Question

The electrolysis of \(\mathrm{BiO}^{+}\) produces pure bismuth. How long would it take to produce \(10.0 \mathrm{g}\) Bi by the electrolysis of a \(\mathrm{BiO}^{+}\) solution using a current of \(25.0 \mathrm{A} ?\)

Short answer

It would take approximately \(184.46 \mathrm{s}\) to produce \(10.0 \mathrm{g}\) of bismuth by the electrolysis of a \(\mathrm{BiO}^{+}\) solution using a current of \(25.0 \mathrm{A}\).

Step-by-step solution

Calculate the moles of bismuth to be produced

First, we need to find the moles of bismuth (\(n_{Bi}\)) to be produced. To do this, divide the mass of bismuth (\(m_{Bi} = 10 g\)) by its molar mass (\(M_{Bi} = 208.98 g/mol\)): \[n_{Bi} = \frac{m_{Bi}}{M_{Bi}}\]

Determine the moles of electrons used in the process

In the electrolysis process, BiO⁺ ion loses one electron to form pure bismuth (Bi). Therefore, one mole of bismuth requires one mole of electrons. So, the moles of electrons needed (\(n_{e}\)) are equal to the moles of bismuth: \[n_{e} = n_{Bi}\]

Calculate the charge needed for the process

Now, we need to find the total charge (in coulombs) required for the electrolysis process (\(Q\)). Faraday's constant (\(F\)) states that one mole of electrons carries a charge of 96,485 C/mol. Therefore, to find the charge needed, multiply the moles of electrons by Faraday's constant: \[Q = n_{e} \times F\]

Calculate the time required for electrolysis

Finally, to find the time required for electrolysis (\(t\)), we will use the formula for electric current (\(I\)), which is the charge per unit time: \[I = \frac{Q}{t}\] Rearranging the equation to find the time, we get: \[t = \frac{Q}{I}\] Now, let's find the time by plugging in the given values: 1. Calculate the moles of bismuth to be produced: \[n_{Bi} = \frac{10.0 \mathrm{g}}{208.98 \mathrm{g/mol}} \approx 0.0478 \mathrm{mol}\] 2. Determine the moles of electrons used in the process: \[n_{e} = 0.0478 \mathrm{mol}\] 3. Calculate the charge needed for the process: \[Q = 0.0478 \mathrm{mol} \times 96485 \mathrm{C/mol} \approx 4611.47 \mathrm{C}\] 4. Calculate the time required for electrolysis: \[t = \frac{4611.47 \mathrm{C}}{25.0 \mathrm{A}} \approx 184.46 \mathrm{s}\] So, it would take approximately 184.46 seconds to produce 10.0 grams of bismuth by the electrolysis of a BiO⁺ solution using a current of 25.0 A.

Key concepts

Faraday's constant

Faraday's constant is an essential concept in electrochemistry, representing the charge of one mole of electrons. It is a derived value that makes calculations involving electrons more accessible. Faraday's constant is approximately 96,485 coulombs per mole (C/mol). This means that

• 1 mole of electrons carries a charge of 96,485 C.
• This value is crucial for converting between moles of electrons and electrical charge in an electrochemical cell.

If you're wondering why Faraday's constant is so vital, consider it as a bridge connecting the macroscopic world (moles of a substance) with the microscopic behavior of electrons in a chemical reaction. This constant ensures that when you calculate reactions or processes involving electrons, like electrolysis, you can smoothly translate those into measurable quantities of electric charge. In the context of our exercise, it is used to determine the total charge needed to produce a specified amount of bismuth by multiplying by the moles of electrons involved in the reaction.

Moles calculation

Calculating moles is fundamental in chemistry, as it allows you to quantify the number of particles, such as atoms or molecules, participating in a chemical reaction. Moles give chemists a consistent means of measuring amount of substance. The formula to calculate moles is:

• Moles = Mass / Molar Mass

In our example:

• The mass of bismuth, Bi, intended for production is 10 g.
• The molar mass of bismuth is 208.98 g/mol, meaning each mole weighs 208.98 g.

By applying the moles formula, you can understand that:

• Moles of bismuth (\(n_{Bi}\)) produced is \(\frac{10 \text{ g}}{208.98 \text{ g/mol}} \approx 0.0478 \text{ mol}\).

Calculating moles this way not only helps you determine how much bismuth can be produced, but it also allows further calculations like determining the moles of electrons needed for the electrolysis process.

Electric current calculation

Electric current is a measure of the flow of electric charge through a conductor, and it plays a crucial role in electrolysis. It is defined by the formula:

• Current (I) = Charge (Q) / Time (t)

Where:

• I is the current in amperes (A),
• Q is the charge in coulombs (C),
• t is the time in seconds (s).

This relationship shows that the amount of charge moved through the system over time determines the current. In electrolysis:

• For our task, we need to calculate how long it takes to produce 10 g of bismuth using a current of 25 A.
• Given the total charge needed is 4611.47 C, the formula rearranges to find time: \( t = \frac{Q}{I} \).

By inserting the known values, \( t = \frac{4611.47 C}{25.0 A} \approx 184.46 \text{ seconds} \).Understanding this equation helps to grasp the relationship between charge, time, and current in the process of electrolysis, ensuring that calculations align with real-world measurements.