Faraday's Law of Electrolysis Calculator

Compute the mass of substance deposited or dissolved at an electrode, calculate required current, determine electrolysis time, or find evolved gas volume based on Faraday's first and second laws of electrolysis.

Comprehensive Guide to Faraday's Laws of Electrolysis

Electrolysis represents the non-spontaneous chemical decomposition of a compound brought about by passing an electric current through an ionic liquid or aqueous electrolyte solution. In 1834, English polymath Michael Faraday published groundbreaking quantitative investigations into electrochemistry, establishing the fundamental relationships linking electric charge, molecular stoichiometry, and material mass. Today, Faraday's laws of electrolysis provide the mathematical cornerstone for global metal refining (including electrolytic copper and zinc), industrial chlor-alkali production, electroplating, anodization, and green hydrogen water electrolysis.

Unlike galvanic or voltaic systems evaluated in our Cell EMF Calculator – Electromotive Force of a Cell where spontaneous reactions generate current, an electrolytic cell uses an external direct current (DC) power supply to drive non-spontaneous redox reactions. Electrons enter the cell at the cathode (where reduction occurs) and exit through the anode (where oxidation occurs). Faraday proved that the physical mass of an element altered or liberated at an electrode depends exclusively on the total quantity of electricity passed through the system and the substance's chemical equivalent weight.

Mathematical Formulation of Faraday's Electrolytic Laws

1. Faraday's First Law of Electrolysis

Faraday's first law states that the mass of a substance liberated or deposited at any electrode during electrolysis is directly proportional to the total electric charge (quantity of electricity) that passes through the electrolyte solution:

m = Z * Q

Where:

m: Mass of the substance deposited or dissolved (grams, g)
Q: Total electric charge passed through the solution in coulombs (C)
Z: Electrochemical equivalent (ECE) of the substance (g/C)

Because electric current (I) in amperes measures the rate of charge flow per unit time in seconds (Q = I * t), the equation expands to:

m = Z * I * t

2. Faraday's Second Law of Electrolysis

Faraday's second law states that when the same quantity of electric charge passes through different electrolyte solutions connected in series, the masses of the different substances deposited or liberated at the electrodes are directly proportional to their respective chemical equivalent weights (equivalent masses, E):

m1 / m2 = E1 / E2 = (M1 / z1) / (M2 / z2)

The chemical equivalent weight (E) is defined as the substance's molar mass (M) divided by its valency number (z), which represents the number of electrons transferred per ion during the half-reaction (E = M / z). Convert molar quantities across elements using our Molar Mass Calculator.

3. The Unified Universal Electrolysis Equation

Combining both laws through Faraday's constant (F) yields the modern unified formulation used universally in electrochemistry and electroplating engineering:

m = (I * t * M * eta) / (z * F)

Where:

m: Mass of deposited or dissolved element (g)
I: Steady electrical current (Amperes, A)
t: Total duration of current application (seconds, s)
M: Atomic or molar mass of the deposited element (g/mol)
z: Valence factor (number of electrons exchanged per ion reduced or oxidized)
F: Faraday's constant = 96,485.33212 C/mol of electrons (equivalent to the charge of one mole of electrons: F = e * N_A, where e is the elementary charge and N_A is Avogadro's number, as modeled in our Avogadro's Number Calculator)
eta (η): Current efficiency (expressed as a fraction between 0 and 1, or as a percentage)

4. Gaseous Electrolytic Products at STP

For reactions generating gaseous products, such as hydrogen gas evolution during water splitting (2H⁺ + 2e⁻ → H₂) or chlorine evolution during brine electrolysis (2Cl⁻ → Cl₂ + 2e⁻), the molar volume of an ideal gas at standard temperature and pressure (STP: 0 °C, 1 atm) is 22.414 L/mol. The volume of gas liberated is calculated via:

V = (I * t * 22.414 * eta) / (z * F)

Explore gas volume and temperature dynamics further using our Ideal Gas Law Calculator and STP Calculator.

Step-by-Step Practical Calculation Examples

Worked Example 1: Electroplating Copper onto an Automotive Component

An automotive repair shop electroplates copper onto a steel fixture using an acidic copper(II) sulfate bath (Cu²⁺ + 2e⁻ → Cu). The electroplating cell operates at a constant current of 12.0 Amperes for exactly 45.0 minutes. Assuming 100% current efficiency, calculate the total mass of copper deposited.

Identify Electrochemical Parameters:
Molar mass of Copper (M) = 63.546 g/mol
Electrons transferred per Cu atom (z) = 2
Electric current (I) = 12.0 A
Faraday constant (F) = 96,485 C/mol
Convert Time into Seconds:
t = 45.0 minutes * 60 seconds/minute = 2,700 seconds
Calculate Total Electric Charge (Q):
Q = I * t = 12.0 A * 2,700 s = 32,400 Coulombs
Determine Moles of Electrons Transferred:
Moles of e⁻ = Q / F = 32,400 C / 96,485 C/mol = 0.3358 mol e⁻
Apply the Unified Faraday Equation:
m = (32,400 C * 63.546 g/mol) / (2 * 96,485 C/mol)
m = 2,058,890.4 / 192,970 = 10.669 grams

Conclusion: The process deposits precisely 10.67 grams of pure metallic copper onto the substrate. Verify conversion to moles using our Grams to Moles Calculator.

Worked Example 2: Determining Required Time for Silver Electrodeposition

An artisan jeweler wishes to coat a decorative silver ornament with exactly 5.00 grams of pure silver from a silver nitrate solution (Ag⁺ + e⁻ → Ag, M = 107.868 g/mol, z = 1). The rectifier delivers a steady current of 1.50 A at a current efficiency of 92.0%. How long must the electrolysis run?

Identify Values:
Target mass (m) = 5.00 g
Molar mass (M) = 107.868 g/mol, valency (z) = 1
Current (I) = 1.50 A, efficiency (η) = 0.92
Rearrange Faraday's Equation for Time (t):
t = (m * z * F) / (I * M * η)
Substitute and Solve:
t = (5.00 * 1 * 96,485) / (1.50 * 107.868 * 0.92)
t = 482,425 / 148.858 = 3,240.85 seconds
Convert Seconds into Minutes:
t = 3,240.85 / 60 = 54.01 minutes (approx. 54 minutes and 1 second)

Conclusion: The jeweler must operate the plating bath for 54.0 minutes to achieve the desired coating thickness.

Electrochemical Equivalents of Important Industrial Elements

The table below summarizes molar masses, valency states, chemical equivalents, and electrochemical equivalents (ECE, Z in mg/C) for common elements processed by electrolysis:

Element & Cathode Half-Reaction	Molar Mass (g/mol)	Electrons (z)	Equivalent Weight (E)	ECE, Z (mg/Coulomb)
Silver: Ag⁺ + e⁻ → Ag	107.87	1	107.87	1.1180 mg/C
Gold: Au³⁺ + 3e⁻ → Au	196.97	3	65.66	0.6805 mg/C
Copper (II): Cu²⁺ + 2e⁻ → Cu	63.55	2	31.77	0.3293 mg/C
Nickel: Ni²⁺ + 2e⁻ → Ni	58.69	2	29.35	0.3041 mg/C
Zinc: Zn²⁺ + 2e⁻ → Zn	65.38	2	32.69	0.3388 mg/C
Iron (II): Fe²⁺ + 2e⁻ → Fe	55.85	2	27.92	0.2894 mg/C
Aluminum: Al³⁺ + 3e⁻ → Al	26.98	3	8.99	0.0932 mg/C
Hydrogen: 2H⁺ + 2e⁻ → H₂	1.008	1	1.008	0.0104 mg/C
Oxygen: 2H₂O → O₂ + 4H⁺ + 4e⁻	16.00 (as O)	2	8.00	0.0829 mg/C

Understanding Current Efficiency and Parasitic Side Reactions

In idealized textbook calculations, current efficiency is assumed to be 100%, meaning every electron passing through the circuit participates exclusively in the desired electrode reaction. In commercial and industrial reality, current efficiency often falls between 80% and 98% due to parasitic side reactions, such as:

Hydrogen Evolution: In aqueous electroplating of metals with reduction potentials more negative than hydrogen (such as zinc, nickel, or chromium), protons (H⁺) or water molecules reduce competitively at the cathode (2H₂O + 2e⁻ → H₂ + 2OH⁻), siphoning current away from metal deposition.
Dissolved Oxygen Reduction: Dissolved atmospheric oxygen in the bath can consume electrons at the cathode surface.
Ohmic Heating & Solution Resistance: Internal electrolyte resistance converts electrical energy into waste heat, shifting operating voltages. Evaluate cell fundamentals in our Electrolysis Calculator.

Frequently Asked Questions (FAQ)

1. What is the exact value of Faraday's constant?

Faraday's constant (F) represents the total electric charge carried by one mole of electrons. Under current CODATA definitions, its value is 96,485.33212 Coulombs per mole (C/mol). In routine educational and engineering calculations, it is commonly rounded to 96,500 C/mol.

2. What is the electrochemical equivalent (Z)?

The electrochemical equivalent (Z) is the mass of a substance deposited or liberated by the passage of exactly one Coulomb of electric charge (one Ampere for one second). It is calculated as Z = M / (z * F), with units of grams per Coulomb (g/C) or milligrams per Coulomb (mg/C).

3. Why does aluminum require significantly more electricity to plate than silver?

Aluminum has a valency of z = 3 and a small molar mass (26.98 g/mol), meaning depositing one mole of aluminum (27 g) requires 3 moles of electrons (3 * 96,485 C). In contrast, silver has z = 1 and a heavy molar mass (107.87 g/mol), depositing 108 grams of silver with only 1 mole of electrons. Consequently, depositing an equivalent mass of aluminum consumes roughly 12 times more electric charge than depositing silver.

4. What is the difference between an Ampere, a Coulomb, and a Faraday?

A Coulomb (C) is the SI unit of electric charge. An Ampere (A) measures the rate of charge flow, where 1 Ampere equals 1 Coulomb per second (1 A = 1 C/s). A "Faraday" (older unit of charge) equals one mole of electrons, or approximately 96,485 Coulombs.

5. Can Faraday's laws be applied to alternating current (AC)?

No. Standard Faraday electrolysis requires direct current (DC). Alternating current continually reverses the polarity of the electrodes 50 or 60 times per second. Material deposited during the negative half-cycle dissolves back into solution during the subsequent positive half-cycle, resulting in no net deposition.

6. How does temperature affect the mass of deposited material?

According to Faraday's law, temperature has zero direct mathematical effect on the theoretical mass deposited because mass depends solely on total charge (I * t). However, temperature affects electrolyte conductivity, ion mobility, and the rates of competing side reactions, which can alter the practical current efficiency (η).

7. How does a coulometer work?

A coulometer is an electrochemical device that determines total electric charge by measuring the mass of metal deposited on an electrode. In a silver coulometer, the cathode is weighed before and after current passage; using Faraday's first law with silver's known ECE, the charge Q is calculated with high precision: Q = m / Z.

8. What is the relationship between Faraday's laws and Avogadro's number?

Faraday's constant connects macroscopic electrical charge to microscopic atomic particles via the fundamental relation F = e * N_A, where e is the elementary charge of a single electron (1.602176634 x 10⁻¹⁹ C) and N_A is Avogadro's number (6.02214076 x 10²³ mol⁻¹).

9. Why is current efficiency often less than 100% in industrial zinc or nickel plating?

Zinc and nickel have standard reduction potentials more negative than hydrogen. During aqueous plating, hydrogen ions discharge simultaneously with metal ions at the cathode, generating hydrogen gas bubbles. This parasitic side reaction consumes a fraction of the current, reducing the current efficiency for metal deposition to typically 85% to 95%.

10. Can Faraday's laws predict the volume of gases produced during water electrolysis?

Yes. For pure water electrolysis, producing 1 mole of oxygen gas (O₂) at the anode requires 4 moles of electrons (z = 4), while producing 2 moles of hydrogen gas (H₂) at the cathode requires 4 moles of electrons (z = 2 per mole of H₂). Faraday's laws accurately predict the 2:1 volume ratio of evolved hydrogen to oxygen gas.