D$12.$ A distillation column is separating toluene and xylene, $=3\mathrm{.}03.$ The feed is a saturated liquid, and reflux is returned as a saturated liquid. $p=1\mathrm{.}0$atm.$F=100\mathrm{.}0$kmo$\frac{l}{h}.$ Distillate mole fraction is $x_D=0\mathrm{.}996,$ and bottoms $x_B=0\mathrm{.}008.$ Use the Underwood equation to find $(\frac{L}{D}{)}_{\text{m}\text{i}\text{n}}$ and $V_{\text{m}\text{i}\text{n}}$ at feed mole fractions of $z=0\mathrm{.}1,0\mathrm{.}3,0\mathrm{.}5,0\mathrm{.}7,$ and $0\mathrm{.}9.$ Check your result at $z=0\mathrm{.}5$ with a McCabe$-$Thiele diagram. What are the trends for $|Q_{c,\text{m}\text{i}\text{n}}|$ and $Q_{R,\text{m}\text{i}\text{n}}$ as the toluene feed concentration increases? Hint: If you write the Underwood equation and solve algebraically for $,$ the problem is easier than it looks.