A \(50.0 \mathrm{~cm}\) long column is packed with a strong acid cation-exchange resin \(\left(c_{R T}=2.2 \mathrm{eq} / \mathrm{L}\right.\), \(\left.\varepsilon_{\mathrm{e}}=0.42\right)\). Fluid superficial velocity is \(25.0 \mathrm{~cm} / \mathrm{min}\). 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 \(\mathrm{K}_{\mathrm{Na}-\mathrm{Li}}=2.0\).

a. Resin is initially in the \(\mathrm{H}^{+}\)form in equilibrium with a \(0.50 \mathrm{eq} / \mathrm{L}\) solution of \(\mathrm{HCl}\). Your technician does two experiments: (1) an aqueous feed that is \(0.50 \mathrm{eq} / \mathrm{L} \mathrm{KCl}\) is fed to the column, and (2) after re-equilibrating the column with \(0.50 \mathrm{eq} / \mathrm{L} \mathrm{HCL}\), an aqueous feed that is \(0.50 \mathrm{eq} / \mathrm{L} \mathrm{NaCl}\) is fed to the column. The results show that the \(\mathrm{K}^{+}\)shock wave and the \(\mathrm{Na}^{+}\)shock wave in the two separate experiments exit the column at the same times. The technician believes this result cannot be correct. Determine from calculations if this result is correct.

b. The technician then takes a column in equilibrium with \(\mathrm{x}_{\mathrm{Na}}=0.40\left(\mathrm{C}_{\mathrm{T}}=0.50\right)\) and feeds with a solution with \(\mathrm{x}_{\mathrm{Na}}=0.90\left(\mathrm{C}_{\mathrm{T}}=0.50\right)\). He repeats the experiment with the initial condition \(\mathrm{x}_{\mathrm{K}}=0.40\left(\mathrm{C}_{\mathrm{T}}=0.50\right)\) and feeds with a solution with \(\mathrm{x}_{\mathrm{K}}=0.90\left(\mathrm{C}_{\mathrm{T}}=\right.\) \(0.50)\). At what times do the \(\mathrm{K}^{+}\)shock wave and the \(\mathrm{Na}^{+}\)shock wave exit?

c. The technician then decides to take a column saturated with \(\mathrm{K}^{+}\)and elute it with 0.50 \(\mathrm{N} \mathrm{HCl}\) and to separately take a column saturated with \(\mathrm{Na}^{+}\)and elute it with \(0.50 \mathrm{~N}\) \(\mathrm{HCl}\). Much to the technician's surprise, the center \(\left(\mathrm{x}_{\mathrm{K}}=0.5\right)\) of the \(\mathrm{K}^{+}\)and the center \(\left(\mathrm{x}_{\mathrm{Na}}=0.5\right)\) of the \(\mathrm{Na}^{+}\)waves do not occur at the same time, and the \(\mathrm{K}^{+}\)center exits earlier than the \(\mathrm{Na}^{+}\)center. Calculate when the centers of these diffuse waves exit the column.

d. The technician then looks at when the tails of the two elution curves exit \(\left(\mathrm{x}_{\mathrm{K}} \rightarrow 0.0\right)\) and \(\left(\mathrm{x}_{\mathrm{Na}} \rightarrow 0.0\right)\). The \(\mathrm{Na}^{+}\)tail exits before the \(\mathrm{K}^{+}\)tail. Calculate the times that these two tails exit.

e. Next, the technician looks at the times when \(\left(\mathrm{x}_{\mathrm{K}}=0.90\right)\) and \(\left(\mathrm{x}_{\mathrm{Na}}=0.90\right)\) exit during the elution. The \(\mathrm{K}^{+}\)at this concentration exits before the \(\mathrm{Na}^{+}\)exits. Calculate the times that these two concentrations exit.

f. Finally, the technician notices that for the two diffuse waves there is a concentration of solute \(\left(\mathrm{Na}^{+}\right.\)or \(\left.\mathrm{K}^{+}\right)\)that exits at the same time. What is this concentration \(\left(\mathrm{x}_{\mathrm{Na}}=\right.\)

\(\left.\mathrm{x}_{\mathrm{K}}\right) ?\)
g. The technician is thoroughly confused and does not understand why \(\mathrm{Na}^{+}\)and \(\mathrm{K}^{+}\)come out at the same time in one experiment, the \(\mathrm{K}^{+}\)comes out ahead in some experiments, and the \(\mathrm{Na}^{+}\)comes out ahead in different experiments. Explain the results to the technician.