalent circuit used to fit by EIS experimental data of intercalation.*

*pbgπ* 2 � � � � �<sup>1</sup>

*k R*

circuit similar to the one depicted in **Figure 17** (grain boundary).

¼ ð Þ *σdc* bg þ

*pg* cos

*<sup>ω</sup>hg* � � � �*pg*

*bg* <sup>¼</sup> *<sup>k</sup> R*

> *pbg bg* cos

where the real part of the admittance is

*DOI: http://dx.doi.org/10.5772/intechopen.90883*

of grain established by the following expression:

<sup>1</sup> <sup>þ</sup> ð Þ *ωτ pg*

*ac*ð Þ¼ *ω σdcg* 1 þ ð Þ*τ <sup>g</sup>*

*ac*ð Þ¼ *<sup>ω</sup> <sup>σ</sup>dcg* <sup>1</sup> <sup>þ</sup> *<sup>ω</sup>*

*pg <sup>g</sup>* cos *pg<sup>π</sup>* 2

shape of the following expression:

*σ*0 *ac*ð Þ *<sup>ω</sup>* � �

*σ*0 *ac*ð Þ *<sup>ω</sup>* � �

*σ*0 *ac*ð Þ *<sup>ω</sup>* � �

with *ωhbg* ¼ *τ*

**Figure 17.**

**129**

*σ*0

*σ*0

*σ*0

with *ωhg* ¼ *τ*

*ac*ð Þ¼ *<sup>ω</sup> <sup>k</sup> R* Y0

*Y*0

The charge carriers are the adsorbed nitrate ions, and the protons originate from adsorbed mobile water located on the surface of the clay [4]. On the other hand the charge carriers are responsible for the second jump which is generated by the anions Cl� and the H3O<sup>+</sup> ions intercalated in the interlamellar region according to [38].

The slope changes in the conductivity variation depicted in **Figure 10a** and **b** confirms that the conductivity of our system exhibits two behaviors of frequency dispersion: low-frequency dispersion associated with grains and the other for grain boundaries (**Figure 14a** and **b**).

*Usefulness of figures*: **Figure 10a** of variation suggests the presence of a hopping mechanism in these samples. Such type of conducting behavior is well described by Jonscher's universal power law. On the other hand, our system presents two power laws.

The **Figure 10b** of variation suggests two contributions grains and boundaries grains. This manifest itself in the conductivity diagram, with two hopping conductions which lead to two different slopes.

The figure shows that the joint region of grains is the regions that adsorb nitrate ions, but this variation influences the grain region. On the other hand, this finding is in good agreement with the evolution shown in **Figure 16a**.

The figure presents the deconvolution in order to separate contributions grains and grain boundaries as a function of frequency.
