1.14.32: Hildebrand Solubility Parameter
The cohesive energy density (c.e.d.) of a liquid is defined by equation (a).
\[\text { c.e.d. }=\Delta_{\text {vap }} \mathrm{U}^{0} / \mathrm{V}^{*}(\ell) \nonumber \]
\(\Delta_{\text {vap }} \mathrm{U}^{0}\) is the change in thermodynamic energy when one mole of a given chemical substance passes from the liquid to the vapor state. The square root of the c.e.d. for liquid \(j\) is the Hildebrand solubility parameter for that liquid.
\[\delta=(\text { c.e.d. })^{1 / 2} \nonumber \]
\(\delta\) can be expressed in many units but following the original definition the customary unit is \(\left(\mathrm{cal}^{1 / 2} \mathrm{~cm}^{-3 / 2}\right)\). Property \(\delta\) provides an estimate of cohesion within a given liquid. The idea goes a little further in terms of understanding solubilities. A clever idea is based on the following argument.
Consider two liquids \(\mathrm{A}\) and \(\mathrm{B}\). We want to take a small sample of liquid \(\mathrm{A}\) (as a solute) and dissolve in liquid \(\mathrm{B}\) as the solvent. Within liquid \(\mathrm{A}\) the intermolecular interactions \(\mathrm{A} \ldots \(\mathrm{A}\) are responsible for the cohesion within this chemical substance. Similarly within liquid \(\mathrm{B}\), \(\mathrm{B} - \mathrm{~B}\) intermolecular forces are responsible for the cohesion within liquid \(\mathrm{B}\). If \(\mathrm{B} - \mathrm{~B}\) interactions are much stronger than \(\mathrm{A} - \mathrm{~A}\) and \(\mathrm{A} - \mathrm{~B}\) intermolecular interactions it is likely that \(\mathrm{A}\) will not be soluble in liquid \(\mathrm{B}\). Similarly if \(\mathrm{A} - \mathrm{~A}\) interactions are stronger than \(\mathrm{B} - \mathrm{~B}\) and \(\mathrm{A} - \mathrm{~B}\) interactions it is likely that \(\mathrm{A}\) will not be soluble in liquid \(\mathrm{B}\). If substance \(\mathrm{A}\) is to be soluble in liquid \(\mathrm{B}\), their cohesive energy densities should be roughly equal.