Problem 91 - Chemistry and Chemical Reactivity

Question

Assume an electrolysis cell that produces chlorine from aqueous sodium chloride (called "brine") operates at \(4.6 \mathrm{V}\) (with a current of \(3.0 \times 10^{5}\) amps). Calculate the number of kilowatt-hours of energy required to produce \(1.00 \mathrm{kg}\) of chlorine \(\left(1 \mathrm{kWh}=1 \text { kilowatt-hour }=3.6 \times 10^{6} \mathrm{J}\right).\)

Step-by-Step Solution

Answer

3.47 kWh required to produce 1 kg of chlorine.

Step 1: Identify Key Variables

First, let's identify the key variables given in the problem. We know the voltage \(V = 4.6 \, \text{V}\), the current \(I = 3.0 \times 10^{5} \, \text{A}\), and the chemical equation for chlorine production. Our goal is to find the energy required to produce \(1.00 \, \text{kg}\) of chlorine.

Step 2: Calculate Moles of Chlorine Produced

Chlorine has a molar mass of \(70.906 \, \text{g/mol}\). To find the moles of chlorine, use the formula: \[ \text{moles of Cl}_2 = \frac{1000 \, \text{g}}{70.906 \, \text{g/mol}} \approx 14.1 \, \text{mol} \]

Step 3: Calculate Charge Required

The reaction \(2\text{Cl}^- \rightarrow \text{Cl}_2 + 2e^-\) shows that 2 moles of electrons are needed per mole of chlorine. Therefore, the total charge \(Q\) needed is: \[ Q = (14.1 \, \text{mol}) \times (2 \, \text{mol e}^-/\text{mol Cl}_2) \times (9.65 \times 10^4 \, \text{C/mol e}^-) \approx 2.72 \times 10^6 \, \text{C} \]

Step 4: Calculate Time Required

Use the relationship \(Q = I \cdot t\) to find the time: \[ t = \frac{Q}{I} = \frac{2.72 \times 10^6 \, \text{C}}{3.0 \times 10^5 \, \text{A}} \approx 9.07 \, \text{s} \]

Step 5: Calculate Energy in Joules

The energy \(E\) required is calculated using \(E = V \times Q\): \[ E = 4.6 \, \text{V} \times 2.72 \times 10^6 \, \text{C} = 1.25 \times 10^7 \, \text{J} \]

Step 6: Convert Energy to Kilowatt-hours

Convert from joules to kilowatt-hours using \(1 \, \text{kWh} = 3.6 \times 10^6 \, \text{J}\): \[ \text{kWh} = \frac{1.25 \times 10^7 \, \text{J}}{3.6 \times 10^6 \, \text{J/kWh}} \approx 3.47 \, \text{kWh} \]

Key Concepts

Energy Calculation in Electrolysis

In the process of electrolysis, calculating the energy required is a fundamental step. Energy in this context is essentially the amount of work done to drive a chemical reaction using an electric current. The energy needed is calculated in joules using the following formula:

\[ E = V \times Q \]where:

• \(E\) is the energy in joules
• \(V\) represents the voltage in volts
• \(Q\) denotes the charge in coulombs

For example, in the production of chlorine via the electrolysis of brine, we first determine the total charge \(Q\), and then use it along with the supplied voltage to find \(E\). In our case study, the energy required was calculated as \(1.25 \times 10^7 \) joules.
This calculation helps understand the efficiency and feasibility of the electrolysis process when scaling for industrial applications.

Chlorine Production via Electrolysis

Chlorine is an essential industrial chemical predominantly produced through the electrolysis of saltwater or brine. The reaction can be represented as:

\[ 2 ext{Cl}^- \rightarrow ext{Cl}_2 + 2e^- \]This equation indicates that two moles of chloride ions are needed to produce one mole of chlorine gas. Hence, comprehending the stoichiometry is crucial.
With a molar mass of approximately 70.906 grams per mole, calculating the moles of chlorine involves dividing the mass of chlorine needed by its molar mass. Once this is known, the next step involves finding the total charge \(Q\) required, which further needs understanding of how many moles of electrons are involved per mole of chlorine produced.
In practical setups, understanding this process allows for optimizing the cell design and operation parameters to produce chlorine efficiently, reducing costs and minimizing energy consumption.

Kilowatt-hour Conversion for Energy Management

The energy calculated in joules during electrolysis often needs conversion into kilowatt-hours (kWh) for practical reasons. This is essential for energy budgeting, cost analysis, and efficiency comparison within industries.
The conversion from joules to kilowatt-hours uses the relationship:
\[ 1 \text{ kWh} = 3.6 \times 10^6 \text{ J} \]Thus, to switch from joules to the common electrical energy unit kWh, divide the energy in joules by this conversion factor. For our example, the energy requirement calculated as \(1.25 \times 10^7\) joules comes to approximately 3.47 kWh when converted.
This conversion is valuable for factories in understanding the operational costs associated with electrolysis-based chemical production, allowing for a straightforward view of energy usage akin to what would appear on an electric bill. It helps in strategic planning and implementation of energy-saving measures across operations.