*3.3. Phase Energetics*

Taken on their own, the thermochemical characteristics do not say anything about the internal organization of a particular phase, but can be used to assess their thermodynamic stability. For a detailed analysis of the energy ratios between the forms realized in the system, the heat capacities of all the modifications observed in the system were measured by the DSC method in the temperature range from 20 to 110 ◦C. Changes in the thermodynamic parameters Δ*H*, Δ*S* and Δ*G* of a system with temperature can be described by Equations (1)–(3).

$$
\Delta H^{T1/T0} = \begin{vmatrix}
\int\_{T0}^{T1} \mathcal{C}\_p^{\text{solid}}(T)dT, T1 < T^f, \\
\int\_{T0}^{T^f} \mathcal{C}\_p^{\text{solid}}(T)dT + \Delta H\_f + \int\_{T^f}^{T1} \mathcal{C}\_p^{\text{lq}}(T)dT, T1 \ge T^f,
\end{vmatrix} \tag{1}
$$

$$
\Delta S^{T1/T0} = \begin{bmatrix}
\int\_{T0}^{T1} \frac{C\_p^{\text{solid}}(T)}{T} dT, T1 < T^f, \\
\int\_{T0}^{T^f} \frac{C\_p^{\text{solid}}}{T}(T) dT + \frac{\Delta H\_f}{T^f} + \int\_{T^f}^{T1} \frac{C\_p^{\text{sl}}(T)}{T} dT, T1 \ge T^f,
\end{bmatrix} \tag{2}
$$

$$
\Delta G^{T1/T0} = \Delta H^{T1/T0} - T1 \cdot \Delta S^{T1/T0}.\tag{3}
$$

Based on these equations and the totality of the experimentally-obtained thermochemical information for diol **1**, we constructed an energy diagram reflecting the relationship between the free energies of various crystalline modifications at different temperatures (Figure 4). The details of the calculations are given by us earlier [29].

It should be noted that we do not know the absolute values of the standard thermodynamic potentials of the considered forms. For this reason, the enthalpy and entropy of the enantiopure phase (*R*)-**1** at 20 ◦C were taken as conventional zero in calculating the thermodynamic potentials. Accordingly, changes in the Gibbs energy were calculated by Equation (4).

$$
\Delta \mathbf{G}\_x^T = (H\_x^T - H\_R^{20^\circ \text{C}}) - T(\mathbf{S}\_x^T - \mathbf{S}\_R^{20^\circ \text{C}}) = (H\_x^T - T \cdot \mathbf{S}\_x^T) - (H\_R^{20^\circ \text{C}} - T \cdot \mathbf{S}\_R^{20^\circ \text{C}}).\tag{4}$$

Thus, the Gibbs energy for each phase is calculated relative to a hypothetical system, the enthalpy and entropy of which coincides with the values of these parameters for the enantiopure phase at

20 ◦C. This approach leaves invariant the relative arrangement of curves and characteristic points on the graph.

**Figure 4.** The temperature dependence of the Gibbs free energy change for the enantiopure sample (*R*)-**1** (blue line), racemic conglomerate (*R*+*S*)-**1** (red line), α-*rac*-**1** (violet line), β-*rac*-**1** (olive line), racemic melt (green line), and single enantiomeric melt (cyan line). Dashed lines correspond to the metastable supercooled state of melts; dot lines correspond to extrapolations of the free energies of crystalline phases in the temperature range above their melting temperature.

The representation of the free energies of the putative racemic and enantiopure forms of the substance in a single scale is based on the fact that mixtures of enantiomers in the molten state well satisfy the conditions of the ideal solution model (zero enthalpy of mixing, constant (*R*ln2) entropy of mixing of enantiomers during the formation of racemic melt), which makes it possible to use the energy of the melt level as a reference point for bringing the free energies of all phases realized in the system to a common scale [44].

The complete coincidence of the thermodynamic characteristics of enantiopure crystals (*R*)-**1** and conglomerate (racemic eutectic of the enantiomers), which finds its expression in the merger of the corresponding curves in Figure 4, means that there are no signs of solid solution formation in the (*R*+*S*)-**1** form.

A useful consequence of the ranking of identified crystalline modifications of diol **1** is the ability to evaluate their relative solubility on this basis. Considering that, at least for dilute solutions, the solvation effects for α-*rac*-**1**, β-*rac*-**1** and (*R*+*S*)-**1** will be the same, their relative solubility will be determined only by differences in the free energies of crystalline modifications. The change in the free energy of the phase *x* upon dissolution, Δ*GT <sup>x</sup>*/*soln*, is described by Equation (5), in which c*sat x* is the equilibrium concentration of the saturated solution (i.e., solubility) for phase *x*.

$$
\Delta G\_{\text{x/solu}}^T = -RT\ln\left[\mathbf{c}\_{\text{x}}^{\text{sat}}\right].\tag{5}
$$

Then the solubility ratio of the two phases *x***1** and *x***2** can be calculated on the basis of the difference of the free energies of these phases as follows:

$$\frac{\left[\mathbf{c}\_{\chi2}^{\rm sat}\right]}{\left[\mathbf{c}\_{\chi1}^{\rm sat}\right]} = \exp \frac{-\left(\Delta G\_{\chi2}^{T} - \Delta G\_{\chi1}^{T}\right)}{\mathbf{RT}}.\tag{6}$$

For α-*rac*-**1** and (*R*+*S*)-**1** forms, the difference calculated from the Gibbs free energies under standard conditions is ΔΔ*G*<sup>0</sup> <sup>α</sup>/(*R*+*S*) <sup>≈</sup> 0.36 *kJ*·*mole*−<sup>1</sup> . Then the ratio of their solubilities calculated by Equation (6) will be [*C*α] [*C*(*R*+*S*)] <sup>≈</sup> 1.16. For <sup>β</sup>-*rac*-**<sup>1</sup>** and (*R*+*S*)-**<sup>1</sup>** the calculated values of the corresponding quantities are ΔΔ*G*<sup>0</sup> <sup>β</sup>/(*R*+*S*) <sup>≈</sup> 1.6 *kJ*·*mole*−<sup>1</sup> and [*C*β] [*C*(*R*+*S*)] <sup>≈</sup> 1.9.
