Calculate the concentrations of \(\mathrm{H}^{+}, \mathrm{HCO}_{3}^{-},\) and \(\mathrm{CO}_{3}^{2-}\) in a \(0.025 \mathrm{M} \mathrm{H}_{2} \mathrm{CO}_{3}\) solution.

To find the concentrations for \( \mathrm{H}^{+}, \mathrm{HCO}_{3}^{-}\), use the ionization constant and the initial concentration of \( \mathrm{H}_{2}\mathrm{CO}_{3}\). Approximating, the concentration of \( \mathrm{HCO}_{3}^{-}\) can be found using the equation \( \mathrm{HCO}_{3}^{-} = x=\sqrt{K_{a1}[H2CO3]}\), and the concentration of \( \mathrm{H}^{+} = 2x = 2\sqrt{K_{a1}[H2CO3]}\). For \( \mathrm{CO}_{3}^{2-}\), use the equation \( \mathrm{CO}_{3}^{2-} = K_{a2}\) because the contribution from the second ionization is much smaller than from the first ionization and can be approximated to \( \mathrm{CO}_{3}^{2-}\approx K_{a2}\). Note that this assumes the values of ionization constants \( K_{a1}\) and \( K_{a2}\) for the first and second ionization of \( \mathrm{H}_{2}\mathrm{CO}_{3}\) are known.