Chapter 5.6, Problem 87P

Sulfuric acid, ${\text{H}}_{\text{2}}{\text{SO}}_{\text{4}}$, readily dissociates into ${\text{H}}^{+}$ and ${\text{HSO}}_4^{-}$ ions: ${\text{H}}_2{\text{SO}}_4\to {\text{H}}^{+}{\text{+HSO}}_4^{-}$.

The hydrogen sulfate ion, in turn, can dissociate again: ${\text{HSO}}_4^{-}\leftrightarrow {\text{H}}^{+}{\text{+SO}}_4^{2-}$.

The equilibrium constants for these reactions, in aqueous solutions at 29 K, are approximately ${10}^2$ and ${10}^{-1.9}$, respectively. (For dissociation of acids it is usually more convenient to look up K than $△G°$. By the way, the negative base–10 logarithm of K for such a reaction is called pK, in analogy to pH. So for the first reaction $\text{pK}=-2$, while for the second reaction $\text{pK}=1.9$ .)

(a) Argue that the first reaction tends so strongly to the right that we might as well consider it to have gone to completion, in any solution that could possibly be considered dilute. At what pH values would a significant fraction of the sulfuric acid not be dissociated?

(b) In industrialized regions where lots of coal is burned, the concentration of sulfate in rainwater is typically $5\times {10}^{-5}\text{mol/kg}$. The sulfate can take any of the chemical forms mentioned above. Show that, at this concentration, the second reaction will also have gone essentially to completion, so all the sulfate is in the form of ${\text{SO}}_4^{2-}$. What is the pH of this rainwater?

(c) Explain why you can neglect dissociation of water into ${\text{H}}^{+}$ and ${\text{OH}}^{-}$ in answering the previous question.

(d) At what pH would dissolved sulfate be equally distributed between ${\text{HSO}}_4^{-}$ and ${\text{SO}}_4^{2-}$ ?