Question

A An electrolysis cell for aluminum production operates at \(5.0 \mathrm{V}\) and a current of \(1.0 \times 10^{5} \mathrm{A}\) Calculate the number of kilowatt-hours of energy required to produce 1 metric ton \(\left(1.0 \times 10^{3} \mathrm{kg}\right)\) of aluminum. \(\left(1 \mathrm{kWh}=3.6 \times 10^{6} \mathrm{J} \text { and } 1 \mathrm{J}=\right.\) \(1 \mathbf{C} \cdot \mathbf{V}\)

Short answer

The cell requires 14939 kWh to produce 1 metric ton of aluminum.

Step-by-step solution

Calculate the Charge Required

First, determine the amount of charge needed to produce 1 metric ton (1000 kg) of aluminum. The molar mass of aluminum is 26.98 g/mol, so 1 metric ton equals \(\frac{1000 \text{ kg} \times 1000 \text{ g/kg}}{26.98 \text{ g/mol}} = 37082.2 \text{ mol}\). Since each mole of aluminum requires 3 moles of electrons (for the reaction \(\text{Al}^{3+} + 3e^- \rightarrow \text{Al}\)), the total charge \(Q\) in coulombs is: \[ Q = 3 \times 37082.2 \text{ mol Al} \times 96485 \text{ C/mol} = 10756317030 \text{ C} \]

Calculate the Energy in Joules

Now use the formula \( E = V \times Q \) to calculate the energy in joules required for the electrolysis. Here, \( V = 5.0 \) V and \( Q = 10756317030 \) C from Step 1. \[ E = 5.0 \text{ V} \times 10756317030 \text{ C} = 53781585150 \text{ J} \]

Convert Joules to Kilowatt-hours

Convert the energy calculated in joules to kilowatt-hours. We know that \( 1 \text{ kWh} = 3.6 \times 10^6 \text{ J}\), thus: \[ \text{kWh} = \frac{53781585150 \text{ J}}{3.6 \times 10^6 \text{ J/kWh}} = 14939 \text{ kWh} \]

Key concepts

Aluminum Production

Aluminum production through electrolysis is a critical industrial process. This method involves dissolving aluminum oxide in molten cryolite and then passing an electric current through it. The aluminum ions are reduced to aluminum metal at the cathode, while oxygen ions are oxidized at the anode, releasing oxygen gas. This process is known as the Hall-Héroult process and is the most common method for extracting aluminum worldwide. Producing aluminum from its ore, bauxite, through electrolysis is energy-intensive due to the significant amount of electricity required. Hence, understanding energy calculations in aluminum production is essential for optimizing efficiency and reducing costs. Efficient aluminum production through electrolysis helps in minimizing environmental impacts and conserving resources.

Electrochemical Cells

Electrochemical cells play a vital role in the electrolysis process. These cells are devices that convert chemical energy into electrical energy or vice versa. In the context of aluminum production, an electrochemical cell operates by applying an external voltage to drive a non-spontaneous chemical reaction. This setup typically includes a positive electrode (anode) and a negative electrode (cathode), immersed in an electrolyte solution that facilitates the movement of ions. Here, the electrochemical cell is specifically designed to reduce aluminum ions into their elemental form, leveraging the flow of electrons to achieve the desired chemical change. Understanding how these cells work helps in refining the process to be more efficient and cost-effective.

Energy Conversion

Energy conversion is fundamental to electrolysis in aluminum production. The process entails turning electrical energy into chemical energy to decompose compounds and obtain pure elements. For the electrolysis of aluminum, energy conversion includes calculating the total charge required to reduce a specific mass of aluminum oxide into elemental aluminum. The process begins with determining the amount of electric charge necessary, which depends on the number of moles of aluminum being reduced. Using the formula \(E = V \times Q\), where \(E\) is energy in joules, \(V\) is the voltage applied, and \(Q\) is the total charge, one can calculate the energy needed. This equation illustrates the direct relationship between electrical energy input and the chemical conversion occurring within the cell.

Molar Mass Calculation

Calculating molar mass is a critical first step in determining the charge required for aluminum production. The molar mass of a substance is the mass of one mole of its elementary entities, typically atoms or molecules. For aluminum, the molar mass is 26.98 grams per mole. To determine how much aluminum is required to be reduced, one starts with the mass needed in kilograms and converts it to moles using the molar mass. For example, 1 metric ton (1000 kg) of aluminum is equivalent to \(\frac{1000 \text{ kg} \times 1000 \text{ g/kg}}{26.98 \text{ g/mol}}\). This conversion to moles is essential, as it helps determine the number of electrons required, subsequently connecting to the needed electrical charge for electrolysis. Ensuring accurate molar mass calculations is crucial for precise energy and materials assessments in industrial applications.