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Why isn't $\ce{CHCl3}$ soluble in water?

Why doesn't $\ce{CHCl3}$ participate in hydrogen bonding in water like $\ce{CH3CN}$ as $\ce{Cl}$ can make the $\ce{C-H}$ bond polar like $\ce{CN}$ does?

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Chloroform is soluble in water, though only slightly (~1g/100ml in cold water forming a slightly sweet liquid whose mild anaesthetic affects made it a recreational substance in Victorian times before its toxicity was fully recognised).

Moreover, its hydrogen will form (weak) hydrogen bonds which is probably a better explanation for the solubility in water than the polarity of the molecule.

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