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Usually, in a nucleophilic substitution reaction, a stronger nucleophile replaces a weaker nucleophile.

In any polar protic solvent, the order of nucleophilicity is $\ce{F-}<\ce{Cl-}<\ce{Br-}<\ce{I-}$ (due to stronger solvation of the smaller anions).

While in any polar aprotic solvent, the order of nucleophilicity is $\ce{F-}>\ce{Cl-}>\ce{Br-}>\ce{I-}$.

Why does the Finkelstein reaction use acetone in spite of the fact that it is a polar aprotic solvent? Wouldn't using a polar protic solvent be better since it would make $\ce{I-}$ the strongest nucleophile of the group.

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The key point is driving the equilibrium where you want it. You could do it by relative reaction rates. The Finkelstein reaction, however, uses a better method: precipitation of one of the products. Acetone is specifically chosen because sodium chloride, bromide and fluoride are insoluble in it while sodium iodide is soluble. So the equation becomes:

$$\ce{R-X + Na+ ({acet.}) + I- ({acet.}) -> R-I + NaX v}\tag{1}$$

Nothing beats the equilibrium-driving force of precipitation.

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