## 3 Results and Discussion
### Solubilities
 \({}^{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\textBecause all calculated solubilities meet \(x_{1}<0.002\), the solvent volume fraction is well approximated by \(\Phi_{2}=1\) so that Equation (1) is simplified and the partially factorized expression for \(x_{1}\) reads:  
\[x_{1}=e^{\frac{\Delta H_{1}}{R}\cdot(\frac{1}{T_{m,1}}-\frac{1}{T})}.e^{-\frac{ V_{1}}{RT}\cdot(\delta_{1}-\delta_{2})^{2}}. \tag{2}\]  
Keeping \(T\) at 25 \({}^{\circ}\)C, both the sensitivity of x\({}_{1}\) to the five material parameters and their individual contributions to the uncertainty of \(x_{1}\) can approximately be assessed by computing the solubility for the maximal and minimal value of any one variable while the other variables stay fixed at their respective mid-range values. Since \(x_{1}\) is a continuous and monotonous function in all variables, the span ratio \(R=(\text{maximum }x_{1})/(\text{minimum }x_{1})\) rates the influence a material parameter has on the target quantity solubility. Values for this uncertainty measure or, in the wording of the GUM, sensitivity measure are listed in Table 5. The exact expressions that were used for the calculation of the span ratio \(R\) are compiled in Table S2. Apparently, the solubility parameter errors \(\Delta\delta_{i}\) cause the principal largest uncertainty equally in \(x_{1}\). Their influence on solubility prognosis is the more dominant it is, then the larger the solvent-solute difference \(\Delta\delta\). Since \(\Delta\delta_{i}=\pm 1\) (MPa)\({}^{0.5}\) already is a rather optimistic error estimate, it is unlikely that solubility uncertainty can be improved to this end. The second strongest influence is exerted by the melting enthalpy. Its effect is independent of the solvent, and it is the only prominent factor for small \(\Delta\delta\). Reducing the relative error of the fusion enthalpy to \(\pm 4\%\) would substantially lower the corresponding (max/min) ratio from 17.8 to 1.6, offering a promising route towards a more precise solubility prediction. In the following, the MC results and experimental literature data presented in Figures 1 and 2 are treated. The solid lines in Figure 1 correspond, from top to bottom, to the 100%, 97.5%, 75%, 25%, 2.5%, and 0% percentile from simulation. The dotted line represents the analytical computation results with nodes (filled squares) for each solvent. Besides the solubility as calculated from Equation (1), Figure 1 also depicts the 100%, 95%, and 50% CI as derived from the MC simulation. The diagram reveals expected parabolic behavior for allCIs, as well as decreasing solubility uncertainty (narrowing CIs) as \(\delta_{1}\) and \(\delta_{2}\) become more similar. The latter point is taken up again in more detail in Figure 2.  
The finite number of simulations makes extreme values improbable to occur and, in line with the GUM approach, the 95% CI or interquartile range should be considered to provide a meaningful measure of uncertainty. The quotients of conjugate confidence limits range from 2.1 to 11 (50% CI) and 2.8 to 483 (95% CI), respectively, indicating substantial imprecision. \begin{table}
\begin{tabular}{l l l l} \hline \hline \multirow{2}{*}{**Variable**} & **Uncertainty** & & **Span Ratio** \\  & **Absolute** & **Relative** & \(R=\text{max}(x_{1})/\text{min}(x_{1})\) \\ \hline \(\Delta H_{1}\) & \(\pm 5\) kJ/mol & (\(\pm 11.9\%\)) & 17.8 \\ \(T_{m,1}\) & \(\pm 5\) K & (\(\pm 1.1\%\)) & 1.3 \\ \(V_{1}\) & \(\pm 4\) cm\({}^{3}\)/mol & (\(\pm 3.6\%\)) & 1.6 to 4.0 \({}^{\text{b}}\) \\ \(\delta_{1}\) & \(\pm 1\) (MPa)\({}^{0.5}\) & (\(\pm 2.8\%\)) & 1.5 to 40.0 \({}^{\text{b}}\) \\ \(\delta_{2}\) & \(\pm 1\) (MPa)\({}^{0.5}\) & (\(\pm 6.8\%\)) \({}^{\text{a}}\) & 1.5 to 40.0 \({}^{\text{b}}\) \\ \hline \hline \end{tabular}  
* : error ranges from \(\pm 2.7\%\) for \(\delta_{2}=32.95\) (ethylene glycol) to \(\pm 6.8\%\) for \(\delta_{2}=14.50\) (pentane). \({}^{\text{b}}\): ratio ranges from lowest for ethylene glycol to largest for pentane.  
\end{table}