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Chapter 20: Problem 107

(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?

Short Answer

Expert verified
(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 by step solution

01

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: \(Thickness = 0.15 \: mm = 0.015 \: cm\) \(Area = 0.40 \: m^{2} = 4000 \: cm^{2}\) \(Volume = Thickness × Area = 0.015 \: cm * 4000 \: cm^{2} = 60 \: cm^{3}\) \(Mass = Density × Volume = 7.20 \: \dfrac{g}{cm^{3}} * 60 \: cm^{3} = 432 \: g\)
02

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: \(Moles\: of\: Chromium = \dfrac{Mass}{Molar \: Mass} = \dfrac{432 \: g}{51.996 \: \dfrac{g}{mol}} = 8.308 \: mol\)
03

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: \(Charge = Moles × Faraday \: Constant = 8.308 \: mol * 96,485 \: \dfrac{C}{mol} = 801,960 \: Cubl\)
04

Find the current flow

Now, let's find the current flow required for this electroplating process to happen in 20 seconds: \(Current = \dfrac{Charge}{Time} = \dfrac{801,960 \: C}{20.0 \: s} = 40,098 \: A\)
05

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.

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