Have you ever thought about this fact...
If you add an equal amount of acid to an equal amount alkali, the solution you make isn't necessarily neutral.
i.e. when you add the same amount of OH- to H+ the solution isn't always pH 7.
i.e. neutralisation isn't (necessarily) neutral.
Makes no sense, until you start to think about what is in the "neutral" solution.
Lets use an example,
adding ethanoic acid to sodium hydroxide
ethanoic acid + sodium hydroxide --> Sodium ethanoate + water
Ionically, this is -
ethanoic acid + hydroxide ions --> ethanoate ions + water.
So at "neutralisation", the only things in the conical flask are ethanoate ions and water.
Water is obviously neutral so it must be the ethanoate that is making it (in this case) basic.
Ethanoate ions will absorb protons from water making hydroxides, turning the solution basic.
Why does ethanoate do this? Well, that comes down to the fact that ethanoate ions and ethanoic acid are conjugate acid/base pairs. If one is rubbish, the other is awesome.
Ethanoic acid is a weak acid (rubbish) so ethanoate is a relatively strong base (awesome).
So neutralisation is not the best name for this situation where acid cancels out base. A better name is equivalence as we have an equivalent amount of acid and base (but it isn't neutral).
We shall keep the name neutralisation for when the pH is 7. Which is fairly irrelevant from a chemical point of view. Which is a shame considering your chemistry teacher has been obsessing on it since year 7.
(for an explanation of the pig reference - google, animal farm)
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