## 2 Modified Porosity of Smectite as a Function of Temperature
### Kinetic Dehydration of Interlayer Water  
[PERSON] et al. [41] performed a kinetic study of smectite dehydration and reported that a smectite structure exhibits various hydration states with the intercalation of zero, one, or two planes of water molecules in the interlayer. In their study, according to the mechanisms and kinetics of the transition processes for two planes to one and one to zero planes of water molecules, a solid-state reaction can represent the reaction rate as a product of a function of the transformed fraction (\(\alpha\)) and a rate constant (\(k\)):  
\[d\alpha/dt=kf(\alpha) \tag{4}\]  
The integration of (4), from time = 0 to time = \(t\) leads to Equation (5):  
\[g(\alpha)=kt \tag{5}\]  
where the function \(g(\alpha)\) depends on the reaction mechanism.  
The activation energy \(Ea\) (in J/mol) can be determined using Equation (6):  
\[k=A\exp(-Ea/RT) \tag{6}\]where \(A\) is the frequency or pre-exponential factor (in s\({}^{-1}\)), R denotes the gas constant [8.31 J/(K\(\cdot\)mol)], and \(T\) represents the temperature in kelvin. The reaction rate (\(\alpha\)) can be determined using Equation (7):  
\[\frac{d\alpha}{dt}=k^{m}t^{m-1}(1-\alpha) \tag{7}\]  
The integration of (7) leads to the expression of the transformed fraction \(\alpha\), as follows:  
\[\alpha=1-\exp\left[-(kt)\right]^{m} \tag{8}\]  
Table 3 shows the values of the rate constant (\(k\)) and the parameter (\(m\)) for 2W-1W (two layers to one layer of water molecules) and 1W-0W transitions of smectite interlayer water kinetic dehydration, respectively [41].  
The interlayer molar volume (\(V_{il}\)) can be determined using the following equation:  
\[V_{il}=\left(\frac{abc\cdot\cos(90-\beta)}{10^{24}Z}\right)N_{0} \tag{9}\]  
Figure 4: Schematic for the structure of the smectite mineral and two layers of water molecules in the interlayer spacing. Here, T and O denote the silicon tetrahedral sheet and alumina octahedral sheet, respectively. Cation denotes the cation ion sorption on the interlayer.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dehydration Stage** & **T** (\({}^{\circ}\)**C**) & \(m\) & \(k\) \\ \hline
2W–1W & 35 & 0.846 & \(6.45\times 10^{-3}\) \\
1W–0W & 90 & 0.733 & \(1.55\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Value of the rate constant (\(k\)) and parameter (\(m\)) for 2W–1W and 1W–0W transitions.  
where \(a=5.17\) A, \(b=8.99\) A, and \(\beta=100^{\circ}\), which are parameters associated with the smectite unit cell; \(c\) is the interlayer thickness, \(Z=2\) is the number of units in the chemical formula of the smectite unit cell, and \(N_{0}\) is Avogadro's constant [39].  
The number of moles of water (\(n\)) released through the dehydration of hydrous smectite is calculated thus:  
\[n=\frac{\rho_{il}V_{il}}{M_{w}} \tag{10}\]  
where \(\rho_{il}\) is the density of the interlayer water; and \(M_{w}\) is the molecular weight of water and equals 18.05. [PERSON] and [PERSON] [42] measured the average interlayer water density as 1.05 g/cm. The n is 2.03 and 4.51 for approximately basal spacing of 12.6 A (one layer of water molecules) and 15.7 A (two layers of water molecules) computed using (10), respectively.