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Chapter 21: Problem 46

The overall reaction for the electrolytic production of aluminum by means of the Hall process may be represented as $$\mathrm{Al}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 2 \mathrm{Al}(l)+3 \mathrm{CO}(g)$$ At \(1000^{\circ} \mathrm{C}\), the standard free-energy change for this process is \(594 \mathrm{~kJ} / \mathrm{mol} .\) (a) Calculate the minimum voltage required to produce 1 mole of aluminum at this temperature. (b) If the actual voltage applied is exactly three times the ideal value, calculate the energy required to produce \(1.00 \mathrm{~kg}\) of the metal.

Short Answer

Expert verified
Part (a) The minimum voltage required to produce 1 mole of aluminum is approximately 6.15 Volts. Part (b) The energy required to produce 1.00 kg of Aluminium is approximately 66700 kJ.

Step by step solution

01

Calculate the minimum voltage

In electrochemistry, we use the relationship between free energy (\(ΔG\)) and the electrical work (\(w_{elec}\)), and its connection to the cell potential (\(E_{cell}\)), given by the formula, \(ΔG=-nFE_{cell}\) where \(n\) is the number of moles of electrons transferred (3 in this case, from the balanced equation), \(F\) is the Faraday's constant (96485 C/mol), and \(ΔG\) is the standard free-energy change (594 kJ/mol). To find \(E_{cell}\), you can rearrange the equation to \(E_{cell} = - ΔG/(nF)\). Replace the values in the formula and calculate.
02

Convert energy unit

The initial calculation will give the cell potential in volts, which is also joules per coulomb. To make it easier to use in the next step, convert the energy to kilojoules per mole by multiplying the cell potential by the number of moles of electrons transferred and divide by 1000.
03

Calculate the energy required to produce 1.00 kg of Aluminium

The question states the actual voltage applied is three times the ideal. Multiply the result from step 2 by 3 to get the energy used per mole. As there are 2 moles of aluminium produced in the reaction, divide the energy per mole by 2 to get the energy used to produce one mole of aluminium. Lastly, as the molar mass of aluminium is about 26.98 g/mol, multiply the energy per mole by \(1000/26.98\) to find the energy required to produce 1.00 kg of Aluminium.

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