An Introduction to Thermal Physics · Problem 5.87

Free Energy and Chemical Thermodynamics — Daniel Schroeder

Physics — Statistical Mechanics · sulfuric-acid-dissociation-pH

Problem

Sulfuric acid, H2SO4\mathrm{H}_2\mathrm{SO}_4, readily dissociates into H+\mathrm{H}^+ and HSO4−\mathrm{HSO}_4^- ions: H2SO4⟶H++HSO4−.\mathrm{H}_2\mathrm{SO}_4 \longrightarrow \mathrm{H}^+ + \mathrm{HSO}_4^-. The hydrogen sulfate ion, in turn, can dissociate again: HSO4−⟷H++SO42−.\mathrm{HSO}_4^- \longleftrightarrow \mathrm{H}^+ + \mathrm{SO}_4^{2-}. The equilibrium constants for these reactions, in aqueous solutions at 298 K, are approximately 10210^2 and 10−1.910^{-1.9}, respectively. (For dissociation of acids it is usually more convenient to look up KK than ΔG∘\Delta G^\circ. By the way, the negative base-10 logarithm of KK for such a reaction is called pK\mathbf{pK}, in analogy to pH. So for the first reaction pK=−2\mathrm{pK} = -2, while for the second reaction pK=1.9\mathrm{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 5×10−55 \times 10^{-5} 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 SO42−\mathrm{SO}_4^{2-}. What is the pH of this rainwater? ((c)) Explain why you can neglect dissociation of water into H+\mathrm{H}^+ and OH−\mathrm{OH}^- in answering the previous question. ((d)) At what pH would dissolved sulfate be equally distributed between HSO4−\mathrm{HSO}_4^- and SO42−\mathrm{SO}_4^{2-}?
Intuitive explanation

This problem uses equilibrium constants to track how sulfuric acid dissociates in water through two sequential steps. The key insight is that KK tells you which direction a reaction strongly favors: when K≫1K \gg 1, the reaction goes nearly to completion, and when K≪1K \ll 1, it barely proceeds. By comparing KK values and using the Henderson–Hasselbalch relationship (or the definition of pK\mathrm{pK}), you can predict the dominant chemical form at any given pH and calculate the resulting acidity.

Intuitive explanation

This problem uses equilibrium constants to track how sulfuric acid dissociates in water through two sequential steps. The key insight is that KK tells you which direction a reaction strongly favors: when K≫1K \gg 1, the reaction goes nearly to completion, and when K≪1K \ll 1, it barely proceeds. By comparing KK values and using the Henderson–Hasselbalch relationship (or the definition of pK\mathrm{pK}), you can predict the dominant chemical form at any given pH and calculate the resulting acidity.

Related theory

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