Question

A metallurgist wants to gold-plate an object with a surface area of 17.21 in \(^{2}\). The gold plating must be 0.00200 in. thick (assume uniform thickness). (a) How many grams of gold \(\left(d=10.5 \mathrm{~g} / \mathrm{cm}^{3}\right)\) are required? (b) How many minutes will it take to plate the object from a solution of AuCN using a current of \(7.00 \mathrm{~A} ?\) Assume \(100 \%\) efficiency.

Short answer

Answer: 5.922 grams of gold are required for the plating, and it will take 6.90 minutes to plate the object with 100% efficiency.

Step-by-step solution

Calculating the volume of gold plating

To calculate the volume of gold plating, multiply the surface area by the thickness: Volume = Surface Area * Thickness = 17.21 in^2 * 0.00200 in But, the density is given in cm\(^3\), so it's necessary to convert in\(^2\) to cm\(^2\). Use 1 in = 2.54 cm: 17.21 in\(^2\) * (2.54 cm/in)\(^2\) = 111.12 cm\(^2\) Now, Volume = 111.12 cm\(^2\) * 0.00200 in * 2.54 cm/in = 0.56399 cm\(^3\)

Calculating the mass of gold required

To calculate the mass of gold required, multiply the volume by the density: Mass = Volume * Density = 0.56399 cm\(^3\) * 10.5 g/cm\(^3\) = 5.922 g So, 5.922 grams of gold are required for the plating.

Using Faraday's Law of electrolysis to find the time

Faraday's Law of electrolysis states that mol of electrons required is given by mol of electrons = mol of substance * n where n is the stoichiometry coefficient For gold, \(Au^{+} + e^{-} \rightarrow Au\) so, n = 1 Now, we need to find the mol of gold, Moles of gold = Mass of gold / Molar mass of gold = 5.922 g / 197 g/mol = 0.03004 mol Now, mol of electrons = 0.03004 * 1 = 0.03004 mol Charge = mol of electrons * Faraday's constant = 0.03004 mol * 96485 C/mol = 2899 C Now, we can find the time using the formula Time = Charge / Current Time = 2899 C / 7 A = 414.14 seconds Convert seconds into minutes: 414.14 s * (1 min/60 s) = 6.90 min So, it will take 6.90 minutes to plate the object with 100% efficiency.

Key concepts

Faraday's Law of Electrolysis

Understanding Faraday's Law of electrolysis is essential when diving into electrochemistry. Fundamentally, it relates the amount of substance that is deposited or dissolved at an electrode during electrolysis to the electric charge passing through the electrolyte. The law can be defined in two parts: First, the amount of chemical change is proportional to the amount of electricity that passes through the cell; second, for a given amount of electric charge, the amount of substance that undergoes electrolysis is directly proportional to its equivalent weight.

With these principles in hand, we apply them to practical scenarios like gold plating by calculating the mass of gold deposited on an object using the provided current and the known equivalence of charge to mass of a chemical substance. In this exercise, we needed to determine how long it would take to plate gold on an object, requiring a calculation of the total charge, which then was divided by the constant current to find the time needed for plating.

Stoichiometry in Electroplating

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. In the context of electroplating, stoichiometry involves determining how much of a reactant is needed to produce a desired amount of product—in this case, how much gold is required to achieve a specific thickness of gold plating.

For example, the stoichiometric coefficient, represented by 'n' in Faraday's law, indicates the number of moles of electrons required per mole of substance deposited. In gold electroplating, the stoichiometry of the reaction is simple–one electron is needed to reduce each gold ion to a gold atom, so the coefficient 'n' is 1. This stoichiometric relationship is crucial for calculating the mass of gold needed for plating and the time required for the electroplating process.

Density and Mass Calculations

When you're working with materials, understanding the relationship between density, mass, and volume is key. Density is the mass of a substance per unit volume, and this property helps us calculate the mass of material required for a given volume or conversely, the volume that a certain mass of material will occupy.

In the context of gold plating, the density of gold (given as 10.5 g/cm³) allows us to determine how much gold we need to cover a specific surface area to a certain thickness. Once we have the volume of the plating calculated, we can multiply it by the density to find the mass of gold necessary for the plating. This step is fundamental when preparing the materials for the electroplating process and ensures proper coverage and the desired properties in the plated layer.