Can you find [SO₃²⁻] with just the two acid dissociation constants Kₐ₁ and Kₐ₂?

Sulfurous acid, $\ce{H2SO3}$, is a diprotic acid. \begin{align} K_{a1} &= 1.6 \cdot 10^{-2}\\ K_{a2} &= 6.4 \cdot 10^{-8}\\ \end{align}

Is it possible to find the concentration of $\ce{SO3^2-}$ from this?

I would think you would need to know the concentration of at least one thing. It is like trying to solve for 3 unknowns with 2 equations!

So then [$\ce{SO3^{2−}}$] $= 6.4 \cdot 10^{-8}$?
• There should be a fair warning involved in this. It is approximating, that the dissociation $\ce{HA- + H2O <=> H3O+ + A^2-}$ does not contribute to the hydronium ions. In general it can be assumed that $\ce{[H3O+] \geq [HA- ]}$ and therefore $K_{a2}\geq [\ce{A^2-}]$. Mar 31 '15 at 5:46
You can use approximations like $K_\mathrm{a1}$ is much greater than $K_\mathrm{a2}$ and that first dissociation goes to almost completion. Accurate values cannot be calculated mathematically though.