Q$3(40\%).$ About $80\%$ of world $Mg$ production is from electrolytic reduction of $MgC{l}_2.$ The

electrolytic cell $($IG cell$)$ operates at $-800C$ and the feed is anhydrous $MgC{l}_2.$ The anode$-$cathode

spacing is $55mm.$ The conductivity of the molten salt is $2\mathrm{.}3\frac{S}{c}m.$ The cell operates at a current

density of $0\mathrm{.}70$Amp$\frac{s}{c}{m}^2.$ The anode overvoltage is $0\mathrm{.}50V,$ the cathode overvoltage is $0\mathrm{.}05V,$

and the voltage drop in the busbar$/$electrodes is $1\mathrm{.}30V($note: voltage drop in busbar is independent

from Ohmic voltage drop due to conductivity of electrolyte and has to be accounted separately$).$

The IG cell operates for a week at $35kA$ and $2400kg$ of $Mg$ is recovered.

The overall cell reaction is: as $\{$:$M{g}^{2+}+2C{l}^{-})=Mg(l)+C{l}_2(g)$

The $C{l}_2$ gas is collected from the top. The specific gravity of magnesium at $\{$:$800C)$ is less

than electrolyte so the liquid $Mg$ also floats to the top for collection. The composition of the molten

salt electrolyte is$,$ by weight: $30\%$NaCl$+12\%MgC{l}_2+13\%KCl+45\%CaC{l}_2.$

Write the anodic and cathodic reactions per mole of magnesium

Calculate $E$ for the cell. Assume pure magnesium; $a_{MgC{l}_2}=x_{MgC{l}_2}($mole fraction$)$;

$P_{C{l}_2}=0\mathrm{.}34$atm

Determine the sodium, potassium and calcium concentrations in the liquid magnesium at

electrochemical equilibrium. $($hint: These parasitic reduction side$-$reactions are not likely

to significantly reduce the current efficiency$)$

Calculate the overall applied voltage

Calculate current efficiency and the energy efficiency

Calculate how much heat is transferred from the cell to the surroundings $($in $kW)$

Hint: the amount of work done on the system per unit of time $($in $kW)$ can be calculated by

multiplying $I_{\text{a}\text{p}\text{p}\text{l}\text{i}\text{c}\text{e}\text{d}}$ by $V_{\text{a}\text{p}\text{p}\text{l}\text{i}\text{s}\text{d}-}$

All thermodynamic data:

$MgC{l}_{2,298}=M{g}_{1073}+C{l}_{2,1073},H=704,010\frac{J}{m}$ole

Atomic weights: $Na$:$22\mathrm{.}98$;$Mg$:$24\mathrm{.}31$;$Cl$:$35\mathrm{.}45$;$K$:$39\mathrm{.}1$;$Ca$:$40\mathrm{.}1$