11.4: Enthalpies of Solution and Dilution
The processes of solution (dissolution) and dilution are related. The IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry , 3rd edition, RSC Publishing, Cambridge, 2007, Sec. 2.11.1) recommends the abbreviations sol and dil for these processes.
For an electrolyte solute, a plot of \(\Del H\m\solmB\) versus \(m\B\) has a limiting slope of \(+\infty\) at \(m\B{=}0\), whereas the limiting slope of \(\Del H\m\solmB\) versus \(\sqrt{m\B}\) is finite and can be predicted from the Debye–Hückel limiting law. Accordingly, a satisfactory procedure is to plot \(\Del H\m\solmB\) versus \(\sqrt{m\B}\), perform a linear extrapolation of the experimental points to \(\sqrt{m\B}{=}0\), and then shift the origin to the extrapolated intercept. The result is a plot of \(\varPhi_L\) versus \(\sqrt{m\B}\). An example for aqueous NaCl solutions is shown in Fig. 11.10(a).
We can also evaluate \(\varPhi_L\) from experimental enthalpies of dilution. From Eqs. 11.4.10 and 11.4.22, we obtain the relation \begin{equation} \varPhi_L(m\B'')-\varPhi_L(m\B') = \Del H\m(\tx{dil, \(m\B'{\ra}m\B''\)}) \tag{11.4.25} \end{equation} We can measure the enthalpy changes for diluting a solution of initial molality \(m\B'\) to various molalities \(m\B''\), plot the values of \(\Del H\m(\tx{dil, \(m\B'{\ra}m\B''\)})\) versus \(\sqrt{m\B}\), extrapolate the curve to \(\sqrt{m\B}{=}0\), and shift the origin to the extrapolated intercept, resulting in a plot of \(\varPhi_L\) versus \(\sqrt{m\B}\).
In order to be able to use Eq. 11.4.23, we need to relate the derivative \(\dif\varPhi_L/\dif m\B\) to the slope of the curve of \(\varPhi_L\) versus \(\sqrt{m\B}\). We write \begin{equation} \dif \sqrt{m\B} = \frac{1}{2\sqrt{m\B}}\dif m\B \qquad \dif m\B = 2\sqrt{m\B} \dif\sqrt{m\B} \tag{11.4.26} \end{equation} Substituting this expression for \(\dif m\B\) into Eq. 11.4.23, we obtain the following operational equation for evaluating \(L\B\) from the plot of \(\varPhi_L\) versus \(\sqrt{m\B}\): \begin{gather} \s{ L\B = \varPhi_L + \frac{\sqrt{m\B}}{2} \frac{\dif\varPhi_L}{\dif\sqrt{m\B}} } \tag{11.4.27} \cond{(constant \(T\) and \(p\))} \end{gather}
The value of \(\varPhi_L\) goes to zero at infinite dilution. When the solute is an electrolyte, the dependence of \(\varPhi_L\) on \(m\B\) in solutions dilute enough for the Debye–Hückel limiting law to apply is given by \begin{gather} \s{ \varPhi_L = C_{\varPhi_L}\sqrt{m\B} } \tag{11.4.28} \cond{(very dilute solution)} \end{gather} For aqueous solutions of a 1:1 electrolyte at \(25\units{\(\degC\)}\), the coefficient \(C_{\varPhi_L}\) has the value \begin{equation} C_{\varPhi_L} = 1.988\timesten{3}\units{J kg\(^{1/2}\) mol\(^{-3/2}\)} \tag{11.4.29} \end{equation} (The fact that \(C_{\varPhi_L}\) is positive means, according to Eq. 11.4.25, that dilution of a very dilute electrolyte solution is an exothermic process.) \(C_{\varPhi_L}\) is equal to the limiting slope of \(\varPhi_L\) versus \(\sqrt{m\B}\), of \(\Del H\m\solmB\) versus \(\sqrt{m\B}\), and of \(\Del H\m(\tx{dil, \(m\B'{\ra}m\B''\)})\) versus \(\sqrt{m'\B}\). The value given by Eq. 11.4.29 can be used for extrapolation of measurements at \(25\units{\(\degC\)}\) and low molality to infinite dilution.
Equation 11.4.28 can be derived as follows. For simplicity, we assume the pressure is the standard pressure \(p\st\). At this pressure \(H\B^\infty\) is the same as \(H\B\st\), and Eq. 11.4.17 becomes \(L\B=H\B-H\B\st\). From Eqs. 12.1.3 and 12.1.6 in the next chapter, we can write the relations \begin{equation} H\B=-T^2\bPd{(\mu\B/T)}{T}{p,\allni} \qquad H\B\st=-T^2\frac{\dif(\mu\mbB\st/T)}{\dif T} \tag{11.4.30} \end{equation} Subtracting the second of these relations from the first, we obtain \begin{equation} H\B-H\B\st = -T^2\bPd{(\mu\B-\mu\mbB\st)/T}{T}{p,\allni} \tag{11.4.31} \end{equation} The solute activity on a molality basis, \(a\mbB\), is defined by \(\mu\B-\mu\mbB\st=RT\ln a\mbB\). The activity of an electrolyte solute at the standard pressure, from Eq. 10.3.10, is given by \(a\mbB = (\nu_{+}^{\nu_{+}} \nu_{-}^{\nu_{-}}) \g_{\pm}^{\nu} (m\B/m\st)^{\nu}\). Accordingly, the relative partial molar enthalpy of the solute is related to the mean ionic activity coefficient by \begin{equation} L\B=-RT^2\nu\Pd{\ln\g_{\pm}}{T}{\!\!p,\allni} \tag{11.4.32} \end{equation}
We assume the solution is sufficiently dilute for the mean ionic activity coefficient to be adequately described by the Debye–Hückel limiting law, Eq. 10.4.8: \(\ln\g_{\pm} = -A\subs{DH}\left|z_+z_-\right|\sqrt{I_m}\), where \(A\subs{DH}\) is a temperature-dependent quantity defined in Sec. 10.4. Then Eq. 11.4.32 becomes \begin{gather} \s{ L\B=RT^2\nu\left|z_+z_-\right|\sqrt{I_m}\Pd{A\subs{DH}}{T}{\!\!p,\allni} } \tag{11.4.33} \cond{(very dilute solution)} \end{gather} Substitution of the expression given by Eq. 10.4.9 for \(I_m\) in a solution of a single completely-dissociated electrolyte converts Eq. 11.4.33 to \begin{gather} \s{ L\B = \left[ \frac{RT^2}{\sqrt{2}}\Pd{\rho\A^*A\subs{DH}}{T}{p,\allni} \left(\nu\left|z_+z_-\right|\right)^{3/2} \right]\sqrt{m\B} = C_{L\B}\sqrt{m\B} } \tag{11.4.34} \cond{(very dilute solution)} \end{gather} The coefficient \(C_{L\B}\) (the quantity in brackets) depends on \(T\), the kind of solvent, and the ion charges and number of ions per solute formula unit, but not on the solute molality.
Let \(C_{\varPhi_L}\) represent the limiting slope of \(\varPhi_L\) versus \(\sqrt{m\B}\). In a very dilute solution we have \(\varPhi_L = C_{\varPhi_L}\sqrt{m\B}\), and Eq. 11.4.27 becomes \begin{equation} L\B = \varPhi_L + \frac{\sqrt{m\B}}{2} \frac{\dif\varPhi_L}{\dif\sqrt{m\B}} = C_{\varPhi_L}\sqrt{m\B} + \frac{\sqrt{m\B}}{2} C_{\varPhi_L} \tag{11.4.35} \end{equation} By equating this expression for \(L\B\) with the one given by Eq. 11.4.34 and solving for \(C_{\varPhi_L}\), we obtain \(C_{\varPhi_L}=(2/3)C_{L\B}\) and \(\varPhi_L = (2/3)C_{L\B}\sqrt{m\B}\).