$(1)$ In the electrolytic refining of nickel the anode is a Cu$-$Ni alloy with $80$ atom $\%Ni,$ whereas the cathode is pure nickel.

$($a$)$ Calculate the reversible emf at $25C$ when the alloy is assumed to be an ideal solid solution.

$($b$)$ In practice both $Ni$ and Cu dissolve from the anode. The anolyte is withdrawn and copper is removed by cementation as pure $Cu$ on $Ni$ sponge. Calculate the equilibrium ratio $C\frac{u}{N}i$ in the purified electrolyte at $25C,$ when the ionic activities are taken equal to their molalities.

$($c$)$ In practice the applied cell voltage is $2\mathrm{.}0$ and the cathodic current efficiency is $95$ percent. Calculate the electric energy consumption in kilowatthours per kilogram of refined nickel.