Question

A current of 6.00 A passes through an electrolytic cell containing dilute sulfuric acid for \(3.40 \mathrm{~h}\). If the volume of \(\mathrm{O}_{2}\) gas generated at the anode is \(4.26 \mathrm{~L}\) (at STP), calculate the charge (in coulombs) on an electron.

Short answer

e = 1.602 x 10^{-19} C (approx.)

Step-by-step solution

Calculate Total Charge Passed

To find the total charge passed through the electrolytic cell, use the formula for charge \( Q = I imes t \), where \( I \) is the current in amperes and \( t \) is the time in seconds. Convert the time from hours to seconds: \( 3.40 ext{ hours} = 3.40 imes 3600 ext{ seconds} \). Thus, the total charge is \( Q = 6.00 imes (3.40 imes 3600) \).

Calculate the Faraday's Constant

The volume of oxygen gas \( \mathrm{O}_2 \) generated at the anode can be used to determine the total amount of charge passed. At standard temperature and pressure (STP), 1 mole of an ideal gas occupies 22.4 L. The oxygen produced can hence be related to moles by \( n = \frac{4.26}{22.4} \). For the electrolysis of water, to evolve 1 mole of \( \mathrm{O}_{2} \), 4 moles of electrons are required. Therefore, the total moles of electrons involved are \( 4 \times n \). Since we know that \( 1 ext{ mole of electrons} = 96500 ext{ C} \), the charge associated with these moles is \( Q = 4 \times n \times 96500 \).

Calculate the Charge on an Electron

The charge on an electron \( e \) can be calculated using the total charge and the number of moles of electrons. We use the relation \( e = \frac{Q}{N_A} \), where \( N_A \) is Avogadro's number \( 6.022 \times 10^{23} \).

Finalize the Calculation

After calculating the total charge from steps 1 and 2, substitute the values to find \( e \). Check for consistency between calculated total charge from the electrolytic process and the total charge calculated from Faraday's constant, then use \( e = \frac{Q}{N_A} \) to solve for the specific charge on a single electron.

Key concepts

Faraday's Constant

Faraday's constant is an important value in chemistry and physics, defined as the charge of one mole of electrons. Its approximate value is 96500 coulombs per mole. This constant allows us to connect the amount of electrical charge to the number of moles of electrons involved in electrochemical reactions. In the context of electrolysis, by knowing the Faraday's constant, we can determine how much charge is needed to drive a chemical reaction, like the production of oxygen gas at the anode.
Faraday's constant is used across various fields to relate the macroscopic flow of electrical current to the microscopic flow of electrons. It's especially crucial in determining the efficiencies of electrochemical cells.

Avogadro's Number

Avogadro's number is a fundamental constant that defines the number of constituent particles, usually atoms or molecules, found in one mole of a substance. The value of this constant is approximately \(6.022 \times 10^{23}\). In electrochemistry, Avogadro's number helps us understand the conversion between the macroscopic and microscopic worlds.
In our exercise, it's used to calculate the charge on a single electron. By dividing the total charge by Avogadro's number, we can find how much charge is associated with just one particle (i.e., the electron), which is crucial for understanding fundamental concepts in chemistry and physics.

Charge Calculation

The calculation of charge in an electrochemical setup involves multiplying the current by time. Using the formula \( Q = I \times t \), where \( I \) represents current in amperes and \( t \) represents time in seconds, the total charge is determined. For instance, if a current of \(6 \text{ A}\) flows for \(3.4 \text{ hours}\), we first convert time from hours to seconds to ensure consistency with SI units.

• First, convert \(3.4 \text{ hours}\) into seconds: \(3.4 \times 3600 = 12240 \text{ seconds}\).
• Then, plug in the values: \( Q = 6 \times 12240 = 73440 \text{ coulombs}\).

This total charge computation is fundamental for further calculations in the electrolysis process and gives insight into how many moles of electrons are transferred in the reaction.

Electrolytic Cell

An electrolytic cell is an essential apparatus in electrochemistry that uses electrical energy to drive a non-spontaneous chemical reaction. Within the cell, positive and negative electrodes, known as anode and cathode respectively, facilitate this process.
In our exercise, a current of 6 A is passed through an electrolytic cell with sulfuric acid, resulting in the production of oxygen gas at the anode. Electrolytic cells are not only used for gas production but also for applications like electroplating, electrorefining, and the production of chemicals. The setup usually involves a power source, electrodes immersed in an electrolyte solution, and is designed to ensure the controlled flow of electrons during the reaction.

Oxygen Gas Production

During electrolysis, specifically of water or dilute sulfuric acid, oxygen gas is produced at the anode. This occurs because water (or the solution) decomposes into its elemental components under the influence of an electric current.
Knowing the volume of oxygen gas produced allows us to determine the number of moles of oxygen, using the ideal gas law conditions at standard temperature and pressure (STP), where 1 mole of gas equals 22.4 liters. This calculation informs us how many moles of electrons were involved given that 4 moles of electrons are required to produce 1 mole of oxygen gas. Thus, using the moles of electrons involved, we can back-calculate and link it to the charge and subsequently find out details like the charge associated with single molecules, such as electrons.