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What would happen if I added $\ce{Na2CO3 \cdot n(H2O)}$ to a solution of $\ce{HCl}$? Could I consider the reaction the same as just $\ce{Na2CO3 + HCl -> NaCl + CO2 + H2O}$?

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There is no difference in properties of salt hydrates and non-hydrates after they are dissolved in water, as they are hydrated to same state. So, yes, you can.

However, there is difference in reaction of salt hydrates and non-hydrates outside water solutions.

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