Question

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

Short answer

A current of 9.750 × 10⁴ A must be applied for 4.00 h to produce 1.00 × 10³ kg of barium from the electrolysis of molten barium chloride (BaCl2).

Step-by-step solution

Determine the molar mass of barium

From the periodic table, we know that the molar mass of barium (Ba) is 137.33 g/mol.

Convert the mass of barium into moles

Next, we will find the number of moles of barium (Ba) present in 1000 kg using the following formula: Number of moles (n) = mass (m) / molar mass (M) n(Ba) = $\frac{1.00\times {10}^3\mathrm{kg}}{137.33\mathrm{g}/\mathrm{mol}}$ Before solving, we need to convert kg to g: 1.00 × 10³ kg = 1.00 × 10³ × 10³ g = 1.00 × 10⁶ g Now, we can calculate the number of moles: n(Ba) = $\frac{1.00\times {10}^6\mathrm{g}}{137.33\mathrm{g}/\mathrm{mol}}$ = 7.280 × 10³ mol

Calculate the total charge needed

To find the total charge (Q) needed, we will use Faraday's law of electrolysis: Q = n × F × z where: - n: number of moles - F: Faraday's constant (96,485 C/mol) - z: Charge number (for Ba, z = 2 e-) Q = (7.280 × 10³ mol) × (96,485 C/mol) × (2 e-) = 1.404 × 10⁹ C

Convert the time to seconds

The given time is 4.00 h. We need to convert this time to seconds: 4.00 h = 4.00 × 60 min/h × 60 s/min = 1.440 × 10⁴ s

Calculate the current needed

Lastly, we calculate the current (I) needed using the formula: Current (I) = $\frac{Charge(Q)}{time(t)}$ I = $\frac{1.404\times {10}^9\mathrm{C}}{1.440\times {10}^4\mathrm{s}}$ = 9.750 × 10⁴ A Thus, a current of 9.750 × 10⁴ A must be applied for 4.00 h to produce 1.00 × 10³ kg of barium from the electrolysis of molten barium chloride (BaCl2).

Key concepts

Faraday's Law

Faraday's Law of electrolysis is a fundamental principle in the field of electrochemistry that connects the process of electrical energy conversion to chemical reactions. At its core, Faraday's Law states that the amount of substance produced at each electrode during electrolysis is directly proportional to the total electric charge passed through the electrolyte. This means that if you increase the amount of electricity, more chemical product will be formed.
The law can be summarized in the equation:
$$Q=n\times F\times z$$
where:

• $Q$ is the total charge in coulombs (C),
• $n$ is the number of moles of electrons exchanged,
• $F$ is Faraday’s constant $(96,485C/mol)$, which represents the charge of one mole of electrons,
• $z$ is the number of electrons involved in the electrochemical reaction.

This law is crucial when calculating the charge required to produce a specific mass of substance during electrolysis, as demonstrated in the exercise to produce barium from barium chloride.

Molar Mass

Molar mass is a key concept in chemistry and is defined as the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It acts as a bridge between the mass of a substance and the amount in moles, allowing us to convert between these two quantities.
To find the molar mass of an element like barium (Ba), one can refer to the periodic table. For barium, the molar mass is listed as 137.33 g/mol. This means that one mole of barium atoms weighs 137.33 grams. When determining how much barium can be produced, we must first convert the desired mass from kilograms to grams and then use the molar mass to find the number of moles.
In the exercise, by converting 1000 kg of barium to grams and using its molar mass, students calculate that they require approximately 7.280 $\times$ 10³ moles of barium to complete the electrolysis process.

Current Calculation

The calculation of current, required during electrolysis, involves the relationship between charge, time, and current. Current, denoted by $I$, is the rate at which charge flows through a conductor and is measured in amperes (A).
According to the formula:
$$I=\frac{Q}{t}$$
where:

• $I$ is the current in amperes,
• $Q$ is the total charge in coulombs,
• $t$ is the time for which the current flows, measured in seconds.

In the exercise, the total charge needed to electrolyze the barium chloride is calculated to be 1.404 $\times$ 10⁹ C. Given that the reaction occurs over 4 hours (converted to seconds as 1.440 $\times$ 10⁴ s), the required current to complete the process is determined to be 9.750 $\times$ 10⁴ A. This calculation demonstrates how electrochemical processes can efficiently convert electrical energy to chemical energy, vital for industrial applications.

Electrochemistry

Electrochemistry is a fascinating branch of chemistry that explores the intersection between chemical reactions and electricity. It involves the study of how electric currents can drive chemical reactions, a process commonly referred to as electrolysis.
Electrochemical processes have practical applications, from metal refining and electroplating to powering batteries and fuel cells. During electrolysis, an electrolytic cell is used to pass an electric current through a liquid or molten ionic substance, causing the ions to move and undergo redox reactions.
In the context of the original problem, the electrolysis of molten barium chloride is used to produce barium metal. This process demonstrates the real-world application of electrochemistry in producing elements by breaking down compounds into their constituents using an electric current. Understanding electrochemistry helps in grasping how industries achieve large-scale chemical transformations and the role that electrical energy plays in these processes.