## 2 Theoretical Aspects  
The absorbance of a dye depends both on the wavelength of the incident light, \(\lambda\), and on its mole fraction, \(x_{D}\), i.e., \(A=A(\lambda,x_{D})\). Such a function is mathematically complex, so that the molecular properties of the liquid solutions of dyes are investigated by monitoring only the peak of maximum intensity as the dye concentration varies. We point out that a single experimental point hardly captures the structure-related, and for this reason, we introduced the integral absorbance [21], defined as follows:  
\[\overline{A}=\int_{\lambda_{1}}^{\lambda_{2}}A(\lambda,x_{D})d\lambda \tag{1}\]  
where \([\lambda_{1},\lambda_{2}]\) is the wavelength interval of the absorption band under investigation. The complexity of a dye absorption spectrum can be interpreted in terms of _association_ and _solution_. By the former, we refer to the tendency of the dye to form polymers. By solvation we refer to the tendency of the dye to form complexes with solvent molecules. The ability of a dye molecule to solvate or associate is closely related to its electronic structure. Hence, the absorption spectrum of the dye, experimentally recorded, is the superposition of these two effects, i.e.,  
\[\overline{A}=\overline{A}^{(1)}\varepsilon_{1}+\overline{A}^{(2)}\varepsilon _{2} \tag{2}\]  
where \(\overline{A}^{(1)}\) and \(\overline{A}^{(2)}\) are the contributions to the absorption spectrum due to the dye-solvent (_state_ (1)) and dye-dye (_state_ (2)) interactions, respectively, while \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the fractions of dye molecules that are in states (1) or (2), respectively. Only dye molecules absorb the light, so that the mole fraction of adsorbent molecules in states (1) and (2) are:  
\[\varepsilon_{1}=\frac{x_{1}}{x_{D}},\ \ \varepsilon_{2}=1-\varepsilon_{1} \tag{3}\]  
Substitution of Equation (3) into Equation (2) results in:  
\[\overline{A}=\overline{A}^{(2)}+\Delta^{(12)}\frac{x_{1}}{x_{D}} \tag{4}\]  
where:  
\[\Delta^{(1,2)}=\overline{A}^{(1)}-\overline{A}^{(2)} \tag{5}\]In order to obtain the function \(x_{1}(x_{D})\), we stress that as the dye concentration varies, drastic variations in aqueous solutions are observed. Such changes have been interpreted in terms of multiple dye molecule associations interacting through the electron system of \(\pi\)-electron orbital system of the aromatic rings, stacking in a sandwich-like mode [22; 23; 24]. On the contrary, in no aqueous medium, very limited changes are shown by the absorption spectra. Indeed, even in the most concentrated solutions, the monomer dye species prevails, and the only relevant equilibrium is the dimerization,  
\[2D\ \rightleftharpoons D_{2} \tag{6}\]  
The equilibrium constant of this equilibrium is expressed in a thermodynamically consistent way for liquid-phase reactions [25]. Activities should be used rather than concentrations [26]. Nonetheless, we use a mole fraction-based equilibrium constant in the present work. The mole fraction-based equilibrium constant:  
\[K_{x}=\frac{x_{2}}{x_{1}^{2}} \tag{7}\]  
is related to the activity-based equilibrium constant \(K\) by:  
\[K=K_{x}K_{\Gamma} \tag{8}\]  
where \(K_{\Gamma}\) is defined analogous to \(K_{x}\), but with the corresponding activity coefficients. The thermodynamic theory yields that \(K\) is independent of the composition, while \(K_{x}\) is, in general, composition dependent. Nevertheless, for very diluted solutions, \(K_{\Gamma}\approx 1\) so that the equilibrium is completely characterized by:  
\[K_{x}\ \approx K \tag{9}\]  
On the other hand, the total dye mole fraction is connected to \(x_{1}\) e \(x_{2}\) by:  
\[x_{D}=x_{1}+2x_{2}=x_{1}+2 Kx_{1}^{2} \tag{10}\]  
where Equations (7) and (8) have been used.