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This is a simple concept that I can't seem to understand. Why will a strong acid neutralize as much base as a weak acid, if the acids are of the same volume and concentration? A strong acid will dissociate more in solution and thus have a greater number of $\ce{H^+}$ ions, as far as my understanding goes. The opposite if true for the weak base. So wouldn't the strong acid neutralize more base than the weak acid, because it has more $\ce{H^+}$ ions to neutralize the $\ce{OH^-}$ ions of the base with?

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Strong acid is dissociated all but completely: you have a certain amount of $\ce H^+$ and start neutralizing them, one drop at a time. As you do so, their numbers steadily decrease. It feels like sawing a huge log with a hand saw: you see the amount of job, and you do it until it is all done.

Weak acid is dissociated only partially. You see a relatively small amount of $\ce H^+$, but as you start neutralizing them, more molecules of acid get dissociated and their $\ce H^+$ ions stand in place of those you've neutralized. It feels like sawing a thin log, but a peculiar one: it seems to regenerate under the saw.

So at the end of the day, it is the sheer amount of acid that matters, rather than immediate concentration of $\ce H^+$.

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The hydroxide ions will take hydrogens from any type of substance that can donate a proton, whether it is dissociated or not. In fact, even amphoteric substances that tend to have a more basic character will neutralize a solution of strong base to some degree. Therefore, it doesn't really matter whether the acid is weak or strong.

This also means that polyprotic acids will neutralize more of a strong a base in lower concentrations than strong monoprotic acids, since they have more hydrogens donate.

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