A \(50.0 \mathrm{~cm}\) long column is packed with a resin that immobilizes a liquid stationary phase. The column is initially clean, \(\mathrm{c}_{\mathrm{A}}=0\). At \(\mathrm{t}=0\), we input a feed that is \(\mathrm{c}_{\mathrm{A} \text {.feed }}=1.5 \mathrm{~g} / \mathrm{L}\). Superficial velocity is \(20.0 \mathrm{~cm} / \mathrm{min}\). The packing has \(\varepsilon_{\mathrm{e}}=0.40, \varepsilon_{\mathrm{p}}=0.54, \mathrm{~K}_{\mathrm{d}}=1.0, ho_{\mathrm{s}}=\) \(1.124 \mathrm{~kg} / \mathrm{L}\), and equilibrium for component \(\mathrm{A}\) is an unfavorable isotherm, \(\mathrm{q}=1.2 \mathrm{c}_{\mathrm{A}} /\left(1-0.46 \mathrm{c}_{\mathrm{A}}\right)\) where \(\mathrm{q}\) is in \(\mathrm{g} / \mathrm{kg}\) and \(\mathrm{c}_{\mathrm{A}}\) is in \(\mathrm{g} / \mathrm{L}\).

Use solute movement theory to predict the outlet concentration profile of A ( \(\mathrm{c}_{\text {out }}\) vs. time).