# 12.5: Solid-Liquid Equilibria


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We can readily derive a relation between $$K\subs{s}$$ and the molalities of the ions in the saturated solution by treating the dissolved salt as a single solute substance, B. We write the equilibrium in the form B$$^*$$(s)$$\arrows$$B(sln), and write the expression for the solubility product as a proper quotient of activities: $$K\subs{s} = \frac{a\mbB}{a\B^*} \tag{12.5.24}$$ From Eq. 10.3.16, we have $$a\mbB= \G\mbB \g_{\pm}^\nu(m_+/m\st)^{\nu_+}(m_-/m\st)^{\nu_-}$$. This expression is valid whether or not the ions M$$^{z_+}$$ and X$$^{z_-}$$ are present in solution in the same ratio as in the solid salt. When we replace $$a\mbB$$ with this expression, and replace $$a\B^*$$ with $$\G\B^*$$ (Table 9.5), we obtain $$K\subs{s} = \left(\frac{\G\mbB}{\G\B^*}\right) \g_{\pm}^\nu \left( \frac{m_+}{m\st} \right)^{\nu_+} \left( \frac{m_-}{m\st} \right)^{\nu_-} \tag{12.5.25}$$ where $$\nu=\nu_+ + \nu_-$$ is the total number of ions per formula unit. $$\g_{\pm}$$ is the mean ionic activity coefficient of the dissolved salt in the saturated solution, and the molalities $$m_+$$ and $$m_-$$ refer to the ions M$$^{z_+}$$ and X$$^{z_-}$$ in this solution.

The first factor on the right side of Eq. 12.5.25, the proper quotient of pressure factors for the reaction B$$^*$$(s)$$\ra$$B(sln), will be denoted $$\G\subs{r}$$ (the subscript “r” stands for reaction). The value of $$\G\subs{r}$$ is exactly $$1$$ if the system is at the standard pressure, and is otherwise approximately $$1$$ unless the pressure is very high.

If the aqueous solution is produced by allowing the salt to dissolve in pure water, or in a solution of a second solute containing no ions in common with the salt, then the ion molalities in the saturated solution are $$m_+=\nu_+m\B$$ and $$m_-=\nu_-m\B$$ where $$m\B$$ is the solubility of the salt expressed as a molality. Under these conditions, Eq. 12.5.25 becomes \begin{gather} \s{ K\subs{s} = \G\subs{r} \g_{\pm}^{\nu} \left(\nu_+^{\nu_+}\nu_-^{\nu_-}\right) \left( \frac{m\B}{m\st} \right)^{\nu} } \tag{12.5.26} \cond{(no common ion)} \end{gather} We could also have obtained this equation by using the expression of Eq. 10.3.10 for $$a\mbB$$.

If the ionic strength of the saturated salt solution is sufficiently low (i.e., the solubility is sufficiently low), it may be practical to evaluate the solubility product with Eq. 12.5.26 and an estimate of $$\g_{\pm}$$ from the Debye–Hückel limiting law (see Prob. 12.19). The most accurate method of measuring a solubility product, however, is through the standard cell potential of an appropriate galvanic cell (Sec. 14.3.3).

Since $$K\subs{s}$$ is a thermodynamic equilibrium constant that depends only on $$T$$, and $$\G\subs{r}$$ depends only on $$T$$ and $$p$$, Eq. 12.5.26 shows that any change in the solution composition at constant $$T$$ and $$p$$ that decreases $$\g_{\pm}$$ must increase the solubility. For example, the solubility of a sparingly-soluble salt increases when a second salt, lacking a common ion, is dissolved in the solution; this is a salting-in effect.

Equation 12.5.25 is a general equation that applies even if the solution saturated with one salt contains a second salt with a common ion. For instance, consider the sparingly-soluble salt M$$_{\nu_+}$$X$$_{\nu_-}$$ in transfer equilibrium with a solution containing the more soluble salt M$$_{\nu'_+}$$Y$$_{\nu'_-}$$ at molality $$m\C$$. The common ion in this example is the cation M$$^{z_+}$$. The expression for the solubility product is now \begin{gather} \s{ K\subs{s} = \G\subs{r} \g_{\pm}^{\nu} (\nu_+m\B+\nu'_+m\C)^{\nu_+}(\nu_-m\B)^{\nu_-}/(m\st)^{\nu} } \tag{12.5.27} \cond{(common cation)} \end{gather} where $$m\B$$ again is the solubility of the sparingly-soluble salt, and $$m\C$$ is the molality of the second salt. $$K\subs{s}$$ and $$\G\subs{r}$$ are constant if $$T$$ and $$p$$ do not change, so any increase in $$m\C$$ at constant $$T$$ and $$p$$ must cause a decrease in the solubility $$m\B$$. This is called the common ion effect.

From the measured solubility of a salt in pure solvent, or in an electrolyte solution with a common cation, and a known value of $$K\subs{s}$$, we can evaluate the mean ionic activity coefficient $$\g_{\pm}$$ through Eq. 12.5.26 or 12.5.27. This procedure has the disadvantage of being limited to the value of $$m\B$$ existing in the saturated solution.

We find the temperature dependence of $$K\subs{s}$$ by applying Eq. 12.1.12: $$\frac{\dif\ln K\subs{s}}{\dif T} = \frac{\Delsub{sol,B}H\st}{RT^2} \tag{12.5.28}$$ At the standard pressure, $$\Delsub{sol,B}H\st$$ is the same as the molar enthalpy of solution at infinite dilution, $$\Delsub{sol,B}H^{\infty}$$.

This page titled 12.5: Solid-Liquid Equilibria is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Howard DeVoe via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.