The feed to a continuous distillation column, at steady$-$state, is a liquid mixture containing $60\%$ ethanol, $5\%$ contaminant and the rest water by mass. The distillate is $90\%$ ethanol and the rest water. The bottoms product is a mixture of water, ethanol and contaminant. The column is operated such that the mass flow rates of the distillate and bottoms product are equal to each other.

$1.$ Perform a degree of freedom analysis and show that the unknowns $($mass flow rates and compositions if labeled$)$ can be calculated.

$2.$ Write down the mass balance equations and process specification equations if any.

$3.$ Calculate $($a$)$ the contaminant mass fraction in the bottoms, and $($b$)$ fraction of the ethanol in the feed that is lost in the bottoms $($mass of ethanol in bottoms$/$mass of ethanol in feed$).$