Reversed size dependence in ion solubility

My book says that solubility increases with interionic distance, because the attractive forces between ions gets smaller. However, if one of the ions in a binary ionic system is very large compared to the other, the trend is reversed. For example, CsF has greater solubility than CsCl. Why is this?

• Lattice enthalpy inversely depends on size difference.also lattice enthalpy inversely related to solubility. Thus as size difference increases solubility increases
– Bbr
Nov 27 '17 at 14:04

This can be explained with some crude thermodynamic models. The lattice enthalpy $$\Delta H_\mathrm L$$ of an alkali metal halide $$\ce{MX}$$ is given by

$$\Delta H_\mathrm L = \frac{k_1}{r_\ce{M} + r_\ce{X}}$$

and the hydration enthalpies by

$$\Delta H_\mathrm S(\ce{M}) = -\frac{k_2}{r_\ce{M}}\qquad \qquad \Delta H_\mathrm S(\ce{X}) = -\frac{k_2}{r_\ce{X}}$$

where $$k_1$$ and $$k_2$$ are some positive constants and $$r_\ce{M}$$ and $$r_\ce{X}$$ are the ionic radii of $$\ce{M+}$$ and $$\ce{X-}$$ respectively.

The solubility is therefore a balance between these two quantities: larger (magnitudes of) solvation enthalpies increase solubility, but larger lattice enthalpies reduce solubility. The solubility $$S$$ is thus related to the difference of these terms. We ignore the entropic contribution, or to be precise, we assume that $$\Delta S$$ does not vary too much between different salts $$\ce{MX}$$.

$$S \sim \frac{k_2}{r_\ce{M}} + \frac{k_2}{r_\ce{X}} - \frac{k_1}{r_\ce{M} + r_\ce{X}}$$

Let us now assume that

$$r_\ce{M} = a+b \qquad \qquad r_\ce{X} = a - b$$

where $$a,b$$ are more positive constants. Here we have implicitly assumed that $$r_\ce{M} \geq r_\ce{X}$$, but it works the other way round too, so there is no loss of generality. Thus

\begin{align} S &\sim \frac{k_2}{a+b} + \frac{k_2}{a-b} - \frac{k_1}{2a} \\ &= \frac{2ak_2}{a^2-b^2} - \frac{k_1}{2a} \end{align}

This is clearly a minimum when $$b = 0$$, i.e. $$r_\ce{M} = r_\ce{X}$$. So, salts with similar cationic and anionic radii are less soluble, and vice versa.