12.5: Solid-Liquid Equilibria
A freezing-point curve (freezing point as a function of liquid composition) and a solubility curve (composition of a solution in equilibrium with a pure solid as a function of temperature) are different ways of describing the same physical situation. Thus, strange as it may sound, the composition \(x\A\) of an aqueous solution at the freezing point is the mole fraction solubility of ice in the solution.
12.5.1 Freezing points of ideal binary liquid mixtures
Section 12.2.1 described the use of freezing-point measurements to determine the solvent chemical potential in a solution of arbitrary composition relative to the chemical potential of the pure solvent. The way in which freezing point varies with solution composition in the limit of infinite dilution was derived in Sec. 12.4.1. Now let us consider the freezing behavior over the entire composition range of an ideal liquid mixture.
Let \(T'\subs{f}\) be the freezing point of a liquid mixture of composition \(x'\A\) and \(x'\B=1-x'\A\), and let \(T''\subs{f}\) be the melting point of the solid compound of composition \(x''\A=a/(a+b)\) and \(x''\B=b/(a+b)\). Figure 12.7 shows an example of a molten metal mixture that solidifies to an alloy of fixed composition. The freezing-point curve of this system is closely approximated by Eq. 12.5.23.
12.5.5 Solubility of a solid electrolyte
Consider an equilibrium between a crystalline salt (or other kind of ionic solid) and a solution containing the solvated ions: \[ \tx{M\(_{\nu_+}\)X\(_{\nu_-}\)(s)} \arrows \nu_+\tx{M\(^{z_+}\)(aq)} + \nu_-\tx{X\(^{z_-}\)(aq)} \] Here \(\nu_+\) and \(\nu_-\) are the numbers of cations and anions in the formula unit of the salt, and \(z_+\) and \(z_-\) are the charge numbers of these ions. The solution in equilibrium with the solid salt is a saturated solution. The thermodynamic equilibrium constant for this kind of equilibrium is called a solubility product , \(K\subs{s}\).
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: \begin{equation} K\subs{s} = \frac{a\mbB}{a\B^*} \tag{12.5.24} \end{equation} 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 \begin{equation} 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} \end{equation} 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: \begin{equation} \frac{\dif\ln K\subs{s}}{\dif T} = \frac{\Delsub{sol,B}H\st}{RT^2} \tag{12.5.28} \end{equation} At the standard pressure, \(\Delsub{sol,B}H\st\) is the same as the molar enthalpy of solution at infinite dilution, \(\Delsub{sol,B}H^{\infty}\).