Problem 107 - Chemistry: The Central Science

Question

(a) How many coulombs are required to plate a layer of chromium metal \(0.25 \mathrm{~mm}\) thick on an auto bumper with a total area of \(0.32 \mathrm{~m}^{2}\) from a solution containing \(\mathrm{CrO}_{4}{\underline{\phantom{xx}}}^{2-}\) ? The density of chromium metal is \(7.20 \mathrm{~g} / \mathrm{cm}^{3}\), (b) What current flow is required for this electroplating if the bumper is to be plated in \(10.0 \mathrm{~s}\) ? (c) If the external source has an emf of \(+6.0 \mathrm{~V}\) and the electrolytic cell is \(65 \%\) efficient, hew much electrical power is expended to electroplate the bumper?

Step-by-Step Solution

Answer

The short answer is: (a) To plate the auto bumper, \(3.21 \times 10^{6}\) Coulombs are required. (b) The current flow required for electroplating in 10 seconds is \(3.21 \times 10^{5}\) Amps. (c) The electrical power expended for electroplating is \(2.96 \times 10^{6}\) Watts, considering 65% efficiency of the electrolytic cell.

Step 1: Determine the mass of chromium needed

First, we need to determine the mass of chromium required to plate the auto bumper. Calculate the volume of chromium required using thickness and area, and then find the mass using density. - Thickness \(= 0.25 \mathrm{~mm} = 0.025 \mathrm{~cm}\) - Area \(= 0.32 \mathrm{~m}^{2} = 3200 \mathrm{~cm}^{2}\) - Density \(= 7.20 \mathrm{~g/cm^{3}}\) Volume \(= 0.025 \mathrm{~cm} \times 3200 \mathrm{~cm^{2}} = 80 \mathrm{~cm^{3}}\) Mass of chromium \(= \mathrm{Density} \times \mathrm{Volume}\) \( = 7.20 \mathrm{~g/cm^{3}} \times 80 \mathrm{~cm^{3}} = 576 \mathrm{~g}\)

Step 2: Find the moles of chromium needed

Now, we need to find the moles of chromium to plate the auto bumper. The molar mass of chromium is 51.996 g/mol. Moles of chromium \(= \frac{\mathrm{Mass}}{M_{\mathrm{Cr}}}\) \(= \frac{576 \mathrm{~g}}{51.996 \mathrm{~g/mol}} = 11.08 \mathrm{~mol}\)

Step 3: Calculate the number of coulombs required

Use Faraday's law of electrolysis to find the number of coulombs required to plate the chromium. Faraday's constant (F) \(= 96485 \mathrm{~C/mol}\) Chromium has an oxidation state of +3 in the electrolytic cell Coulombs \(= \mathrm{Moles} \times \mathrm{Charge} \times F\) \( = 11.08 \mathrm{~mol} \times 3 \times 96485 \mathrm{~C/mol} = 3.21 \times 10^{6} \mathrm{~C}\)

Step 4: Calculate the current flow required for plating

We are given the time required for plating \(= 10.0~\mathrm{s}\). To find the current, we can use the formula: Current (I) \(= \frac{\mathrm{Coulombs}}{\mathrm{Time}}\) \(= \frac{3.21 \times 10^{6} \mathrm{~C}}{10.0 \mathrm{~s}} = 3.21 \times 10^{5} \mathrm{~A}\)

Step 5: Calculate the electrical power expended

We know the electrolytic cell is \(65\%\) efficient and the emf is \(+6.0 \mathrm{~V}\). To find the electrical power expended, use the formula: Power \(= \frac{\mathrm{emf} \times I}{\mathrm{Efficiency}}\) \(= \frac{6.0 \mathrm{~V} \times 3.21 \times 10^{5} \mathrm{~A}}{0.65} = 2.96 \times 10^{6} \mathrm{~W}\) So, the answers are (a) \(3.21 \times 10^{6}\) Coulombs, (b) \(3.21 \times 10^{5}\) Amps, and (c) \(2.96 \times 10^{6}\) Watts.

Key Concepts

Faraday's Law of Electrolysis

Faraday's law of electrolysis is a fundamental principle that quantifies the relationship between the electric charge passed through an electrolyte and the amount of substance that is deposited at an electrode. According to this law, the amount of chemical change produced by an electric current during electrolysis is directly proportional to the quantity of electricity (coulombs) that passes through the electrolyte.

This principle helps to calculate the number of coulombs needed for a plating process, such as in the exercise scenario with chromium plating. The law also indicates that the number of equivalents of a substance deposited at an electrode is directly proportional to the number of equivalents of electrons (or electric charge) that pass through the electrochemical cell.

An 'equivalent' is a measure of the reactive capacity of a molecule and is based on the number of electrons that are transferred in the redox reaction. For chromium electroplating, chromium ions typically gain three electrons (since Cr has an oxidation state of +3), meaning that three equivalents of electrons are required to deposit one equivalent of chromium atoms on the bumper.

Electrochemical Cell Efficiency

Electrochemical cell efficiency is a measure of how effectively an electrochemical cell converts electrical energy into chemical energy during an electrolysis process. This efficiency can be affected by various factors, including the electrical resistance of the solution, the quality of the electrodes, and the voltage applied across the cell.

Efficiency is generally expressed as a percentage and is calculated by dividing the actual yield of the electrochemical process by the theoretical yield and then multiplying by 100. For example, if an electrochemical cell has an efficiency of 65%, this implies that only 65% of the electrical energy supplied to the cell is converted into the desired chemical reaction; the remainder is lost, for instance, as heat due to the resistance in the electrolytic solution.

In the case of the bumper electroplating exercise, this concept is applied to determine the electrical power expended, considering the efficiency of the cell. If the cell's efficiency is not taken into account, the calculated power required would be higher than what's actually necessary, leading to an overestimation of the cost and energy consumption of the electroplating process.

Calculating Moles in Electroplating

In electroplating, understanding how to calculate moles can be crucial for determining the amount of metal that will be deposited onto a substrate. This process involves converting mass to moles using the molar mass of the plating metal, as demonstrated in the textbook exercise with chromium.

The molar mass is a key value, which represents the mass of one mole of a given substance. In this context, one mole of a substance contains Avogadro's number of atoms, ions, or molecules. For chromium, which has a molar mass of approximately 51.996 grams per mole, converting the mass of chromium needed for the plating (in grams) to moles allows for the use of Faraday's law to calculate the total charge required to deposit that amount of metal.

To sum up, calculating moles is an essential step in estimating the required electrical charge for the deposition process. From the moles, we then determine the charge, by considering the charge of the ions involved and Faraday's constant, eventually leading to the calculation of the total coulombs required in the electroplating of the auto bumper.