15.3 Determine the concentrations of hydronium and hydroxide ions in \( 1.25 \times 10^{-3} \mathrm{M} \) aqueous perchloric acid.

To determine the concentrations of hydronium ions \([H_3O^+]\) and hydroxide ions \([OH^-]\) in a solution of perchloric acid (\(HClO_4\)), we start by recognizing that this is a strong acid, meaning it fully dissociates in water. Therefore, the concentration of hydronium ions will equal the concentration of the acid: \[ [H_3O^+] = 1.25 \times 10^{-3} \, \mathrm{M} \] Next, we can find the concentration of hydroxide ions using the ion product of water at 25°C, which is: \[ K_w = [H_3O^+][OH^-] = 1.0 \times 10^{-14} \] By rearranging this equation, we can solve for \([OH^-]\): \[ [OH^-] = \frac{K_w}{[H_3O^+]} = \frac{1.0 \times 10^{-14}}{1.25 \times 10^{-3}} \approx 8.0 \times 10^{-12} \, \mathrm{M} \] So, the concentrations of hydronium and hydroxide ions in the solution are \(1.25 \times 10^{-3} \, \mathrm{M}\) and approximately \(8.0 \times 10^{-12} \, \mathrm{M}\), respectively.