# Effect of Water Adsorption on Cation-Surface Interaction Energy in the Na-Mordenite of 5.5 : 1 Si/Al Ratio

Authors: Sekou Diaby
Journal: Journal of Chemistry
 (2016)

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## Abstract

The mobility of the Na+ cations localized at the inner surface of the studied mordenite zeolite depends on the material surface properties. In this work, we show that the activation energy, ΔEhop, relating to the Na+ cation hopping displacement is associated to the surface potential and therefore can be used to get a better insight into the zeolite surface properties. Indeed, when molecules as water are adsorbed at the surface, they modify the surface potential energy and hence influence the value of ΔEhop. If the adsorbed molecules are polar they directly interact with the cations which become more mobile. The more the ΔEhop value is, the less the amount of adsorbed water molecules is. Alterations of the ΔEhop value with respect to the amount of adsorbed water molecules are interpreted using the Dubinin model which is based on simple adsorption principle.


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## Body

## 1. Introduction

Zeolites are crystalline aluminosilicate minerals with highly interconnected latticeworks of nanopores. These materials are thus characterized by an important surface area which gives them very interesting properties to be utilized in adsorption and catalysis. There are hundreds of zeolites showing differences in the geometric shape of the nanopore and the chemical features of the nanopore surface. The latter is directly related to the zeolite Si/Al ratio since the substitution of an Al atom for a Si one yields an excess of negative electrical charges which is compensated by cations localized at the nanopore surface. These cations are so-called extra-framework (later abbreviated EFC) or compensation cations. The type and density of EFC at the surface play a crucial role in the surface properties. For instance, the presence of H+ or of some metallic cations is known to be the most relevant feature that controls the adsorption and catalysis efficiency of a given zeolite. Therefore, the accurate knowledge of the EFC localization at the surface as well as their nature and density is imperative for a better understanding of the zeolite surface properties and hence for a better control of their practical uses.Owing to electrostatic interaction, the polarity of water molecules strongly favours their adsorption on or near the cations. Hydration and dehydration (which are important practical issues) of the zeolites are therefore much dependent on the EFC. Basically, it is thought that the solvation of the EFC by the adsorbed water molecules reduces the direct interaction between the EFC and the zeolite framework and consequently enhances the EFC mobility. The change in the EFC mobility with the amount of water molecules adsorbed into the zeolite can thus be seen as a significant outcome for characterizing the water adsorption process and the zeolite surface properties.In this paper, we investigate the water adsorption into a mordenite zeolite with 5.5 : 1 Si/Al ratio and its influence on the EFC Na+. The amount of adsorbed water is determined using thermogravimetric analysis (TGA). Besides, Dielectric Relaxation Spectroscopy (DRS) which is well known to be a very convenient tool to probe electrical charge displacement or dipole reorientation is used to measure the evolution of the EFC mobility as a function of the amount of adsorbed water molecules and of the temperature. The activation energy for the Na+ mobility, ΔEhop, can be then determined from the Arrhenius treatment of the DRS data. The experimental modification of ΔEhop with respect to the amount of adsorbed water molecules is finally interpreted using the Dubinin model which is based on simple adsorption principle.The paper is constructed as follows. In Section2.1, we recall some of the structural features of the mordenite zeolite studied here. In Section 2.2, the experimental methods used for measuring the amount of adsorbed water (TGA) and the EFC mobility (DRS) are briefly described. The Dubinin model is given in Section 2.3 whereas the analysis of the experimental results with this model is reported in Section 3.1. In Section 3.2, we discuss the data and their analysis. Conclusion is in Section 4.


## 2. Materials and Methods

### 2.1. The Na+ Mordenite

The sodium mordenite is an aluminosilicate of the formula(1)Na+xAlxSi48-xO96,yH2O.Its structure consists of a linkage of tetrahedral compounds (SiO4)4- and tetrahedral compounds (AlO4)5-, joined between them through their vertexes. Like the zeolite, the sodium mordenite is characterized by its structural unit otherwise called unit cell (u.c). It is comprised of tetrahedrons whose centres are occupied by silicon cations (Si4+), or aluminium cations (Al3+) whereas the oxygen atoms strongly attached to these centre cations occupy the vertexes of the tetrahedrons that link together in various ways. Each of the four tetrahedron oxygen atoms can exchange with the centre cation, an electron which causes a negative charge to arise when the concerned cation is trivalent as is Al3+. Then this charge is balanced by a Na+ cation referred to as a compensation cation or EFC (for the alkaline group or alkaline earth group). The crystalline arrangement of the tetrahedral compounds, shaped like rings between them, is imparted to the mordenite zeolite porous properties. The compensation cations while being responsible for the catalysis do not form part of the open structure of the mineral, and under certain conditions one can replace them by other cations without altering the crystal structure. Thus, channels large enough to let the ions in and out appear throughout this crystal lattice. We showed a unit cell of mordenite with a diagram in Figure 1, and we indicated the different cationic sites I, II, III, IV, and VI denoted following Mortier’s classification system for the crystallographic sites [1].Figure 1
Showing the cationic sites within a unit cell of Na-mordenite [1].The pore size varies according to the nature of the additional cation, electrical neutrality being maintained. The compensation cations contribute a lot to many properties of the mordenite zeolites. They particularly attract polar molecules like those of water. So some mordenite zeolites hold, besides the alkaline ions and alkaline earth ions, water molecules in the ducts, and cavities located within the framework of the aluminosilicates. Thus, under the effect of heat energy, a Na-mordenite crystal is expelled from its framework, water molecules. However, the crystal does not become smaller. It conserves its initial measurements and shape. The places that occupied the water molecules remain unoccupied, making the dehydrated Na-mordenite into a desiccant or an adsorbent for the water vapour or for other gases. Numerous industrial methods involving heterogeneous catalytic reactions like oil refining generally employ zeolites. These reactions use elemental processes that take place at the interfacial area separating the solid catalyst from the fluid phases containing the reactants and products. So it is essential to have sufficient knowledge of reactive molecule fixation on the solid and then expect to master the successive steps in this heterogeneous catalysis.


### 2.2. Experimental Methods

#### 2.2.1. Thermogravimetric Analysis (TGA) Measurements

The Na-mordenite sample placed in the capacitor as a dielectric is prepared in the following way: first, the zeolite is finely ground in an agate mortar and then it is compacted in the form of a disc, 15 mm thick and 1mm in diameter, under a pressure of 108 Pa. Then, it is placed for 24 hours under an atmosphere saturated with water vapour, at temperature T=298K for the purpose of being completely hydrated. A series of measurements performed by using thermogravimetric analysis (TGA) allows the number of water molecules fastened per unit cell (u.c) to be determined for a series of treatment temperatures (TT). These measurements are carried out when inputting a known quantity of powdered zeolite fully hydrated following the steps earlier outlined, into an apparatus (SETARAM,Labsys™). The mordenite zeolite is submitted to heating, at the fixed rate of 4°C per hour, under a flow of perfectly dried argon up to the equilibrium characterized by the treatment temperature (TT). The apparatus measures uninterruptedly the mass losses due to the departure of water molecules and makes it possible to determine the remaining quantity of water, at equilibrium, in the sample, at each treatment temperature (TT) ranging from 100°C to 220°C. As shown in Figure 2, we plotted the variations in the number (Na) of the adsorbed water molecules per unit cell of Na-mordenite, as a function of the treatment temperature (TT), at which the zeolite is held.Figure 2
Number(Na) of adsorbed water molecules per unit cell (u.c) of Na-mordenite as a function of treatment temperature (TT) to which the zeolite was heated.


#### 2.2.2. Study by Means of Dielectric Relaxation Spectroscopy (DRS)

(1) Electric Polarization. Unlike the conductors that contain free charges moving around the medium, under the effect of an electric field, the dielectrics are substances whose electrons are strongly linked to their atom so that one can observe neither a courant due to the free electrons nor a courant induced by the displacements of ions. By applying an electric field, however, one observes in this poor electricity conductor, that is, the dielectric, tiny displacements of unlike charges in opposite directions, and this relative movement causes dipoles to appear; the dielectric is said to be polarized. Many different mechanisms can be responsible for the development of these electric dipoles, but the experiments show that an induced polarization arises in any dielectric submitted to an electric field. The microscopic laws that describe this polarization, in the case of the polar dielectrics, allow the relative permittivity to be defined as a complex quantity (εr∗):(2)εr∗=εr′-j×εr′′.Then one shows [2–6] that, under the effect of an alternating electric field of angular frequency ω, this relative permittivity varies according to the relaxation time (τ) that characterizes the movement that brings dipoles into alignment with this field: (3)εr∗=εr,∞+εr,0-εr,∞1+j×ω×τ.By using (2) to separate the imaginary and real parts of (3), one gets(4)εr′=εr,∞+εr,0-εr,∞1+ω×τ2,εr′′=εr,0-εr,∞×ω×τ1+ω×τ2.It is noted, regarding these equations, that the curves schematizing the value of εr′′ as a function of ω, for a given value of τ, pass through a peak when ω=1/τ. In actual practice, it is noticeable that the peaks of these curves move towards the high values of ω when the temperature (T) of the sample rises.(2) Jumping Mechanism. This last remark inspired us to bring in a microscopic modelling for relating the relaxation time (τ) to the temperature (T). In this model, each of the component particles of the studied sample is placed at the minimum of a potential well when it is in its equilibrium position. It can climb out of there only when the thermal motion [7, 8] enables it to reach an energy level high enough to make it possible to cross over a potential barrier characteristic of its structural environment. If one imagines its state when the system is held at T=0K, each charge carrier is trapped into a site whose geometry solely determines its frequency of oscillations (τ0-1) in this fundamental state [9, 10]. The thermal motion provides additional energy for it, which is excited to a state of higher energy, and increases the likelihood of it crossing over the potential barrier (ΔEhop) separating the two neighbouring positions characterized by the same fundamental energy. This operation enables it to move between two neighbouring equilibrium positions. This description of an elemental jumping also indicates that the value of the barrier (ΔEhop) gives a measure of the energy associated with the pseudo dissociation reaction of the matching pair of charge carrier-site. Then, one can use Maxwell-Boltzmann’s distribution law to calculate the number of charge carriers likely to climb out of this potential well, under the effect of the thermal motion, which allows us to express the relaxation time of the sodium cations as(5)τ=τ0×exp⁡ΔEhopk×T,where k=1.38×10-23 J·K-1 is the Boltzmann constant. This equation can be written in the following form:(6)Log1τ=Log1τ0-ΔEhopk×T.The dipoles that came into being in the phenomenon of polarization within the studied Na-mordenite of 5.5 : 1 Si/Al ratio are oriented from the side of the rigid network of the aluminium silicate where the negative pole materializes, towards the side of the compensation cations (Na+), located in the sites I, IV, and VI [11–13], that is, in turn the positive pole. The calculations show that, in the system, τ0 varies slightly from one site to another and always remains very close to a value such that Log(1/τ0) = 30.61.(3) Measuring Method. The Na-mordenite sample compacted under a pressure of 108 Pa, in the form of a disc, is placed between the conducting surfaces separated by a distance (d) of a plane capacitor whose vacuum capacitance is denoted by C0. When one applies across the capacitor a sinusoidal tension Uω, with an angular frequency (ω), an electric polarization (p→) occurs, with a phase difference relative to the electric field E→. Under these conditions, the electric current (I(ω)), flowing through the capacitor, is expressed as a complex quantity (7) and with the use of (2) can be rearranged as a sum of a real part and an imaginary one (8). Consider the following:(7)Iω=Uω×j×ω×εr∗×C0,(8)Iω=Uω×ω×εr′′×C0+j×ω×εr′×C0.The capacitor filled with Na-mordenite can also be described by an equivalent circuit, formed of a parallel connection of a resistance (R) with a capacitor (with capacitance C). The admittance of such circuit is written as (1/R)+j×C×ω and allows calculating the electric current I(ω) which passes in this capacitor when it is subjected to a tension U(ω): (9)Iω=Uω×1R+j×C×ω.The comparison of (8) and (9) leads to (10)εr′′=1R×C0×ω,εr′=CC0.In actual practice, by the aid of an impedance-meter (NOVOCONTROL), we determined the characteristics of a capacitor with the Na-mordenite as dielectric by measuring its capacitance (C) and its conductance (G=1/R), when it is submitted to a 0.5V alternating electric field, whose angular frequency ranges from 6.28×10-3 rad ×s-1 to 6.28×10+7 rad ×s-1.


### 2.3. Gas Adsorption in Solids and the Dubinin Model

#### 2.3.1. Reminder of Some Modelling of Gas Adsorption on Some Solids

As far back as 1916, Langmuir [14, 15] went through a model likely to account for a fully gaseous monomolecular layer adsorption on a solid surface, by assuming that the molecules making up this layer do not interact between them. This model was designed for interpreting the formation of the monolayer. This assumes that the evaporation rate (ve) is proportional to the fraction (θ=Na/N0) of the surface area covered at any given time by the adsorbed molecules, Na being the number of adsorbing molecules or adsorbates fastened on the surface and N0 the total number of the adsorption sites. The condensation rate (vc) of the molecules on the surface is proportional to the surface not yet covered (1-θ) and the pressure P on the gas. At equilibrium, the condensation rate (vc=kc×P×(1-θ)) of the molecules on the surface is equal to the evaporation rate (ve=ke×θ) of the molecules from the surface, given with(11)kc×P×1-θ=ke×θ.One can write(12)PNa=PN0+1KT×N0.Denoting by K(T)=kc/ke, equilibrium is constant. Thus, when one knows the surface covered by an adsorbing molecule, one can deduce from the Langmuir isotherms, corresponding to (12), the specific surface area of the adsorbent.In 1938, Brunauer et al. [16] proposed a new model of the adsorption phenomenon, taking into account the formation of several molecule layers adsorbed and gas-gas interaction on the one hand and gas-solid ones on the other hand. They have got the following equation called BET equation, which enables a convenient representation of the adsorption isotherms to be obtained: (13)PP-P0×Na=1c×N0+c-1c×N0×PP0,where(14)c=exp⁡E1-ElR×T.The quantity c provides information on the affinity of the adsorbed molecules with the solid. The energy E1 constitutes an estimate of the heat of adsorption by the prime layer, and energy El is a measure of the adsorbate condensing heat, R is the universal gas constant, and Tis the isotherm temperature. The graphic representation of (13), for the values of the ratio P/P0, such that 0.05 ≤ P/P0 ≤ 0.3, allows on the one hand the quantity c and on the other hand the specific area of the adsorbent to be determined.In 1940, Dubinin, by analyzing a whole series of experimental results, contributed interestingly to describing the gaseous molecule adsorption phenomenon on microporous solids. He showed clearly a correlation by relating the amount of adsorbed gas to the energy involved in the molecule fixation mechanism on the surface. In cooperation with many other authors [17–22], he compared this energy to −ΔG, ΔG being Gibbs free energy change of the transformation reaction from the adsorbed phase to the gaseous one, which is described by the following chemical equation:(15)MoleculeAdsorbed,atTandP0⟺Moleculein gas phase,atTandPApplying the laws of thermodynamics on equilibrium states, then one obtains(16)ΔG+R×T×LnPP0=0hence-ΔG=R×T×LnPP0.For a given system of adsorbate-adsorbent, Dubinin showed that the graphic representation of the adsorption rate (θ=Na/N0), as a function of P/P0, yields only one curve, referred to as a characteristic. In the case of a given solid, the characteristic curves of different adsorbates can be fitted upon each other by means of using a specific parameter (β) called affinity whose value is dependent upon the nature of the adsorbate. The value of an energy E0 characterizing the structure of the solid and related to the width L of the pores existing on its surface is a measure of the adsorbent influence that is given with(17)L=10.8E0-11.4.In 1947, Dubinin et al. [23, 24] put forward an equation allowing the description of the organic molecule adsorption on the activated carbons. Then, in 1971, Dubinin and Astakhov [25] expanded it and made it applicable to the other solids including the zeolite by adding an exponent η that characterizes the solid’s heterogeneousness given with(18)Na=N0×exp⁡--ΔGβ×E0η=N0×exp⁡-B×-ΔGη.We must repeat that E0 and B are characteristic quantities of the studied solid.


## 2.1. The Na+ Mordenite

The sodium mordenite is an aluminosilicate of the formula(1)Na+xAlxSi48-xO96,yH2O.Its structure consists of a linkage of tetrahedral compounds (SiO4)4- and tetrahedral compounds (AlO4)5-, joined between them through their vertexes. Like the zeolite, the sodium mordenite is characterized by its structural unit otherwise called unit cell (u.c). It is comprised of tetrahedrons whose centres are occupied by silicon cations (Si4+), or aluminium cations (Al3+) whereas the oxygen atoms strongly attached to these centre cations occupy the vertexes of the tetrahedrons that link together in various ways. Each of the four tetrahedron oxygen atoms can exchange with the centre cation, an electron which causes a negative charge to arise when the concerned cation is trivalent as is Al3+. Then this charge is balanced by a Na+ cation referred to as a compensation cation or EFC (for the alkaline group or alkaline earth group). The crystalline arrangement of the tetrahedral compounds, shaped like rings between them, is imparted to the mordenite zeolite porous properties. The compensation cations while being responsible for the catalysis do not form part of the open structure of the mineral, and under certain conditions one can replace them by other cations without altering the crystal structure. Thus, channels large enough to let the ions in and out appear throughout this crystal lattice. We showed a unit cell of mordenite with a diagram in Figure 1, and we indicated the different cationic sites I, II, III, IV, and VI denoted following Mortier’s classification system for the crystallographic sites [1].Figure 1
Showing the cationic sites within a unit cell of Na-mordenite [1].The pore size varies according to the nature of the additional cation, electrical neutrality being maintained. The compensation cations contribute a lot to many properties of the mordenite zeolites. They particularly attract polar molecules like those of water. So some mordenite zeolites hold, besides the alkaline ions and alkaline earth ions, water molecules in the ducts, and cavities located within the framework of the aluminosilicates. Thus, under the effect of heat energy, a Na-mordenite crystal is expelled from its framework, water molecules. However, the crystal does not become smaller. It conserves its initial measurements and shape. The places that occupied the water molecules remain unoccupied, making the dehydrated Na-mordenite into a desiccant or an adsorbent for the water vapour or for other gases. Numerous industrial methods involving heterogeneous catalytic reactions like oil refining generally employ zeolites. These reactions use elemental processes that take place at the interfacial area separating the solid catalyst from the fluid phases containing the reactants and products. So it is essential to have sufficient knowledge of reactive molecule fixation on the solid and then expect to master the successive steps in this heterogeneous catalysis.


## 2.2. Experimental Methods

### 2.2.1. Thermogravimetric Analysis (TGA) Measurements

The Na-mordenite sample placed in the capacitor as a dielectric is prepared in the following way: first, the zeolite is finely ground in an agate mortar and then it is compacted in the form of a disc, 15 mm thick and 1mm in diameter, under a pressure of 108 Pa. Then, it is placed for 24 hours under an atmosphere saturated with water vapour, at temperature T=298K for the purpose of being completely hydrated. A series of measurements performed by using thermogravimetric analysis (TGA) allows the number of water molecules fastened per unit cell (u.c) to be determined for a series of treatment temperatures (TT). These measurements are carried out when inputting a known quantity of powdered zeolite fully hydrated following the steps earlier outlined, into an apparatus (SETARAM,Labsys™). The mordenite zeolite is submitted to heating, at the fixed rate of 4°C per hour, under a flow of perfectly dried argon up to the equilibrium characterized by the treatment temperature (TT). The apparatus measures uninterruptedly the mass losses due to the departure of water molecules and makes it possible to determine the remaining quantity of water, at equilibrium, in the sample, at each treatment temperature (TT) ranging from 100°C to 220°C. As shown in Figure 2, we plotted the variations in the number (Na) of the adsorbed water molecules per unit cell of Na-mordenite, as a function of the treatment temperature (TT), at which the zeolite is held.Figure 2
Number(Na) of adsorbed water molecules per unit cell (u.c) of Na-mordenite as a function of treatment temperature (TT) to which the zeolite was heated.


### 2.2.2. Study by Means of Dielectric Relaxation Spectroscopy (DRS)

(1) Electric Polarization. Unlike the conductors that contain free charges moving around the medium, under the effect of an electric field, the dielectrics are substances whose electrons are strongly linked to their atom so that one can observe neither a courant due to the free electrons nor a courant induced by the displacements of ions. By applying an electric field, however, one observes in this poor electricity conductor, that is, the dielectric, tiny displacements of unlike charges in opposite directions, and this relative movement causes dipoles to appear; the dielectric is said to be polarized. Many different mechanisms can be responsible for the development of these electric dipoles, but the experiments show that an induced polarization arises in any dielectric submitted to an electric field. The microscopic laws that describe this polarization, in the case of the polar dielectrics, allow the relative permittivity to be defined as a complex quantity (εr∗):(2)εr∗=εr′-j×εr′′.Then one shows [2–6] that, under the effect of an alternating electric field of angular frequency ω, this relative permittivity varies according to the relaxation time (τ) that characterizes the movement that brings dipoles into alignment with this field: (3)εr∗=εr,∞+εr,0-εr,∞1+j×ω×τ.By using (2) to separate the imaginary and real parts of (3), one gets(4)εr′=εr,∞+εr,0-εr,∞1+ω×τ2,εr′′=εr,0-εr,∞×ω×τ1+ω×τ2.It is noted, regarding these equations, that the curves schematizing the value of εr′′ as a function of ω, for a given value of τ, pass through a peak when ω=1/τ. In actual practice, it is noticeable that the peaks of these curves move towards the high values of ω when the temperature (T) of the sample rises.(2) Jumping Mechanism. This last remark inspired us to bring in a microscopic modelling for relating the relaxation time (τ) to the temperature (T). In this model, each of the component particles of the studied sample is placed at the minimum of a potential well when it is in its equilibrium position. It can climb out of there only when the thermal motion [7, 8] enables it to reach an energy level high enough to make it possible to cross over a potential barrier characteristic of its structural environment. If one imagines its state when the system is held at T=0K, each charge carrier is trapped into a site whose geometry solely determines its frequency of oscillations (τ0-1) in this fundamental state [9, 10]. The thermal motion provides additional energy for it, which is excited to a state of higher energy, and increases the likelihood of it crossing over the potential barrier (ΔEhop) separating the two neighbouring positions characterized by the same fundamental energy. This operation enables it to move between two neighbouring equilibrium positions. This description of an elemental jumping also indicates that the value of the barrier (ΔEhop) gives a measure of the energy associated with the pseudo dissociation reaction of the matching pair of charge carrier-site. Then, one can use Maxwell-Boltzmann’s distribution law to calculate the number of charge carriers likely to climb out of this potential well, under the effect of the thermal motion, which allows us to express the relaxation time of the sodium cations as(5)τ=τ0×exp⁡ΔEhopk×T,where k=1.38×10-23 J·K-1 is the Boltzmann constant. This equation can be written in the following form:(6)Log1τ=Log1τ0-ΔEhopk×T.The dipoles that came into being in the phenomenon of polarization within the studied Na-mordenite of 5.5 : 1 Si/Al ratio are oriented from the side of the rigid network of the aluminium silicate where the negative pole materializes, towards the side of the compensation cations (Na+), located in the sites I, IV, and VI [11–13], that is, in turn the positive pole. The calculations show that, in the system, τ0 varies slightly from one site to another and always remains very close to a value such that Log(1/τ0) = 30.61.(3) Measuring Method. The Na-mordenite sample compacted under a pressure of 108 Pa, in the form of a disc, is placed between the conducting surfaces separated by a distance (d) of a plane capacitor whose vacuum capacitance is denoted by C0. When one applies across the capacitor a sinusoidal tension Uω, with an angular frequency (ω), an electric polarization (p→) occurs, with a phase difference relative to the electric field E→. Under these conditions, the electric current (I(ω)), flowing through the capacitor, is expressed as a complex quantity (7) and with the use of (2) can be rearranged as a sum of a real part and an imaginary one (8). Consider the following:(7)Iω=Uω×j×ω×εr∗×C0,(8)Iω=Uω×ω×εr′′×C0+j×ω×εr′×C0.The capacitor filled with Na-mordenite can also be described by an equivalent circuit, formed of a parallel connection of a resistance (R) with a capacitor (with capacitance C). The admittance of such circuit is written as (1/R)+j×C×ω and allows calculating the electric current I(ω) which passes in this capacitor when it is subjected to a tension U(ω): (9)Iω=Uω×1R+j×C×ω.The comparison of (8) and (9) leads to (10)εr′′=1R×C0×ω,εr′=CC0.In actual practice, by the aid of an impedance-meter (NOVOCONTROL), we determined the characteristics of a capacitor with the Na-mordenite as dielectric by measuring its capacitance (C) and its conductance (G=1/R), when it is submitted to a 0.5V alternating electric field, whose angular frequency ranges from 6.28×10-3 rad ×s-1 to 6.28×10+7 rad ×s-1.


## 2.2.1. Thermogravimetric Analysis (TGA) Measurements

The Na-mordenite sample placed in the capacitor as a dielectric is prepared in the following way: first, the zeolite is finely ground in an agate mortar and then it is compacted in the form of a disc, 15 mm thick and 1mm in diameter, under a pressure of 108 Pa. Then, it is placed for 24 hours under an atmosphere saturated with water vapour, at temperature T=298K for the purpose of being completely hydrated. A series of measurements performed by using thermogravimetric analysis (TGA) allows the number of water molecules fastened per unit cell (u.c) to be determined for a series of treatment temperatures (TT). These measurements are carried out when inputting a known quantity of powdered zeolite fully hydrated following the steps earlier outlined, into an apparatus (SETARAM,Labsys™). The mordenite zeolite is submitted to heating, at the fixed rate of 4°C per hour, under a flow of perfectly dried argon up to the equilibrium characterized by the treatment temperature (TT). The apparatus measures uninterruptedly the mass losses due to the departure of water molecules and makes it possible to determine the remaining quantity of water, at equilibrium, in the sample, at each treatment temperature (TT) ranging from 100°C to 220°C. As shown in Figure 2, we plotted the variations in the number (Na) of the adsorbed water molecules per unit cell of Na-mordenite, as a function of the treatment temperature (TT), at which the zeolite is held.Figure 2
Number(Na) of adsorbed water molecules per unit cell (u.c) of Na-mordenite as a function of treatment temperature (TT) to which the zeolite was heated.


## 2.2.2. Study by Means of Dielectric Relaxation Spectroscopy (DRS)

(1) Electric Polarization. Unlike the conductors that contain free charges moving around the medium, under the effect of an electric field, the dielectrics are substances whose electrons are strongly linked to their atom so that one can observe neither a courant due to the free electrons nor a courant induced by the displacements of ions. By applying an electric field, however, one observes in this poor electricity conductor, that is, the dielectric, tiny displacements of unlike charges in opposite directions, and this relative movement causes dipoles to appear; the dielectric is said to be polarized. Many different mechanisms can be responsible for the development of these electric dipoles, but the experiments show that an induced polarization arises in any dielectric submitted to an electric field. The microscopic laws that describe this polarization, in the case of the polar dielectrics, allow the relative permittivity to be defined as a complex quantity (εr∗):(2)εr∗=εr′-j×εr′′.Then one shows [2–6] that, under the effect of an alternating electric field of angular frequency ω, this relative permittivity varies according to the relaxation time (τ) that characterizes the movement that brings dipoles into alignment with this field: (3)εr∗=εr,∞+εr,0-εr,∞1+j×ω×τ.By using (2) to separate the imaginary and real parts of (3), one gets(4)εr′=εr,∞+εr,0-εr,∞1+ω×τ2,εr′′=εr,0-εr,∞×ω×τ1+ω×τ2.It is noted, regarding these equations, that the curves schematizing the value of εr′′ as a function of ω, for a given value of τ, pass through a peak when ω=1/τ. In actual practice, it is noticeable that the peaks of these curves move towards the high values of ω when the temperature (T) of the sample rises.(2) Jumping Mechanism. This last remark inspired us to bring in a microscopic modelling for relating the relaxation time (τ) to the temperature (T). In this model, each of the component particles of the studied sample is placed at the minimum of a potential well when it is in its equilibrium position. It can climb out of there only when the thermal motion [7, 8] enables it to reach an energy level high enough to make it possible to cross over a potential barrier characteristic of its structural environment. If one imagines its state when the system is held at T=0K, each charge carrier is trapped into a site whose geometry solely determines its frequency of oscillations (τ0-1) in this fundamental state [9, 10]. The thermal motion provides additional energy for it, which is excited to a state of higher energy, and increases the likelihood of it crossing over the potential barrier (ΔEhop) separating the two neighbouring positions characterized by the same fundamental energy. This operation enables it to move between two neighbouring equilibrium positions. This description of an elemental jumping also indicates that the value of the barrier (ΔEhop) gives a measure of the energy associated with the pseudo dissociation reaction of the matching pair of charge carrier-site. Then, one can use Maxwell-Boltzmann’s distribution law to calculate the number of charge carriers likely to climb out of this potential well, under the effect of the thermal motion, which allows us to express the relaxation time of the sodium cations as(5)τ=τ0×exp⁡ΔEhopk×T,where k=1.38×10-23 J·K-1 is the Boltzmann constant. This equation can be written in the following form:(6)Log1τ=Log1τ0-ΔEhopk×T.The dipoles that came into being in the phenomenon of polarization within the studied Na-mordenite of 5.5 : 1 Si/Al ratio are oriented from the side of the rigid network of the aluminium silicate where the negative pole materializes, towards the side of the compensation cations (Na+), located in the sites I, IV, and VI [11–13], that is, in turn the positive pole. The calculations show that, in the system, τ0 varies slightly from one site to another and always remains very close to a value such that Log(1/τ0) = 30.61.(3) Measuring Method. The Na-mordenite sample compacted under a pressure of 108 Pa, in the form of a disc, is placed between the conducting surfaces separated by a distance (d) of a plane capacitor whose vacuum capacitance is denoted by C0. When one applies across the capacitor a sinusoidal tension Uω, with an angular frequency (ω), an electric polarization (p→) occurs, with a phase difference relative to the electric field E→. Under these conditions, the electric current (I(ω)), flowing through the capacitor, is expressed as a complex quantity (7) and with the use of (2) can be rearranged as a sum of a real part and an imaginary one (8). Consider the following:(7)Iω=Uω×j×ω×εr∗×C0,(8)Iω=Uω×ω×εr′′×C0+j×ω×εr′×C0.The capacitor filled with Na-mordenite can also be described by an equivalent circuit, formed of a parallel connection of a resistance (R) with a capacitor (with capacitance C). The admittance of such circuit is written as (1/R)+j×C×ω and allows calculating the electric current I(ω) which passes in this capacitor when it is subjected to a tension U(ω): (9)Iω=Uω×1R+j×C×ω.The comparison of (8) and (9) leads to (10)εr′′=1R×C0×ω,εr′=CC0.In actual practice, by the aid of an impedance-meter (NOVOCONTROL), we determined the characteristics of a capacitor with the Na-mordenite as dielectric by measuring its capacitance (C) and its conductance (G=1/R), when it is submitted to a 0.5V alternating electric field, whose angular frequency ranges from 6.28×10-3 rad ×s-1 to 6.28×10+7 rad ×s-1.


## 2.3. Gas Adsorption in Solids and the Dubinin Model

### 2.3.1. Reminder of Some Modelling of Gas Adsorption on Some Solids

As far back as 1916, Langmuir [14, 15] went through a model likely to account for a fully gaseous monomolecular layer adsorption on a solid surface, by assuming that the molecules making up this layer do not interact between them. This model was designed for interpreting the formation of the monolayer. This assumes that the evaporation rate (ve) is proportional to the fraction (θ=Na/N0) of the surface area covered at any given time by the adsorbed molecules, Na being the number of adsorbing molecules or adsorbates fastened on the surface and N0 the total number of the adsorption sites. The condensation rate (vc) of the molecules on the surface is proportional to the surface not yet covered (1-θ) and the pressure P on the gas. At equilibrium, the condensation rate (vc=kc×P×(1-θ)) of the molecules on the surface is equal to the evaporation rate (ve=ke×θ) of the molecules from the surface, given with(11)kc×P×1-θ=ke×θ.One can write(12)PNa=PN0+1KT×N0.Denoting by K(T)=kc/ke, equilibrium is constant. Thus, when one knows the surface covered by an adsorbing molecule, one can deduce from the Langmuir isotherms, corresponding to (12), the specific surface area of the adsorbent.In 1938, Brunauer et al. [16] proposed a new model of the adsorption phenomenon, taking into account the formation of several molecule layers adsorbed and gas-gas interaction on the one hand and gas-solid ones on the other hand. They have got the following equation called BET equation, which enables a convenient representation of the adsorption isotherms to be obtained: (13)PP-P0×Na=1c×N0+c-1c×N0×PP0,where(14)c=exp⁡E1-ElR×T.The quantity c provides information on the affinity of the adsorbed molecules with the solid. The energy E1 constitutes an estimate of the heat of adsorption by the prime layer, and energy El is a measure of the adsorbate condensing heat, R is the universal gas constant, and Tis the isotherm temperature. The graphic representation of (13), for the values of the ratio P/P0, such that 0.05 ≤ P/P0 ≤ 0.3, allows on the one hand the quantity c and on the other hand the specific area of the adsorbent to be determined.In 1940, Dubinin, by analyzing a whole series of experimental results, contributed interestingly to describing the gaseous molecule adsorption phenomenon on microporous solids. He showed clearly a correlation by relating the amount of adsorbed gas to the energy involved in the molecule fixation mechanism on the surface. In cooperation with many other authors [17–22], he compared this energy to −ΔG, ΔG being Gibbs free energy change of the transformation reaction from the adsorbed phase to the gaseous one, which is described by the following chemical equation:(15)MoleculeAdsorbed,atTandP0⟺Moleculein gas phase,atTandPApplying the laws of thermodynamics on equilibrium states, then one obtains(16)ΔG+R×T×LnPP0=0hence-ΔG=R×T×LnPP0.For a given system of adsorbate-adsorbent, Dubinin showed that the graphic representation of the adsorption rate (θ=Na/N0), as a function of P/P0, yields only one curve, referred to as a characteristic. In the case of a given solid, the characteristic curves of different adsorbates can be fitted upon each other by means of using a specific parameter (β) called affinity whose value is dependent upon the nature of the adsorbate. The value of an energy E0 characterizing the structure of the solid and related to the width L of the pores existing on its surface is a measure of the adsorbent influence that is given with(17)L=10.8E0-11.4.In 1947, Dubinin et al. [23, 24] put forward an equation allowing the description of the organic molecule adsorption on the activated carbons. Then, in 1971, Dubinin and Astakhov [25] expanded it and made it applicable to the other solids including the zeolite by adding an exponent η that characterizes the solid’s heterogeneousness given with(18)Na=N0×exp⁡--ΔGβ×E0η=N0×exp⁡-B×-ΔGη.We must repeat that E0 and B are characteristic quantities of the studied solid.


## 2.3.1. Reminder of Some Modelling of Gas Adsorption on Some Solids

As far back as 1916, Langmuir [14, 15] went through a model likely to account for a fully gaseous monomolecular layer adsorption on a solid surface, by assuming that the molecules making up this layer do not interact between them. This model was designed for interpreting the formation of the monolayer. This assumes that the evaporation rate (ve) is proportional to the fraction (θ=Na/N0) of the surface area covered at any given time by the adsorbed molecules, Na being the number of adsorbing molecules or adsorbates fastened on the surface and N0 the total number of the adsorption sites. The condensation rate (vc) of the molecules on the surface is proportional to the surface not yet covered (1-θ) and the pressure P on the gas. At equilibrium, the condensation rate (vc=kc×P×(1-θ)) of the molecules on the surface is equal to the evaporation rate (ve=ke×θ) of the molecules from the surface, given with(11)kc×P×1-θ=ke×θ.One can write(12)PNa=PN0+1KT×N0.Denoting by K(T)=kc/ke, equilibrium is constant. Thus, when one knows the surface covered by an adsorbing molecule, one can deduce from the Langmuir isotherms, corresponding to (12), the specific surface area of the adsorbent.In 1938, Brunauer et al. [16] proposed a new model of the adsorption phenomenon, taking into account the formation of several molecule layers adsorbed and gas-gas interaction on the one hand and gas-solid ones on the other hand. They have got the following equation called BET equation, which enables a convenient representation of the adsorption isotherms to be obtained: (13)PP-P0×Na=1c×N0+c-1c×N0×PP0,where(14)c=exp⁡E1-ElR×T.The quantity c provides information on the affinity of the adsorbed molecules with the solid. The energy E1 constitutes an estimate of the heat of adsorption by the prime layer, and energy El is a measure of the adsorbate condensing heat, R is the universal gas constant, and Tis the isotherm temperature. The graphic representation of (13), for the values of the ratio P/P0, such that 0.05 ≤ P/P0 ≤ 0.3, allows on the one hand the quantity c and on the other hand the specific area of the adsorbent to be determined.In 1940, Dubinin, by analyzing a whole series of experimental results, contributed interestingly to describing the gaseous molecule adsorption phenomenon on microporous solids. He showed clearly a correlation by relating the amount of adsorbed gas to the energy involved in the molecule fixation mechanism on the surface. In cooperation with many other authors [17–22], he compared this energy to −ΔG, ΔG being Gibbs free energy change of the transformation reaction from the adsorbed phase to the gaseous one, which is described by the following chemical equation:(15)MoleculeAdsorbed,atTandP0⟺Moleculein gas phase,atTandPApplying the laws of thermodynamics on equilibrium states, then one obtains(16)ΔG+R×T×LnPP0=0hence-ΔG=R×T×LnPP0.For a given system of adsorbate-adsorbent, Dubinin showed that the graphic representation of the adsorption rate (θ=Na/N0), as a function of P/P0, yields only one curve, referred to as a characteristic. In the case of a given solid, the characteristic curves of different adsorbates can be fitted upon each other by means of using a specific parameter (β) called affinity whose value is dependent upon the nature of the adsorbate. The value of an energy E0 characterizing the structure of the solid and related to the width L of the pores existing on its surface is a measure of the adsorbent influence that is given with(17)L=10.8E0-11.4.In 1947, Dubinin et al. [23, 24] put forward an equation allowing the description of the organic molecule adsorption on the activated carbons. Then, in 1971, Dubinin and Astakhov [25] expanded it and made it applicable to the other solids including the zeolite by adding an exponent η that characterizes the solid’s heterogeneousness given with(18)Na=N0×exp⁡--ΔGβ×E0η=N0×exp⁡-B×-ΔGη.We must repeat that E0 and B are characteristic quantities of the studied solid.


## 3. Measurement Method and Interpretation

### 3.1. Experimental Data and Analysis

#### 3.1.1. Utilization of the Polarization Measurements

We reproduced the same hydration conditions and then placed a Na-mordenite sample, held for three hours at a treatment temperature (TT) at equilibrium, in a measuring cell of an impedance-meter (NOVOCONTROL). Then we observed the variations in the imaginary part of the dielectric permittivity (εr′′) in relation to the angular frequency (ω) of the applied electric field (E→) and to the temperature (T) imposed on the sample. First of all, we ensure that the number of water molecules fastened on the sample does not increase by means of an external action whereby it is kept under an atmosphere in equilibrium with its hydration rate and that with the use of a device operating with a flow of perfectly dry argon. Thus we know that, under these conditions, a process wherein water moves can partake only if the sample undergoes an increase in temperature. Then, we imposed on it a temperature series, each of which was inferior to TT-10, for determining the activation energy of the polarization phenomenon. Complying with these requirements, we measured the imaginary part of the permittivity (εr′′), according to the angular frequency (ω) of the applied electric field. As shown in Figure 3, we obtained a series of curves by drawing the variations in εr′′ as a function of log(ω/2π), at TT = 373 K and T varying from 323 K to 263 K.Figure 3
Loss factorεr′′ in Na-mordenite of 5.5 : 1 Si/Al ratio (previously hydrated at the treatment temperature (TT)), as a function of log(ω/2π) at each temperature T such that T < TT − 10K.Referring to the curves, we localizedlog(ωm/2π) corresponding to the abscissa of the maximum of εr′′. Based on (3), we know that this point (εr′′ peak maximum) is characterized by the relationship ωm × τ = 1, whereof one derives straightforwardly τ = 1/ωm, for each measuring temperature T. Then, one plots the values of Log(1/τ)=Log(ωm) thus obtained as a function of 1000/T, where T is the temperature that was maintained constant throughout the determination of the curve: εr′′=f(log(ω/2π)).Then, one makes sure that the relationship (5) can be used while drawing the representative curve of Log(1/τ)=f(1000/T), and checking, as illustrated in Figure 4, that one actually obtains a straight line whose slope allows ΔEhop to be determined from (6) [26–28].Figure 4
Using a Na-mordenite sample of 5.5 : 1Si/Al ratio previously hydrated at the treatment temperature TT = 373 K, to ascertain experimentally the relationship Logωm=Log1/τ0-ΔEhop/(k×T).The value of the activation energy (ΔEhop) calculated from the slope of the straight line drawn, as shown in Figure 4, is ΔEhop = 0.64 eV for TT = 373 K. Thereafter we repeated the method that has just been described for determining the variations in the value of ΔEhop, when the Na-mordenite sample of 5.5 : 1 Si/Al ratio is submitted to the treatment temperatures (TT) ranging from 333 K to 467 K. The output of the measurements by differential thermal analysis (DTA) we previously performed is given with Figure 5.Figure 5
Change in the jumping energy (ΔEhop) in a Na-mordenite of 5.5 : 1 Si/Al ratio as a function of the number (Na) of adsorbed water molecules per unit cell (u.c).Thus we showed that each treatment temperature of the Na-mordenite of 5.5 : 1Si/Al ratio corresponds to an amount of adsorbed water molecules per unit cell that we already denoted by Na. Figure 5 illustrates the graph of the value of ΔEhop as a function of the number (Na) of adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio.


### 3.2. Discussion

The adsorbed water molecules fastened on the Na-mordenite are in equilibrium with those in vapour phase, in contact with the solid. The laws obeyed by the mechanism of this equilibrium can be studied when regarding it as a pseudo reaction given with(19)MoleculeAdsorbed atTandP0⟺Moleculein gas phase atTandPThe laws of thermodynamics of the equilibrium states enable one to write(20)-ΔG=R×T×LogPP0.Many authors including Dubinin et al. [29–35] assimilated the energy which sustains the mechanism fixing the molecules adsorbed on the surface to the opposite of Gibbs free energy change (-ΔG) of the pseudo reaction. To interpret the observed behaviour, it is necessary to provide an estimate of the interaction energy between the adsorbed water molecules and the exchanged cations on the solid surface.


#### 3.2.1. Making Use of a Lennard-Jones Potential to Describe the Interaction between a Compensation Cation and the Adsorbed Water Molecules

We assume that each interaction energy involved in the adsorption phenomena can be described by a Lennard-Jones potential [36–38]. By disregarding the term of repulsion whose influence is negligible in an adsorption process, we can write the relationship already suggested by Hasley, Hill, and Frenkel [39–41]: (21)u=∑jκrjn,where u is the interaction energy between an adsorbed water molecule and the surface, rj is the distance between the adsorbed molecule and an atom on the surface that interacts with it, κ is a constant, and n is an empirical exponent.Then one considers the stage of the adsorption process where the number of the adsorbed water molecules is lower than that which makes it possible for the monolayer to be formed, that is to say less than 10 adsorbed water molecules per unit cell of Na-mordenite. Previous works have shown that these water molecules are distributed on the solid surface to form a latticework [42–44]. Denoting by rww the distance between two water molecules, by rkw the distance between a sodium cation and its closest neighbouring water molecules, by Na the number of adsorbed water molecules per unit cell of Na-mordenite, and by Aσ the solid’s total surface accessible to the water molecules, one can write the two following equations: (22)Aσ=α×Na×rww2,(23)rkw=β×rww,where α and β are parameters that describe the lattice geometry. By combining (22) and (23), one derives(24)rkw=β×Aσα×Na.Substituting the values of rkw, obtained from (24), into (21), one can sum up the energies of a cation interaction with the adsorbed water molecules which are its closest neighbours. Thus one obtains an estimate of the variation in the energy (ΔEhop) caused by the first adsorbed water molecules, implying that Na ≤ 10. We incorporated the expression of rkw (24) into (21), and we obtained after due calculations the following expression of the variations in the jumping energy (ΔEhop) under the effect of the Na water molecule adsorption:(25)ΔEhopNa=ΔEhop0-Na×κ×α×N0β×Aσn.For greater convenience, we rearranged this equation as (26)ΔEhopNa=ΔEhop0-B×Na.In expression (26), ΔEhop0 stands for the cation extraction energy from its site under the conditions where the Na-mordenite sample is fully dehydrated and B is a coefficient of variation. Then we used (26), and try to find out the change in the value of ΔEhop according to the number (Na) of adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio. The result is shown in Figure 6 whereon one sees that (26) allows the observed adsorption behaviour to be interpreted properly.Figure 6
Fit between the curve schematizing the equationΔEhop(Na)=ΔEhop0-B×Na and the plot of the experimental values of ΔEhop(Na) as a function of the number (Na) of adsorbed water molecules per unit cell (u.c) of a Na-mordenite of 5.5 : 1 Si/Al ratio.This well fit between the curves enables us to determine the following values:(27)ΔEhop0=0.81eV,B=0.0164eV·mole´cule-1.


#### 3.2.2. Making Use of Dubinin, Raduskevich, and Kaganer’s Empirical Approach to Describe the Interaction between a Compensation Cation and the Adsorbed Water Molecules

The second aspect of the interpretation of the observed behaviours, based on the experimentally determined values ofΔEhop(Na) we plotted as a function of the number (Na) of the adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio, involves using the empirical approach suggested by Dubinin, Raduskevich, and Kaganer [29–32]. Indeed, these authors proposed an equation relating the ratio (Na/N0) and the energy that is involved in the process of the molecule fixation on the surface. The equation is written as follows: (28)LogNaN0=-DDub×u2.Assume that(29)u=R×T×LogPP0.By using reasoning similar to that we developed in the previous section, we derive the following equation aiming at recovering the change in the value of ΔEhop as a function of the number (Na) of water molecules adsorbed per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio:(30)ΔEhopNa=ΔEhopT+LogN0/NaDDub.In that equation, ΔEhopT denotes the extraction energy of the compensation sodium cation from its site, under the conditions where the Na-mordenite surface is coated with a water molecule monolayer. It is shown in Figure 7 that the plot of the values of ΔEhop(Na) experimentally determined follows closely the shape of the curve schematizing ΔEhop(Na)=ΔEhopT+LogN0/Na/DDub. Making that curve fit suitably upon the experimental plot then provides the following parameters:(31)ΔEhopT=0.58eV,DDub=41.3223eV-2and N0 = 12.17water molecules per unit cell of Na-mordenite.Figure 7
Comparison between the experimental plot of the variations inΔEhop(Na) and the curve schematizing the equation ΔEhop(Na)=ΔEhopT+(LogN0/Na)/DDub, elicited from the empirical approach of Dubinin.


## 3.1. Experimental Data and Analysis

### 3.1.1. Utilization of the Polarization Measurements

We reproduced the same hydration conditions and then placed a Na-mordenite sample, held for three hours at a treatment temperature (TT) at equilibrium, in a measuring cell of an impedance-meter (NOVOCONTROL). Then we observed the variations in the imaginary part of the dielectric permittivity (εr′′) in relation to the angular frequency (ω) of the applied electric field (E→) and to the temperature (T) imposed on the sample. First of all, we ensure that the number of water molecules fastened on the sample does not increase by means of an external action whereby it is kept under an atmosphere in equilibrium with its hydration rate and that with the use of a device operating with a flow of perfectly dry argon. Thus we know that, under these conditions, a process wherein water moves can partake only if the sample undergoes an increase in temperature. Then, we imposed on it a temperature series, each of which was inferior to TT-10, for determining the activation energy of the polarization phenomenon. Complying with these requirements, we measured the imaginary part of the permittivity (εr′′), according to the angular frequency (ω) of the applied electric field. As shown in Figure 3, we obtained a series of curves by drawing the variations in εr′′ as a function of log(ω/2π), at TT = 373 K and T varying from 323 K to 263 K.Figure 3
Loss factorεr′′ in Na-mordenite of 5.5 : 1 Si/Al ratio (previously hydrated at the treatment temperature (TT)), as a function of log(ω/2π) at each temperature T such that T < TT − 10K.Referring to the curves, we localizedlog(ωm/2π) corresponding to the abscissa of the maximum of εr′′. Based on (3), we know that this point (εr′′ peak maximum) is characterized by the relationship ωm × τ = 1, whereof one derives straightforwardly τ = 1/ωm, for each measuring temperature T. Then, one plots the values of Log(1/τ)=Log(ωm) thus obtained as a function of 1000/T, where T is the temperature that was maintained constant throughout the determination of the curve: εr′′=f(log(ω/2π)).Then, one makes sure that the relationship (5) can be used while drawing the representative curve of Log(1/τ)=f(1000/T), and checking, as illustrated in Figure 4, that one actually obtains a straight line whose slope allows ΔEhop to be determined from (6) [26–28].Figure 4
Using a Na-mordenite sample of 5.5 : 1Si/Al ratio previously hydrated at the treatment temperature TT = 373 K, to ascertain experimentally the relationship Logωm=Log1/τ0-ΔEhop/(k×T).The value of the activation energy (ΔEhop) calculated from the slope of the straight line drawn, as shown in Figure 4, is ΔEhop = 0.64 eV for TT = 373 K. Thereafter we repeated the method that has just been described for determining the variations in the value of ΔEhop, when the Na-mordenite sample of 5.5 : 1 Si/Al ratio is submitted to the treatment temperatures (TT) ranging from 333 K to 467 K. The output of the measurements by differential thermal analysis (DTA) we previously performed is given with Figure 5.Figure 5
Change in the jumping energy (ΔEhop) in a Na-mordenite of 5.5 : 1 Si/Al ratio as a function of the number (Na) of adsorbed water molecules per unit cell (u.c).Thus we showed that each treatment temperature of the Na-mordenite of 5.5 : 1Si/Al ratio corresponds to an amount of adsorbed water molecules per unit cell that we already denoted by Na. Figure 5 illustrates the graph of the value of ΔEhop as a function of the number (Na) of adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio.


## 3.1.1. Utilization of the Polarization Measurements

We reproduced the same hydration conditions and then placed a Na-mordenite sample, held for three hours at a treatment temperature (TT) at equilibrium, in a measuring cell of an impedance-meter (NOVOCONTROL). Then we observed the variations in the imaginary part of the dielectric permittivity (εr′′) in relation to the angular frequency (ω) of the applied electric field (E→) and to the temperature (T) imposed on the sample. First of all, we ensure that the number of water molecules fastened on the sample does not increase by means of an external action whereby it is kept under an atmosphere in equilibrium with its hydration rate and that with the use of a device operating with a flow of perfectly dry argon. Thus we know that, under these conditions, a process wherein water moves can partake only if the sample undergoes an increase in temperature. Then, we imposed on it a temperature series, each of which was inferior to TT-10, for determining the activation energy of the polarization phenomenon. Complying with these requirements, we measured the imaginary part of the permittivity (εr′′), according to the angular frequency (ω) of the applied electric field. As shown in Figure 3, we obtained a series of curves by drawing the variations in εr′′ as a function of log(ω/2π), at TT = 373 K and T varying from 323 K to 263 K.Figure 3
Loss factorεr′′ in Na-mordenite of 5.5 : 1 Si/Al ratio (previously hydrated at the treatment temperature (TT)), as a function of log(ω/2π) at each temperature T such that T < TT − 10K.Referring to the curves, we localizedlog(ωm/2π) corresponding to the abscissa of the maximum of εr′′. Based on (3), we know that this point (εr′′ peak maximum) is characterized by the relationship ωm × τ = 1, whereof one derives straightforwardly τ = 1/ωm, for each measuring temperature T. Then, one plots the values of Log(1/τ)=Log(ωm) thus obtained as a function of 1000/T, where T is the temperature that was maintained constant throughout the determination of the curve: εr′′=f(log(ω/2π)).Then, one makes sure that the relationship (5) can be used while drawing the representative curve of Log(1/τ)=f(1000/T), and checking, as illustrated in Figure 4, that one actually obtains a straight line whose slope allows ΔEhop to be determined from (6) [26–28].Figure 4
Using a Na-mordenite sample of 5.5 : 1Si/Al ratio previously hydrated at the treatment temperature TT = 373 K, to ascertain experimentally the relationship Logωm=Log1/τ0-ΔEhop/(k×T).The value of the activation energy (ΔEhop) calculated from the slope of the straight line drawn, as shown in Figure 4, is ΔEhop = 0.64 eV for TT = 373 K. Thereafter we repeated the method that has just been described for determining the variations in the value of ΔEhop, when the Na-mordenite sample of 5.5 : 1 Si/Al ratio is submitted to the treatment temperatures (TT) ranging from 333 K to 467 K. The output of the measurements by differential thermal analysis (DTA) we previously performed is given with Figure 5.Figure 5
Change in the jumping energy (ΔEhop) in a Na-mordenite of 5.5 : 1 Si/Al ratio as a function of the number (Na) of adsorbed water molecules per unit cell (u.c).Thus we showed that each treatment temperature of the Na-mordenite of 5.5 : 1Si/Al ratio corresponds to an amount of adsorbed water molecules per unit cell that we already denoted by Na. Figure 5 illustrates the graph of the value of ΔEhop as a function of the number (Na) of adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio.


## 3.2. Discussion

The adsorbed water molecules fastened on the Na-mordenite are in equilibrium with those in vapour phase, in contact with the solid. The laws obeyed by the mechanism of this equilibrium can be studied when regarding it as a pseudo reaction given with(19)MoleculeAdsorbed atTandP0⟺Moleculein gas phase atTandPThe laws of thermodynamics of the equilibrium states enable one to write(20)-ΔG=R×T×LogPP0.Many authors including Dubinin et al. [29–35] assimilated the energy which sustains the mechanism fixing the molecules adsorbed on the surface to the opposite of Gibbs free energy change (-ΔG) of the pseudo reaction. To interpret the observed behaviour, it is necessary to provide an estimate of the interaction energy between the adsorbed water molecules and the exchanged cations on the solid surface.


### 3.2.1. Making Use of a Lennard-Jones Potential to Describe the Interaction between a Compensation Cation and the Adsorbed Water Molecules

We assume that each interaction energy involved in the adsorption phenomena can be described by a Lennard-Jones potential [36–38]. By disregarding the term of repulsion whose influence is negligible in an adsorption process, we can write the relationship already suggested by Hasley, Hill, and Frenkel [39–41]: (21)u=∑jκrjn,where u is the interaction energy between an adsorbed water molecule and the surface, rj is the distance between the adsorbed molecule and an atom on the surface that interacts with it, κ is a constant, and n is an empirical exponent.Then one considers the stage of the adsorption process where the number of the adsorbed water molecules is lower than that which makes it possible for the monolayer to be formed, that is to say less than 10 adsorbed water molecules per unit cell of Na-mordenite. Previous works have shown that these water molecules are distributed on the solid surface to form a latticework [42–44]. Denoting by rww the distance between two water molecules, by rkw the distance between a sodium cation and its closest neighbouring water molecules, by Na the number of adsorbed water molecules per unit cell of Na-mordenite, and by Aσ the solid’s total surface accessible to the water molecules, one can write the two following equations: (22)Aσ=α×Na×rww2,(23)rkw=β×rww,where α and β are parameters that describe the lattice geometry. By combining (22) and (23), one derives(24)rkw=β×Aσα×Na.Substituting the values of rkw, obtained from (24), into (21), one can sum up the energies of a cation interaction with the adsorbed water molecules which are its closest neighbours. Thus one obtains an estimate of the variation in the energy (ΔEhop) caused by the first adsorbed water molecules, implying that Na ≤ 10. We incorporated the expression of rkw (24) into (21), and we obtained after due calculations the following expression of the variations in the jumping energy (ΔEhop) under the effect of the Na water molecule adsorption:(25)ΔEhopNa=ΔEhop0-Na×κ×α×N0β×Aσn.For greater convenience, we rearranged this equation as (26)ΔEhopNa=ΔEhop0-B×Na.In expression (26), ΔEhop0 stands for the cation extraction energy from its site under the conditions where the Na-mordenite sample is fully dehydrated and B is a coefficient of variation. Then we used (26), and try to find out the change in the value of ΔEhop according to the number (Na) of adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio. The result is shown in Figure 6 whereon one sees that (26) allows the observed adsorption behaviour to be interpreted properly.Figure 6
Fit between the curve schematizing the equationΔEhop(Na)=ΔEhop0-B×Na and the plot of the experimental values of ΔEhop(Na) as a function of the number (Na) of adsorbed water molecules per unit cell (u.c) of a Na-mordenite of 5.5 : 1 Si/Al ratio.This well fit between the curves enables us to determine the following values:(27)ΔEhop0=0.81eV,B=0.0164eV·mole´cule-1.


### 3.2.2. Making Use of Dubinin, Raduskevich, and Kaganer’s Empirical Approach to Describe the Interaction between a Compensation Cation and the Adsorbed Water Molecules

The second aspect of the interpretation of the observed behaviours, based on the experimentally determined values ofΔEhop(Na) we plotted as a function of the number (Na) of the adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio, involves using the empirical approach suggested by Dubinin, Raduskevich, and Kaganer [29–32]. Indeed, these authors proposed an equation relating the ratio (Na/N0) and the energy that is involved in the process of the molecule fixation on the surface. The equation is written as follows: (28)LogNaN0=-DDub×u2.Assume that(29)u=R×T×LogPP0.By using reasoning similar to that we developed in the previous section, we derive the following equation aiming at recovering the change in the value of ΔEhop as a function of the number (Na) of water molecules adsorbed per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio:(30)ΔEhopNa=ΔEhopT+LogN0/NaDDub.In that equation, ΔEhopT denotes the extraction energy of the compensation sodium cation from its site, under the conditions where the Na-mordenite surface is coated with a water molecule monolayer. It is shown in Figure 7 that the plot of the values of ΔEhop(Na) experimentally determined follows closely the shape of the curve schematizing ΔEhop(Na)=ΔEhopT+LogN0/Na/DDub. Making that curve fit suitably upon the experimental plot then provides the following parameters:(31)ΔEhopT=0.58eV,DDub=41.3223eV-2and N0 = 12.17water molecules per unit cell of Na-mordenite.Figure 7
Comparison between the experimental plot of the variations inΔEhop(Na) and the curve schematizing the equation ΔEhop(Na)=ΔEhopT+(LogN0/Na)/DDub, elicited from the empirical approach of Dubinin.


## 3.2.1. Making Use of a Lennard-Jones Potential to Describe the Interaction between a Compensation Cation and the Adsorbed Water Molecules

We assume that each interaction energy involved in the adsorption phenomena can be described by a Lennard-Jones potential [36–38]. By disregarding the term of repulsion whose influence is negligible in an adsorption process, we can write the relationship already suggested by Hasley, Hill, and Frenkel [39–41]: (21)u=∑jκrjn,where u is the interaction energy between an adsorbed water molecule and the surface, rj is the distance between the adsorbed molecule and an atom on the surface that interacts with it, κ is a constant, and n is an empirical exponent.Then one considers the stage of the adsorption process where the number of the adsorbed water molecules is lower than that which makes it possible for the monolayer to be formed, that is to say less than 10 adsorbed water molecules per unit cell of Na-mordenite. Previous works have shown that these water molecules are distributed on the solid surface to form a latticework [42–44]. Denoting by rww the distance between two water molecules, by rkw the distance between a sodium cation and its closest neighbouring water molecules, by Na the number of adsorbed water molecules per unit cell of Na-mordenite, and by Aσ the solid’s total surface accessible to the water molecules, one can write the two following equations: (22)Aσ=α×Na×rww2,(23)rkw=β×rww,where α and β are parameters that describe the lattice geometry. By combining (22) and (23), one derives(24)rkw=β×Aσα×Na.Substituting the values of rkw, obtained from (24), into (21), one can sum up the energies of a cation interaction with the adsorbed water molecules which are its closest neighbours. Thus one obtains an estimate of the variation in the energy (ΔEhop) caused by the first adsorbed water molecules, implying that Na ≤ 10. We incorporated the expression of rkw (24) into (21), and we obtained after due calculations the following expression of the variations in the jumping energy (ΔEhop) under the effect of the Na water molecule adsorption:(25)ΔEhopNa=ΔEhop0-Na×κ×α×N0β×Aσn.For greater convenience, we rearranged this equation as (26)ΔEhopNa=ΔEhop0-B×Na.In expression (26), ΔEhop0 stands for the cation extraction energy from its site under the conditions where the Na-mordenite sample is fully dehydrated and B is a coefficient of variation. Then we used (26), and try to find out the change in the value of ΔEhop according to the number (Na) of adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio. The result is shown in Figure 6 whereon one sees that (26) allows the observed adsorption behaviour to be interpreted properly.Figure 6
Fit between the curve schematizing the equationΔEhop(Na)=ΔEhop0-B×Na and the plot of the experimental values of ΔEhop(Na) as a function of the number (Na) of adsorbed water molecules per unit cell (u.c) of a Na-mordenite of 5.5 : 1 Si/Al ratio.This well fit between the curves enables us to determine the following values:(27)ΔEhop0=0.81eV,B=0.0164eV·mole´cule-1.


## 3.2.2. Making Use of Dubinin, Raduskevich, and Kaganer’s Empirical Approach to Describe the Interaction between a Compensation Cation and the Adsorbed Water Molecules

The second aspect of the interpretation of the observed behaviours, based on the experimentally determined values ofΔEhop(Na) we plotted as a function of the number (Na) of the adsorbed water molecules per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio, involves using the empirical approach suggested by Dubinin, Raduskevich, and Kaganer [29–32]. Indeed, these authors proposed an equation relating the ratio (Na/N0) and the energy that is involved in the process of the molecule fixation on the surface. The equation is written as follows: (28)LogNaN0=-DDub×u2.Assume that(29)u=R×T×LogPP0.By using reasoning similar to that we developed in the previous section, we derive the following equation aiming at recovering the change in the value of ΔEhop as a function of the number (Na) of water molecules adsorbed per unit cell of Na-mordenite of 5.5 : 1 Si/Al ratio:(30)ΔEhopNa=ΔEhopT+LogN0/NaDDub.In that equation, ΔEhopT denotes the extraction energy of the compensation sodium cation from its site, under the conditions where the Na-mordenite surface is coated with a water molecule monolayer. It is shown in Figure 7 that the plot of the values of ΔEhop(Na) experimentally determined follows closely the shape of the curve schematizing ΔEhop(Na)=ΔEhopT+LogN0/Na/DDub. Making that curve fit suitably upon the experimental plot then provides the following parameters:(31)ΔEhopT=0.58eV,DDub=41.3223eV-2and N0 = 12.17water molecules per unit cell of Na-mordenite.Figure 7
Comparison between the experimental plot of the variations inΔEhop(Na) and the curve schematizing the equation ΔEhop(Na)=ΔEhopT+(LogN0/Na)/DDub, elicited from the empirical approach of Dubinin.


## 4. Conclusion

In this work, we carried out dielectric relaxation measurements to investigate experimentally the change in the jumping energy of the sodium cations in a Na-mordenite of 5.5 : 1Si/Al ratio, according to the degree of advancement of the water molecule adsorption. We also suggested two types of interpretations to account for the behaviours we observed:(1)
The first one issuing from the theory of Frenkel, Halsey, and Hill makes it possible to give an exact representation of the results we got experimentally. It allows determining the extraction energy of the sodium cations in a dehydrated zeolite.(2)
The second one, based on the empirical approach suggested by Dubinin, Raduskevich, and Kaganer, also leads to a very satisfactory representation of the experimental results. An adequate handling of them provides an estimate of the extraction energy of the sodium cations in a zeolite when its surface is coated with a monolayer of adsorbing molecules and also an estimate of the number of the molecules that form the monolayer when the concerned molecules are those of water.This paper suggests a new experimental approach that gives original information on elemental processes involved in the adsorption phenomenon enabling one to account quantitatively and in detail for the interactions between the solid surface, the compensating cations, and the molecules adsorbed onto the aluminosilicates characterized by the presence of pores wherein the exchangeable cations are trapped.



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*Source: 2152180-2016-02-21.xml*