Problem 70 - Chemistry The Molecular Science

Question

The vanadium(II) ion can be produced by electrolysis of a vanadium(III) salt in solution. Calculate how long you must carry out an electrolysis if you wish to convert completely \(0.125 \mathrm{~L}\) of \(0.0150-\mathrm{M} \mathrm{V}^{3+}(\mathrm{aq})\) to \(\mathrm{V}^{2+}(\mathrm{aq})\) using a current of \(0.268 \mathrm{~A}\).

Step-by-Step Solution

Answer

Electrolysis must be conducted for approximately 11.25 minutes.

Step 1: Determine the number of moles of V3+

First, calculate the moles of \( V^{3+} \). Use the concentration and volume of the solution.\[\text{Moles of } V^{3+} = 0.0150 \, \text{M} \times 0.125 \, \text{L} = 0.001875 \, \text{mol}\]

Step 2: Understand the reaction in electrolysis

During electrolysis, each \( V^{3+} \) ion is reduced to \( V^{2+} \) by gaining one electron.\[\mathrm{V}^{3+} + ext{e}^- \rightarrow \mathrm{V}^{2+} \]This indicates that the number of moles of electrons needed equals the number of moles of \( V^{3+} \).

Step 3: Calculate the total charge required

We need to calculate the total charge by using the formula:\[\text{Total charge (Coulombs)} = \text{moles of electrons} \times \text{Faraday's constant (96485 C/mol)}\]Substitute the moles of \( V^{3+} \) (or electrons):\[0.001875 \, \text{mol} \times 96485 \, \text{C/mol} = 180.640625 \, \text{C}\]

Step 4: Calculate the time needed for electrolysis

Use the formula relating current, charge, and time:\[\text{Time (seconds)} = \frac{\text{Total charge (Coulombs)}}{\text{Current (Amperes)}}\]Substitute the values into the formula:\[\text{Time (seconds)} = \frac{180.640625 \, \text{C}}{0.268 \, \text{A}} = 674.77 \, \text{s}\]

Step 5: Convert the time to minutes

To present the time in a more standard unit, convert seconds into minutes by dividing by 60:\[\text{Time (minutes)} = \frac{674.77 \, \text{s}}{60} \approx 11.25 \, \text{minutes}\]

Key Concepts

Vanadium Ion Reduction

Vanadium ion reduction is a fundamental process in electrochemistry where a vanadium(III) ion, denoted as \( V^{3+} \), gains an electron to become a vanadium(II) ion, represented as \( V^{2+} \). This transformation occurs during electrolysis. In this reaction, each \( V^{3+} \) ion needs precisely one electron to convert into \( V^{2+} \). The equation of the reaction is:
\[\mathrm{V}^{3+} + \mathrm{e}^- \rightarrow \mathrm{V}^{2+}\]
This reduction process is key because it showcases how electrons move during a redox reaction. Understanding this concept is crucial for predicting how substances will react during electrolysis and other chemical processes. When observing the amount of reduction needed for an entire solution, the number of \( V^{3+} \) ions dictates how many electrons are required, and hence, how much electricity needs to pass through the solution to cause the conversion.

Faraday's Constant

Faraday's Constant is a pivotal number in electrochemistry and is used to calculate the total electric charge required to carry out reactions involving ions. It is named after Michael Faraday, a scientist who made significant contributions to the field of electromagnetism and electrochemistry. The value of Faraday's Constant is approximately 96485 Coulombs per mole of electrons.

• This constant helps in determining how much charge in Coulombs is needed to convert a certain number of moles of ions during electrolysis.
• It links the macroscopic concept of moles with the microscopic world of electrons and provides a bridge to calculate the amount of electricity in chemical processes.

For example, the reduction of \( V^{3+} \) to \( V^{2+} \) requires electrons, and their total charge can be calculated using Faraday’s Constant. By multiplying the number of moles of electrons needed by the constant, one derives the total charge required to drive the electrochemical reaction.

Current and Charge Relationship

The relationship between current and charge is foundational in understanding electrolysis and electrochemical processes. The formula linking these two quantities can be expressed as:
\[\text{Charge (Coulombs)} = \text{Current (Amperes)} \times \text{Time (seconds)}\]
This equation means that the total charge transferred during an electrochemical reaction is determined by the amount of current flowing and the time over which it flows.

• Current is the rate at which charge flows through a conductor or in this case, the electrolytic solution.
• Knowing either the current or the charge can help solve for the missing variable, essential for problems involving electrolysis time or necessary current.

In practical applications, adjusting the current affects how quickly a reaction can happen during electrolysis. For instance, a higher current would mean a faster reaction time, assuming the total charge necessary remains the same. Understanding this relationship is key to controlling electrolysis processes effectively.