Question

A factory wants to produce \(1.00 \times 10^{3} \mathrm{kg}\) barium from the electrolysis of molten barium chloride. What current must be applied for \(4.00 \mathrm{h}\) to accomplish this?

Short answer

The current that must be applied for 4.00 hours to produce \(1.00 \times 10^3 \, \mathrm{kg}\) of barium from the electrolysis of molten barium chloride is approximately \(9.79 \times 10^4 \, \mathrm{A}\).

Step-by-step solution

Calculate the number of moles of barium to be produced

For barium, the molar mass (Ba) is approximately 137 g/mol. We're given the target production of barium is 1.00 x 10³ kg, so we need to first convert this mass to grams and then to moles: Mass of barium (g) = 1.00 x 10³ kg x (1000 g/kg) = 1.00 x 10⁶ g Now, convert the mass to moles: Moles = (Mass of barium) / (Molar mass of barium) Moles = \( \frac{1.00 \times 10^6 \, \mathrm{g}}{137 \, \mathrm{g/mol}} \approx 7.30 \times 10^3 \, \mathrm{mol} \)

Calculate the total charge required to produce the desired amount of barium

In electrolysis, two moles of electrons are required to produce one mole of barium (Ba²⁺ + 2e⁻ → Ba). From Faraday's law of electrolysis, 1 mole of electrons carries a charge, called a Faraday (F) which is approximately 96485 C/mol (Coulombs per mole). Total charge (C) = moles of barium x (moles of electrons per mole of barium) x Faraday's constant Total charge (C) = \( 7.30 \times 10^3 \, \mathrm{mol} \times 2 \, \mathrm{mol} \, \mathrm{e}^{-}/\mathrm{mol} \, \mathrm{Ba} \times 96485 \, \mathrm{C}/\mathrm{mol} \, \mathrm{e}^{-} \approx 1.41 \times 10^9 \, \mathrm{C} \)

Calculate the required current for the given time

We are given the total time required to produce 1.00 x 10³ kg of barium as 4.00 hours. To find the required current, first, we need to convert the time to seconds: Total time (s) = 4.00 hours x (3600 s/h) = 14400 s Now, from the formula: current (I) = total charge (C) / total time (s), we have: Current (I) = \( \frac{1.41 \times 10^9 \, \mathrm{C}}{14400 \, \mathrm{s}} \approx 9.79 \times 10^4 \, \mathrm{A} \) So, the current that must be applied for 4.00 hours to produce 1.00 x 10³ kg of barium from the electrolysis of molten barium chloride is approximately 9.79 x 10⁴ A.

Key concepts

Faraday's Law of Electrolysis

Electrolysis is a fascinating process where electricity is used to drive a chemical reaction. A key principle in understanding electrolysis is Faraday's Law. Essentially, Faraday's Law of Electrolysis tells us how much material will be deposited or dissolved at an electrode during electrolysis based on the quantity of electricity used. This law states that the amount of substance altered at an electrode during electrolysis is proportional to the quantity of electricity that passes through the system.
Faraday's Law is summarized by the formula: \[ m = \frac{Q}{Fz} \] where:

• \( m \) is the mass of the substance produced at the electrode in grams,
• \( Q \) is the total electric charge passed through the system in Coulombs (C),
• \( F \) is Faraday's constant, approximately 96485 C/mol, which is the charge of one mole of electrons,
• \( z \) is the number of electrons involved in the reaction for each ion of the substance.

Understanding Faraday’s Law is fundamental in determining how much of a compound can be produced during electrolysis, helping predict the quantities in large-scale manufacturing processes such as in factories.

Moles Calculation

One of the first steps in the electrolysis process involves calculating the number of moles of the substance to be produced or required for the reaction. The concept of a mole is crucial because it allows chemists to convert between mass and the number of particles in a substance.
To find the number of moles (\( n \)), you use the following formula: \[ n = \frac{m}{M} \] where:

• \( m \) is the mass of the substance in grams,
• \( M \) is the molar mass of the substance in grams per mole.

In our exercise, we need to produce barium where the target mass is given, and the molar mass of barium is known (~137 g/mol). Using these values, we can calculate that for 1.00 x 10³ kg (or 1.00 x 10⁶ g), roughly 7300 moles of barium are required.
This step is vital because knowing the moles helps determine how many electrons are involved in the electrochemical reaction, thus guiding the subsequent calculation of the current needed.

Current Calculation in Electrolysis

Once you have determined the total charge required from Faraday's law and the moles of the material to be produced, the next step is to calculate the current required for the electrolysis process. Current is the flow of electric charge, and for electrolysis, knowing the current helps control the rate at which the chemical reaction occurs.
The formula for calculating the current (\( I \)) required is: \[ I = \frac{Q}{t} \] where:

• \( Q \) is the total charge in Coulombs (obtained from the product of moles of electrons, Faraday's constant, and moles of the material),
• \( t \) is the total time in seconds for which the current is applied.

In the exercise, it was determined that a charge of approximately 1.41 x 10⁹ Coulombs is needed over 4 hours (14400 seconds).
By substituting these values, the required current is calculated to be approximately 9.79 x 10⁴ Amperes (A).
Understanding how to calculate current is crucial for planning and executing electrolysis in industrial settings, ensuring the efficient and safe operation of electrochemical processes.