Since the solid solute and its corresponding molten solid must be in a state of equilibrium at the melting point, it follows that:
where the enthalpy of fusion (A Hm) is equal to Tm A Sm, where A f is the entropy of fusion at the melting temperature. Under these circumstances, equation (18) may also be written as:
The enthalpy and entropy of fusion, and the melting temperature may all be measured through the use of differential scanning calorimetry (DSC), and therefore equations (18) and (20) provide a simple way to predict the solubility of a solute in an ideal solution.
To achieve a better prediction of the solubility of a solute, one must consider the temperature dependence of the enthalpy of fusion, which is described by the Kirchoff equation
where A Cp is the difference between the heat capacities of the supercooled liquid and that of the corresponding solid. Therefore:
With the assumption that ACp is independent of temperature, integration of equation (16) and the replacement of A H by A HfT, yields the Hildebrand equation
AHm/1 1\ ACp Tm - T ACp Tm ln Xb = ln aB =--—---+-^^---p ln — (23)
Equation (23) provides a better prediction of the solubility of a solute in an ideal solution.
Prediction of solubility in an ideal solution can also be performed using the entropy approach developed by Hildebrand and Scott (Hildebrand and Scott, 1962). Assuming that A H ^ TA f & TACp, they found that:
A Sfm T
Equation (24) is similar to equation (20), except that ln(XB) is correlated to ln(T) instead of 1/T. The solubility prediction using equation (24) was found to have a better tolerance for the non-ideality of the solution than that obtained using equation (20).
Several approaches have been used to predict the entropy of fusion required for the prediction of solubility. According to Walden's rule, the entropy of fusion (ASf) is approximately equal to 13 cal/Kmol for most organic compounds (Walden, 1908). Use of this approximation reduces equation (20) to:
where 0m is the melting point of the solute in degrees centigrade.
Yalkowsky proposed that the entropy of fusion of an organic compound is the sum of translational, rotational, and internal entropy changes when it is released from the crystal lattice (Yalkowsky, 1979):
while, the translational entropy change consists of the components associated with the expansion and change of position as the solid melts.
Yalkowsky also proposed empirical values and limits for these components. Both the Walden and Yalkowsky models provide ways by which one can predict the entropy of fusion, and therefore predict the solubility of the solute in an ideal solution.
Over a small temperature range, the enthalpy of solution of a solid can be assumed to be independent of temperature. The van't Hoff equation shows that ln( XB) increases with temperature, until the solid melts at T = Tm. At this condition, the solid forms a liquid in the absence of solvent, and since XB = 1, the slope of the van't Hoff plot is equal to (AHS/R). The degree of ideality associated with a given solution may therefore be tested by evaluating the degree of linear correlation between ln(XB) and 1/T. Figure 1 shows the ideal behavior of naphthalene dissolved in benzene and xylene, which is due to the similar nature of the molecules involved, and the strength of intermolecular interactions such as polarity, polarizability, molecular volume, and hydrogen-bonding characteristics (Grant and Higuchi, 1990). On the other hand, the molecular properties of ethanol are very different from those of naphthalene. Thus one finds that for solutions of naphthalene in ethanol, ln( XB) does not exhibit a linear dependence on 1/T, which is taken as an indication of the non-ideal character of the solution.
Typically, one finds that the solubility that would be predicted assuming the model of an ideal solution is normally much higher than the solubility that is actually measured for a non-ideal solution.
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