2.9 Relation Between the Faraday Constant, Avogadro Constant, and the Charge on the Electron

In the previous section, we learned about electrolysis and the factors that determine the products formed at the electrodes. Now, we need to understand the quantitative relationship between the amount of electricity passed through the cell and the amount of substance produced or consumed. This relationship is built on three fundamental constants: the Faraday constant, the Avogadro constant, and the charge on a single electron.

Main Idea

The main idea is that the total charge passed during electrolysis can be calculated from the current and time. This charge is carried by electrons. By knowing the charge of a single electron and the number of electrons in a mole (Avogadro's number), we can determine the total charge of one mole of electrons, which is called the Faraday constant. This constant is the bridge between the macroscopic world of measurable quantities (grams, coulombs) and the microscopic world of atoms and electrons.

Core Concept

The core concept is the relationship between charge, moles of electrons, and mass of substance. One mole of electrons carries a charge of one Faraday (F). The charge on a single electron is the elementary charge (e), and one mole contains Avogadro's number (N_A) of electrons. Therefore, the total charge of one mole of electrons (the Faraday constant) is the product of the charge on a single electron and Avogadro's number: F = N_A × e

Detailed Explanation

The Relationship Between Current, Time, and Charge

The amount of a substance produced at an electrode during electrolysis depends on the period of time for which a constant current is passed and the quantity of charge in coulombs that passes through the electrolyte. The relation between current and time is: Q = I × t

• Q = Quantity of charge in coulombs (C)
• **I** = Current in Ampere (A)
• t = Time of electrolysis in seconds (s)

The Faraday Constant

The quantity of electricity can be expressed by the Faraday constant (F). One faraday is the amount of electric charge carried by one mole of electrons.

The charge on a single electron is: e = 1.60217662 × 10^-19 C

The charge on one mole of electrons is: Charge = e × N_A = (1.60217662 × 10^-19 C) × (6.022 × 10²³ mol^-1) = 96485.332 C/mol

It is therefore concluded that 96485.332 C is the charge on one mole of electrons. This quantity of charge is referred to as one Faraday. Thus, the quantity of charge in electrolysis can be determined from the number of Faradays of charge that passes. For most calculations, the value of the Faraday will be taken as 96500 C.

The Equation Relating F, N_A, and e

The relationship between the Faraday constant (F), Avogadro constant (N_A), and the charge on the electron (e) can be expressed by the equation: F = N_A × e

• F = Faraday constant (charge per mole of electrons) = 96485.332 C/mol
• N_A = Avogadro constant (number of entities per mole) = 6.022 × 10²³ mol^-1
• e = Elementary charge (charge on a single electron) = 1.60217662 × 10^-19 C

Application to Electrolysis Calculations

The amount of electricity needed to deposit one mole of a substance depends on the number of electrons required for its reduction.

Example 1: Deposition of Sodium Na^+ + e^- \longrightarrow Na To deposit one mole of Na, one mole of electrons is required. The amount of electricity needed is 1 F (96500 C).

Example 2: Deposition of Copper Cu^{2+} + 2e^- \longrightarrow Cu To deposit one mole of Cu, two moles of electrons are required. The amount of electricity needed is 2 F.

Solved Example from the Textbook

Example 2.8: Electrolysis of Molten ZnCl_2

Question: In the electrolysis of molten ZnCl_2, how much Zn can be deposited at the cathode by the passage of a 0.01-ampere current for one hour?

Solution:

Step 1: Calculate the total charge passed. Q = I × t Q = 0.01 A × (1 × 60 × 60) s Q = 0.01 × 3600 = 36 C

Step 2: Convert the charge to Faradays. 1 F = 96500 C Number of Faradays = 3696500 = 3.73 × 10^-4 F

Step 3: Determine the moles of Zn deposited. The cathode reaction is: Zn^{2+} + 2e^- \longrightarrow Zn This means that 2 Faradays of charge are required to deposit 1 mole of Zn. Moles of Zn = 3.73 × 10^-4 F2 = 1.865 × 10^-4 mol

Step 4: Calculate the mass of Zn. Mass of Zn = moles × molar mass Mass = (1.865 × 10^-4 mol) × 65.38 g/mol Mass ≈ 0.012 g

Final Answer: The mass of zinc deposited is approximately 0.012 g.

Intermediate Level

The relationship F = N_A × e is fundamental. It connects the macroscopic measurement of charge (coulombs) to the microscopic world of particles (electrons). In an electrolysis calculation, you first calculate the total charge Q = It. Then, you convert this to moles of electrons by dividing by F (n_e^- = QF). The balanced half-reaction tells you the ratio of moles of electrons to moles of product. Finally, you multiply by the molar mass to get the mass of the product. This systematic approach allows you to solve problems involving electroplating, electrolysis, and other electrochemical processes.

Important Points

• - Q = I × t: Charge (C) = Current (A) × Time (s).
• - Faraday constant (F): The charge of one mole of electrons.
• - F ≈ 96500 C/mol (or more precisely 96485.332 C/mol).
• - F = N_A × e: Faraday constant = Avogadro constant × charge on one electron.
• - Number of moles of electrons = QF.
• - The number of Faradays required to deposit one mole of a substance equals the number of electrons in its half-reaction.
• - Na^+ + e^- \rightarrow Na (1 F per mole)
• - Cu^{2+} + 2e^- \rightarrow Cu (2 F per mole)

Remember

• - Q = I × t
• - F = 96500 C/mol
• - F = N_A × e
• - n_e^- = QF
• - Mass = moles × molar mass
• - 1 mole of electrons = 1 Faraday = 96500 C.
• - To deposit 1 mole of a metal, you need as many Faradays as the charge on the metal ion.

Common Mistakes

• - Forgetting to convert time to seconds. Current is in amperes (coulombs per second), so time must be in seconds. If time is given in minutes or hours, it must be multiplied by 60 or 3600, respectively.
• - Using the wrong value of n. The number of Faradays required to deposit one mole of a substance is not always 1. It is equal to the number of electrons transferred in the half-reaction. For example, for Al^{3+}, n=3, so 3 Faradays are needed to deposit 1 mole of Al.
• - Confusing the charge on one electron with the Faraday constant. The charge on one electron is 1.602 × 10^-19 C, while the Faraday constant is 96500 C/mol. Make sure you use the correct value for the correct calculation.
• - Forgetting to divide by n when calculating moles of product. After finding the moles of electrons, you must divide by the number of electrons per mole of product to get the moles of product.
• - Using the atomic mass instead of the molar mass. In chemical calculations, always use the molar mass (g/mol), not the atomic mass (amu).

Quick Revision

The relationship between the Faraday constant, Avogadro constant, and the charge on the electron is given by F = N_A × e. The Faraday constant (F) is the charge of one mole of electrons, approximately 96500 C/mol. During electrolysis, the total charge passed is Q = I × t. The number of moles of electrons used is QF. The balanced half-reaction tells you how many moles of electrons are required to produce one mole of the product. This allows you to calculate the mass of the substance deposited or liberated at the electrode.

60-Second Revision

The Faraday constant (F) is the charge of one mole of electrons, equal to the product of Avogadro's number and the elementary charge (F = N_A × e). It is approximately 96500 C/mol. The total charge passed in electrolysis is Q = I × t. The number of moles of electrons is QF. The number of electrons in the balanced half-reaction determines the number of Faradays needed to produce one mole of the product.

Frequently Asked Questions (FAQs)

Q1. What is the relationship between the Faraday constant, Avogadro constant, and the charge on the electron? Answer: The relationship is given by the equation: F = N_A × e where F is the Faraday constant, N_A is the Avogadro constant (6.022 × 10²³ mol^-1), and e is the elementary charge (1.602 × 10^-19 C). This equation shows that the total charge of one mole of electrons is the product of the charge on a single electron and the number of electrons in a mole.

Q2. What is the Faraday constant, and what is its value? Answer: The Faraday constant (F) is the amount of electric charge carried by one mole of electrons. Its value is approximately 96485.332 C/mol, but for most calculations, it is taken as 96500 C/mol.

• Q3. How is the quantity of charge passed during electrolysis calculated?
• Answer: The quantity of charge (Q) is calculated using the formula:
• Q = I × t
• where I is the current in amperes and t is the time in seconds. The result is in coulombs (C).

• Q4. How many Faradays are required to deposit one mole of copper?
• Answer: The reduction half-reaction for copper is:
• Cu^{2+} + 2e^- \rightarrow Cu
• To deposit one mole of copper, two moles of electrons are required. Therefore, 2 Faradays of charge are needed.

• Q5. How many Faradays are required to deposit one mole of sodium?
• Answer: The reduction half-reaction for sodium is:
• Na^+ + e^- \rightarrow Na
• To deposit one mole of sodium, one mole of electrons is required. Therefore, 1 Faraday of charge is needed.

• Q6. Calculate the mass of zinc deposited when a current of 0.01 A is passed through molten ZnCl_2 for one hour.
• Answer:
• First, calculate charge: Q = I × t = 0.01 A × 3600 s = 36 C.
• Convert to Faradays: 36 C / 96500 C/mol = 3.73 × 10^-4 F.
• The half-reaction is Zn^{2+} + 2e^- \rightarrow Zn, so 2 Faradays deposit 1 mole of Zn.
• Moles of Zn = 3.73 × 10^-4 / 2 = 1.865 × 10^-4 mol.
• Mass of Zn = 1.865 × 10^-4 mol × 65.38 g/mol ≈ 0.012 g.

Q7. Why is the Faraday constant important in electrochemistry? Answer: The Faraday constant is important because it allows us to relate the macroscopic quantity of electricity (charge in coulombs) to the microscopic amount of substance (moles of electrons). It is the bridge between the electrical and chemical worlds, enabling quantitative calculations in electrolysis and electrochemical cells.

Q8. What is the charge on a single electron? Answer: The charge on a single electron (elementary charge) is approximately -1.602 × 10^-19 C. The magnitude is used in the calculation of the Faraday constant.

• Q9. How is the number of moles of electrons calculated from the total charge?
• Answer: The number of moles of electrons is calculated by dividing the total charge (Q) by the Faraday constant (F):
• Moles of electrons = QF

• Q10. What is the relationship between the Faraday constant and the charge on one mole of electrons?
• Answer: The Faraday constant is exactly the charge on one mole of electrons. It is equal to the product of the elementary charge and Avogadro's number.

Multiple Choice Questions (MCQs)

• Q1. What is the relationship between the Faraday constant (F), Avogadro's number (N_A), and the charge of an electron (e)?
• A. F = N_A × e
• B. F = eN_A
• C. F = N_fracAe
• D. F = e² × N_A
• Correct Answer: A
• Explanation: The Faraday constant is the product of Avogadro's number and the charge on a single electron.

• Q2. What is the value of the Faraday constant?
• A. 6.022 × 10²³ C/mol
• B. 1.602 × 10^-19 C/mol
• C. 96500 C/mol
• D. 8.314 C/mol
• Correct Answer: C
• Explanation: The Faraday constant is approximately 96500 C/mol.

• Q3. How is the total charge (Q) calculated?
• A. Q = It
• B. Q = I × t
• C. Q = tI
• D. Q = I + t
• Correct Answer: B
• Explanation: The total charge is the product of current and time, Q = I × t.

• Q4. How many Faradays are required to deposit one mole of aluminum from Al^{3+}?
• A. 1
• B. 2
• C. 3
• D. 4
• Correct Answer: C
• Explanation: The half-reaction is Al^{3+} + 3e^- \rightarrow Al, so 3 moles of electrons (3 Faradays) are required to deposit 1 mole of aluminum.

• Q5. What is the charge on one mole of electrons called?
• A. Coulomb
• B. Ampere
• C. Faraday
• D. Volt
• Correct Answer: C
• Explanation: The charge of one mole of electrons is called a Faraday.

• Q6. If a current of 2 A is passed for 10 seconds, what is the total charge?
• A. 20 C
• B. 5 C
• C. 0.2 C
• D. 12 C
• Correct Answer: A
• Explanation: Q = I × t = 2 A × 10 s = 20 C.

• Q7. What is the charge on a single electron?
• A. 1.602 × 10^-19 C
• B. 9.11 × 10^-31 C
• C. 6.022 × 10²³ C
• D. 96500 C
• Correct Answer: A
• Explanation: The elementary charge (charge on one electron) is approximately 1.602 × 10^-19 C.

• Q8. How many Faradays are required to deposit one mole of silver from Ag^+?
• A. 1
• B. 2
• C. 3
• D. 4
• Correct Answer: A
• Explanation: The half-reaction is Ag^+ + e^- \rightarrow Ag, so 1 Faraday is required.

• Q9. How many moles of electrons are there in 193000 C of charge?
• A. 0.5 mol
• B. 1 mol
• C. 2 mol
• D. 4 mol
• Correct Answer: C
• Explanation: Number of moles of electrons = QF = 19300096500 = 2 mol.

• Q10. What is the relationship between the Faraday constant and the charge on one mole of electrons?
• A. The Faraday constant is half the charge of one mole of electrons.
• B. The Faraday constant is twice the charge of one mole of electrons.
• C. The Faraday constant is equal to the charge of one mole of electrons.
• D. There is no relationship.
• Correct Answer: C
• Explanation: The Faraday constant is defined as the charge of one mole of electrons.

Multiple Choice Questions (MCQs)

• Q1. The relationship F = N_A × e connects which three quantities?
• A. Force, mass, acceleration
• B. Faraday constant, Avogadro constant, elementary charge
• C. Energy, frequency, Planck's constant
• D. Current, voltage, resistance
• Correct Answer: B
• Explanation: The equation F = N_A × e connects the Faraday constant, Avogadro constant, and the elementary charge.

• Q2. What is the approximate value of the Faraday constant?
• A. 6.022 × 10²³ C/mol
• B. 1.602 × 10^-19 C/mol
• C. 96500 C/mol
• D. 8.314 C/mol
• Correct Answer: C
• Explanation: The Faraday constant is approximately 96500 C/mol.

• Q3. What is the formula for calculating the total charge passed during electrolysis?
• A. Q = It
• B. Q = I × t
• C. Q = tI
• D. Q = I + t
• Correct Answer: B
• Explanation: The total charge is the product of current and time, Q = I × t.

• Q4. How many Faradays are needed to deposit one mole of copper from Cu^{2+}?
• A. 1
• B. 2
• C. 3
• D. 4
• Correct Answer: B
• Explanation: The half-reaction is Cu^{2+} + 2e^- \rightarrow Cu, so 2 Faradays are needed.

• Q5. How many Faradays are needed to deposit one mole of sodium from Na^+?
• A. 1
• B. 2
• C. 3
• D. 4
• Correct Answer: A
• Explanation: The half-reaction is Na^+ + e^- \rightarrow Na, so 1 Faraday is needed.

• Q6. What is the charge on a single electron?
• A. 1.602 × 10^-19 C
• B. 9.11 × 10^-31 C
• C. 6.022 × 10²³ C
• D. 96500 C
• Correct Answer: A
• Explanation: The elementary charge is approximately 1.602 × 10^-19 C.

• Q7. If a current of 5 A is passed for 60 seconds, what is the total charge?
• A. 300 C
• B. 12 C
• C. 0.083 C
• D. 65 C
• Correct Answer: A
• Explanation: Q = I × t = 5 A × 60 s = 300 C.

• Q8. What is the relationship between the Faraday constant and the charge of one mole of electrons?
• A. They are the same.
• B. The Faraday constant is half the charge.
• C. The Faraday constant is twice the charge.
• D. They are unrelated.
• Correct Answer: A
• Explanation: The Faraday constant is defined as the charge of one mole of electrons.

• Q9. How many Faradays are required to deposit one mole of aluminum from Al^{3+}?
• A. 1
• B. 2
• C. 3
• D. 4
• Correct Answer: C
• Explanation: The half-reaction is Al^{3+} + 3e^- \rightarrow Al, so 3 Faradays are required.

• Q10. How many moles of electrons are present in 48250 C of charge?
• A. 0.5 mol
• B. 1 mol
• C. 2 mol
• D. 4 mol
• Correct Answer: A
• Explanation: Moles of electrons = QF = 4825096500 = 0.5 mol.