## 1 Introduction  
The production and processing of chemicals normally involves solute-solvent systems at some stage. Thus, solubility plays a key role in all sorts of industrial applications such as synthesis, extraction, or purification, as it also does, for instance, in packaging, painting, food technology, or pharmacology. In commercial environments, attention is placed on rational product design and process intensification. The development of solid amorphous dispersions in the pharmaceutical industry (e.g., drug containing polymers), the functionalization of food packaging films with antioxidants, or the formulation of multicomponent solvents for membrane production, separation purposes, or other applications, are examples of practical problems that benefit clearly from the possibility of reliable solubility prediction. Consequently, it is of great industrial value to predict or, at the least, reasonably estimate the solubility from basic physicochemical solute and solvent property data. A frequently adopted approach using the solubility parameter concept has been proposed early on for nonpolar solvents by [PERSON] and [PERSON] [1; 2; 3; 4]. Despite its various assumptions, this approach is considered useful in many practical areas, not least because of its conceptual simplicity, even though its predictive accuracy has not been quantitatively assessed in the literature. This study evaluates the numerical uncertainty typically encountered when applying the Hildebrand-Scott approach to solubility prognosis of the commercially important natural antioxidant \(\mathrm{l}\)-ascorbic acid (LAA). The equation giving the mole fraction solubility \(x_{1}\) of a solid solute reads in its basic form for a regular solution [5; 6]:  
\[\ln(x_{1})=\frac{\Delta H_{1}}{R}\bigg{(}\,\frac{1}{T_{m,1}}-\frac{1}{T}\, \bigg{)}-\frac{V_{1}\cdot\Phi_{2}^{2}\cdot(\delta_{1}-\delta_{2})^{2}}{RT} \tag{1}\]  
where \(\Delta H_{1}\) the heat of fusion in units of J/mol, \(T_{m,1}\) the melting temperature, and \(V_{1}\) the molar volume; for the solute, \(\Phi_{2}\) the solvent volume fraction, and the Hildebrand solubility parameters are \(\delta_{1}\) (solute) and \(\delta_{2}\) (solvent). \(R\) and \(T\) have their usual meaning. Throughout this paper, subscript \"1\" (\"2\") designates the solute (solvent). The derivation of Equation (1) rests on assuming a vanishing term containing the molar heat capacity difference of the solid and supercooled liquid state of the solute. Both regular solution theory as well as the Hildebrand solubility parameter concept originally treat the mixing of two liquids. To extend this approach to dissolving crystalline solids in liquids at \(T<T_{m,1}\) the molar volume \(V_{1}\) of the (hypothetical) subcooled liquid solute needs to enter Equation (1). Generally, knowledge of the value of \(V_{1}\) necessitates approximations, for instance, by group contribution methods [7] or by the extrapolation of solute melt density data to the solution temperature. Applying Equation (1) to various solute-solvent systems, it was rated vaguely to provide correlations or predictions that were \"quite good,\" \"fair,\" or \"unreliable\" for nonpolar, polar aprotic, and, protic solvents, respectively [5] (p. 344). This rating stays unchanged if the Hildebrand solubility parameter \(\delta\) is replaced by the totalHansen solubility parameter (HSP) as defined by \(\delta_{t}=\left(\delta_{d}^{2}+\delta_{p}^{2}+\delta_{h}^{2}\right)^{0.5}\) with, in extension to Hildebrand, now three contributions to the material cohesive energy accounting for nonpolar dispersive, polar, and hydrogen bonding interactions (subscripts 'd', 'p', and 'h'). The HSP concept has been explained in detail in the literature [8; 9]. In principle, the theoretical basis for Equation (1) limits its use to nonpolar systems, but any remedy for this drawback makes necessary a considerably more complex treatment leading to intricate equations.