Problem 92 - Chemistry: The Central Science in SI Units, Global Edition

Question

Metallic magnesium can be made by the electrolysis of molten \(\mathrm{MgCl}_{2}\) (a) What mass of \(\mathrm{Mg}\) is formed by passing a current of 4.55 A through molten \(\mathrm{MgCl}_{2}\), for 4.50 days? (b) How many minutes are needed to plate out \(25.00 \mathrm{~g} \mathrm{Mg}\) from molten \(\mathrm{MgCl}_{2}\) using \(3.50 \mathrm{~A}\) of current?

Step-by-Step Solution

Answer

For part (a), the mass of magnesium produced is 222.92 g. For part (b), it will take 947.77 minutes to plate out 25 g of magnesium using a current of 3.50 A.

Step 1: Identify relevant information

Given data for (a): - Current (I): 4.55 A - Time (t): 4.50 days - Molar mass of Mg: 24.31 g/mol - Faraday's constant (F): 96485 C/mol Given data for (b): - Mass of Mg: 25.00 g - Current (I): 3.50 A

Step 2: Calculate the charge passed through the electrolyte for part (a)

First, convert time from days to seconds: Time (t) = 4.50 days * (24 hours/1 day) * (60 minutes/1 hour) * (60 seconds/1 minute) = 388800 seconds Next, calculate the charge (Q) using Q = It: Q = (4.55 A) * (388800 seconds) = 1769040 C

Step 3: Calculate the moles of Mg produced for part (a)

Use Faraday's Law to find the moles of Mg produced (n): n = Q / (2 * F) n = 1769040 C / (2 * 96485 C/mol) = 9.17 mol

Step 4: Calculate the mass of Mg produced for part (a)

Use the molar mass of Mg to find the mass of Mg produced: Mass of Mg = n * Molar Mass of Mg Mass of Mg = 9.17 mol * 24.31 g/mol = 222.92 g So, the mass of magnesium produced is 222.92 g.

Step 5: Calculate the charge passed through the electrolyte for part (b)

We know the mass of Mg to be produced is 25 g. First, we'll find the moles of Mg: n = Mass of Mg / Molar Mass of Mg n = 25 g / 24.31 g/mol = 1.03 mol Now we'll use Faraday's Law to find the charge required: Q = n * (2 * F) Q = 1.03 mol * (2 * 96485 C/mol) = 199032.7 C

Step 6: Calculate time needed to plate out Mg for part (b)

Use the relationship Q = It to find the time (t): t = Q / I t = 199032.7 C / 3.50 A = 56866.47 seconds Now convert the time to minutes: t = 56866.47 seconds * (1 minute / 60 seconds) = 947.77 minutes So, it will take 947.77 minutes to plate out 25 g of Mg using a current of 3.50 A.

Key Concepts

Faraday's Law

Faraday's Law plays a crucial role in understanding the process of electrolysis. It essentially allows us to determine the amount of substance produced at an electrode during electrolysis. The essence of Faraday's Law is captured in the formula:\[ n = \frac{Q}{zF} \]where:

• \( n \) is the amount of substance (in moles) produced,
• \( Q \) is the total electric charge passed through the substance (in coulombs),
• \( z \) is the number of electrons involved in the reaction (which is 2 for magnesium since it forms \( \text{Mg}^{2+} \)),
• \( F \) is the Faraday's constant, approximately 96485 C/mol.

This component of the law is essential when calculating the mass of magnesium formed during electrolysis. By knowing the charge and using Faraday's Law, you can deduce the number of moles of magnesium produced. This approach is vital for accurately predicting the outcomes of electrochemical reactions.

Current and Charge Calculation

Calculating the charge that passes through the electrolyte is an important step in electrolysis problems. The charge \( Q \) can be obtained through the relationship:\[ Q = It \]where:

• \( I \) is the current in amperes (A), and
• \( t \) is time in seconds (s).

To convert time given in days to seconds, use:\[ t = \text{days} \times 24 \times 60 \times 60 \]As a result, if 4.55 A of current flows for 4.5 days, the total charge is calculated by substituting these values into the equation for \( Q \). It involves a straightforward multiplication to determine the total coulombs passing through the circuit.This computed charge is then used to find out how many moles of magnesium can be produced, making it a critical calculation for understanding the efficiency and completeness of the electrolysis process.

Moles and Molar Mass

The concept of moles and molar mass is fundamental to chemical calculations. In the context of electrolysis, we use these concepts to link the microscopic world of atoms to measurable macroscopic quantities.To find the moles of substance produced, the formula:\[ n = \frac{\text{Mass}}{\text{Molar Mass}} \]comes into play. Molar mass is the weight of one mole of a substance and is typically expressed in grams per mole (g/mol).For magnesium, given that its molar mass is 24.31 g/mol, we can calculate the number of moles from a given mass. For example, if you need to produce 25 g of Mg, the calculation would be:\[ n = \frac{25 \text{ g}}{24.31 \text{ g/mol}} \approx 1.03 \text{ mol} \]This calculated moles of magnesium are then essential for determining the charge required during electrolysis, thereby linking Faraday's Law, current, charge, and the physical quantity of magnesium required.