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What is it about barium chloride specifically that makes it able to form a precipitate with sulfate ions.

What is the chemistry behind this?

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Not necessarily about $\ce{BaCl2}$ itself, but when treated with a more reactive compound (containing $\ce{SO4^2-}$) ions such as sulfuric acid, the ($\ce{Cl^-}$) ions in $\ce{BaCl2}$ are displaced with the $\ce{SO4^2-}$ essentially forming an insoluble precipitate:

$$\ce{BaCl2(aq) + [SO4]^2-(aq) -> BaSO4(s) +2Cl^-(aq)}$$

$\ce{BaSO4}$ is a sparingly soluble salt ($K_{\mathrm{sp}} = 1.07 \times 10^{-10}$).

(The solubility of the ionic salt is extremely small, such that the saturated solution has very few ions)

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