$(2)$ An electrolytic cell operates at $700C$ with a $Pb-Mg$ alloy as the anode and pure $Mg$ as the cathode. The electrolyte is a molten $MgCl-$NaCl mixture.

$($a$)$ Calculate the reversible emf when the $Mg$ activity in the alloy, relative to pure liquid magnesium, is $0\mathrm{.}04.$

$($b$)$ Calculate the practical cell voltage at a current density of $1\frac{A}{c}{m}^2$ when the conductance of the electrolyte is $2\mathrm{.}5oh{m}^{-1}c{m}^{-1},$ and the anode$-$to$-$cathode distance is $10cm.$ Overvoltages are insignificant.

$($c$)$ Calculate the electric energy consumption in kilowatthours per kilogram of $Mg,$ as well as the production rate in kilograms of $Mg$ per square meter $*$ hour for case $($b$)$ when the current efficiency is $90$ percent.