Question

Question:A current of 75,000 A is passed through an electrolysis cell containing molten MgCl2 for 7.0 days. Calculate the maximum theoretical mass of magnesium that can be recovered.

Short answer

The maximum theoretical mass of magnesium is $5.7\times {10}^6\mathrm{g}$.

Step-by-step solution

Calculation of charge

Given the current,$\mathrm{I}=75,000\mathrm{A}$

Time,

$\begin{array}{c}\mathrm{t}=7\mathrm{d}\left(\begin{array}{c}1\mathrm{d}=24\mathrm{h}\\ =24\mathrm{h}\times 60\min \times 60\mathrm{s}\\ =8.64\times {10}^4\mathrm{s}\end{array}\right)\\ =7\times 8.64\times {10}^4\mathrm{s}\\ =6.05\times {10}^5\mathrm{s}\end{array}$

A charge is the product of current and time.

Charge,

$\begin{array}{c}\mathrm{Q}=\mathrm{It}\\ =75,000\mathrm{A}\times 6.05\times {10}^5\mathrm{s}\\ =4.54\times {10}^{10}\mathrm{As}\end{array}$

Step 2: Calculation of the number of moles of electrons

The charge is the product of the number of moles of electrons and the Faraday constant, which is calculated by the following formula:

$$\begin{gathered} \mathrm{Q}=\mathrm{nF} \\ \begin{array}{c}\mathrm{n}=\frac{\mathrm{Q}}{\mathrm{F}}\\ =\frac{4.54\times {10}^{10}\mathrm{As}}{9.65\times {10}^4{\mathrm{Asmol}}^{-1}}\\ =4.705\times {10}^5\mathrm{mol}\end{array} \end{gathered}$$

There are $4.705\times {10}^5\mathrm{mol}$of electrons.

Calculation of the mass of aluminum deposited

Electrolysis of magnesium occurs as follows:

${\mathrm{Mg}}^{2+}+2{\mathrm{e}}^{-}\to \mathrm{Mg}$

Two moles of electrons produce one mole of magnesium metal. So, $4.705\times {10}^5\mathrm{mol}$ of electrons produce $\frac{4.705\times {10}^5\mathrm{mol}}{2}=2.35\times {10}^5\mathrm{mol}$ of magnesium metal.

The mass of the magnesium metal is calculated as follows:

The number of moles of magnesium, ${\mathrm{n}}_{\mathrm{Mg}}=\frac{{\mathrm{m}}_{\mathrm{Mg}}}{{\mathrm{M}}_{\mathrm{Mg}}}$, where ${\mathrm{m}}_{\mathrm{Mg}}$ is the mass of magnesium metal and ${\mathrm{M}}_{\mathrm{Mg}}$ is the molar mass of magnesium.

$\begin{array}{c}2.35\times {10}^5\mathrm{mol}=\frac{{\mathrm{m}}_{\mathrm{Al}}}{27{\mathrm{gmol}}^{-1}}\\ {\mathrm{m}}_{\mathrm{Al}}=2.35\times {10}^5\mathrm{mol}\times 24.305{\mathrm{gmol}}^{-1}\\ =5.7\times {10}^6\mathrm{g}\end{array}$

The mass of magnesium metal is $5.7\times {10}^6\mathrm{g}$.