Activities and the salt effect

Every solubility calculation so far rested on an unspoken assumption: ions do not notice each other, so concentration is the effective concentration. Put it to the test. Gypsum, $\ce{CaSO4*2H2O}$, has $K_\mathrm{sp} = 2.4\times10^{-5}$, so $s = \sqrt{K_\mathrm{sp}}$ predicts 4.9 mmol/L; at 25 °C it actually dissolves to 15.3 mmol/L, three times as much. Add potassium nitrate — which shares no ion with gypsum — and more dissolves still: 21.2 mmol/L in 0.103 M KNO₃. The electrostatic forces between ions cannot be ignored.

The equilibrium constants of thermodynamics ($\Delta G^\circ = -RT\ln K$) are written in activities: $a_i = \gamma_i m_i/m^\circ$, where $\gamma_i$ is the activity coefficient. For a sparingly soluble salt MX, $K_\mathrm{sp} = a_{\ce{M}}a_{\ce{X}} = \gamma_\pm^2[\ce{M}][\ce{X}]$. In ionic solutions $\gamma < 1$, so more salt must dissolve before the activity product reaches $K_\mathrm{sp}$.

$\gamma$ depends on the total “charge concentration” of every ion in the solution — the ionic strength $I$, to which each ion contributes with the square of its charge: 0.10 mol/kg NaCl has $I = 0.10$; $\ce{MgSO4}$ at the same concentration has $I = 0.40$. The activity coefficient of a single ion cannot be measured on its own (a solution is always neutral); experiments give the mean activity coefficient $\gamma_\pm$ of the salt.

Every ion slightly attracts ions of opposite charge and repels those of its own. Thermal motion smears this preference into a diffuse ionic atmosphere whose total charge equals the central ion’s and has the opposite sign, with a thickness given by the Debye length $\kappa^{-1}$: in water at 25 °C, $\kappa^{-1} = 3.04\ \text{Å}/\sqrt{I}$ — 96 Å at 1 mmol/kg, 9.6 Å at 0.1 mol/kg. The ion and its atmosphere attract each other, lowering the ion’s energy and so its chemical potential: that is where $\gamma < 1$ comes from.

Debye and Hückel (1923) turned this picture into formulas. Written with lengths they are especially transparent: $\ln\gamma_\pm = -|z_+z_-|\,l_\text{B}/2(\kappa^{-1} + \mathring{a})$ — the ion feels its atmosphere as an opposite charge placed one Debye length beyond the distance of closest approach $\mathring{a}$. $l_\text{B} = 7.16$ Å is the Bjerrum length, the distance at which two unit charges interact with an energy of exactly $kT$. Every constant follows from the permittivity of water and the temperature; none is fitted.

Back to gypsum. In pure water its own ions give $I = 0.041$ and $\gamma_\pm = 0.48$, which raises the free ions to $\sqrt{K_\mathrm{sp}}/\gamma_\pm = 10.2$ mmol/L — still a third short of the measurement. The rest is ion pairs: $\ce{Ca^2+}$ and $\ce{SO4^2-}$ held together electrostatically as neutral $\ce{CaSO4(aq)}$, with $K = 10^{2.36}$. Having no charge, the pair has an activity coefficient of 1, so its concentration, $K\cdot K_\mathrm{sp} = 5.5$ mmol/L, does not depend on the ionic strength. Together that is 15.7 mmol/L, 2.5 % from the measured 15.3.

In KNO₃ solutions the free ions grow with the ionic strength while the pairs stay put; with nothing fitted, the model reproduces the measured solubility to within 6 % up to 0.10 M. For $\ce{BaSO4}$ the same kind of pair amounts to only $K\cdot K_\mathrm{sp} = 5.5\times10^{-8}$ M, which is negligible: the smaller $K_\mathrm{sp}$, the less ion pairs matter.