## 2 Materials and Methods
### Theoretical Model of the Refractive Index of Gasoline Admixture Solutions  
Since the composition of a mixture typically includes constituents belonging to different classes of compounds, a model suitable to adequately describe its refractive index must take into account a specific type of molecular interactions and specific mixing rules between the constituents. For example, it is known that a molar volume, \(V\), of an idealized binary solution should follow the linear relation involving molar fractions of both elements, \(\phi_{1,2}\), their molar masses, \(m_{1,2}\), and molar densities, \(\rho_{1,2}\), [27]:  
\[V=\frac{m_{1}}{\rho_{1}}\phi_{1}+\frac{m_{2}}{\rho_{2}}\phi_{2}. \tag{5}\]Numerous other mixing rules can be found in the literature [28; 29]. For example, the Lorentz-Lorenz mixing rule, given by Equation (7), is known to be particularly suitable for the description of binary solutions of organic solvents [27; 30; 31]. The refractive index of 95-octane gasoline pseudo-binary mixtures with isopropanol has recently been studied in [32; 33]. Four different mixing rules, described by four equations, namely, the Eykman (Equation (6)), Lorentz-Lorenz (Equation (7)), Newton (Equation (8)) and Gladstone-Dale (Equation (9)):  
\[\frac{n_{D}^{2}-1}{n_{D}^{2}+0.4}=\phi_{1}\Bigg{(}\frac{n_{D1}^{2}-1}{n_{D1}^{ 2}+0.4}\Bigg{)}+\phi_{2}\Bigg{(}\frac{n_{D2}^{2}-1}{n_{D2}^{2}+0.4}\Bigg{)}, \tag{6}\]  
\[\frac{n_{D}^{2}-1}{n_{D}^{2}+2}=\phi_{1}\Bigg{(}\frac{n_{D1}^{2}-1}{n_{D1}^{2} +2}\Bigg{)}+\phi_{2}\Bigg{(}\frac{n_{D2}^{2}-1}{n_{D2}^{2}+2}\Bigg{)}, \tag{7}\]  
\[n_{D}^{2}-1=\phi_{1}\Big{(}n_{D1}^{2}-1\Big{)}+\phi_{2}\Big{(}n_{D2}^{2}-1 \Big{)}, \tag{8}\]  
\[n_{D}-1=\phi_{1}(n_{D1}-1)+\phi_{2}(n_{D2}-1). \tag{9}\]  
Were tested in the quoted above publications in the temperature range from 293.15 K to 313.5 K. In Equations (6)-(9), the symbols \(n_{D1}\) and \(n_{D2}\) stand for the refractive index of isopropanol-gasoline mixture components \(\phi_{1}\) and \(\phi_{2}\), respectively. Using these equations, the refractive index of isopropanol-gasoline mixtures was found to increase linearly from \(n_{D1}\) to \(n_{D2}\), with \(n_{D2}\) belonging to the constituent of the larger absolute value of \(n\)[33]. The best correspondence was found for the Eykman and Lorentz-Lorenz model; however, the differences between the measured and calculated \(n_{D}\) using the Gladstone-Dale equation (Equation (9)) were also reasonably small with the absolute values \(\Delta n_{D}=|\,n_{D,exp}-(\phi_{1}n_{D1}-\phi_{2}n_{D2})|\) not larger than \(\sim\)0.0016. They applied to the situation when the gasoline and isopropanol concentrations were approximately equal (\(\phi_{12}\approx 0.5\)). With smaller solvent or admixture concentrations, the differences were even  
Figure 4: Graphical representation of the \(\Delta t\) parameter determination: (**a**) for samples containing pure solvent and pure admixture when the reference pulse delay is measured with respect to an empty cuvette described by Equation (3); (**b**) for samples containing increasing concentration of admixture in solution as described by Equation (11). smaller, approaching \(\Delta n_{D}\cong 0\). The mixing rules were tested for the visible light of the wavelength \(\lambda=589.3\) nm (sodium D line). In order to verify if the Gladstone-Dale mixing rule can be useful for modeling the measured pulse delays in the THz spectral range, we have employed Equation (9) using the mass fractions of isopropanol and gasoline, \(\phi_{1}\) and \(\phi_{2}\), respectively. With this new representation, the refractive index of the solution, \(n_{D}\), is described by the relation:  
\[n_{D}=\phi_{1}n_{D1}+\phi_{2}n_{D2}. \tag{10}\]  
Equation (10) defines the refractive index of a mixture as the weighted sum of the refractive indices of solvent and admixture with the weights \(\phi_{1,2}\), given as the mass fractions (%wt.) with respect to the total mass of the solution.