Problem 91 - Chemistry: The Central Science in SI Units, Global Edition

Question

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

Step-by-Step Solution

Answer

(a) The mass of \(\mathrm{Cr}\) plated out after 2.00 days is 744.33 g. (b) The amperage required to plate out 0.250 mol \(\mathrm{Cr}\) from a \(\mathrm{Cr}^{3+}\) solution in a period of 8.00 h is 25.061 A.

Step 1: (a) Calculate the total charge

: First, we need to determine the total charge (in coulombs) that has passed through the solution. Since we have the current (7.60 A) and the time (2.00 days), we can calculate the charge using the formula: \[ Q = I \times t \] Convert the time to seconds: \[ 2.00\,\mathrm{days} \times \frac{24\,\mathrm{h}}{1\,\mathrm{day}} \times \frac{3600\,\mathrm{s}}{1\,\mathrm{h}} = 172800\,\mathrm{s} \] Now, calculate the total charge (Q): \[ Q = 7.60\,\mathrm{A} \times 172800\,\mathrm{s} = 1313280\,\mathrm{C} \]

Step 2: (a) Apply Faraday's law

: Now, apply Faraday's law to calculate the amount of substance (moles of \(\mathrm{Cr}\)) produced during electrolysis: \[ n = \frac{Q}{z \times F} \] Where \(n\) is the amount (in moles) of substance (in this case, \(\mathrm{Cr}\)), \(Q\) is the total charge, \(z\) is the number of electrons transferred (3 for \(\mathrm{Cr}^{3+}\)), and \(F\) is Faraday's constant (\(96485\,\mathrm{C/mol}\)). \[ n = \frac{1313280\,\mathrm{C}}{3 \times 96485\,\mathrm{C/mol}} = 14.315\,\mathrm{mol} \]

Step 3: (a) Convert moles to mass

: To find the mass of \(\mathrm{Cr}\), we need its molar mass, which is \(51.996\,\mathrm{g/mol}\): \[ m = n \times M \] Where \(m\) is the mass, \(n\) is the amount of substance and \(M\) is the molar mass. \[ m = 14.315\,\mathrm{mol} \times 51.996\,\mathrm{g/mol} = 744.33\,\mathrm{g} \]

Step 4: (b) Calculate moles of Cr for given time

: For the second part of the problem, we are given the amount of substance (\(0.250\,\mathrm{mol}\,\mathrm{Cr}\)) and the desired time frame (\(8.00\,\mathrm{h}\)). Let's convert the time to seconds: \[ 8.00\,\mathrm{h} \times \frac{3600\,\mathrm{s}}{1\,\mathrm{h}} = 28800\,\mathrm{s} \]

Step 5: (b) Use Faraday's law to find the required current

: We will once again use Faraday's law, but this time, we will rearrange the formula to find the required current (I): \[ I = \frac{n \times z \times F}{t} \] We already have the values for \(n\), \(z\), and \(t\), so plug in the values: \[ I = \frac{0.250\,\mathrm{mol} \times 3 \times 96485\,\mathrm{C/mol}}{28800\,\mathrm{s}} = 25.061\,\mathrm{A} \] #Answer#: (a) The mass of \(\mathrm{Cr}\) plated out after 2.00 days is 744.33 g. (b) The amperage required to plate out 0.250 mol \(\mathrm{Cr}\) from a \(\mathrm{Cr}^{3+}\) solution in a period of 8.00 h is 25.061 A.

Key Concepts

Faraday's Law

Faraday's Law of Electrolysis is a foundational principle in electrochemistry. It states that the amount of substance produced at each electrode during electrolysis is directly proportional to the quantity of electricity passed through the electrolyte. This law helps us connect electricity and chemistry in a meaningful way. To predict the amount of a chemical change that occurs, we use the formula:

\[ n = \frac{Q}{z \times F} \]where:

• \(n\) is the amount of substance, measured in moles
• \(Q\) is the total charge passed through the system, in Coulombs
• \(z\) is the number of electrons transferred in the reaction
• \(F\) is Faraday’s constant, approximately \(96485 \, \mathrm{C/mol}\)

This formula is essential for quantifying changes in chemical reactions during electrolysis. By knowing these values, we can calculate how much of a substance is transformed during the electrolysis process.

Coulombs

The unit of charge in the International System of Units (SI) is the Coulomb. Named after Charles-Augustin de Coulomb, it represents the quantity of electricity transported in one second by a current of one ampere. In calculations involving electrolysis, the Coulomb is crucial because it measures the total electrical charge involved.
To find the Coulombs during a process, you need the current (in amperes) and the time (in seconds). The relationship is simple:
\[ Q = I \times t \]where:

• \(Q\) is the charge in Coulombs
• \(I\) is the current in amperes
• \(t\) is the time in seconds

For example, a current of \(7.60\) amperes running for \(2.00\) days (converted to seconds as \(2.00 \times 24 \times 3600 = 172800 \text{ seconds}\)) will pass a total charge of \(1313280 \text{ Coulombs}\). This is essential for determining the quantity of substance produced during electrolysis.

Current (Amperes)

Current, measured in amperes (A), is the rate at which electric charge flows past a point in a circuit. It defines how fast electrons are moving through the conductor. For electrolysis, knowing the current is crucial because it directly impacts the total charge that enables chemical reactions to occur inside the electrolyte.
The formula you use is:
\[ Q = I \times t \]Thus, for electrolysis applications, you need to:

• Determine the desired chemical reaction outcome (in terms of moles of product)
• Use Faraday's Law to identify the required total charge (Coulombs)
• Calculate the necessary current to achieve this goal within a given timeframe

For instance, if you need to plate out \(0.250\, \text{mol}\) of chrome in \(8\) hours, you determine the current by rearranging Faraday's Law:\[ I = \frac{n \times z \times F}{t} \]Inserting values gives a current I of \(25.061 \text{ amperes}\), highlighting the relationship between current, time, and the production of a desired amount of substance.