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7.7: Solubility

Last updated

Jul 19, 2021
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- 7.6: Colligative Properties
- 7.8: Non-ideality in Solutions - Activity

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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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