You wish to extract aniline from an aqueous phase solution by contacting it with toluene in a

counter$-$current solvent extraction process. The tower has $10$ stages and is illustrated below. State

any assumptions that you make.

i$)$ By performing a material balance around each stage, derive a set of $10$ simultaneous equations

$($with unknown $x_i)$ that must be solved. Assume that toluene and water are completely

immiscible. As a first approximation, assume that equilibrium is achieved at each stage and is

described by the relationship,

$m=\frac{Y_i}{x_i}=9$

Where, $Y_i=$ mass of aniline $\frac{?}{?}$ mass of toluene in the solvent$-$rich phase leaving stage $i$

and, $x_i=$ mass of aniline $\frac{?}{?}$ mass of water in the aqueous phase leaving stage $i$

ii$)$ Solve the equations derived in $($i$)$ using in$-$built functions in python.

iii$)$ A better approximation of the equilibrium relationship is $m=9+20x_i.$ Modify your equations

and explain how you can recognize that the system is now non$-$linear.

iv$)$ Solve the system and compare your results to the linear approximation.