$(20$ pts$)$ A water solution containing $10\mathrm{.}0\%$ acetic acid is added to a water solution containing $30\mathrm{.}0\%$ acetic acid flowing at the rate of $20\mathrm{.}0k\frac{g}{m}in.$ The product $P$ of the combination leaves at the rate of $100k\frac{g}{m}in.$

$($a$)$ Sketch the process and insert the provided information.

$($b$)$ Determine how many independent balances can be written.

$($c$)$ Determine the composition of $P.$

$(20$ pts$)$ A thickner in a waste disposal unit of a plant removes water from wet sewage sludge as shown in the figure below. How many kilograms of water leave the thickner per $100.kg$ of wet sludge that enter the thickner? Assume that the process is in the steady state.

$(20$ pts$)$ A gas containing $80\mathrm{.}0\%C{H}_4$ and $20\mathrm{.}0\%He$ is sent through a quartz diffusion tube to recover the helium. Twenty percent by weight of the original gas is recovered, and its composition is $50\mathrm{.}0\%He.$ Calculate the composition of the waste gas if $100kg$ moles of gas are processed per minute.