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Are there any sodium salts (e.g. NaCl) that are insoluble in water? If so, what are their approximate $K_\mathrm{sp}$ values?

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If your definition of a "sodium salt" is any solid compound that contains monovalent cationic sodium, then all you need to do is to look below your feet. The earth has abundant sodium bearing minerals (aka "salts").

Scroll down in The Mineralogy of Sodium to where it says Most widespread minerals containing Sodium. Out of all of these, only halite is soluble. The rest are mostly insoluble. I'm not sure their ksp values will mean anything here, because the dissolution of these minerals is up to kinetics and not thermodynamics at STP.

You can also have a look at Mineral Species containing Sodium (Na). Other than a few that are soluble (particularly some of the first on the list), the rest are not that soluble. Two reasons for that - they are silicates or phosphates which are in general insoluble, and they have other things in them.

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Sodium tends to form water-soluble compounds such as halides, sulfate, nitrate, carboxylate and carbonates. There are only isolated examples of sodium compounds precipitating from water solution. However, nature provides examples of many insoluble sodium compounds such as the feldspars (aluminum silicates of sodium, potassium and calcium), sodium bismuthate ($\ce{NaBiO3}$), sodium octamolybdate ($\ce{Na2Mo8O25·4H2O}$), sodium thioplatinate ($\ce{Na4Pt3S6}$), sodium uranate ($\ce{Na2UO4}$). Both form of sodium antimonate (meta - $\ce{NaSbO3.7H2O}$, pyro - $\ce{Na2H2Sb2O7.H2O}$) has a solubility 0.7g/L. Sodium metaphosphate, $\ce{NaPO3}$ has both soluble and insoluble form.

Ref.: The Elements, PediaPress

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  • $\begingroup$ Also, sodium fluorosilicate $\endgroup$
    – user85778
    Commented Aug 16, 2023 at 19:47
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Cryolite, $\ce{Na3AlF6}$, is one such compound. https://en.wikipedia.org/wiki/Cryolite

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The salt $\ce{NaK2AlF6}$ ("elpasolite") is insoluble. Its exothermic precipitation is used as the basis for the thermometric titrimetric quantitative determination of sodium in a wide range of sample matrices. The determination is fast, accurate and highly precise.

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$\ce{Na3[FeF6]}$ and $\ce{Na2H2Sb2O7}$ are white precipitates when $\ce{Na+}$ is dissolved with $\ce{[FeF6]^{3-}}$ and $\ce{K2H2Sb2O7}$ respectively.

Hence, these reagents are used in the qualitative analysis of $\ce{Na+}$ ions.

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