Thermodynamic analysis of solubility data 1: phase diagrams of systems salt hydrate + water

Abstract

Solubility equilibria between solid salts, salt hydrates, and water have been analyzed on a thermodynamic basis. Fitting equations with three or four adjustable parameters suffice to describe the temperature dependence of solubility adequately when the latter is expressed in ionic (species) mole fractions. Phase diagrams, temperature versus mole fractions, summarize and deal concisely with equilibrium solubility properties of the systems. Comparison of experimental and estimated ratios of slopes to liquidus curves intersecting at isobaric invariants, such as, for example, eutectic or peritectic points, provides a thermodynamic consistency test.

Graphical Abstract

Appendix

The slope ratio for a peritectic point has been given in Eq. (25). Equations for other cases are given here.

Slope ratio at a eutectic point 1 (freezing curve and solubility curve intersect)

$$ \left( {\frac{{\sigma_{{{\text{sln}} /{\text{SrW}}}} }}{{\sigma_{{{\text{sln}} /{\text{ice}}}} }}} \right)_{\text{eut 1}} = - \left( {\frac{ 1- (r + 1 )x}{x}} \right)\left\langle {\frac{{\frac{{\Updelta_{\text{fus}} S_{\text{W}}^{{-}\!\!\!\!\hbox{o}} (T)}}{R} - \ln \left[ {\frac{x}{1 + (\nu - 1)x}} \right] - \frac{\text{d}}{{{\text{d}}T}}[T\ln (f_{\text{W}} )]_{\text{ice}} }}{{\frac{{\Updelta_{\text{sln}} S_{\text{SrW}}^{{-}\!\!\!\!\hbox{o}} (T)}}{R} - \ln \left\{ {\frac{{(\nu_{\pm} x)^{v} (1 - x)^{r} }}{{[1 + (\nu - 1)x]^{v + r} }}} \right\} - \frac{\text{d}}{{{\text{d}}T}}[T\ln (f^{v}_{\pm} f_{\text{W}}^r )]_{\text{SrW}} }}} \right\rangle $$

(35)

When experimental and theoretical slopes neglecting the f terms differ considerably but their ratios agree, the following relationship must hold:

$$ \left( {\frac{{\sigma_{\text{ice/sln}} }}{{\sigma_{\text{sln/SrW}} }}} \right)_{\text{eut 1}} \approx - \frac{{\left[ {\frac{x}{1 - (1 + r)x}} \right]\left\langle {\frac{{\Updelta_{{{\text{sln}} }} S_{\text{SrW}}^{{-}\!\!\!\!\hbox{o}} (T)}}{R} - \ln \left\{ {\frac{{(\nu_{\pm} x)^{v} (1 - x)^{r} }}{{[1 + (\nu - 1)x]^{v + r} }}} \right\}} \right\rangle }}{{\frac{{\Updelta_{\text{fus}} S_{\text{W}}^{{-}\!\!\!\!\hbox{o}} }}{R} - \ln \left[ {\frac{(1 - x)}{1 + (\nu - 1)x}} \right]}} $$

(36)

$$ \left( {\frac{{\sigma_{\text{ice/sln}} }}{{\sigma_{\text{sln/SrW}} }}} \right)_{\text{eut 1}} \approx \frac{{\frac{\text{d}}{{{\text{d}}T}}\left[ {T\ln (f^{v}_{\pm} f_{\text{W}}^{r} )} \right]}}{{\frac{\text{d}}{{{\text{d}}T}}(T\ln f_{\text{W}} )}} $$

(37)

A quite similar result is obtained when the slope ratio $\frac{{\sigma_{\text{ice/sln}} }}{{\sigma_{\text{sln/SrW}} }}$ is estimated by Eq. (38), the enthalpy version of Eq. (36). Again agreement between the experimental and theoretical slope ratio is much better than between experimental and theoretical individual slopes.

$$ \left( {\frac{{\sigma_{\text{ice/sln}} }}{{\sigma_{\text{sln/SrW}} }}} \right)_{\text{eut 1}} \approx - \frac{{x\Updelta_{\text{sln}} H_{\text{SrW}}^{{-}\!\!\!\!\hbox{o}} }}{{[1 - (r + 1)x]\Updelta_{\text{fus}} H_{\text{W}}^{{-}\!\!\!\!\hbox{o}} }} $$

(38)

Slope ratio at a eutectic point 2 (solubility curves of anhydrous solid and salt hydrate intersect)

In this case the binary compound has a stable melting point, and the intersection occurs on the high mole fraction side of the composition of the binary compound.

$$ \left( {\frac{{\sigma_{{{\text{sln}}/{\text{SrW}}}}}}{{\sigma_{{{\text{sln}}/{\text{S}}}} }}} \right)_{\text{eut 2}} = \left( {\frac{ 1- (r + 1 )x}{1-x}} \right)\left\langle {\frac{{\frac{{\Updelta_{{{\text{sln }}}}S_{\text{S}}^{{-}\!\!\!\!\hbox{o}} (T)}}{R} - \nu \ln \left[ {\frac{(\nu_{\pm} x)}{1 + (\nu - 1)x}} \right] - \frac{\text{d}}{{{\text{d}}T}}[T\ln (f_{\pm})]_{\text{s}} }}{{\frac{{\Updelta_{{{\text{sln}} }} S_{\text{SrW}}^{{-}\!\!\!\!\hbox{o}} (T)}}{R} - \ln \left\{ {\frac{{(\nu_{\pm} x)^{v} (1 - x)^{r} }}{{[1 + (\nu - 1)x]^{v + r} }}} \right\} - \frac{\text{d}}{{{\text{d}}T}}[T\ln (f^{v}_{\pm} f^{r}_{\text{W}} )]_{\text{SrW}} }}} \right\rangle $$

(39)

When experimental and theoretical slopes neglecting the f terms differ considerably but their ratios agree at least approximately, the error is partly compensated by Eqs. (40) and (41), which are analogous to Eqs. (36) and (37):

$$ \left( {\frac{{\sigma_{{{\text{sln}} /{\text{SrW}}}} }}{{\sigma_{{{\text{sln}} /{\text{S}}}} }}} \right)_{\text{eut}} \approx \left( {\frac{ 1- (r + 1 )x}{1-x}} \right)\left\langle {\frac{{\frac{{\Updelta_{{{\text{sln }}}} S_{S}^{{-}\!\!\!\!\hbox{o}} (T)}}{R} - \nu \ln \left[ {\frac{(\nu_{\pm} x)}{1 + (\nu - 1)x}} \right]}}{{\frac{{\Updelta_{{{\text{sln }}}} S_{\text{SrW}}^{{-}\!\!\!\!\hbox{o}} (T)}}{R} - \ln \left\{ {\frac{{(\nu_{\pm} x)^{v} (1 - x)^{r} }}{{[1 + (\nu - 1)x]^{v + r} }}} \right\}}}} \right\rangle $$

(40)

$$ \left( {\frac{{\sigma_{{{\text{sln}} /{\text{SrW}}}} }}{{\sigma_{{{\text{sln}} /{\text{S}}}} }}} \right)_{\text{eut}} \approx \frac{\nu{\frac{\text{d}}{{{\text{d}}T}}[T\ln ({{f_{\pm}}} )]_{\text{S}} }}{{\frac{\text{d}}{{{\text{d}}T}}[T\ln (f^{\nu}_{\pm} f_{\text{W}}^r )]_{\text{SrW}} }} $$

(41)

For the system LiNO₃ + H₂O Eq. (42), the enthalpy version of Eq. (40) is not applicable, because $\Updelta_{\text{sln}} H_{\text{S}}^{{-}\!\!\!\!\hbox{o}}$ is exothermic (see Table 1), predicting the same sign for $\sigma_{\text{sln/SrW}}$ and $\sigma_{\text{sln/S}}$.

$$ \frac{{\sigma_{\text{sln/SrW}} }}{{\sigma_{\text{sln/S}} }} \approx - \frac{{[1 - (r + 1)x]\Updelta_{\text{sln}} H_{\text{S}}^{{-}\!\!\!\!\hbox{o}} }}{{(1 - x)\Updelta_{\text{sln}} H_{\text{SrW}}^{{-}\!\!\!\!\hbox{o}} }} $$

(42)