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 takes approximately 3.08 minutes 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 Bi produced

First, convert the mass of bismuth into moles using its molar mass (Molar mass of Bi \(= 208.98 \mathrm{~g/mol}\)). Number of moles = \(\frac{mass}{molar~mass}\) Number of moles of Bi = \(\frac{10.0 \mathrm{~g}}{208.98 \mathrm{~g/mol}}\)

Calculate the moles of electrons transferred

The balanced equation for the electrolysis of \(\mathrm{BiO}^+\) is: \[\mathrm{BiO}^+ + e^− \rightarrow \mathrm{Bi} +\mathrm{O}^{2-}\] From the equation, we can see that one mole of electrons is required to produce one mole of bismuth. Hence, the number of moles of electrons transferred is equal to the number of moles of Bi produced. Moles of electrons transferred = Number of moles of Bi Moles of electrons transferred ≈ 0.0478 moles

Calculate the charge transferred (in Coulombs)

Use Faraday's constant (\(1 \mathrm{~mol ~of ~electrons} = 96500 \mathrm{~C}\)) to convert moles of electrons into charge: Charge transferred = Moles of electrons transferred × Faraday's constant Charge transferred ≈ 0.0478 moles × 96500 C/mol ≈ 4616.1 C

Calculate the time taken for electrolysis

Now, we need to find the time it takes for the electrolysis to produce \(10.0 \mathrm{~g}\) of bismuth. The formula for calculating time in this case is: Time = \(\frac{charge ~transferred}{current}\) Time ≈ \(\frac{4616.1 \mathrm{~C}}{25.0 \mathrm{~A}}\) Time ≈ 184.6 seconds

Convert the time to minutes

Finally, we can convert the time from seconds to minutes to get a more convenient unit. Time = \(\frac{184.6 \mathrm{~seconds}}{60 \mathrm{~seconds/minute}}\) ≈ 3.08 minutes So, it takes approximately 3.08 minutes to produce 10.0 grams of bismuth by the electrolysis of a \(\mathrm{BiO}^{+}\) solution using a current of \(25.0 \mathrm{~A}\).

Key concepts

Molar Mass

Understanding molar mass is crucial in chemical calculations, especially in electrochemistry. Molar mass, measured in grams per mole (\text{g/mol}), is the weight of one mole of a substance. One mole corresponds to Avogadro's number, which is approximately 6.022 \(\times\) 10²³ atoms, molecules, or ions of that substance.

In the context of electrolysis, molar mass helps us determine the amount of a substance produced or consumed at an electrode. For instance, bismuth (Bi) has a molar mass of 208.98 \(\text{g/mol}\). To find out how many moles of Bi are in 10.0 grams, we divide the mass by the molar mass. This enables us to link the mass of material electrodeposited to the quantity of electric charge required to release or deposit that mass.

Faraday's Constant

Faraday's constant is a fundamental value in electrochemistry and represents the amount of electric charge per mole of electrons. It is approximately 96,500 Coulombs per mole (C/mol). Understanding Faraday's constant is crucial for electrolysis calculations because it allows us to connect the chemical quantity (moles of electrons) with the physical quantity (electric charge).

This constant is derived from two laws known as Faraday's laws of electrolysis, which state that the amount of substance that undergoes oxidation or reduction at each electrode during electrolysis is directly proportional to the quantity of electricity that passes through the electrolyte. When you know the number of moles of electrons transferred during a reaction, multiplying by Faraday's constant gives you the total charge in Coulombs. This charge is used to calculate the duration of the electrolytic process, helping to establish the practicality of the electrochemical reaction for industrial applications.

Moles of Electrons

The mole concept extends to particles beyond atoms and molecules, including subatomic particles like electrons. In electrochemistry, 'moles of electrons' is a term that describes the amount of electrons that are transferred in an electrochemical reaction. Moles of electrons are central to stoichiometry in electrolysis calculations since the transfer of electrons drives the chemical change.

Each mole of electrons represents the same number of electrons as there are atoms in 12 grams of carbon-12, which is Avogadro's number. To calculate the moles of electrons transferred in an electrolytic process, we must understand the stoichiometry of the electrochemical reaction. This involves balancing the chemical equation and identifying the number of electron transfers for each reaction element. From there, the relationship between moles of electrons and the mass of substances involved in the electrochemical reaction can be established.

Electrochemical Reaction

Electrochemical reactions are at the heart of electrolysis, a process where electrical energy is used to drive a non-spontaneous chemical reaction. These reactions involve the transfer of electrons between species via an external circuit and often result in the decomposition of compounds.

An electrochemical reaction always involves two half-reactions – oxidation and reduction. In our example, the reduction of \(\mathrm{BiO}^{+}\) ions to Bi metal involves gaining an electron (reduction). The balanced equation \(\mathrm{BiO}^+ + e^- \rightarrow \mathrm{Bi} + \mathrm{O}^{2-}\) shows us that for every mole of \(\mathrm{BiO}^{+}\) ions reduced, one mole of electrons is required. This stoichiometric relationship allows us to perform calculations on the time and amount of substance produced during electrolysis. Understanding the precise and balanced nature of electrochemical reactions is essential to predict the outcomes of electrolytic processes in various applications, such as metal plating, purification, and battery operations.