8. t-plot method

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]:

> Z ¼ ∑ m N¼0

<sup>N</sup> <sup>¼</sup> <sup>∂</sup> ln <sup>Θ</sup>

interactions between the adsorbed molecules for N < m given by:

<sup>N</sup>!ð Þ <sup>m</sup> � <sup>N</sup> ! <sup>Z</sup><sup>j</sup>

ð Þ <sup>Λ</sup> <sup>N</sup> <sup>Z</sup><sup>j</sup> a � �<sup>N</sup>

<sup>N</sup> <sup>¼</sup> <sup>∂</sup> ln <sup>Z</sup>

<sup>θ</sup> <sup>¼</sup> <sup>N</sup>

<sup>θ</sup> <sup>¼</sup> <sup>N</sup>

1 RTΛ � �

> b RT � �

ZIð Þ¼ <sup>N</sup> <sup>m</sup>!

N!

KI <sup>¼</sup> <sup>Z</sup><sup>j</sup> Zj g

KM <sup>¼</sup> <sup>Z</sup><sup>j</sup> Zj g

" #

" #

ZMð Þ¼ <sup>N</sup> <sup>w</sup><sup>N</sup>

<sup>θ</sup> <sup>¼</sup> <sup>N</sup>

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

<sup>∂</sup><sup>λ</sup> <sup>¼</sup> RT <sup>∂</sup> ln <sup>Θ</sup>

mM <sup>¼</sup> MN Mm N 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

> a � �<sup>N</sup>

<sup>∂</sup><sup>λ</sup> <sup>¼</sup> <sup>λ</sup> <sup>∂</sup> ln <sup>Z</sup>

<sup>m</sup> <sup>¼</sup> KIP

<sup>m</sup> <sup>¼</sup> KMP 1 þ KMIP

exp

exp

Eg <sup>0</sup> � Ea 0 � � <sup>þ</sup> <sup>Ω</sup><sup>θ</sup> RT � �

Eg <sup>0</sup> � Ea 0 � � <sup>þ</sup> <sup>Φ</sup><sup>θ</sup> RT � �

exp �N

∂μ � �

exp � N Ea

<sup>∂</sup><sup>λ</sup> <sup>¼</sup> <sup>A</sup> B

� �

<sup>0</sup> þ ηEi � � RT � �

E<sup>0</sup> þ αð Þ N=w <sup>i</sup> � � RT � � �

<sup>1</sup> <sup>þ</sup> KIP (19)

<sup>Z</sup>ð Þþ <sup>2</sup> :…… <sup>þ</sup> <sup>λ</sup>nZ nð Þ � �<sup>M</sup> <sup>¼</sup> <sup>Z</sup><sup>M</sup> (15)

<sup>λ</sup>NZ Nð Þ (16)

(17)

(18)

(20)

<sup>Θ</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>λ</sup>Zð Þþ <sup>1</sup> <sup>λ</sup><sup>2</sup>

where

Applied Surface Science

Potential. Thereafter, as:

Now since:

Consequently:

With:

82

The t-plot method suppose that the adsorbed phase as a liquid adhered film over 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 will be [38]:

$$n\_a = \rho\_L t$$

Attraction by the adsorption field is given by V zð Þ¼ <sup>A</sup> <sup>z</sup><sup>9</sup> � <sup>B</sup> zm. Thereafter, following 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 enthalpy, we have <sup>μ</sup> � <sup>μ</sup><sup>L</sup> <sup>¼</sup> RT ln <sup>P</sup> P0 . Hence, supposing now that [38]:

$$
\mu - \mu\_L = V(\mathbf{z}) \tag{22}
$$

Subsequently, since the adsorption process considered in the present model is a multilayer one, then V zð Þ<sup>≈</sup> � <sup>B</sup> zm and consequently RT ln <sup>P</sup> P0 � � ¼ � <sup>B</sup> zm ¼ � <sup>C</sup> tm; hence, the thickness could be evaluated after normalizing an adsorption isotherm for an adsorbent that does not possess micropores, or mesopores. Subsequently, the multiplayer thickness, t, can be calculated by the following relation:

$$t = \frac{n\_a}{N\_m} d\_0$$

where d0 is the thickness of a monolayer.

Thereafter, supposing again, that the surface liquid film is assumed of uniform width, t, along with having a density equal to the bulk liquid adsorbate, ρL, hence, we have [32]:

$$d\_0 = \frac{M}{\sigma N\_A \rho\_L}$$

where NA is the number of Avogado and σ is the cross-sectional area, that is, the normal area that each molecule occupies in a completed monolayer. For instance, if σ(N2) = 0.162 nm<sup>2</sup> for N2 at 77 K, M(N2) = 28.1 g/mol, and ρL(N2) = 0.809 g/cm<sup>3</sup> then, d0 = 0.354 nm.

The following relation t versus, (1/x), is used between others to carry out the t-plot [32, 38]:

$$t = 3.54 \left( \frac{5}{2.303 \log \left( \frac{P\_0}{P} \right)} \right)^{1/3} \tag{23}$$

is as well used for the application of the t-plot methodology. Now it is necessary

/g)] the outer surface applying the t–plot method is as follows: after the elimi-

y ¼ na ¼ Rt þ Na ¼ mx þ b The linear regression is made, then the intercept, b=Na, and the slope, m=R, are calculated, given that the intercept is related to the micropore volume, WMP

is the molar volume of the adsorptive at the adsorption temperature,T, [37] is then calculated micropore volume; <sup>W</sup>MP <sup>¼</sup> <sup>116</sup>:1x32:<sup>565</sup> <sup>¼</sup> <sup>5</sup>:17mmol=gL; together with,

Adsorption is a general tendency of matter, and during its occurrence, a decrease in the surface tension is experienced by the solid. For this reason, adsorption is a spontaneous process, where the Gibbs free energy decreases, i.e., ΔG < 0. Besides, in the course of physical adsorption, molecules from a chaotic bulk phase are transferred to a relatively ordered adsorbed state, since in the adsorbed state molecules can only move within the surface or a pore; consequently, in the course of adsorption by the entire system, a reduction of entropy takes place, i.e., ΔS < 0;

ΔG ¼ ΔH � TΔS

ΔG ¼ ΔH � TΔS < 0

As a matter of fact, the main thermodynamic relation for a bulk mixture system

i

μidni (25)

Consequently, the adsorption process releases heat, i.e., it is an exothermic process. Consequently, it is stimulated by a reduction of the adsorption experiment

dU ¼ TdS � PdV þ ∑

where S is their entropy, U is the internal energy, V is the volume, T is the temperature, μ<sup>i</sup> is the chemical potential, and ni is the constituent number of moles included in the system [18]. Now it is supposed that the adsorbent together with the adsorbed gas is a solid solution, labeled, system aA; thereafter, using the proposed scheme, it is possible to deduce the following equation, which thermodynamically

dUaA ¼ TdSaA � PdVaA þ μAdnA þ μadna where UaA, SaA, and VaA are the internal energy, entropy, and volume of the

system aA, respectively, and μ<sup>a</sup> and μ<sup>A</sup> are the chemical potentials of the

or

<sup>M</sup> <sup>¼</sup> <sup>32</sup>:565 (cm<sup>3</sup>

/g)] and S [in

/

;

mol)

(24)

<sup>M</sup> =32.565 cm<sup>3</sup>

to state that the molar volume of liquid nitrogen at 77 K is <sup>V</sup>ð Þ <sup>77</sup><sup>K</sup> <sup>N</sup><sup>2</sup>

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials

The method to calculate, W, the microporous volume [in (cm3

thereafter, using the Gurvich rule <sup>W</sup>MP <sup>¼</sup> NaVL, where VL, (Vð Þ <sup>77</sup><sup>K</sup> <sup>N</sup><sup>2</sup>

the outer surface, <sup>S</sup> <sup>¼</sup> RVL <sup>¼</sup> <sup>63</sup>:08x32:<sup>565</sup> <sup>¼</sup> <sup>2</sup>, <sup>054</sup>m<sup>2</sup>=<sup>g</sup> [39].

9. Thermodynamics of adsorption

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

subsequently [2, 32]:

temperature.

is given by:

85

describes the aA system [2, 32]:

nation of the points that do not fit a linear plot (Figure 10):

mol) [37].

(m<sup>2</sup>

which is the Halsey equation, valid for N2 at 77 K or the equation used by De Boer. Moreover, the following expression:

$$t = \left(\frac{13.9}{0.034 + \log\left(\frac{p\_0}{P}\right)}\right)^{1/2}$$

Figure 10. t-plot for the adsorption of N<sup>2</sup> at 77 K in a high specific surface silica labeled 70bs-2-25C [39].

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

is as well used for the application of the t-plot methodology. Now it is necessary to state that the molar volume of liquid nitrogen at 77 K is <sup>V</sup>ð Þ <sup>77</sup><sup>K</sup> <sup>N</sup><sup>2</sup> <sup>M</sup> <sup>¼</sup> <sup>32</sup>:565 (cm<sup>3</sup> / mol) [37].

The method to calculate, W, the microporous volume [in (cm3 /g)] and S [in (m<sup>2</sup> /g)] the outer surface applying the t–plot method is as follows: after the elimination of the points that do not fit a linear plot (Figure 10):

$$y = n\_a = Rt + N\_a = m\mathbf{x} + b$$

The linear regression is made, then the intercept, b=Na, and the slope, m=R, are calculated, given that the intercept is related to the micropore volume, WMP ; thereafter, using the Gurvich rule <sup>W</sup>MP <sup>¼</sup> NaVL, where VL, (Vð Þ <sup>77</sup><sup>K</sup> <sup>N</sup><sup>2</sup> <sup>M</sup> =32.565 cm<sup>3</sup> mol) is the molar volume of the adsorptive at the adsorption temperature,T, [37] is then calculated micropore volume; <sup>W</sup>MP <sup>¼</sup> <sup>116</sup>:1x32:<sup>565</sup> <sup>¼</sup> <sup>5</sup>:17mmol=gL; together with, the outer surface, <sup>S</sup> <sup>¼</sup> RVL <sup>¼</sup> <sup>63</sup>:08x32:<sup>565</sup> <sup>¼</sup> <sup>2</sup>, <sup>054</sup>m<sup>2</sup>=<sup>g</sup> [39].

## 9. Thermodynamics of adsorption

Subsequently, since the adsorption process considered in the present model is a

the thickness could be evaluated after normalizing an adsorption isotherm for an adsorbent that does not possess micropores, or mesopores. Subsequently, the

> <sup>t</sup> <sup>¼</sup> na Nm d0

Thereafter, supposing again, that the surface liquid film is assumed of uniform width, t, along with having a density equal to the bulk liquid adsorbate, ρL, hence,

> <sup>d</sup><sup>0</sup> <sup>¼</sup> <sup>M</sup> σNAρ<sup>L</sup>

where NA is the number of Avogado and σ is the cross-sectional area, that is, the normal area that each molecule occupies in a completed monolayer. For instance, if σ(N2) = 0.162 nm<sup>2</sup> for N2 at 77 K, M(N2) = 28.1 g/mol, and ρL(N2) = 0.809 g/cm<sup>3</sup>

The following relation t versus, (1/x), is used between others to carry out the t-plot

2:303 log <sup>P</sup><sup>0</sup>

which is the Halsey equation, valid for N2 at 77 K or the equation used by De

<sup>0</sup>:<sup>034</sup> <sup>þ</sup> log <sup>P</sup><sup>0</sup>

!<sup>1</sup>=<sup>2</sup>

!<sup>1</sup>=<sup>3</sup>

P � �

P � �

<sup>t</sup> <sup>¼</sup> <sup>3</sup>:<sup>54</sup> <sup>5</sup>

<sup>t</sup> <sup>¼</sup> <sup>13</sup>:<sup>9</sup>

t-plot for the adsorption of N<sup>2</sup> at 77 K in a high specific surface silica labeled 70bs-2-25C [39].

multiplayer thickness, t, can be calculated by the following relation:

zm and consequently RT ln <sup>P</sup>

P0 � �

¼ � <sup>B</sup>

zm ¼ � <sup>C</sup>

tm; hence,

(23)

multilayer one, then V zð Þ<sup>≈</sup> � <sup>B</sup>

Applied Surface Science

we have [32]:

then, d0 = 0.354 nm.

[32, 38]:

Figure 10.

84

where d0 is the thickness of a monolayer.

Boer. Moreover, the following expression:

Adsorption is a general tendency of matter, and during its occurrence, a decrease in the surface tension is experienced by the solid. For this reason, adsorption is a spontaneous process, where the Gibbs free energy decreases, i.e., ΔG < 0. Besides, in the course of physical adsorption, molecules from a chaotic bulk phase are transferred to a relatively ordered adsorbed state, since in the adsorbed state molecules can only move within the surface or a pore; consequently, in the course of adsorption by the entire system, a reduction of entropy takes place, i.e., ΔS < 0; subsequently [2, 32]:

$$\begin{aligned} \Delta G &= \Delta H - T\Delta S \\ \text{or} \\ \Delta G &= \Delta H - T\Delta S \circ \mathbf{0} \end{aligned} \tag{24}$$

Consequently, the adsorption process releases heat, i.e., it is an exothermic process. Consequently, it is stimulated by a reduction of the adsorption experiment temperature.

As a matter of fact, the main thermodynamic relation for a bulk mixture system is given by:

$$d\mathbf{U} = \mathbf{TdS} - \mathbf{PdV} + \sum\_{i} \mu\_{i} dn\_{i} \tag{25}$$

where S is their entropy, U is the internal energy, V is the volume, T is the temperature, μ<sup>i</sup> is the chemical potential, and ni is the constituent number of moles included in the system [18]. Now it is supposed that the adsorbent together with the adsorbed gas is a solid solution, labeled, system aA; thereafter, using the proposed scheme, it is possible to deduce the following equation, which thermodynamically describes the aA system [2, 32]:

$$dU\_{aA} = TdS\_{aA} - PdV\_{aA} + \\\mu\_A dn\_A + \\\mu\_a dn\_a$$

where UaA, SaA, and VaA are the internal energy, entropy, and volume of the system aA, respectively, and μ<sup>a</sup> and μ<sup>A</sup> are the chemical potentials of the

adsorbate, a, and the adsorbent, A, while na and nA are the number of moles of the adsorbate and the adsorbent in the system aA, respectively. Now if we define Γ ¼ na=nA; then:

$$
\mu\_a = \mu\_a(T, P, \Gamma) \text{ and } \mu\_A = \mu\_A(T, P) \tag{26}
$$

In Figure 12, the measurement of the differential heat of adsorption for the adsorption of carbon dioxide in the natural mordenite labeled MP is reported, which is a mordenite from the Palmarito, Santiago de Cuba, Cuba, deposit (mordenite (80 wt.%), clinoptilolite (5 wt.%), together with montmorillonite (2–10 wt.%), quartz (1–5 wt.%), calcite (1–6 wt.%), feldspars (0–1 wt.%), and volcanic glass) [2]. In Figure 12, the obtained data are shown indicating that CO2 adsorption process in this zeolite is energetically heterogeneous; i.e., the heat of adsorption is a diminishing function of the zeolite micropore volume recovery, i.e., θ = na/nmax. That is, the plot of qdiff versus θ shows the following: two steps, one at 90 kJ/mol and the other at 70 kJ/mol, where was released, moderately high values of the adsorption heats, indicating that CO2 molecules powerfully interact through their quadrupole moments with the mordenite framework; after that, a reduction of

Synthesis, Characterization, and Adsorption Properties of Nanoporous Materials

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

Figure 12.

87

Figure 11.

Adsorption volumetric equipment.

Plot of qdiff vs. θ for the adsorption of CO2 at 300 K in the MP zeolite [2].

As a result:

$$
\left[\frac{d\ln P}{dT}\right]\_{\Gamma} = \frac{\overline{H}\_{\text{g}} - \overline{H}\_{a}}{RT^{2}} = \frac{q\_{iso}}{RT^{2}}\tag{27}
$$

where Hg and Ha are the partial molar enthalpies of the adsorbate in the gas phase and in the aA system, respectively; now, applying Eq. (16), it is possible to define the isosteric enthalpy of adsorption [31]

$$q\_{\rm iso} = \overline{H}\_{\rm g} - \overline{H}\_{\rm a} \tag{28}$$

where qiso is the enthalpy of desorption, or the isosteric heat of adsorption, which is calculated with the help of adsorption isotherms. An additional significant adsorption heat is the differential heat of adsorption, since when an adsorbate contacts an adsorbent, heat is released. In our case, the thermal effect produced was measured with the help of a thermocouple placed inside the adsorbent and referred at room temperature (Figure 11), a variety of the Tian–Calvet heat-flow calorimeter [32]. This calorimetric methodology is characterized by the fact that the difference of temperature among the tested adsorbent and a thermostat is determined; therefore, in this heat-flow calorimeter, the produced thermal energy in the adsorption cell is permitted to flow with no limitations to the thermostat. In the calorimeter constructed by us, heat flows throughout a thermocouple; thereafter, the voltage produced by the thermocouple, which is proportional to the thermal power, is amplified and recorded in an x-y plotter (see Figure 11); in which, the actual thermal effect generated is the integral heat of adsorption, measured using the Eq. (2)

$$
\Delta Q = k \int\_0^{t\_{\text{max}}} \Delta T dt \tag{29}
$$

where ΔQ is the integral heat of adsorption released during the finite increment, κ is a calibration constant, ΔT is the difference between thermostat temperature and the sample temperature during adsorption, and t is time, the differential heat of adsorption being calculated as follows [32, 33]:

$$q\_{d\overline{\mathcal{J}}\overline{\mathcal{J}}} = \frac{\Delta Q}{\Delta n\_a} = \frac{k \int\_0^{t\_{\text{max}}} \Delta T dt}{\Delta n\_a}$$

The heat-flow calorimeter used consisted of the high vacuum line for adsorption measurements applying the volumetric method; as reported in Figure 11, the equipment comprises the following: a Pyrex glass, vacuum system including a sample holder, a dead volume, a dose volume, a U-tube manometer, and a thermostat, including now a thermocouple immersed in the adsorbent bed, which was coupled to an x-y plotter.

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

In Figure 12, the measurement of the differential heat of adsorption for the adsorption of carbon dioxide in the natural mordenite labeled MP is reported, which is a mordenite from the Palmarito, Santiago de Cuba, Cuba, deposit (mordenite (80 wt.%), clinoptilolite (5 wt.%), together with montmorillonite (2–10 wt.%), quartz (1–5 wt.%), calcite (1–6 wt.%), feldspars (0–1 wt.%), and volcanic glass) [2]. In Figure 12, the obtained data are shown indicating that CO2 adsorption process in this zeolite is energetically heterogeneous; i.e., the heat of adsorption is a diminishing function of the zeolite micropore volume recovery, i.e., θ = na/nmax. That is, the plot of qdiff versus θ shows the following: two steps, one at 90 kJ/mol and the other at 70 kJ/mol, where was released, moderately high values of the adsorption heats, indicating that CO2 molecules powerfully interact through their quadrupole moments with the mordenite framework; after that, a reduction of

Figure 11. Adsorption volumetric equipment.

adsorbate, a, and the adsorbent, A, while na and nA are the number of moles of the adsorbate and the adsorbent in the system aA, respectively. Now if we define

<sup>¼</sup> Hg � Ha

where Hg and Ha are the partial molar enthalpies of the adsorbate in the gas phase and in the aA system, respectively; now, applying Eq. (16), it is possible to

where qiso is the enthalpy of desorption, or the isosteric heat of adsorption, which is calculated with the help of adsorption isotherms. An additional significant adsorption heat is the differential heat of adsorption, since when an adsorbate contacts an adsorbent, heat is released. In our case, the thermal effect produced was measured with the help of a thermocouple placed inside the adsorbent and referred at room temperature (Figure 11), a variety of the Tian–Calvet heat-flow calorimeter [32]. This calorimetric methodology is characterized by the fact that the difference of temperature among the tested adsorbent and a thermostat is determined; therefore, in this heat-flow calorimeter, the produced thermal energy in the adsorption cell is permitted to flow with no limitations to the thermostat. In the calorimeter constructed by us, heat flows throughout a thermocouple; thereafter, the voltage produced by the thermocouple, which is proportional to the thermal power, is amplified and recorded in an x-y plotter (see Figure 11); in which, the actual thermal effect generated is the integral heat of adsorption, measured using

ΔQ ¼ k

qdiff <sup>¼</sup> <sup>Δ</sup><sup>Q</sup> Δna ¼

tmax ð

0

where ΔQ is the integral heat of adsorption released during the finite increment, κ is a calibration constant, ΔT is the difference between thermostat temperature and the sample temperature during adsorption, and t is time, the differential heat of

> <sup>k</sup> <sup>Ð</sup><sup>t</sup>max 0

The heat-flow calorimeter used consisted of the high vacuum line for adsorption

measurements applying the volumetric method; as reported in Figure 11, the equipment comprises the following: a Pyrex glass, vacuum system including a sample holder, a dead volume, a dose volume, a U-tube manometer, and a thermostat, including now a thermocouple immersed in the adsorbent bed, which was

ΔTdt

Δna

RT<sup>2</sup> <sup>¼</sup> qiso

d ln P dT � �

define the isosteric enthalpy of adsorption [31]

adsorption being calculated as follows [32, 33]:

coupled to an x-y plotter.

86

Γ

μ<sup>a</sup> ¼ μað Þ T; P; Γ and μ<sup>A</sup> ¼ μAð Þ T; P (26)

qiso ¼ Hg � Ha (28)

ΔTdt (29)

RT<sup>2</sup> (27)

Γ ¼ na=nA; then:

Applied Surface Science

As a result:

the Eq. (2)

the adsorption heat is found, up to a value corresponding to the bulk heat of condensation of the CO2 molecules [26–28].
