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Step-by-Step Solution
Step 1: Write the Balanced Chemical Equation
The Hall process for the extraction of aluminium involves the electrolysis of alumina using carbon anodes. The overall simplified reaction is:
$2Al_{2}O_{3} + 3C \to 4Al + 3CO_{2}$
Step 2: Determine the Stoichiometric Ratio
From the balanced equation:
2 moles of $Al_{2}O_{3}$ (which produce 4 moles of Al) require 3 moles of C.
4 moles of Al have a mass of $4 \times 27 = 108\,\text{g}$ of Al.
3 moles of C have a mass of $3 \times 12 = 36\,\text{g}$ of C.
Thus, $108\,\text{g}$ of Al is produced by $36\,\text{g}$ of C.
Step 3: Calculate the Carbon Required for 270 kg of Aluminium
We have $270\,\text{kg}$ of Al, which is $270,000\,\text{g}$ of Al. If $108\,\text{g}$ of Al needs $36\,\text{g}$ of C, then:
$ \text{Mass of C needed}
= \left(\dfrac{36}{108}\right) \times 270000
= 90000\,\text{g}
= 90\,\text{kg}
$
Therefore, the mass of carbon anode consumed to produce $270\,\text{kg}$ of aluminium by the Hall process is $90\,\text{kg}$.
