A \(500.0 \mathrm{~cm}\) long column is packed with a strong acid resin \(\left(c_{R T}=2.2 \mathrm{eq} / \mathrm{L}, \varepsilon_{\mathrm{e}}=0.42\right)\). Superficial velocity is \(25.0 \mathrm{~cm} / \mathrm{min}\). Counter-ions are not excluded. Co-ions are excluded. Dechow (1989) lists \(\mathrm{K}_{\mathrm{H}-\mathrm{Li}}=1.26\) and \(\mathrm{K}_{\mathrm{K}-\mathrm{Li}}=2.63\). Note: The questions ask for three exit times - if there is a shock wave, these times will be identical.

a. If the resin is initially in equilibrium with a \(1.0 \mathrm{eq} / \mathrm{L}\) solution of \(\mathrm{HCl}\) and \(\mathrm{KCl}\left(\mathrm{x}_{\mathrm{H}}=\right.\) \(\left.0.8, \mathrm{x}_{\mathrm{K}}=0.2\right)\) and an aqueous feed that is \(1.0 \mathrm{eq} / \mathrm{L}\) solution of \(\mathrm{HCl}\) and \(\mathrm{KCl}\left(\mathrm{x}_{\mathrm{H}}=\right.\) \(\left.0.15, \mathrm{x}_{\mathrm{K}}=0.85\right)\) is fed to the column, calculate the predicted times the \(\mathrm{K}^{+}\)wave exits the column (give exit times for \(\mathrm{K}^{+}\)concentrations of \(0.01,0.50\), and \(0.85 \mathrm{eq} / \mathrm{L}\) ).

b. If the resin is initially in equilibrium with a \(1.0 \mathrm{eq} / \mathrm{L}\) solution of \(\mathrm{HCl}\) and \(\mathrm{KCl}\left(\mathrm{x}_{\mathrm{H}}=\right.\) \(\left.0.20, \mathrm{x}_{\mathrm{K}}=0.80\right)\) and an aqueous feed that is \(1.0 \mathrm{eq} / \mathrm{L}\) solution of \(\mathrm{HCl}\) and \(\mathrm{KCl}\left(\mathrm{x}_{\mathrm{H}}=\right.\) \(\left.0.85, \mathrm{x}_{\mathrm{K}}=0.15\right)\) is fed to the column, calculate predicted times the \(\mathrm{K}^{+}\)wave exits the column (give exit times for \(\mathrm{K}^{+}\)concentrations of \(0.15,0.5\), and \(0.85 \mathrm{eq} / \mathrm{L}\) ).