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I'm a bit confused about amphiprotic species – specifically why some species can be amphiprotic and others can't.

For instance, $\ce{SH-}$ is an amphiprotic species (in aqueous solution).

$$ \begin{align} &\text{Acid:} &\ce{SH- + H2O &<=> S^2- + H3O+}\\ &\text{Base:} &\ce{SH- + H2O &<=> H2S + OH-} \end{align} $$

On the other hand, isn't $\ce{HNO3}$ an amphiprotic species too? I don't understand why $\ce{HNO3}$ isn't amphiprotic whilst $\ce{HS-}$ is; they seem to both be able to have an acid and a base reaction:

$$ \begin{align} &\text{Acid:} &\ce{HNO3 + H2O &<=> NO3- + H3O+}\\ &\text{Base:} &\ce{HNO3 + H2O &<=> H2NO3+ +OH-} \end{align} $$

I hope this makes sense.

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    $\begingroup$ I guess it ultimately comes down to circumstance, consider superacids, some of which can protonate methane. Usually, one thinks about diluted aqueous solutions when talking about amphiprotic species. $\ce{HNO3}$ undergoes a base reaction like you listed in the course of nitration of aromatics such as benzene. $\endgroup$ – TAR86 Jan 25 '18 at 11:55
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Amphiprotic are species that have both acidic and basic properties. A classic example of an amphiprotic ion is dihydrogen phosphate $\ce{H2PO4^-}$, which reacts in the presence of a $\ce{H3O+}$ as $$\ce{H2PO4^- + H_3O^+ <=> H_3PO4},$$ where $\ce{H3PO4}$ is the conjugate acid of the original base. In the presence of an $\ce {OH-} $ the $\ce{H2PO4-}$ reacts in the following way $$\ce {H2PO4^- + OH- <=> HPO4^2- + H2O}.$$

Aminoacids are amphiprotic, as they contain both, a weak acid and a weak base functional group. Also water behaves amphiprotic.

The second reaction you wrote, even if in theory possible does not happen since $\ce{HNO3}$ is such a strong acid, that in water is totally dissociated and has no tendency to behave as a base.

You can check this by writing down the general equilibria and calculate the equilibrium constant $K$ of each reaction you wrote. The $K_\mathrm{a}$ of nitric acid is $2.4 \times 10^1$. This means that, since $$K_\mathrm{a} \times K_\mathrm{b} = K_\mathrm{w} = \pu{1.0*10^{-14}}$$ the $K_\mathrm{b}$ of nitric acid is at least in the order of $10^{-15},$ which gives you an idea why this reaction won't happen.

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