6. The Dubinin adsorption isotherm equation

The Dubinin adsorption isotherm equation can be inferred by the application of the Dubinin's Theory of Volume Filling, i.e., (where volume filling is a process which takes place by the filling of the adsorption space rather than the surface coverage), together with, the Polanyi's adsorption potential [32]. In this regard, in 1914, Polanyi created perhaps the first convincing physical adsorption model. Mijail. M. Dubinin, a former pupil of Polanyi applied this model that essentially entailed a link between the adsorption space volume, Vi, and the adsorption energy field, εi, (see Figure 3): i.e.,ε<sup>i</sup> ¼ F Vð Þ<sup>i</sup> , termed by Polanyi as the characteristic function, a temperature independent function. For the deduction of the Dubinin isotherm equation, the next consideration was that [33]:

Figure 3. Polanyi adsorption model.

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials DOI: http://dx.doi.org/10.5772/intechopen.83355

$$
\mu\_{\rm g} = \mu\_L + \varepsilon\_i = \mu\_a \tag{7}
$$

In which, μ<sup>g</sup> is the chemical potential of the gas phase adsorbate, μ<sup>L</sup> is the chemical potential of the pure liquid adsorbate, μ<sup>a</sup> is the adsorbed phase chemical potential, while ε<sup>i</sup> is the potential energy of the adsorption field. Consequently, applying of Eq. (7), it is possible to demonstrate that:

$$
\varepsilon\_i = RT \ln \left(\frac{P\_0}{P\_i}\right) \tag{8}
$$

where Po is the vapor pressure of the adsorptive at the temperature,T is the adsorption experiment, while Pi is the equilibrium adsorption pressure (ε could be also designed as the differential work of adsorption). Now, following the so-called Gurvich rule, it is possible to obtain the subsequent relation Vi <sup>¼</sup> <sup>V</sup>Lna between the volume of the adsorption space, Vi, and the amount adsorbed, where VL is the molar volume of the liquid phase that conforms to the adsorbed phase. Combining Eq. (8) with the characteristic function, the relation between the volume of the adsorption space, and the amount adsorbed, we will get [32]:

$$F(V\_i) = f(n\_a) = \varepsilon\_i = RT \ln\left(\frac{P\_0}{P\_i}\right) \tag{9}$$

Now, applying the Weibull distribution function, the relation between the amount adsorbed, na, and the differential work of adsorption, ε, is defined by the following relation [33]:

$$n\_{\rm at} = N\_{\rm at} \exp\left(-\frac{\varepsilon}{E}\right)^{n} \tag{10}$$

where E is a parameter termed the characteristic energy of adsorption; meanwhile, Na is the maximum amount adsorbed in the volume of the micropore, n (1 < n < 5) being an empirical parameter. Now combining Eqs. (9) and (10) is feasible to construe the Dubinin adsorption isotherm equation [32] as follows:

$$n\_{a} = N\_{a} \exp\left(-\frac{RT}{E} \ln\left[\frac{P\_{0}}{P}\right]\right)^{n} \tag{11}$$

It is possible, as well, to express the Dubinin adsorption isotherm equation in linear form:

$$
\ln\left(n\_d\right) = \ln\left(N\_d\right) - \left(\frac{RT}{E}\right)^n \ln\left(\frac{P\_0}{P}\right)^n \tag{12}
$$

which is a very powerful tool for the description of the experimental data of adsorption in microporous material.

In Figure 4 is shown the Dubinin plot of N2 adsorption data at 77 K in the pressure range: 0.001 < P/Po < 0.03, in a high silica commercial H-Y zeolite, precisely, the acid Y zeolite labeled CBV-720, manufactured by PQ corporation; where, adsorption data was gathered in an Autosorb-1 automatic volumetric gas adsorption system [26]; been evidently the experimental data correctly fitted by Eq. (12)

Recapitulating, the concrete form to made the linear Dubinin plot as represented in Figure 4 was as follows:

$$y = \ln\left(n\_a\right) = \ln\left(N\_a\right) - \left(\frac{RT}{E}\right)^n \ln\left(\frac{P\_0}{P}\right)^n = b - m\infty$$

This allows the calculation of Va, the so-called dead volume. Nevertheless the really expanded volume should consider the volume occupied by the adsorbent

Vmuestra cm3 <sup>¼</sup> m g½ �

And ρ is the apparent density of the tested adsorbent material. Consequently:

Va � Vmuestra ¼ V<sup>0</sup>

T K½ �

ni � nj ¼ naj which is the amount adsorbed in the tested sample, which plotted against the

The Dubinin adsorption isotherm equation can be inferred by the application of

the Dubinin's Theory of Volume Filling, i.e., (where volume filling is a process which takes place by the filling of the adsorption space rather than the surface coverage), together with, the Polanyi's adsorption potential [32]. In this regard, in 1914, Polanyi created perhaps the first convincing physical adsorption model. Mijail. M. Dubinin, a former pupil of Polanyi applied this model that essentially entailed a link between the adsorption space volume, Vi, and the adsorption energy field, εi, (see Figure 3): i.e.,ε<sup>i</sup> ¼ F Vð Þ<sup>i</sup> , termed by Polanyi as the characteristic function, a temperature independent function. For the deduction of the Dubinin

T K½ �

Pi½ � Atm <sup>2</sup>πr2Xi mm<sup>3</sup> ½ �x10<sup>3</sup> <sup>þ</sup> Vd½ � ml <sup>þ</sup> Pj�1V<sup>0</sup>

� Pj½ � Atm <sup>2</sup>πr2Xj mm3 ½ �x10<sup>3</sup> <sup>þ</sup> Vd½ �þ ml <sup>V</sup><sup>0</sup>

<sup>R</sup> atm:ml mmol:K 

<sup>R</sup> atm:ml mmol:K 

equilibrium pressure Pj provides the adsorption isotherm.

6. The Dubinin adsorption isotherm equation

isotherm equation, the next consideration was that [33]:

ρ <sup>g</sup> cm3 

a

a

a

¼ ni � nj

sample, i.e.:

Applied Surface Science

where:

Figure 3.

76

Polanyi adsorption model.

According ideal gas low:

"osmotic" equilibrium between two solutions, i.e., vacancy plus molecules of different concentrations and the molecules in the gas phase. The solutions are formed in the micropores, and the gas phase, where the solvents are the vacancies, i.e., the vacuum [33] been, these solutions in equilibrium, in the case when one of the

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials

Where the main supposition, made within the frame of this adsorption theory is that the role of the adsorption field could be simulated by; Π.; the so called "osmotic pressure" that is, the role of the energy of adsorption existing within zeolite or related materials channels and cavities, can be replicated by the pressure variation among the adsorbed and gas phases [32]; consequently, considering that the adsorption space is an inert volume; hence, the adsorption effect is produced by a virtual pressure compressing the adsorbed phase in this volume; in this case, we will have only a volume, i.e., a void adsorption space, in which an external pressure, Π took the role of the adsorption field [33]. Therefore, applying the hypothesizes of the osmotic theory of adsorption is feasible to affirm that the volume occupied by the adsorbate, Va, and the vacancies, Vx, or free volume is [29] Va + Vx = V; now in view of the fact that the volume occupied by an adsorbed molecule, b, and a

solutions is submerged in an external field.

DOI: http://dx.doi.org/10.5772/intechopen.83355

vacancy is the same, therefore:

Va b þ

Vx Nab <sup>¼</sup> na Na þ N<sup>x</sup> Na

Va Nab <sup>þ</sup>

osmotic equation being expressed as follows:

na

adsorption isotherm for volume filling:

where <sup>y</sup> <sup>¼</sup> <sup>P</sup><sup>B</sup>, <sup>x</sup> <sup>¼</sup> <sup>P</sup><sup>B</sup>

79

<sup>y</sup> <sup>¼</sup> PB <sup>¼</sup> Na

Vx

Consequently, if we multiply the previous equation by <sup>1</sup>

<sup>b</sup> <sup>¼</sup> na <sup>þ</sup> <sup>N</sup><sup>x</sup> <sup>¼</sup> <sup>V</sup>

<sup>¼</sup> <sup>V</sup>

where Xa and. X<sup>x</sup> correspondingly are the molar fractions of adsorbed molecules and vacancies. Then considering that the adsorption process in a micropore system can be described as an osmotic process; in which vacuum, that is, the vacancies is the solvent, whereas the adsorbed molecules is the solute [3]; hence, using the Osmosis Thermodynamics methodology applied to the above described model, it is

possible to obtain the following adsorption isotherm equation [29, 32, 33]:

na <sup>¼</sup> NaK0PB

PB na 

, m=Na, is the slope and <sup>b</sup> <sup>¼</sup> <sup>1</sup>

necessary to state that for B = 1 the Osmotic isotherm reduces to a Langmuir-type

na <sup>¼</sup> NaK0,LP 1 þ K0,L, P

In Figure 6, the linear plot of Eq. (14) is shown, using B = 0.5, fitting NH3 at 300 K adsorption results in Mg-CMT, i.e., homoionic magnesium natural zeolite,

It is termed the osmotic isotherm of adsorption, or the Sips, or Bradleys isotherm equation, where this isotherm equation, fairly well, describes the experimental data of adsorption in zeolites, and other micoroporous materials; the linear form of the

> þ 1

<sup>b</sup> <sup>¼</sup> Na

Nab <sup>¼</sup> Xa <sup>þ</sup> <sup>X</sup><sup>x</sup> <sup>¼</sup> <sup>1</sup>

Na

<sup>1</sup> <sup>þ</sup> <sup>K</sup>0PB (13)

<sup>K</sup> <sup>¼</sup> mx <sup>þ</sup> <sup>b</sup> (14)

<sup>K</sup> is the intercept. Now it is

, we obtain:

Figure 4. Dubinin plot sample CBV-720 N2 at 77 K [32].

Figure 5.

Dubinin plot of carbon dioxide adsorption on nickel nitroprusside [34].

where <sup>y</sup> <sup>¼</sup> ln ð Þ na , <sup>b</sup> <sup>¼</sup> ln ð Þ Na , <sup>m</sup> <sup>¼</sup> RT E <sup>n</sup> , and <sup>x</sup> <sup>¼</sup> ln <sup>P</sup><sup>0</sup> P <sup>n</sup>

Nevertheless, this fitting process can be as well made using a non-linear regression method; in which, the fitting process is carried out with a program based on a least square procedure, allowing the calculation of the best fitting parameters of the Eq. (12), that is, Na, E, been n, taken as a constant, for example, n = 2; additionally the program compute the regression coefficient, along with the standard errors.

As another example, in Figure 4, the use of the Dubinin equation in the measurement of the micropore volume, of a nickel nitroprusside (Ni-NP), is reported [34](Figure 5).

#### 7. Osmotic and Langmuir-type adsorption isotherms

Within the frame of the osmotic theory of adsorption, the adsorption phenomenon in a microporous adsorbent, for example a zeolite, is considered as the

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials DOI: http://dx.doi.org/10.5772/intechopen.83355

"osmotic" equilibrium between two solutions, i.e., vacancy plus molecules of different concentrations and the molecules in the gas phase. The solutions are formed in the micropores, and the gas phase, where the solvents are the vacancies, i.e., the vacuum [33] been, these solutions in equilibrium, in the case when one of the solutions is submerged in an external field.

Where the main supposition, made within the frame of this adsorption theory is that the role of the adsorption field could be simulated by; Π.; the so called "osmotic pressure" that is, the role of the energy of adsorption existing within zeolite or related materials channels and cavities, can be replicated by the pressure variation among the adsorbed and gas phases [32]; consequently, considering that the adsorption space is an inert volume; hence, the adsorption effect is produced by a virtual pressure compressing the adsorbed phase in this volume; in this case, we will have only a volume, i.e., a void adsorption space, in which an external pressure, Π took the role of the adsorption field [33]. Therefore, applying the hypothesizes of the osmotic theory of adsorption is feasible to affirm that the volume occupied by the adsorbate, Va, and the vacancies, Vx, or free volume is [29] Va + Vx = V; now in view of the fact that the volume occupied by an adsorbed molecule, b, and a vacancy is the same, therefore:

$$\frac{V\_a}{b} + \frac{V\_\chi}{b} = n\_a + N^\chi = \frac{V}{b} = N\_a$$

Consequently, if we multiply the previous equation by <sup>1</sup> Na , we obtain:

$$\frac{V\_a}{N\_a b} + \frac{V\_\chi}{N\_a b} = \frac{n\_a}{N\_a} + \frac{N^\chi}{N\_a} = \frac{V}{N\_a b} = X\_a + X^\chi = 1$$

where Xa and. X<sup>x</sup> correspondingly are the molar fractions of adsorbed molecules and vacancies. Then considering that the adsorption process in a micropore system can be described as an osmotic process; in which vacuum, that is, the vacancies is the solvent, whereas the adsorbed molecules is the solute [3]; hence, using the Osmosis Thermodynamics methodology applied to the above described model, it is possible to obtain the following adsorption isotherm equation [29, 32, 33]:

$$m\_d = \frac{N\_d K\_0 P^B}{1 + K\_0 P^B} \tag{13}$$

It is termed the osmotic isotherm of adsorption, or the Sips, or Bradleys isotherm equation, where this isotherm equation, fairly well, describes the experimental data of adsorption in zeolites, and other micoroporous materials; the linear form of the osmotic equation being expressed as follows:

$$y = P^B = N\_d \left(\frac{P^B}{n\_d}\right) + \frac{1}{K} = m\infty + b \tag{14}$$

where <sup>y</sup> <sup>¼</sup> <sup>P</sup><sup>B</sup>, <sup>x</sup> <sup>¼</sup> <sup>P</sup><sup>B</sup> na , m=Na, is the slope and <sup>b</sup> <sup>¼</sup> <sup>1</sup> <sup>K</sup> is the intercept. Now it is necessary to state that for B = 1 the Osmotic isotherm reduces to a Langmuir-type adsorption isotherm for volume filling:

$$n\_{\boldsymbol{a}} = \frac{N\_{\boldsymbol{a}} K\_{0,L} P}{1 + K\_{0,L} P}$$

In Figure 6, the linear plot of Eq. (14) is shown, using B = 0.5, fitting NH3 at 300 K adsorption results in Mg-CMT, i.e., homoionic magnesium natural zeolite,

where <sup>y</sup> <sup>¼</sup> ln ð Þ na , <sup>b</sup> <sup>¼</sup> ln ð Þ Na , <sup>m</sup> <sup>¼</sup> RT

Dubinin plot of carbon dioxide adsorption on nickel nitroprusside [34].

7. Osmotic and Langmuir-type adsorption isotherms

[34](Figure 5).

78

Figure 4.

Applied Surface Science

Figure 5.

Dubinin plot sample CBV-720 N2 at 77 K [32].

E <sup>n</sup>

Nevertheless, this fitting process can be as well made using a non-linear regression method; in which, the fitting process is carried out with a program based on a least square procedure, allowing the calculation of the best fitting parameters of the Eq. (12), that is, Na, E, been n, taken as a constant, for example, n = 2; additionally the program compute the regression coefficient, along with the standard errors. As another example, in Figure 4, the use of the Dubinin equation in the measurement of the micropore volume, of a nickel nitroprusside (Ni-NP), is reported

Within the frame of the osmotic theory of adsorption, the adsorption phenom-

enon in a microporous adsorbent, for example a zeolite, is considered as the

, and <sup>x</sup> <sup>¼</sup> ln <sup>P</sup><sup>0</sup>

P <sup>n</sup>

Figure 6.

Linear osmotic plot, with B = 0.5 of the adsorption data of NH3 at 300 K in magnesium homoionic CMT zeolite [32].

concretely a blend of mordenite (39 wt. %), clinoptilolite (42 wt. %) along with additional phases (15 wt. %), where these supplementary phases are: montmorillonite (2–10 wt. %), calcite (1–6 wt. %), feldspars (0–1 wt. %), volcanic glass together with quartz (1–5 wt. %). These results were measured in a Pyrex glass volumetric adsorption vacuum system, consisting of sample holder, dead volume, dose volume, U-tube manometer, and thermostat [32]; this plot allowed us to calculate the maximum adsorption capacity of this zeolite, which is m=Na <sup>=</sup> 5.07 mmol/g, and <sup>b</sup> <sup>¼</sup> <sup>1</sup> <sup>K</sup> ¼ �0:92, [(Torr)0.5]; as a conclusion, it is possible to affirm that the experimental data are correctly fitted by Eq. (14).

The fitting process of the osmotic isotherm equation could be also carried out with the help of a non-linear regression method, where the fitting process allows us to calculate the best fitting parameters of the Eq. (13), i.e., Na, K0, and B, if this parameter is not taken as a constant, for example, B = 1; besides the program calculates the regression coefficient and the standard errors.

Further adsorption isotherms are reported below [35] (Figures 7 and 8):

Additionally, the adsorption of oxygen (O2) and nitrogen (N2) in modified natural mordenite from the Palmarito, Santiago de Cuba, Cuba, deposit, composed of mordenite (80 wt.%), clinoptilolite (5 wt.%), and other phases (15 wt.%), where the other phases included montmorillonite (2–10 wt.%), quartz (1–5 wt.%), calcite (1–6 wt.%), feldspars (0–1 wt.%), and volcanic glass [2]. Labeled MP are reported in Table 1, where H means acid; NH, ammonia; Li, Lithium; Na, Sodium; K, Potassium; Mg, Magnesium; Ca, Calcium; Sr., Strontium; Ba, Barium zeolite; where, the oxygen nitrogen selectivity is given by [36].

$$a = Y\_a X\_a / Y\_g X\_g$$

along with the following composition: Na8Al8Si40 O96 � 24H2O; composed fundamentally of a channel system composed of a 5-membered ring system parallel to [001], having a free diameter of 6.6 Å, interconnected by smaller channels, parallel to [010], of 2.8 Å free diameter. However, the existence of stacking faults in the framework reduces the effective diameter of the channels to about 4 Å [37]. Moreover, the volume of the unit cell is 2794 ̊A3, while the kinetic diameters of nitrogen and oxygen are dN ¼ 3:6 Å and dO ¼ 3:5 Å. On the other hand, the quadrupole interaction (Q) of nitrogen is larger than that of oxygen, i.e., QN ¼ 0.31 Debye > QO ¼ 0:10 Debye; meanwhile, the polarizability (P) of nitrogen is also bigger than that of oxygen, that is, PN ¼ 4:31 ̊A3>PO ¼ 3:96 ̊A3, facts that explain the higher values measured for the Langmuir constants in the case of nitrogen in comparison

Maximum adsorption oxygen (NMAX.OXIGENO) and nitrogen (NMAX.NITROGENO), Langmuir constants (KOXYGEN and KNITROGEN) and oxygen- nitrogen selectivity (αNITROG- OXIG) on modified natural mordenite

The application of the grand canonical ensemble (GCE) allows us to handle the adsorption process in microporous materials such as zeolites and related materials. In this case, the whole zeolite is considered a GCE, i.e., the zeolite cavities or channels are considered in the frame of this model as independent open systems constituting the ensemble, additionally the adsorption field within the cavities is energetically homogeneous, i.e., the adsorption field is the same at any site within

with those measured for oxygen adsorption [36].

Figure 8.

Table 1.

MP [36].

81

Ethane adsorption on zeolite 13 X (Na-FAU) at 291 K.

DOI: http://dx.doi.org/10.5772/intechopen.83355

NMAX.NITROGEN (mmol/g)

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials

H-MP 0.24 0.5 1.3 2.3 3.7 NH-MP 0.10 0.25 0.12 2.0 42 Li-MP 0.21 0.86 1.8 3.0 6.8 Na-MP 0.29 0.63 9.3 2.0 4.7 K-MP 0.07 0.26 3.1 3.7 4.4 Mg-MP 0.13 0.56 0.21 3.9 84 Ca-MP 0.27 0.56 1.4 4.8 7.1 Sr-MP 0.32 0.59 1.2 4.6 7.1 Ba-MP 0.23 0.75 1.5 1.9 4.1

KOXYGEN (Torr)�<sup>1</sup>

�10�<sup>3</sup>

KNITROGEN (Torr)�11�10�<sup>3</sup> αNITROG- OXIG

(mmol/g)

ZEOLITES NMAX.OXYGEN

where Xa and Ya are the molar fractions of the adsorbates in the adsorbed phase, while Xg and Yg are the molar fractions of both adsorbates in the gas phase.

It is necessary now to state that the framework of mordenite shows an orthorhombic unit cell displaying the space group Cmcm or Cmc21; in this sese, the unit cell of the Na cationic form has dimensions a = 18.13 Å, b = 20.49 Å, and c = 7.52 Å,

Figure 7. Nitrogen adsorption on zeolite 5 a (Ca, Na-LTA) at 291 K.

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials DOI: http://dx.doi.org/10.5772/intechopen.83355

Figure 8.

concretely a blend of mordenite (39 wt. %), clinoptilolite (42 wt. %) along with additional phases (15 wt. %), where these supplementary phases are: montmorillonite (2–10 wt. %), calcite (1–6 wt. %), feldspars (0–1 wt. %),

to affirm that the experimental data are correctly fitted by Eq. (14).

calculates the regression coefficient and the standard errors.

where, the oxygen nitrogen selectivity is given by [36].

Nitrogen adsorption on zeolite 5 a (Ca, Na-LTA) at 291 K.

m=Na <sup>=</sup> 5.07 mmol/g, and <sup>b</sup> <sup>¼</sup> <sup>1</sup>

Figure 6.

Applied Surface Science

Figure 7.

80

[32].

volcanic glass together with quartz (1–5 wt. %). These results were measured in a Pyrex glass volumetric adsorption vacuum system, consisting of sample holder, dead volume, dose volume, U-tube manometer, and thermostat [32]; this plot allowed us to calculate the maximum adsorption capacity of this zeolite, which is

Linear osmotic plot, with B = 0.5 of the adsorption data of NH3 at 300 K in magnesium homoionic CMT zeolite

The fitting process of the osmotic isotherm equation could be also carried out with the help of a non-linear regression method, where the fitting process allows us to calculate the best fitting parameters of the Eq. (13), i.e., Na, K0, and B, if this parameter is not taken as a constant, for example, B = 1; besides the program

Further adsorption isotherms are reported below [35] (Figures 7 and 8): Additionally, the adsorption of oxygen (O2) and nitrogen (N2) in modified natural mordenite from the Palmarito, Santiago de Cuba, Cuba, deposit, composed of mordenite (80 wt.%), clinoptilolite (5 wt.%), and other phases (15 wt.%), where the other phases included montmorillonite (2–10 wt.%), quartz (1–5 wt.%), calcite (1–6 wt.%), feldspars (0–1 wt.%), and volcanic glass [2]. Labeled MP are reported in Table 1, where H means acid; NH, ammonia; Li, Lithium; Na, Sodium; K, Potassium; Mg, Magnesium; Ca, Calcium; Sr., Strontium; Ba, Barium zeolite;

α ¼ YaXa=YgXg

It is necessary now to state that the framework of mordenite shows an orthorhombic unit cell displaying the space group Cmcm or Cmc21; in this sese, the unit cell of the Na cationic form has dimensions a = 18.13 Å, b = 20.49 Å, and c = 7.52 Å,

while Xg and Yg are the molar fractions of both adsorbates in the gas phase.

where Xa and Ya are the molar fractions of the adsorbates in the adsorbed phase,

<sup>K</sup> ¼ �0:92, [(Torr)0.5]; as a conclusion, it is possible

Ethane adsorption on zeolite 13 X (Na-FAU) at 291 K.


#### Table 1.

Maximum adsorption oxygen (NMAX.OXIGENO) and nitrogen (NMAX.NITROGENO), Langmuir constants (KOXYGEN and KNITROGEN) and oxygen- nitrogen selectivity (αNITROG- OXIG) on modified natural mordenite MP [36].

along with the following composition: Na8Al8Si40 O96 � 24H2O; composed fundamentally of a channel system composed of a 5-membered ring system parallel to [001], having a free diameter of 6.6 Å, interconnected by smaller channels, parallel to [010], of 2.8 Å free diameter. However, the existence of stacking faults in the framework reduces the effective diameter of the channels to about 4 Å [37]. Moreover, the volume of the unit cell is 2794 ̊A3, while the kinetic diameters of nitrogen and oxygen are dN ¼ 3:6 Å and dO ¼ 3:5 Å. On the other hand, the quadrupole interaction (Q) of nitrogen is larger than that of oxygen, i.e., QN ¼ 0.31 Debye > QO ¼ 0:10 Debye; meanwhile, the polarizability (P) of nitrogen is also bigger than that of oxygen, that is, PN ¼ 4:31 ̊A3>PO ¼ 3:96 ̊A3, facts that explain the higher values measured for the Langmuir constants in the case of nitrogen in comparison with those measured for oxygen adsorption [36].

The application of the grand canonical ensemble (GCE) allows us to handle the adsorption process in microporous materials such as zeolites and related materials. In this case, the whole zeolite is considered a GCE, i.e., the zeolite cavities or channels are considered in the frame of this model as independent open systems constituting the ensemble, additionally the adsorption field within the cavities is energetically homogeneous, i.e., the adsorption field is the same at any site within

the adsorption space; besides, each cage can accommodatem ¼ w=b, molecules, where w and b are the volumes of the cavity and the adsorbed molecule, respectively. Hence if the ensemble is constituted by M independent open cavities, i.e., systems, the grand canonical partition function of the zeolite is given by [32]:

where

$$\boldsymbol{\Theta} = \left[\mathbf{1} + \lambda \mathbf{Z}(\mathbf{1}) + \lambda^2 \mathbf{Z}(\mathbf{2}) + \dots + \lambda^n \mathbf{Z}(n)\right]^M = \overline{\mathbf{Z}}^M \tag{15}$$

$$\overline{Z} = \sum\_{N=0}^{m} \lambda^N Z(N) \tag{16}$$

Equation (19) is of the Fowler-Guggenheim type (FGT), but describing volume filling rather than surface covering, where both equations reduce to Langmuir-type (LT) isotherm equations, as well describing volume filling. Moreover, the osmotic isotherm equation and the FGT types have the same mathematical form in the case where B = 1 in the osmotic equation, as well as are equivalent to the Langmuir T

To test these isotherm types, the FGT and LT type equations can be written as

¼ LnK þ

While the linear form of the LT type isotherms is P ¼ Nað Þþ P=na 1=K ¼ ax þ b. In Figure 9, the plot of carbon dioxide adsorption on Ni-NP at 273 and 300 K is

The t-plot method suppose that the adsorbed phase as a liquid adhered film over

na ¼ ρLt

ing a model similar to that proposed by Polanyi, where it is considered that the entropy contribution to the free energy is small in comparison with the change of

P0

<sup>z</sup><sup>9</sup> � <sup>B</sup>

μ � μ<sup>L</sup> ¼ V zð Þ (22)

. Hence, supposing now that [38]:

zm. Thereafter, follow-

Attraction by the adsorption field is given by V zð Þ¼ <sup>A</sup>

enthalpy, we have <sup>μ</sup> � <sup>μ</sup><sup>L</sup> <sup>¼</sup> RT ln <sup>P</sup>

the solid surface, the model was proposed by Halsey, De Boer, and coworkers following ideas previously proposed by Frenkel-Halsey-&-Hill which stated that it is possible to calculate, t, the width in (Angstrom) of the adsorbed layer, or multilayer thickness [32] and plot it as a function of: x = P/Po. The methodology is effective for a multilayer adsorption; in which, the surface liquid film is supposed to show a unvarying width, t, density equal to the bulk liquid adsorbate, ρL, and be in contact with a uniform surface that produces an attraction adsorption field over the solid surface; hence, based on the aforementioned arguments, the adsorbed amount

kθ

RT (21)

ln <sup>θ</sup> 1 � θ 

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials

type equations, when Ω ¼ 0; Φ ¼ 0.

Carbon dioxide adsorption on Ni-NP at 273 and 300 K [34].

DOI: http://dx.doi.org/10.5772/intechopen.83355

follows [2]:

Figure 9.

reported [34].

will be [38]:

83

8. t-plot method

Is the channel or cavity grand canonical partition function; been, Z(N) the Canonical Partition Function for N molecules in the channel or cavity (0 < N < m), while the absolute activity is given by λ ¼ exp ð Þ μ=RT while, μ is the Chemical Potential. Thereafter, as:

$$\overline{N} = \frac{\partial \ln \Theta}{\partial \lambda} = RT \left(\frac{\partial \ln \Theta}{\partial \mu}\right) \tag{17}$$

$$\theta = \frac{\overline{N}}{mM} = \frac{M \overline{N} \,\overline{N}}{Mm \, m}$$

Now it is necessary to state that within the frame of the adsorption process in zeolites and related materials two cases are possible; that is, inmobile (I) or mobile (M) adsorption, been in the first and second cases, the canonical partition functions for the inmobile (Zi) and mobile (ZM) cases, in a homogeneous field without lateral interactions between the adsorbed molecules for N < m given by:

$$Z\_{l}(N) = \frac{m!}{N!(m-N)!} \left(Z\_{a}^{l}\right)^{N} \exp\left(-\frac{N\left(E\_{0}^{d} + \eta E\_{i}\right)}{RT}\right) \tag{18}$$

$$Z\_{M}(N) = \frac{\omega^{N}}{N!} \left(\Lambda\right)^{N} \left(Z\_{a}^{l}\right)^{N} \exp\left(-N\left[\frac{\left(E\_{0} + a(N/w)\_{i}\right)}{RT}\right]\right)$$

Now since:

$$\overline{N} = \frac{\partial \ln \overline{Z}}{\partial \lambda} = \lambda \left(\frac{\partial \ln \overline{Z}}{\partial \lambda} = \frac{A}{B}\right).$$

Consequently:

$$\theta = \frac{\overline{N}}{m} = \frac{K\_I P}{1 + K\_I P} \tag{19}$$

$$\theta = \frac{\overline{N}}{m} = \frac{K\_M P}{1 + K\_{MI} P}$$

With:

$$K\_{I} = \left[\frac{Z^{j}}{Z\_{\text{g}}^{i}}\right] \left(\frac{1}{RT\Lambda}\right) \exp\left\{\frac{\left[E\_{0}^{\text{g}} - E\_{0}^{\text{a}}\right] + \Omega\theta}{RT}\right\} \tag{20}$$

$$K\_{M} = \left[\frac{Z^{j}}{Z\_{\text{g}}^{i}}\right] \left(\frac{b}{RT}\right) \exp\left\{\frac{\left[E\_{0}^{\text{g}} - E\_{0}^{\text{a}}\right] + \Phi\theta}{RT}\right\}$$

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials DOI: http://dx.doi.org/10.5772/intechopen.83355

Figure 9. Carbon dioxide adsorption on Ni-NP at 273 and 300 K [34].

Equation (19) is of the Fowler-Guggenheim type (FGT), but describing volume filling rather than surface covering, where both equations reduce to Langmuir-type (LT) isotherm equations, as well describing volume filling. Moreover, the osmotic isotherm equation and the FGT types have the same mathematical form in the case where B = 1 in the osmotic equation, as well as are equivalent to the Langmuir T type equations, when Ω ¼ 0; Φ ¼ 0.

To test these isotherm types, the FGT and LT type equations can be written as follows [2]:

$$\ln\left(\frac{\theta}{1-\theta}\right) = LnK + \frac{k\theta}{RT} \tag{21}$$

While the linear form of the LT type isotherms is P ¼ Nað Þþ P=na 1=K ¼ ax þ b. In Figure 9, the plot of carbon dioxide adsorption on Ni-NP at 273 and 300 K is reported [34].
