Question $1$:

$[10]$

An aqueous solution containing a valuable solute is colored by small amount of an impurity. An impurity is to be removed by adsorption on decolorizing carbon. A series of laboratory tests were made, until equilibrium was established, yielding the following data at constant temperature.

$\backslash$table$[[Kg$ carbon $\frac{?}{kg}$ of solution,$0,0\mathrm{.}001,0\mathrm{.}004,0\mathrm{.}008,0\mathrm{.}02,0\mathrm{.}04],[$Equilibrium Color,$9\mathrm{.}6,8\mathrm{.}6,6\mathrm{.}3,4\mathrm{.}3,1\mathrm{.}7,0\mathrm{.}7]]$

The color intensity was measured on an arbitrary scale, proportional to the concentration of the colored substance. It is desired to reduce the color to $10\%$ of its original value, $9\mathrm{.}6.$ Determine the quantity of fresh carbon required per $1000kg$ of solution for a single$-$stage operation, for two stage crosscurrent operations using the minimum total amount of carbon, and two$-$stage countercurrent operations.