## 2 Materials and Methods  
The development of this study used reported thermodynamic data in the literature of the acetone-butanol mixture (12). Table 1 shows data for acetone-butanol mixture regarding different pressures. \(x_{1}\) and \(y_{1}\) represent the molar concentration in liquid and vapor phases for component 1 (Acetone), respectively. Considering the physical-chemical properties and chemical affinity of the mixture, the [PERSON] model used to estimate the thermodynamic parameters as follows in Equations (1) and (2) [14]:  
\[\mathrm{Ln}\gamma_{1}=-\mathrm{Ln}(x_{1}+x_{2}\Lambda_{12})+x_{2}\bigg{(}\frac{ \Lambda_{12}}{x_{1}+x_{2}\Lambda_{12}}-\frac{\Lambda_{21}}{x_{2}+x_{1}\Lambda_ {21}}\bigg{)} \tag{1}\]  
\[\mathrm{Ln}\gamma_{2}=-\mathrm{Ln}(x_{2}+x_{1}\Lambda_{21})-x_{1}\bigg{(} \frac{\Lambda_{12}}{x_{1}+x_{2}\Lambda_{12}}-\frac{\Lambda_{21}}{x_{2}+x_{1} \Lambda_{21}}\bigg{)} \tag{2}\]  
\(\Lambda_{12}\) is the interaction parameter of component 1 to component 2, x\({}_{2}\) is the mole fraction of component 2 in the liquid phase, \(y_{2}\) is the molar fraction of component 2 in the vapor phase and \(\Lambda_{21}\) is the interaction parameter of component 2 to component 1. Finally, \(\gamma_{1}\) and \(\gamma_{2}\) are the activity coefficients of components 1 and 2, respectively. Considering the Wilson model showed in Equations (1) and (2), the following step was to estimate the activity coefficients (\(\gamma_{1}\) and \(\gamma_{2}\)) of mixture component. For this, the Modified Raoult's Law was used. This equation [see Equation (3)] is applied for binary non-ideal vapor-liquid equilibrium systems, as follows [7]:  
\[\gamma_{\mathrm{i}}=\frac{y_{\mathrm{i}}PF_{\mathrm{i}}}{x_{\mathrm{i}}P_{ \mathrm{i}}^{\mathrm{sat}}} \tag{3}\]  
\(y_{\mathrm{i}}\) and \(x_{\mathrm{i}}\) are the concentration of component \(i\) in the vapor and the liquid phase, respectively. \(F_{\mathrm{i}}\) is the fugacity coefficient, \(P\) and \(P_{\mathrm{i}}^{\mathrm{sat}}\) are the system and saturation pressure, respectively. Next, the modeling of the binary system required the estimation of the fugacity coefficient for both components, according to Equation (3). This estimation was developed by using the virial equation of state described by Equations (4)-(10) as follows [5]:  
\[F_{\mathrm{i}}=\exp\left(\frac{B_{\mathrm{ii}}(P-Psat)+\left(Py_{\mathrm{j}}^{2 }\delta_{\mathrm{ij}}\right)}{RT}\right) \tag{4}\]  
\[\delta_{\mathrm{ij}}=2B_{\mathrm{ij}}-B_{\mathrm{ii}}-B_{\mathrm{jj}} \tag{5}\] \begin{table}
\begin{tabular}{c c c} \hline
**Pressure (P/kPa)** & \(x_{1}\) & \(y_{1}\) \\ \hline  
\hline \end{tabular}
\end{table} 
Table 1: Equilibrium data for acetone–butanol mixture. \[\mathrm{Bij}=\mathrm{Bij}^{0}+\mathrm{w_{ij}}\mathrm{Bij}^{1} \tag{6}\] \[\mathrm{w_{ij}}=\frac{\mathrm{w_{i}}+\mathrm{w_{j}}}{2}\] (7) \[\mathrm{Bij}^{0}=0083-\frac{0422}{\mathrm{T_{rj}^{1,6}}}\] (8) \[\mathrm{Bij}^{1}=0139-\frac{0172}{\mathrm{T_{rj}^{4,2}}}\] (9) \[\mathrm{T_{rj}}=\left(\mathrm{T_{i}}\times\mathrm{T_{j}}\right)^{0,5} \tag{10}\]  
\(\mathrm{T_{rj}},\mathrm{w_{ij}},\mathrm{Bij}^{0},\mathrm{Bij}^{1}\), \(\mathrm{Bij}\) are the crossed reduced temperature, acentric factor, zero-order virial coefficient, 1 st order virial coefficient and virial coefficient for a binary mixture, respectively. \(\mathrm{w_{i}}\), \(\mathrm{T_{i}}\) and \(\mathrm{w_{j}}\), \(\mathrm{T_{j}}\) are the acentric factor and reduced temperature of component \(i\) and component \(j\), respectively. Once all required thermodynamic data were estimated, the solution of the nonlinear system composed of Equations (1) and (2), is formulated from the application of the nonlinear least squares (NLS) [15], taking into consideration that \(\Lambda_{12}\) and \(\Lambda_{21}\) are the adjustable parameter of the nonlinear system.