Question

An acidified solution was electrolyzed using copper electrodes. A constant current of 1.18 A caused the anode to lose \(0.584 \mathrm{~g}\) after \(1.52 \times 10^{3} \mathrm{~s}\). (a) What is the gas produced at the cathode, and what is its volume at STP? (b) Given that the charge of an electron is \(1.6022 \times 10^{-19} \mathrm{C},\) calculate Avogadro's number. Assume that copper is oxidized to \(\mathrm{Cu}^{2+}\) ions.

Short answer

Hydrogen gas is produced; its volume is 0.206 L at STP; Avogadro's number is approximately \(6.10 \times 10^{23} \).

Step-by-step solution

Understanding the Reaction

In the given electrolysis process, copper metal at the anode is oxidized to Cu²⁺ ions as copper loses mass. The mass lost from the copper anode implies Cu(s) → Cu²⁺(aq) + 2e⁻. At the cathode, hydrogen gas is produced from protons in the acidified solution.

Calculate Moles of Copper Oxidized

The molar mass of copper is approximately 63.55 g/mol. The moles of copper oxidized equals the mass lost divided by the molar mass: \[ \text{Moles of Cu} = \frac{0.584 \text{ g}}{63.55 \text{ g/mol}} = 0.00919 \text{ moles} \]

Calculate the Coulombs Used

Using the relation \( Q = I \times t \), where \( Q \) is the charge in coulombs, \( I \) is the current in amperes, and \( t \) is the time in seconds: \[ Q = 1.18 \text{ A} \times 1.52 \times 10^3 \text{ s} = 1793.6 \text{ C} \]

Calculate Moles of Electrons Used

From Step 2, each mole of Cu produces 2 moles of electrons. So, moles of electrons \( = 2 \times 0.00919 = 0.01838 \text{ moles} \).

Find the Volume of Gas at the Cathode

The gas produced at the cathode is hydrogen, H₂. Each mole of electrons at the cathode produces 0.5 moles of H₂. Thus moles of H₂ = 0.5 x 0.01838 = 0.00919 \text{ moles}. At STP, 1 mole of gas occupies 22.4 L. Volume \( = 0.00919 \text{ moles} \times 22.4 \text{ L/mol} = 0.206 \text{ L} \).

Calculate Avogadro's Number

Knowing \( Q = nF \) where \( n \) is moles of electrons and \( F \) is Faraday's constant. Calculate \( F \) as \( F = \frac{Q}{n} = \frac{1793.6 \text{ C}}{0.01838 \text{ moles}} = 97611.4 \text{ C/mol} \). Since \( F = e \times N_A \), \( N_A = \frac{F}{e} \). Thus, \( N_A = \frac{97611.4}{1.6022 \times 10^{-19}} = 6.10 \times 10^{23} \).

Key concepts

Copper Oxidation

During the electrolysis process, copper oxidation is key. Copper at the anode loses electrons and is transformed into copper ions, specifically \( ext{Cu} ightarrow ext{Cu}^{2+} + 2 ext{e}^- \). The copper metal's mass loss at the anode indicates how much copper is converting to copper ions. When 0.584 grams of copper lost its mass, we calculated the number of moles using the molar mass of copper which is 63.55 g/mol. The result was 0.00919 moles. This means that this amount of copper underwent oxidation to form \( ext{Cu}^{2+}\).

• The importance of this oxidation process lies in the transfer of electrons, which drives electrochemical reactions.
• Understanding copper’s role helps in applications such as electroplating and refining where copper ions are critical.

Hydrogen Production

At the cathode in an acidified solution, hydrogen gas is produced. This happens as protons from the solution gain electrons. Here's the reaction: 2\( ext{H}^+ + 2 ext{e}^- ightarrow ext{H}_2\). Every two electrons produce one mole of hydrogen gas because each hydrogen molecule \( ext{H}_2\) needs two electrons. In this exercise, 0.01838 moles of electrons resulted in 0.00919 moles of hydrogen gas, aligning perfectly with our copper calculations. The produced 0.00919 moles of hydrogen gas at STP occupies a volume:\[ = 0.00919 ext{ moles} imes 22.4 ext{ L/mol} = 0.206 ext{ L}. \]

• Understanding hydrogen production is essential in various industries including fuel cells and the hydrogen economy.
• Electrolysis's ability to produce hydrogen cleanly makes it a focus for sustainable energy solutions.

Coulombs Calculation

In electrolysis, the flow of electric charge is measured in coulombs. We calculate the total charge (Q) using the formula:\[ Q = I \times t, \]where I represents current in amperes and t represents time in seconds. In our exercise, a current of 1.18A ran for 1520 seconds, leading to 1793.6 coulombs of charge. This is essential because:

• The calculation of coulombs helps in understanding how much electrical energy is required for a chemical reaction.
• It's crucial for industries to estimate energy costs precisely for processes involving electrolysis.

Avogadro's Number Calculation

Avogadro's number represents the number of atoms, ions, or molecules in one mole of a substance. To find it, we use the relationship between the charge, moles of electrons, and Faraday's constant (F). The formula is:\[ N_A = \frac{F}{e}, \]where Q = nF and F = \frac{Q}{n}. Given 1793.6 C and 0.01838 moles, we find F as 97611.4 C/mol. Using the charge of an electron (1.6022 \times 10^{-19} \, \text{C}), Avogadro's number is calculated to be approximately 6.10 \times 10^{23}.

• This number is fundamental in chemistry, allowing chemists to convert between the microscopic scale of atoms and the macroscopic scale of grams.
• It helps in stoichiometric calculations that are crucial in chemical reactions and equations.