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$$\ce{HCO3- (aq) + H2O(l) <=> H2CO3(aq) +OH- (aq)}$$

I do not understand why this makes an Arrhenius base and why it doesn't form: $$\ce{HCO3- (aq) + H2O(l) <=> CO3^2- (aq) +H3O+ (aq)}$$

Could someone explain this to me?

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Both of these equations are valid, as $\ce{HCO3-}$ is amphoteric (i.e. both an acid and a base).

To answer your question, one has to consider that the $\mathrm{p}K_\text{a}$ value for $\ce{H2CO3}$ is 6.37 and that for $\ce{HCO3-}$ is 10.32. So if we started from "pure" $\ce{HCO3-}$ (let's assume sodium bicarbonate) and add it to water, what would happen?

The $\mathrm{p}K_\text{b}$ for $\ce{HCO3-}$ is of $14 - 6.37 = 7.63$, it is larger than the $\mathrm{p}K_\text{a}$ for the acidic function (10.32). Therefore although both a weak base and a weak acid, it is a stronger base than acid, increasing the pH when added in water.

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To sum it up, we know that $\ce{H2CO3}$ is an acid, so for $\ce{HCO3-}$, just look at the $\ce{pK_{a}}$ and the $\ce{pK_{b}}$ values. Pick the greatest one and do it from there.

$\ce{pK_{a}}$ = ${7.6}$ $\ce{pK_{b}}$ = ${6.4}$

So now we know that the entity gains an $\ce{H+}$.

Thanks

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