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I'm curious but in a double displacement reaction I know that the anions are swapped around and normally a precipitate is formed, but why do the ions swap? Has it got something to do with electrical charges?

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They don’t really swap; the swapping of ions is a way to simplify things for the beginner — and I simplification I do not necessarily consider to be necessary or good.

What actually happens: the ions in a double displacement reaction are typically in solution. In solution, the salts are no longer salts but separated into their ions. For example with sodium chloride:

$$\ce{NaCl(s) ->[H2O] Na+ (aq) + Cl- (aq)}\tag{1}$$

I could draw the same thing for silver nitrate:

$$\ce{AgNO3 (s) ->[H2O] Ag+ (aq) + NO3- (aq)}\tag{2}$$

Now if these two solutions are brought together, we initially have all four ions swimming around. But silver and chloride will form a badly soluble solid under these conditions. So, as soon as silver ions and chloride ions meet, the reverse reaction will happen:

$$\ce{Ag+ (aq) + Cl- (aq) ->[H2O] AgCl (s) v}\tag{3}$$

In the solution there are still the nitrate ions and the sodium ions which just weren’t affected by this process and see no need to take part.

Formally, considering your reactants and products, it seems as if the ions had been swapped. But that is not really the case; the final solution only contains those ions that did not precipitate as a solid together.

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