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When a double displacement reaction occurs with compound AB and CD, what determines that B and C are going to bond instead of A and C?

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Double displacement reactions involve ionic compounds, i.e. salts formed from positive cations and negative anions. Consider the following setup, $$\ce{\color{blue}{A^{+}}\color{red}{B^{-}} + \color{blue}{C^{+}}\color{red}{D^{-}} -> \color{blue}{A^{+}}\color{red}{D^{-}} + \color{blue}{C^{+}}\color{red}{B^{-}}}$$ where the cations are indicated in blue and anions in red (and all charges are $\pm 1$ to reflect your question). All salts are comprised of cations and anions, hence $\ce{\color{blue}{AC}}$ wouldn't form since both are cations. $\ce{\color{blue}{C}\color{red}{B}}$ would form instead.

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