Article — Electrolysis Calculator (Faraday Law)
Electrolysis calculator: Faraday law m = QM/(nF)
Electrolysis deposits metal at a cathode in proportion to the electric charge passed. Faraday law gives m = (M · I · t) / (n · F), where M is molar mass, I is current, t is time, n is electrons per ion, and F = 96485 C/mol. A 2 A current for 1 hour plates 2.37 g of copper.
Michael Faraday discovered the quantitative laws of electrolysis in 1834, decades before electrons were recognized as discrete particles. His insight — that the mass deposited is proportional to charge and inversely proportional to the equivalent weight — predates the modern atomic theory of matter. Today the same equation underlies industrial copper refining, aluminum production, water splitting for hydrogen, and chrome plating.
What is electrolysis?
Electrolysis uses electrical energy to drive a non-spontaneous chemical reaction. A direct current passes between two electrodes immersed in an electrolyte (either a molten salt or a solution of ionic species). Positive ions (cations) migrate to the cathode where they accept electrons and reduce; negative ions migrate to the anode where they release electrons and oxidize.
The classic setup: an aqueous copper sulfate solution with two copper electrodes connected to a battery. Cu²⁺ ions reduce at the cathode (Cu²⁺ + 2e⁻ → Cu), depositing pure copper. Copper from the anode oxidizes back into solution (Cu → Cu²⁺ + 2e⁻). Mass transfers from anode to cathode. This is the basis of industrial copper refining — impure blister copper at the anode, 99.99% pure deposited copper at the cathode.
Faraday discovered the laws of electrolysis in 1834, almost 60 years before J.J. Thomson identified the electron in 1897. Faraday couldn't name the unit of charge "the electron" because the concept didn't exist yet — he called the proportionality constant the "electrochemical equivalent" of each substance.
Faraday law of electrolysis
Faraday's first law states that mass deposited is proportional to charge:
m = (M · I · t) / (n · F)Q = I · t total charge (C)m = (M · Q) / (n · F) same formulan_mol electrons Q / FFaraday's second law states that for the same charge, masses of different substances deposited are proportional to their equivalent weights (molar mass divided by n). One mole of electrons (96485 C) deposits one mole of Ag⁺, half a mole of Cu²⁺, or one-third mole of Al³⁺ — because each requires 1, 2, or 3 electrons respectively to reduce.
The Faraday constant explained
F = 96485 C/mol is the charge carried by one mole of electrons. It equals Avogadro's number times the elementary charge: 6.022 × 10²³ × 1.602 × 10⁻¹⁹. After the 2019 SI redefinition, both N_A and e are fixed exactly, so F is now exactly 96485.33212... C/mol with no measurement uncertainty.
The constant matters because it converts between electrical units (coulombs, the SI charge unit) and chemical units (moles). A current of 1 ampere passing for 1 second delivers 1 coulomb, which equals 1/96485 = 1.0364 × 10⁻⁵ moles of electrons. Knowing how many electrons each ion needs gives the moles of metal deposited; multiplying by molar mass gives grams.
Electrolysis current efficiency
Faraday's law gives the theoretical maximum deposition. Real systems run below this because not all the current goes into the desired reaction. Side reactions consume some current:
- Hydrogen evolution at cathode — 2 H⁺ + 2 e⁻ → H₂ competes with metal reduction.
- Oxygen evolution at anode — 2 H₂O → O₂ + 4 H⁺ + 4 e⁻ in aqueous systems.
- Mediator redox cycling — Fe²⁺/Fe³⁺ shuttles can transfer charge without depositing metal.
- Re-dissolution — freshly plated metal dissolves back at low overpotential.
Industrial efficiencies range widely. Cu electrowinning hits 90–95%. Ni plating reaches 95–99%. Chrome plating from chromic acid runs 10–20% because hydrogen evolution dominates. Aluminum production (Hall-Héroult process) operates at 88–92% in modern cells.
Electrolysis applications in industry
Five industrial-scale uses dominate the global electrolysis market:
Aluminum production (Hall-Héroult). Molten Al₂O₃ in cryolite at 950°C, electrolyzed with carbon anodes. Each kilogram of aluminum needs roughly 13 kWh.
Chlor-alkali process. Electrolysis of brine produces Cl₂ at the anode, H₂ at the cathode, NaOH in the cathode compartment. Global production exceeds 70 million tons of chlorine per year.
Copper electrorefining. Impure copper anode dissolves; pure Cu deposits on the cathode. Gold and silver impurities settle as anode mud. Produces 99.99% pure Cu for electrical wiring.
Electroplating. Deposit a thin metal layer on a base substrate. Chrome on steel for corrosion resistance, gold on connectors, zinc on iron (galvanizing).
Hydrogen by water electrolysis. H₂O → H₂ + ½ O₂. Growing for green hydrogen powered by renewables. Modern PEM electrolyzers reach 70–80% energy efficiency.
Worked electrolysis examples
Three numerical walks:
Copper plating. Deposit 5 g of copper from CuSO₄ solution using 1 A current. Time t = (5 × 2 × 96485) / (63.55 × 1) = 15,184 seconds = 4 hours 13 minutes. Real time accounting for 90% efficiency: 4 h 41 min.
Silver plating. Plate 1 g of silver in 30 minutes (1800 s). Current I = (1 × 1 × 96485) / (107.87 × 1800) = 0.497 A. Easy to set on a bench power supply.
Aluminum. A modern aluminum smelter cell runs at 350,000 A continuously. In 24 hours: Q = 350,000 × 86,400 = 3.024 × 10¹⁰ C. Theoretical mass: m = (26.98 × 3.024 × 10¹⁰) / (3 × 96485) = 2820 kg per day per cell. With 90% efficiency: 2540 kg/day actual.
For quick estimates, remember that 1 ampere-hour deposits 1.118 grams of silver, 1.186 grams of copper (Cu²⁺), or 0.336 grams of aluminum. These are the "electrochemical equivalents" from Faraday's original 1834 tables, accurate to four significant figures.
Electrolysis calculation pitfalls
Three common errors trip up first-time users:
- Wrong n value — pick n from the cathode half-reaction, not the metal's most common oxidation state. Tin can be Sn²⁺ (n=2) or Sn⁴⁺ (n=4); the electrolyte determines which.
- Forgetting unit conversion — Faraday's law uses SI throughout: amperes, seconds, grams, g/mol. Milliamperes and minutes need conversion before substitution.
- Treating predicted mass as actual — without an efficiency correction, you'll overestimate by 5–80% depending on the system.
For lab-scale problem sets, Faraday's law alone is sufficient. For industrial sizing, you must multiply theoretical mass by the current efficiency (typically 0.85–0.99 for clean systems, much lower for chrome). Skipping this step underestimates the power required by 5–20% in routine engineering applications.
Aluminum production by electrolysis consumes about 4% of world electricity — roughly the same as the entire UK national grid. Recycling aluminum uses only 5% of the energy required to make it from bauxite, which is why aluminum recycling rates exceed 75% in most developed countries.