Chromium plating is applied by electrolysis to objects suspended in a dichromate solution, according to the following (unbalanced) half-reaction: $$ \mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}(a q)+e^{-}+\mathrm{H}^{+}(a q) \longrightarrow \mathrm{Cr}(s)+\mathrm{H}_{2} \mathrm{O}(l) $$ How long (in hours) would it take to apply a chromium plating \(1.0 \times 10^{-2} \mathrm{~mm}\) thick to a car bumper with a surface area of \(0.25 \mathrm{~m}^{2}\) in an electrolytic cell carrying a current of \(25.0 \mathrm{~A} ?\) (The density of chromium is \(\left.7.19 \mathrm{~g} / \mathrm{cm}^{3} .\right)\)

An electrolytic cell uses electrical energy to drive a non-spontaneous chemical reaction. It's an essential tool in electrochemistry, where electrodes are immersed in an electrolyte, a solution that contains ions. When an external voltage is applied, ions move towards electrodes with an opposite charge: cations towards the cathode (negative electrode) and anions towards the anode (positive electrode).

For chromium plating, the object to be plated, such as a car bumper, serves as the cathode. When current travels through the cell, chromium ions in the solution are reduced and form a thin layer of chromium metal on the bumper. This process not only enhances the appearance of the bumper but also increases its resistance to corrosion.

Michael Faraday's groundbreaking work in electrolysis gave us the Faraday's law of electrolysis, which quantitatively relates the amount of substance that undergoes oxidation or reduction at each electrode to the amount of electrical charge passing through the electrolyte. Faraday’s first law states that the mass of a substance altered at an electrode during electrolysis is proportional to the quantity of electricity used. The second law says the mass of different substances electrolyzed by a certain quantity of electricity is proportional to their equivalent weights.

This principle is crucial for calculating how long it will take to apply a chromium coating of a certain thickness. The electrical charge needed to deposit chromium is directly related to the number of electrons, and thus atoms, needed to build up the metal layer. By knowing the current and the charge associated with a certain mass of chromium, we can determine the time required for plating.

Stoichiometry is the section of chemistry that deals with the relative quantities of reactants and products in chemical reactions. In an electrolysis setting, stoichiometry helps us understand how electrons are transferred in the chemical reactions taking place at the electrodes. For instance, the stoichiometry of the chromium half-reaction tells us that six electrons are needed to reduce one dichromate ion to two chromium atoms.

In the context of plating, stoichiometry allows us to calculate the exact amount of chromium that will be deposited on the bumper for a given number of electrons. By understanding the reaction's stoichiometry, we can estimate the volume and mass of chromium needed for the desired plating thickness.

Chemical calculations involve using mathematical techniques to quantify aspects of chemical reactions. For practical purposes, such as chromium plating, these calculations enable us to find crucial information, like the mass of chromium needed to coat a bumper to the desired thickness. Using the density of chromium, we can relate this mass to a volume, considering the specific thickness of the plating.

The calculations involve various conversions, like changing millimeters to meters to match the units of surface area, or grams to kilograms when using density. Furthermore, we employ Avogadro's number to correlate the amount of chromium in moles to the number of atoms, and subsequently, the electrons required to reduce them. Finally, using Faraday's law, these chemical calculations reveal the time needed for the electroplating process, ensuring precision in the application of chromium coatings.