# Prediction of Solvatochromic Polarity Parameters for Aqueous Mixed-Solvent Systems
## 2 Models and Methods
### Predictive Model  
A predictive model was originally proposed for estimating Kamlet-Taft dipolarity/polarizability (KT-\(\pi^{\star}\)) [22] of binary nonpolar-polar mixtures with an assumption that the gas-phase dipole moment (\(\mu\)) of the polar component can quantify a trend of mixture KT-\(\pi^{\star}\) values at a fixed mole composition. Due to a linear relationship between KT-\(\pi^{\star}\) and \(E_{\rm{T}}\) values (homomorphism line, Figure 2), the predictive model for KT-\(\pi^{\star}\) values can directly transform into a function form for estimating \(E_{\rm{T}}\) values for nonpolar-polar mixtures reported in our previous work [20], as shown by Equation (2). In this work, the model (Equation (2)) was used to predict the \(E_{\rm{T}}\) values of aqueous mixtures due to similar negative deviation trends in both aqueous mixtures and nonpolar-polar mixtures as mentioned in the introduction (Figure 1). \[\Delta E_{\rm{T,mix}}^{\rm{N}}=\mu_{2}(-x_{1}\ln(x_{1}+1.981x_{2})-x_{2}\ln(x _{2}+0.181x_{1}))\times\rm{CF}^{\rm{HBD}} \tag{2}\]  
where \(\mu_{2}\) is the gas-phase dipole moment of component 2. The 1.981 and 0.181 values are the universal Wilson constant parameters (\(\Lambda_{12}\) and \(\Lambda_{21}\)) for predictions and were evaluated by correlating experimental data with [PERSON] thermodynamic excess function [22]. The \(\Delta E_{\rm{T,mix}}^{\rm{N}}\) is relative normalized electronic transition energy and defined by Equations (3)-(5). \[\Delta E_{\rm{T,mix}}^{\rm{N}}=E_{\rm{T,mix}}^{\rm{N}}-(x_{1}E_{\rm{T,1}}^{ \rm{N}}+x_{2}E_{\rm{T,2}}^{\rm{N}})=E_{\rm{T,mix}}^{\rm{N}}-x_{2} \tag{3}\]  
\[E_{\rm{T,mix}}^{\rm{N}}=\frac{E_{\rm{T,mix}}-E_{\rm{T,1}}^{0}}{E_{\rm{T,2}}^{ 0}-E_{\rm{T,1}}^{0}} \tag{4}\]  
\[E_{\rm{T}}(\rm{kcal}\cdot\rm{mol}^{-1})=28591/\lambda_{\rm{max}}(\rm{nm}) \tag{5}\]  
where \(\lambda_{\rm{max}}\) (nm) is the maximum absorption of the wavelength of a solvatochromic indicator (Reichardt indicator) obtained from UV-Vis spectroscopy. The \(E_{\rm{T,mix}}\) and \(E_{\rm{T,mix}}^{\rm{N}}\) are the electronic transition energy and the normalized electronic transition energy of the binary mixtures. To apply the predictive model (Equation (2)) to aqueous systems studied in this work, component 1 refers to water and component 2 refers to cosolvent (HBA or HBD). According to normalization (Equation (4)), the \(E_{\rm{T,1}}^{\rm{N}}\) and \(E_{\rm{T,2}}^{\rm{N}}\) of pure components 1 and 2 in Equation (3) are set equal to zero and unity, respectively. The \(CF^{\rm{HBD}}\) parameter in Equation (2) is the HBD contribution factor and the parameter is only applied to an aqueous mixture of HBD cosolvents due to specific interaction of HBD solvent with indicator [20] that causes a deviation in the pure \(E_{\rm{T,2}}^{0}\) value (HBD solvent) from the homomorphism line (Figure 2). The \(CF^{\rm{HBD}}\) values in Equation (6) can be estimated by a deviation of the actual \(E_{\rm{T}}\) value of pure HBD cosolvent (\(E_{\rm{T,2}}^{0}\)) from the homomorphism line in Figure 2 as shown in Equation (6). \[CF^{\rm{HBD}}=E_{T}^{0,\rm{Non}}-E_{\rm{T,2}}^{0} \tag{6}\]  
where \(E_{T}^{0,\rm{Non}}\) is a non-HBD bonding \(E_{T}\) value of pure HBD cosolvents defined as a linear function of dipolarity/polarizability (KT-\(\pi^{\star}\)), that is the homomorphism linear line in Figure 2. The linear relationships of \(E_{T}^{0,\rm{Non}}\) were given in detail in our previous work [20], and the \(CF^{\rm{HBD}}\) values of nine HBD cosolvents studied in this work are given in Table 1 along with their Kamlet-Taft acidity (\(\alpha\)). An average \(CF^{\rm{HBD}}\) value from three indicators (Table 1) was used in predictions (Equation (2)) for nine aqueous mixtures of HBD cosolvents. To extend to other HBD cosolvents, a linear relationship of \(CF^{\rm{HBD}}\) with Kamlet-Taft acidity (\(\alpha\)) was proposed in Equation (7) and Figure 3, because the \(\alpha\) values of HBD cosolvents are widely available in the literature [27; 28].