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What kind of chemical reaction is $ AgN{O_3} + NaCl \to AgCl + NaN{O_3} $ $ ? $

Answer
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Hint: First we have to know that a chemical reaction is a process in which one or more chemical substances (reactants) are converted to one or more different substances (products). The chemical reactions are basically divided into six types, namely combination, decomposition, single-replacement, double-replacement, precipitation and combustion.

Complete Step By Step Answer:
Double displacement reaction is a chemical reaction where two compounds react by exchanging ions to produce two new compounds. In double replacement reactions, the positive ions (cations) exchange negative ion partners. Many double displacement reactions occur between ionic compounds that are dissolved in water. The general equation of double replacement reaction is
 $ AB + CD \to AD + CB $
Given reaction is
 $ AgN{O_3} + NaCl \to AgCl + NaN{O_3} $
Since the given reactants $ AgN{O_3} $ , $ NaCl $ are ionic compounds and there is an exchange of ions between the reactants. In this reaction, $ AgCl $ is the precipitate. Thus, the given chemical reaction is a double displacement reaction.
Hence the reaction is both precipitation reaction as well as double displacement reaction.

Additional Information:
Combination reaction is a reaction in which two or more reactants combine to form a single product. It is also known as a synthesis reaction.
Example: $ 2Na + C{l_2} \to 2NaCl $
Decomposition reaction is a reaction in which a single compound breaks into two or more simpler compounds.
Example: $ CaC{O_3} \to CaO + C{O_2} $
Precipitation Reaction is a reaction that involves the formation of an insoluble product.
Their reactants are soluble, but the product formed is insoluble and separates out as a solid.

Note:
Double displacement reaction is also called a metathesis reaction. The product obtained in a double displacement reaction is either a gas or solid which can be separated from the reaction mixture. Also, double displacement reactions can be further classified as neutralization and precipitation reactions.