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For example, if there is one mole of carbonate ions ($\ce{CO3 ^2-}$) and sulfate ions ($\ce{SO4 ^2-}$) in a solution, and one mole of hydrochloric acid ($\ce{HCl}$) is added, will the one mole of hydrochloric acid react just with the sulfate ions as that is stronger, forming hydrogen sulfate ions?
Or will the acid react with both the ions forming hydrogen carbonate ions ($\ce{HCO3-}$) and hydrogen sulfate ions ($\ce{HSO4-}$)?

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    $\begingroup$ The key concept you need to understand is that acid-base reactions are relatively fast, so you'll do whatever the equilibrium for the mixture dictates. $\endgroup$ – Zhe Mar 24 '17 at 1:19
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    $\begingroup$ Sulfate ions are much weaker than carbonate ions. Based on my answer below this implies that even if the acid reacts with both bases, the protons will ultimately end up almost all on the carbonate (or on water molecules from any carbonic acid that forms and decomposes before we can get to full equilibrium). $\endgroup$ – Oscar Lanzi Mar 24 '17 at 14:55
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Both based will indeed react with the acid. But wait, there's more. When the bases react to form their conjugare acids, the conjugate acid from one base could react with the other base. When one base is strong and the other weak, this secondary reaction will favor the direction where the strong base gets protonated. So you see the strong base gets more protons while the weak base tends to be pushed back to the basic form.

What if both bases are similar in strength? Then we apply equilibrium thermodynamics. Say hydrochloric acid is added to a solution containing both sodium acetate and sodium nitrite. The equilibrium between each of the latter two ions is governed by the appropriate acid dissociation constants:

$\ce {HC2H3O2 = H+ + (C2H3O2)-} $ $K_{a1}={\ce {[H+][(C2H3O2)-]}}/{\ce {HC2H3O2}}$

$\ce {HNO2 = H+ + (NO2)-} $ $K_{a2}={\ce {[H+][(NO2)-]}}/{\ce {HNO2}}$

To get the equilibrium that relates the competing weak acids and bases directly, just divide to eliminate the hydrogen ion concentration; in this case

${\ce {[HNO2][(C2H3O2)-]}}/{\ce {[(NO2)-][HC2H3O2]}=K_{a1}/K_{a2}}$

So your acid dissociation constants aren't just about acid dissociation. They also govern how a limited supply of protons gets distributed between competing bases.

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Though your title question would be difficult to answer for a general case, your example is pretty straight forward. Carbonate is a much stronger base than sulfate, so you would convert a mole of carbonate to a mole of bicarbonate before any significant protonation of sulfate would occur. And in the general case, it will be the stronger base that is protonated first, but the picture can get complex with bases of similar basicity and such.

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