A (90.0 mathrm{~cm}) long laboratory column is packed with a strong acid cation-exchange resin ( (mathrm{c}_{mathrm{RT}}) (left.=2.5 mathrm{eq} / mathrm{L}, varepsilon_{mathrm{e}}=0.39 ight)). Cations are not

A \(90.0 \mathrm{~cm}\) long laboratory column is packed with a strong acid cation-exchange resin ( \(\mathrm{c}_{\mathrm{RT}}\) \(\left.=2.5 \mathrm{eq} / \mathrm{L}, \varepsilon_{\mathrm{e}}=0.39\right)\). Cations are not excluded. Anions are excluded. Anderson (1997) lists \(\mathrm{K}_{\mathrm{H}-\mathrm{Li}}=1.3\) and \(\mathrm{K}_{\mathrm{K}-\mathrm{Li}}=2.9\) and \(\mathrm{K}_{\mathrm{Ca}-\mathrm{Li}}=5.2\). The resin is initially in the \(\mathrm{K}^{+}\)form in equilibrium with a solution of \(\mathrm{KCl}\) that is \(0.03 \mathrm{eq} / \mathrm{L}\).

a. First, a \(0.03 \mathrm{eq} / \mathrm{L}\) solution with \(\mathrm{x}_{\mathrm{Ca}+2}=0.70\) is fed to the column at a fluid superficial velocity of \(25.0 \mathrm{~cm} / \mathrm{min}\) at time \(\mathrm{t}=0\). Does a shock or diffuse wave occur? What is the velocity and breakthrough time of this wave?

b. After column is totally saturated with \(0.03 \mathrm{eq} / \mathrm{L}\) solution with \(\mathrm{x}_{\mathrm{Ca}+2}=0.70\), reset clock to \(t=0\) and feed in counterflow a regenerant fluid that is \(1.1 \mathrm{eq} / \mathrm{L}\) of \(\mathrm{KCl}\) (no calcium) with a superficial velocity of \(35.0 \mathrm{~cm} / \mathrm{min}\). How long does it take for total ion wave to breakthrough?

A diffuse calcium wave occurs during elution. Calculate the velocity and breakthrough time of the fastest \(\mathrm{C}_{\mathrm{a}}^{+2}\) wave, the velocity and breakthrough time of the slowest \(\mathrm{C}_{\mathrm{a}}^{+2}\) wave, and the velocity and breakthrough time of the \(\mathrm{C}_{\mathrm{a}}^{+2}\) wave with \(\mathrm{x}_{\mathrm{Ca}+2}\) equal to 0.5 .

c. What concentration of regenerant fluid is required to not have a diffuse wave during the regeneration step?