Schroder-van Laar equations

The liquidus line of a phase diagram can be calculated using the Schroder-Van Laar equation ... 

An example of the type of melting point phase diagram that can be obtained for a conglomerate system is shown in Fig. 6, which illustrates one-half of the phase diagram reported for 4,4 -demethyl-8,9,10-trinor-spiro-2,2 -bornane [40]. Below the eutectic temperature of 67°C, the system exists as a mixture of solid o-enantiomer and L-enantiomer. At the exact composition of the racemic mixture X= 0.5), the system will exist entirely in the liquid phase above the eutectic temperature. At mole fractions where the amount of L-enantiomer exceeds that of the o-enantiomer, the system will exist as an equilibrium mixture of racemic liquid and solid L-enantiomer. As required by the phase diagram of a conglomerate, the eutectic temperature is the lowest temperature attainable at which any liquid phase can exist in equilibrium with any solid phase. An excellent fit of the data according to the Schroder-Van Laar equation was obtained, with XH = 5.9 kcal/mol and Tp = 95°C. 

However, without dissociation, in an ideal solution at infinite dilution the contributions of the enthalpy of mixing and solvation to the dissolution process can considered to be negligible, thus giving ApH AsH. Therefore, the Schroder-van Laar equation using the melting enthalpy also for solution equilibria is often a good approximation to estimate the temperature dependence of the solubility of organic substances. 

Figure 3.18 Linearization of the simplified Schroder-van Laar equation (Equation 3.8) to evaluate the temperature dependence of solubility (van t Hoff plot).

As already introduced in Section 3.3.2, it is always advisable to first calculate the ideal solubility curve from the melting data of a substance using the simplified Schroder-van Laar equation (Equation 3.8). The presentation of the ideal values together with a couple of measured solubilities in a van t Hoff plot (Figures 3.18 and 3.20) is a reasonable way to evaluate the solubility behavior of a substance. Deviations between the ideal and real curves slopes e.g. indicate differences between the heat of melting and the heat of solution. Kinks occurring in the van t Hoff plot are an indication of various soHd phases of the substance present. Furthermore, a reasonable linearity of the plot allows extraction of solubility data in the temperature range studied. 

Figure 12.11 Solubilities of carbamazepine and 19 different cocrystal formers in water (O), 2-propanol ( ), an equimolar mixture of acetic acid, acetone, -dioxane, DMF, DMSO, ethylene glycol, formic acid, sulfolane, and water O), an equimolar mixture of anisole, benzyl alcohol, 2-butanol, w-butyl acetate, butyronitrile, decalin, /7-dioxane, ethylene glycol, methyl isobutyl ketone, NMP, toluene, and water ( ), as well as ideal solubility values calculated from differential scanning calorimetry (DSC) data with the Schroder-van Laar equation (x). Part of the experimental solubility data are taken from Rager and Hilfiker (2010), see Further Reading, ref 24.

Rarely, if ever, does one particular chiral nematic compound display ideal behaviour, which would make it suitable for use in a given application, and normally carefully formulated mixtures of a variety of meso-genic and non-mesogenic components, which may have very different thermodynamic properties, are used. Formulation of mixtures is frequently a complex trade-off of one property over another and is usually achieved by application of the Schroder-van Laar equation [17-20], which allows the thermodynamic characteristics of a par-... 

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