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I know that sulfuric acid is considered a strong acid, and the dissociation of its first $\ce{H+}$ ion is complete and of its second $\ce{H+}$ ion is partial.

Phosphoric acid is weak and all three of its $\ce{H+}$ ions dissociate partially.

I just wanted to clarify: are dibasic and tribasic acids considered strong if their first dissociation is complete (and subsequent ones will naturally be partial)?

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I mean whenever you have a polyprotic acid/base and they say that if it is strong or weak you can only assume the first dissociation is strong or weak.

Say like you have $\ce{H2SO4}$ and we all know that it is a strong acid. This only means that the first dissociation is strong.

$$\ce{H2SO4 -> HSO4- + H3O+}$$

So in this case above only the first dissociation is strong. And then you cannot and do not know if $\ce{HSO4}$ is strong as well.

So overall in any case we cannot assume and the data must be provided. Good Question.

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If you think about each dissociate state separately, than it's easy to answer your question. $\ce{H2SO4}$ is a very strong acid $(\mathrm{p}K_\mathrm{a}\approx -3.0),$ but $\ce{HSO4-}$ is not so strong $(\mathrm{p}K_\mathrm{a}\approx 2.0).$

Phosphoric acid starts with $\mathrm{p}K_\mathrm{a}\approx 2.14,$ which as as you said is not so strong, and $\ce{H2PO4-}$ and $\ce{HPO4^2-}$ have $\mathrm{p}K_\mathrm{a}\approx 7.2$ and $\mathrm{p}K_\mathrm{a}\approx 12.3,$ so definetly weak.

So yes, we say sulfuric acid is strong because we're talking about $\ce{H2SO4},$ and what comes after should be mentioned separately.

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