Question

(a) A \(\mathrm{Cr}^{3+}(a q)\) solution is electrolyzed, using a current of 7.60 A. What mass of \(\mathrm{Cr}(s)\) is plated out after 2.00 days? (b) What amperage is required to plate out \(0.250 \mathrm{~mol}\) Cr from a \(\mathrm{Cr}^{3+}\) solution in a period of \(8.00 \mathrm{~h} ?\)

Short answer

The mass of Chromium plated out after 2.00 days is 235.2 g. The amperage required to plate out 0.250 mol of Cr from a Cr^3+ solution in a period of 8.00 hours is 2.53 A.

Step-by-step solution

(a) Calculate the moles of electrons passed through the cell

First, we need to convert the time from days to seconds: \(2.00\,days × \frac{24\,hours}{1\,day} × \frac{60\,minutes}{1\,hour} × \frac{60\,seconds}{1\,minute} = 172,800\,s\). Then, we can use the equation for moles of electrons passed through the cell: moles of electrons = (7.60 A) × (172,800 s) / (96,485 C/mol) = 13.58 mol.

(a) Find moles of Chromium plated out

The stoichiometry of the reaction is as follows: \[Cr^{3+}(aq) + 3e^- \to Cr(s)\] For every 3 moles of electrons, we have 1 mole of Chromium plated out. Thus, we can find the moles of Chromium as follows: moles of Chromium = 13.58 mol electrons / 3 = 4.526 mol.

(a) Calculate the mass of Chromium plated out

Now, we can convert the moles of Chromium to mass using its molar mass (51.996 g/mol): mass of Chromium = (4.526 mol) × (51.996 g/mol) = 235.2 g. So the mass of Chromium plated out after 2.00 days is 235.2 g.

(b) Find the moles of electrons needed to plate out 0.250 mol Cr

From the stoichiometry of the reaction, we need 3 moles of electrons for 1 mole of Chromium. Thus, the moles of electrons needed = 3 × 0.250 mol = 0.750 mol.

(b) Calculate the time in seconds

We need to convert the given time from hours to seconds: \(8.00\,hours × \frac{60\,minutes}{1\,hour} × \frac{60\,seconds}{1\,minute} = 28,800\,s\).

(b) Calculate the amperage needed to plate out 0.250 mol Cr

Now, we can use the equation for moles of electrons passed through the cell to find the amperage: current (A) = (0.750 mol × 96,485 C/mol) / 28,800 s = 2.53 A. Thus, the amperage required to plate out 0.250 mol of Cr from a Cr^3+ solution in a period of 8.00 hours is 2.53 A.

Key concepts

Current Calculation

In electrolysis, calculating the current is a crucial step in determining how much substance can be processed. Current, measured in amperes (A), refers to the flow of electric charge. In the context of electrolysis, this flow of charge is used to drive a non-spontaneous chemical reaction.
To calculate the current, you need the total charge and the time over which the charge flows. The relationship can be described by the formula:

• Current (A) = Total charge (C) / Time (s)

The total charge is often determined using the Faraday constant, which is approximately 96,485 coulombs per mole of electrons. By knowing the moles of electrons involved in the reaction, you can calculate the total charge.
For example, when electrolyzing a solution to plate out metal, you'd measure the current in order to determine how much of that metal can be deposited over a given period. In our exercise, the solution used 7.60 A of current for two days to deposit chromium. You'll also need to relate time into seconds to use this formula effectively.

Mole Concept

The mole concept is a foundational idea in chemistry that helps us bridge the gap between the microscopic world of atoms and molecules and the macroscopic world we observe. One mole is equivalent to Avogadro's number (\[6.022 \times 10^{23}\] entities), which is a huge number used to represent how many atoms, ions, or molecules are in a sample.
In electrolysis, the mole concept helps us convert between the number of electrons transferred and the amount of substance plated out at the electrodes. By using the known stoichiometry of a chemical reaction, we can determine how many moles of a substance will be involved in a given process.
For our exercise, chromium ions are reduced by gaining electrons, which requires an understanding of how many moles of electrons are needed. When \[Cr^{3+}\] is reduced to Cr(s), for every mole of chromium that is plated out, three moles of electrons are involved. This conversion is vital for calculating many quantities in electrochemical reactions, linking theoretical chemistry with practical applications like electroplating.

Stoichiometry

Stoichiometry refers to the quantitative relationships in a chemical reaction, allowing us to calculate reactants and products based on balanced chemical equations. In electrolysis, understanding stoichiometry is key to determining how much of a reactant is consumed or how much of a product is formed.
The first step is to write a balanced chemical equation, which shows the proportions of reactants and products. In our context, for chromium plating, the equation \[Cr^{3+}(aq) + 3e^- \to Cr(s)\] indicates that three moles of electrons are necessary to reduce one mole of \[Cr^{3+}\] to solid chromium.
Given this stoichiometric relationship, we can use it to solve various questions about the process, such as how many moles of a substance are formed or how much current is needed. Clearly understanding these relationships allows us to efficiently calculate properties such as the mass of chromium that can be plated or the time required for a certain amount of plating to occur. This bridges theoretical principles with practical electrochemical applications effectively.