Water is amazing in many ways and it plays a very significant part in the chemistry of acids and bases.

While water molecules are covalent and highly polar in nature, a tiny number of water molecules form ions, as shown in the following animation:

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[To show clearly the ionisation of water and its dynamic equilibrium. Animation and commentary very powerful for this tricky concept.]

Notice the use of, H₃O⁺, the  hydronium ion, being used in place of H⁺ to represent the hydrated hydrogen ion or proton. Even H₃O⁺ is thought to be a simplification and further water molecules are associated with the hydronium ion to give other species, eg H₉O₄⁺. This can be considered as H₃O⁺ being hydrogen bonded to three neighbouring water molecules.

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H₂O_(l) +  H₂O_(l) ∏  H₃O⁺_(aq)  + OH^-_(aq)  

Is H⁺ wrong?

No, it is a simplification and it works well in many cases. However, in more advanced school study of acids and bases it is necessary to use H₃O⁺ in many examples involving equilibria where this is key to understanding the behaviour of particular acids and bases as well as water.

Sometimes H⁺ will be used in this topic where appropriate, for example, in calculations using pH = -log[H⁺] where its meaning is clear.