Problem 107 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) How many coulombs are required to plate a layer of chromium metal \(0.15 \mathrm{~mm}\) thick on an auto bumper with a total area of \(0.40 \mathrm{~m}^{2}\) from a solution containing \(\mathrm{CrO}_{4}^{2-}\) ? 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 \(20.0 \mathrm{~s} ?(\mathbf{c})\) If the external source has an emf of \(+5.5 \mathrm{~V}\) and the electrolytic cell is \(60 \%\) efficient, how much electrical energy is expended to electroplate the bumper?

Step-by-Step Solution

Answer

(a) 801,960 Coulombs of charge are required to plate the chromium layer. (b) The current flow required for electroplating in 20 seconds is 40,098 A. (c) The electrical energy expended for electroplating, considering the cell's efficiency, is 7,351,300 J.

Step 1: Find the mass of chromium required

We are given the thickness of the chrome layer (0.15 mm) and the total plated surface area (0.40 m²). First, we need to find the volume of chromium and then calculate its mass, using the given density: \(Thickness = 0.15 \: mm = 0.015 \: cm\) \(Area = 0.40 \: m^{2} = 4000 \: cm^{2}\) \(Volume = Thickness × Area = 0.015 \: cm * 4000 \: cm^{2} = 60 \: cm^{3}\) \(Mass = Density × Volume = 7.20 \: \dfrac{g}{cm^{3}} * 60 \: cm^{3} = 432 \: g\)

Step 2: Find the moles of chromium

Now that we have found the mass of chromium, we can determine the number of moles of chromium required using its molar mass: \(Moles\: of\: Chromium = \dfrac{Mass}{Molar \: Mass} = \dfrac{432 \: g}{51.996 \: \dfrac{g}{mol}} = 8.308 \: mol\)

Step 3: Find the amount of charge in Coulombs

Using the Faraday constant (F), which is 96,485 C/mol, we will determine the amount of charge required to plate the chromium: \(Charge = Moles × Faraday \: Constant = 8.308 \: mol * 96,485 \: \dfrac{C}{mol} = 801,960 \: Cubl\)

Step 4: Find the current flow

Now, let's find the current flow required for this electroplating process to happen in 20 seconds: \(Current = \dfrac{Charge}{Time} = \dfrac{801,960 \: C}{20.0 \: s} = 40,098 \: A\)

Step 5: Find the electrical energy expended

The electrolytic cell is 60% efficient, meaning that only 60% of the electrical energy is used for electroplating and the remaining 40% is wasted. To calculate the true energy used to plate the bumper, we will find the theoretical energy required and then divide by the efficiency: \(Theoretical \: Energy = Charge × emf = 801,960 \: C * 5.5 \: V = 4,410,780 \: J\) \(Actual \: Energy = \dfrac{Theoretical \: Energy}{Efficiency} = \dfrac{4,410,780 \: J}{0.60} = 7,351,300 \: J\) To summarize, the electroplating process requires (a) 801,960 Coulombs of charge, (b) a current of 40,098 A, and (c) an expenditure of 7,351,300 J of electrical energy considering the cell's efficiency.

Key Concepts

Coulombs

Coulombs are a unit used to measure electric charge. In electrochemistry, the quantity of electric charge is crucial, especially in processes like electroplating. A coulomb is the amount of charge transferred by a current of one ampere in one second. In the given problem, the task is to plate a bumper with chromium, a metal known for its resistance to corrosion and attractive finish. The need to determine how many coulombs are necessary helps us understand the amount of electric charge required to deposit a specific mass of metal on a surface. This involves calculating the total charge required based on the moles of chromium, utilizing the Faraday constant to convert between moles of electrons and charge in coulombs.

Faraday Constant

The Faraday Constant is a key figure in the realm of electrochemistry. Named after Michael Faraday, it represents the amount of electric charge carried by one mole of electrons (approximately 96,485 coulombs per mole). This constant is crucial when calculating how much charge is needed to perform a specific chemical reaction in electroplating. It provides a bridge between macroscopic quantities like current and microscopic quantities such as the number of molecules or atoms. For this exercise, the Faraday Constant is used to determine the total charge in coulombs required to plate chromium. By multiplying the number of moles of chromium by this constant, we can find the total coulombs needed for the reaction.

Electrical Energy

Electrical energy is the power used to perform work, in this case, the work of electroplating. The energy expended is a function of the total charge and the potential difference (or voltage) applied across the electrolytic cell. This is given by the formula \( \text{Energy} = \text{Charge} \times \text{Voltage} \). However, since the cell in our problem is only 60% efficient, it reminds us that not all the energy supplied is used effectively. The theoretical energy calculation is adjusted to account for this, leading to the actual energy expended being higher than the energy expected in a fully efficient system. This helps highlight how real-world systems often have inefficiencies that must be considered.

Current Flow

Current flow is the rate at which electric charge moves through a conductor, measured in amperes (A). It is central to understanding how quickly the electroplating process occurs. In this scenario, knowing the total charge in coulombs allows us to find the current needed using the formula \( \text{Current} = \dfrac{\text{Charge}}{\text{Time}} \). This gives a direct indication of how fast we can plate the bumper with chromium given a certain amount of time. Current flow has to be precisely controlled: too high a current might lead to poor quality plating, while too low a current would mean the process takes too long. Calculating the right current value is crucial for efficient and quality electroplating.