A distillation column is separating toluene and xylene, \(\alpha=3.03\). The feed is a saturated liquid, and reflux is returned as a saturated liquid. \(p\) \(=1.0 \mathrm{~atm} . \mathrm{F}=100.0 \mathrm{kmol} / \mathrm{h}\). Distillate mole fraction is \(\mathrm{x}_{\mathrm{D}}=0.996\), and bottoms \(\mathrm{x}_{\mathrm{B}}=0.008\). Use the Underwood equation to find \((\mathrm{L} / \mathrm{D})_{\min }\) and \(\mathrm{V}_{\min }\) at feed mole fractions of \(\mathrm{z}=0.1,0.3,0.5,0.7\), and 0.9 . Check your result at \(\mathrm{z}=0.5\) with a McCabe-Thiele diagram. What are the trends for \(\left|Q_{\mathrm{c}, \text { min }}\right|\) and \(Q_{\mathrm{R}, \text { min }}\) as the toluene feed concentration increases? Hint: If you write the Underwood equation and solve algebraically for \(\varphi\), the problem is easier than it looks.