Problem 71 - Chemistry: Principles and Reactions

Question

A baby's spoon with an area of \(6.25 \mathrm{~cm}^{2}\) is plated with silver from \(\mathrm{AgNO}_{3}\) using a current of \(2.00 \mathrm{~A}\) for two hours and 25 minutes. (a) If the current efficiency is \(82.0 \%\), how many grams of silver are plated? (b) What is the thickness of the silver plate formed \(\left(d=10.5 \mathrm{~g} / \mathrm{cm}^{3}\right)\) ?

Step-by-Step Solution

Answer

Answer: The mass of silver plated on the spoon is 15.94 g and the thickness of the silver layer is 0.243 cm.

Step 1: Calculate the time in seconds

Convert 2 hours and 25 minutes to seconds: 2 hours = 2 * 3600 seconds = 7200 seconds 25 minutes = 25 * 60 seconds = 1500 seconds So, Total time = 7200 + 1500 = 8700 seconds.

Step 2: Calculate the charge passed

To calculate the total charge (Q), we will use the formula: Q = I * t, where I is the current (2 A) and t is the total time in seconds (8700 s). Q = 2.00 A * 8700 s = 17400 C

Step 3: Calculate the charge used for silver plating

To find the charge actually used for silver plating, we need to take into account the current efficiency which is given as 82.0%. Therefore, Charge used for silver plating = 17400 C * 0.820 = 14268 C

Step 4: Find the molar equivalents of silver

There is 1 mole of electron(s) involved in the silver plating process from the electrolysis reaction \(\mathrm{Ag^+ + e^- \rightarrow Ag}\). Using Faraday's constant (F = 96485 C/mol), we can calculate the number of molar equivalents of silver (n) by dividing the used charge (14268 C) by F. n = 14268 C / 96485 C/mol = 0.1478 mol

Step 5: Calculate the mass of silver plated

Now, the mass of silver can be found using the product of the moles of silver and its molar mass (m = n * M). The molar mass of silver (M) is 107.87 g/mol. Mass of silver (m) = 0.1478 mol * 107.87 g/mol = 15.94 g

Step 6: Calculate the volume of the silver plated

To find the volume of silver plated, we will use the formula volume (V) = mass (m) / density (d). The density of silver is given as 10.5 g/cm³. Volume of silver (V) = 15.94 g / 10.5 g/cm³ = 1.518 cm³

Step 7: Calculate the thickness of silver plated

We are given the surface area of the spoon (6.25 cm²). To find the thickness of the silver layer (h), we will use the formula: Volume (V) = Area (A) * Thickness (h). Rearranging the formula, we can find the thickness, h = V / A. So, Thickness of the silver layer (h) = 1.518 cm³ / 6.25 cm² = 0.243 cm. The mass of silver plated on the spoon is 15.94 g and the thickness of the silver layer is 0.243 cm.

Key Concepts

Current Efficiency

Current efficiency is a measure of how effectively an electrochemical process converts electrical energy into the desired chemical change, such as plating a metal. In an ideal situation, 100% of the electrical energy would be used for the intended chemical reaction. However, practical efficiencies are often lower due to side reactions and energy losses.

• A common scenario involves calculating how much of the total electric charge introduced into a system actually contributes to the desired outcome.
• This concept is crucial when determining how much substance is deposited during electrolysis.

In our example, the current efficiency is given as 82%, meaning only 82% of the applied electric charge is effective in plating silver onto the spoon.
This efficiency is used to calculate the actual charge responsible for the silver deposition by multiplying the total charge passed by the efficiency percentage.

Faraday's Constant

Faraday's Constant represents the charge of one mole of electrons, approximately 96485 Coulombs per mole (C/mol). It is named after Michael Faraday, one of the pioneers of electrochemistry.

• The constant is essential in relating chemical and electrical quantities in electrochemical reactions.
• It allows for the conversion between the amount of electric charge and amount of substance, which is crucial for understanding processes like electroplating.

In electrolysis, as in silver plating, Faraday's Constant allows us to calculate how many moles of a substance are plated out per mole of electrons transferred. In the example given, by dividing the charge effectively used for silver plating by Faraday's Constant, we determine the amount of substance deposited.

Silver Plating

Silver plating involves the application of a thin layer of silver onto the surface of another metal or material. This is achieved through an electrochemical process where silver ions in a solution are reduced to silver metal and deposited on the target surface.

• The process enhances the appearance and value of items while offering properties like conductivity and corrosion resistance.
• Silver plating is widely used in electronics, jewelry, silverware, and many industrial applications.

In the example provided, silver ions from silver nitrate (\(\mathrm{AgNO} _3\)) are reduced to metallic silver which then sticks to the spoon, forming a thin, lustrous layer. The thickness of this layer depends on factors such as current, time, and efficiency of the process.

Electrolysis Reaction

An electrolysis reaction is a chemical process driven by the application of electrical energy to break bonds of compounds into their constituent elements or ions. This process is used to induce a non-spontaneous reaction.

• Electrolysis is fundamental in extracting and refining metals, electroplating, and producing chemical compounds like chlorine and hydrogen gas.
• During electrolysis, electrical energy forces a transfer of electrons (reduction and oxidation reactions) at the electrodes.

For silver plating, the electrolysis reaction can be represented as\(\mathrm{Ag}^+ + e^- \rightarrow \mathrm{Ag}\). Here, silver ions gain electrons (are reduced) to form solid silver metal on the spoon's surface. Understanding this underlying electrochemical reaction is vital for controlling and optimizing the plating process.