$($A$)$ When the number of components is $C$ and the number of phases is $P,$ explain the degrees of freedom $f$ of the equilibrium system $C+2-P$

by the following procedure.

Show that the total number of variables require to define the system is $C(P-1).$

Chemical potential is determined by the mole fraction of each component and the temperature of the system. The condition for phas

e equilibrium is equal to the chemical potential of all phases in each component. From this result, find the number of equations that

define the concentration of each component.

Derive the above equations using the fact that the degrees of freedom of the system are two additional state variables in addition to

the concentration.

$($B$)$ A cylinder with a piston is filled with a mixture of water and ethanol and its vapor, and the temperature is maintained at $25C.$ Is the $C$

omposition of the liquid and vapor in the cylinder determined independently of the pressure applied to the cylinder? Why or why not?