$172$ kmol$/$h of a liquid mixture of XBR and water enter from the top of a column containing a molar fraction of XBR of $0\mathrm{.}3155.$ The liquid mixture leaves the equipment containing a molar fraction of XBR of $0\mathrm{.}1033.$ On the other hand, the gas used to carry out the separation $($water vapor $+$ XBR$)$ has a molar fraction of XBR of $0\mathrm{.}0476,$ as it comes from a recycling stage. The gas flow to be used $($Vs$,$ in kmol$/$h$)$ should be $1\mathrm{.}5$ times the minimum.

$1.$ What are the values of X$1,$ x$1,$ X$2,$ x$2,$ Y$1,$ y$1,$ & L$2?(7$ Points$)$

$2.$ Determine the value of Ls $(3$ Points$)$

$3.$ Determine the value of the minimum Vs $($show on the graph how you did it$)(15$ Points$)$ Use the graph printed for the exam. $($attached in the question$)$

$4.$Determine the value of Vs $(3$ Points$)$

$5.$Determine the values of Y$2,$ y$2,$ V$2,$ & V$1(7$ Points$)$

$6.$Draw the operating line on the corresponding graph $(5$ Points$)$

$7.$If it is an ideal counter$-$current multistage process, determine $($graphically$)$ the number of stages needed to carry out the separation. $(7$ Points$)$

$8.$Determine the $\%$ removal of XBR$.(6$ Points$)$

$9.$Graphically determine the values of x & y $($lowercase$)$ for each ideal stage and place them on a "box$-$type" flow diagram. Also, show a flow diagram with the values of all process variables calculated $(12$ Points$)$

Note: Work on the entire problem with four decimal places. Write the answers in the assigned space as follows:

$($a$)$ X$1=$ x$.$xxxx$,$ x$1=$x$.$xxxx$,$ x$2=$x$.$xxxxx$,$ X$2=$ x$.$xxxx$,$ y$1=$ x$.$xxxxx$,$ Y$1=$ x$.$xxxxx$,$ L$2=$ x$.$xxxx$,($b$)$ Ls $=$ x$.$xxxx$,($c$).....**$