Is ionisation of acids and bases in water is a decomposition reaction?

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No it is not.

A decomposition reaction is one in which the products on the right side of the chemical equation are different than the reactant on the right side of the chemical equation. There is a chemical difference between the reactants and products. In addition, a key factor in recognizing a decomposition reaction is that there are more products than reactants.

An example of a decomposition reaction is the breakdown of water into hydrogen and oxygen.  2 HOH -->  2 H2  + O2.

Another example is the breakdown of potassium chlorate into potassium chloride and oxygen.   2 KClO3  -->  2 KCl  + 3 O2

In the ionization process, a compound is added to water and the water separates the positive and negative ions in the compound.  This is a physical separation and when the water is removed you still have the same compound you started with.

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