Problem 89 - Chemistry The Molecular Science

Question

You wish to electroplate a copper surface having an area of \(1200 \mathrm{~mm}^{2}\) with a \(1.0-\mu \mathrm{m}\) -thick coating of silver from a solution of \(\mathrm{Ag}(\mathrm{CN})_{2}^{-}\) ions. If you use a current of \(150.0 \mathrm{~mA},\) calculate how much electrolysis time you should use. The density of metallic silver is \(10.5 \mathrm{~g} / \mathrm{cm}^{3} .\)

Step-by-Step Solution

Answer

You should use approximately 125.39 minutes of electrolysis time.

Step 1: Calculate the Volume of Silver Required

The first step is to calculate the total volume of silver required to coat the copper surface. The area of the copper surface is \( 1200\, \text{mm}^2 \) and the thickness of the silver layer is \( 1.0\, \mu\text{m} = 1.0 \times 10^{-3} \text{mm} = 10^{-4} \text{cm} \). Therefore, the volume \( V \) is given by:\[V = \text{area} \times \text{thickness} = 1200 \times 10^{-4} \text{ cm}^2 = 0.12 \text{ cm}^3.\]

Step 2: Calculate the Mass of Silver Needed

Next, we use the density of silver to find the mass required. The density \( \rho \) is given as \( 10.5 \, \text{g/cm}^3 \). Using the formula \( \text{mass} = \text{density} \times \text{volume} \): \[\text{mass} = 10.5 \, \text{g/cm}^3 \times 0.12 \, \text{cm}^3 = 1.26 \, \text{g}.\]

Step 3: Calculate the Number of Moles of Silver

The molar mass of silver (Ag) is approximately \( 107.87 \, \text{g/mol} \). The number of moles \( n \) of silver is calculated as follows:\[n = \frac{\text{mass of silver}}{\text{molar mass of silver}} = \frac{1.26 \, \text{g}}{107.87 \, \text{g/mol}} \approx 0.0117 \, \text{mol}.\]

Step 4: Calculate the Total Charge Required

Since \( \text{Ag}^{+} \) ions are involved, 1 mole of \( \text{Ag} \) corresponds to \( 1 \text{ mole} \) of electrons. The total charge \( Q \) required is calculated using Faraday's constant \( F \approx 96485 \, \text{C/mol} \):\[Q = n \times F = 0.0117 \, \text{mol} \times 96485 \, \text{C/mol} \approx 1128.49 \, \text{C}.\]

Step 5: Calculate the Electrolysis Time

Finally, to find the electrolysis time \( t \), we use the formula \( t = \frac{Q}{I} \), where \( I \) is the current given as \( 0.150 \, \text{A} \):\[t = \frac{1128.49 \, \text{C}}{0.150 \, \text{A}} \approx 7523.27 \, \text{s}.\]Converting the time into minutes, we get approximately \( 125.39 \, \text{minutes} .\)

Key Concepts

Copper Surface Coating

Electroplating involves covering a surface with a thin layer of metal. In this instance, we are applying a silver coating to a copper base. This layer enhances corrosion resistance, improves appearance, and conducts electricity effectively.
To calculate how much silver is needed, we first determine the volume. Given an area of 1200 mm² and a thickness of 1.0 µm (or 10⁻⁴ cm), we multiply these values to find the volume:

• Area: 1200 mm² converted to cm² is 0.12 cm²
• Thickness: 10⁻⁴ cm

The total volume is 0.12 cm³.
This calculation forms the basis for determining the mass and moles of silver required.

Silver Ion Solution

Silver plating uses a solution containing silver ions, specifically (\(\text{Ag(CN)}_2^-\)). These ions provide the source of silver for the electroplating process.
In electroplating, it's vital that the silver ions migrate to the surface of the copper to deposit the silver metal.
Understanding the properties of the silver solution helps control the quality and thickness of the plating.

Electrolysis Time Calculation

Electrolysis time is crucial for determining how long the metal deposition should occur.
This time ensures the correct thickness of silver is achieved. The calculation uses the formula \( t = \frac{Q}{I} \), where:

• \(t\) is the electrolysis time
• \(Q\) is the total charge required
• \(I\) is the current in Amperes

In this scenario, a current of 150 mA (0.150 A) was applied. Calculations showed that a charge of approximately 1128.49 C is needed, resulting in an electrolysis time of about 7523.27 seconds or approximately 125.39 minutes.

Moles of Silver

Moles signify the amount of substance. To compute the moles of silver needed, we use the formula:\[n = \frac{\text{mass of silver}}{\text{molar mass of silver}}\]Given a mass of 1.26 g of silver and its molar mass of 107.87 g/mol:

• The calculation results in approximately 0.0117 mol of silver.

This mole calculation helps in understanding how many ions are required for the plating process.

Faraday's Constant

Faraday's constant is key in electrochemistry, representing the charge of one mole of electrons: approximately 96485 C/mol.
This constant is critical when converting moles of electrons to actual charge.
In our example, using Faraday's constant allows us to determine the total charge needed to deposit 0.0117 mol of silver, resulting in 1128.49 C.
This value ensures that the electroplating process is accurate and efficient.