7.7: Solubility

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The maximum solubility of a solute can be determined using the same methods we have used to describe colligative properties. The chemical potential of the solute in a liquid solution can be expressed

\[ \mu_{B} (solution) = \mu_B^o (liquid) + RT \ln \chi_B \nonumber \]

If this chemical potential is lower than that of a pure solid solute, the solute will dissolve into the liquid solvent (in order to achieve a lower chemical potential!) So the point of saturation is reached when the chemical potential of the solute in the solution is equal to that of the pure solid solute.

\[ \mu_B^o (solid) = \mu_B^o (liquid) + RT \ln \chi_B \nonumber \]

Since the mole fraction at saturation is of interest, we can solve for \(\ln(\chi_B)\).

\[\ln \chi_B = \dfrac{\mu_B^o (solid) = \mu_B^o (liquid)}{RT} \nonumber \]

The difference in the chemical potentials is the molar Gibbs function for the phase change of fusion. So this can be rewritten

\[\ln \chi_B = \dfrac{-\Delta G_{fus}^o}{RT} \nonumber \]

It would be convenient if the solubility could be expressed in terms of the enthalpy of fusion for the solute rather than the Gibbs function change. Fortunately, the Gibbs-Helmholtz equation gives us a means of making this change. Noting that

\[ \left( \dfrac{\partial \left( \dfrac{\Delta G}{T} \right)}{\partial T} \right)_p = \dfrac{\Delta H}{T^2} \nonumber \]

Differentiation of the above expression for \(\ln(\chi_B)\) with respect to \(T\) at constant \(p\) yields

\[ \left( \dfrac{\partial \ln \chi_B}{\partial T} \right)_p = \dfrac{1}{R} \dfrac{\Delta H_{fus}}{T^2} \nonumber \]

Separating the variables puts this into an integrable form that can be used to see how solubility will vary with temperature:

\[ \int_0^{\ln \chi_B} d \ln \chi_B = \dfrac{1}{R} \int_{T_f}^{T} \dfrac{\Delta H_{fus} dT}{T^2} \nonumber \]

So if the enthalpy of fusion is constant over the temperature range of \(T_f\) to the temperature of interest,

\[ \ln \chi_B = \dfrac{\Delta H_{fus}}{R} \left( \dfrac{1}{T_f} - \dfrac{1}{T} \right) \nonumber \]

And \(\chi_B\) will give the mole fraction of the solute in a saturated solution at the temperature \(T\). The value depends on both the enthalpy of fusion, and the normal melting point of the solute.

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