Electrolysis: Faraday's laws, overpotential, and electrodeposition

Intuition Beginner

A galvanic cell uses a spontaneous reaction to produce electricity. An electrolytic cell does the opposite: it uses an external power source to force a non-spontaneous reaction to occur. The process is called electrolysis.

In an electrolytic cell, the external battery pushes electrons through the circuit in the opposite direction to what would happen spontaneously. The battery makes the anode positive and the cathode negative -- the reverse of a galvanic cell. Oxidation still happens at the anode and reduction at the cathode, but the cell requires continuous energy input.

Two laws govern how much product forms during electrolysis. Faraday's first law states that the mass of substance deposited or dissolved at an electrode is directly proportional to the electric charge passed through the cell. Faraday's second law states that for a given amount of charge, the masses of different substances deposited are proportional to their equivalent weights (molar mass divided by the number of electrons transferred per formula unit).

In practice, the voltage needed to drive an electrolytic reaction is always higher than the thermodynamic prediction. The extra voltage is called overpotential, and it accounts for the kinetic barriers at the electrode surface.

Visual Beginner

An electrolytic cell has a battery or power supply connected to two electrodes immersed in an electrolyte solution or molten salt. The battery pumps electrons to the cathode (reduction) and pulls electrons from the anode (oxidation).

For water electrolysis: at the cathode, $2{\text{H}}_2\text{O}(l)+2e^{-}\to {\text{H}}_2(g)+2{\text{OH}}^{-}(aq)$. At the anode, $2{\text{H}}_2\text{O}(l)\to {\text{O}}_2(g)+4{\text{H}}^{+}(aq)+4e^{-}$. The thermodynamic minimum voltage is $1.23\text{V}$, but in practice about $1.7$--$2.0\text{V}$ is needed due to overpotential.

Worked example Beginner

A current of $2.50\text{A}$ is passed through a solution of ${\text{CuSO}}_4$ for 45.0 minutes. How many grams of copper are deposited on the cathode?

Step 1. Calculate total charge. $Q=I\times t=2.50\text{A}\times (45.0\times 60)\text{s}=2.50\times 2700=6750\text{C}$.

Step 2. Calculate moles of electrons. One mole of electrons carries $F=96485\text{C}$. Moles of $e^{-}$ = $6750/96485=0.06995\text{mol}$.

Step 3. Calculate moles of copper. The reduction is ${\text{Cu}}^{2+}+2e^{-}\to \text{Cu}(s)$, so 2 moles of electrons deposit 1 mole of Cu. Moles of Cu = $0.06995/2=0.03498\text{mol}$.

Step 4. Calculate mass. Molar mass of Cu = $63.55\text{g/mol}$. Mass = $0.03498\times 63.55=2.22\text{g}$.

Check your understanding Beginner

Formal definition Intermediate+

Faraday's laws of electrolysis

Faraday's first law. The mass $m$ of substance deposited or dissolved at an electrode is proportional to the total charge $Q$ passed through the electrolyte:

$$m=\frac{Q\cdot M}{nF}$$

where $M$ is the molar mass of the substance, $n$ is the number of electrons transferred per formula unit, and $F=96485\text{C/mol}$. The charge is $Q=It$ for a constant current $I$ over time $t$.

Faraday's second law. When the same quantity of charge is passed through different electrolytes, the masses of the substances deposited are in the ratio of their equivalent weights:

$$\frac{m_1}{m_2}=\frac{M_1/n_1}{M_2/n_2}$$

Both laws follow from the atomic theory of matter and the quantisation of electric charge. Each ion requires a definite number of electrons ($n$) to be reduced, and the charge per mole of electrons is the Faraday constant. The laws are exact; deviations in practice arise from competing side reactions, not from failures of the laws themselves.

Electrolytic vs galvanic cells: a systematic comparison

Property	Galvanic cell	Electrolytic cell
Energy conversion	Chemical to electrical	Electrical to chemical
Spontaneity	Spontaneous ($\Delta G<0$)	Non-spontaneous ($\Delta G>0$)
$E_{\text{cell}}$	Positive	Negative (external voltage overcomes)
Anode charge	Negative	Positive
Cathode charge	Positive	Negative
Electron flow	Anode to cathode (spontaneous)	External supply pushes to cathode
Ion migration	Cations to cathode, anions to anode	Cations to cathode, anions to anode

In both types, oxidation occurs at the anode and reduction at the cathode. Cations always migrate toward the cathode and anions toward the anode. The difference is the direction of energy flow and the sign of the electrode charges.

Overpotential: why real cells need more voltage

The theoretical decomposition potential is the minimum voltage needed to drive a non-spontaneous electrolysis reaction, equal to $|E_{\text{cell}}^{\circ }|$ (or $|E_{\text{cell}}|$ under non-standard conditions, calculated from the Nernst equation). In practice, a higher voltage is required. The difference is the overpotential $\eta$ (Greek eta):

$$E_{\text{applied}}=E_{\text{theoretical}}+\eta$$

Overpotential has three contributions:

Activation overpotential (${\eta }_{\text{act}}$): Energy needed to overcome the activation barrier for electron transfer at the electrode surface. Described by the Butler-Volmer equation (see 14.11.01). Gases evolved at electrodes (especially ${\text{H}}_2$ and ${\text{O}}_2$) have large activation overpotentials that depend strongly on the electrode material.

Concentration overpotential (${\eta }_{\text{conc}}$): Arises when the concentration of reactant at the electrode surface differs from the bulk concentration due to depletion by the electrochemical reaction. At high currents, the surface concentration drops below the bulk value, shifting the Nernst potential.

Resistance overpotential (${\eta }_{\text{res}}$): The $IR$ drop across the electrolyte solution between the electrodes. Minimised by using concentrated electrolytes with high conductivity.

The total overpotential is: $\eta ={\eta }_{\text{act}}+{\eta }_{\text{conc}}+{\eta }_{\text{res}}$.

Electrolysis of water

The overall reaction is $2{\text{H}}_2\text{O}(l)\to 2{\text{H}}_2(g)+{\text{O}}_2(g)$ with $\Delta G^{\circ }=+474\text{kJ/mol}$, corresponding to $E_{\text{cell}}^{\circ }=-1.23\text{V}$.

In acidic solution, the half-reactions are:

Cathode: $2{\text{H}}^{+}(aq)+2e^{-}\to {\text{H}}_2(g)$, $E^{\circ }=0.00\text{V}$.

Anode: $2{\text{H}}_2\text{O}(l)\to {\text{O}}_2(g)+4{\text{H}}^{+}(aq)+4e^{-}$, $E^{\circ }=+1.23\text{V}$.

The theoretical decomposition potential is $1.23\text{V}$, but the oxygen evolution reaction has a large activation overpotential ($0.3$--$0.5\text{V}$ on Pt, up to $1.0\text{V}$ on other materials). In practice, water electrolysis requires $1.7$--$2.0\text{V}$. This extra voltage represents wasted energy (dissipated as heat), which is why efficient electrocatalysts for oxygen evolution are a major research area.

Competing reactions and the electrode potential series

When a solution contains multiple species that can be reduced, the species with the most positive reduction potential is reduced first. For example, an aqueous solution of ${\text{CuSO}}_4$ contains both ${\text{Cu}}^{2+}$ and ${\text{H}}_2\text{O}$. Since $E^{\circ }({\text{Cu}}^{2+}/\text{Cu})=+0.34\text{V}>E^{\circ }({\text{H}}^{+}/{\text{H}}_2)=0.00\text{V}$, copper plates out before hydrogen evolves.

But if the solution contains $\text{NaCl}$, the reduction of ${\text{Na}}^{+}$ ($E^{\circ }=-2.71\text{V}$) is far more negative than the reduction of water ($E^{\circ }\approx -0.83\text{V}$ at pH 7). Water is reduced first, producing hydrogen. Sodium metal cannot be produced by electrolysis of an aqueous solution; it requires electrolysis of the molten salt.

At the anode, the species with the least positive (most negative) oxidation potential is oxidised first. In brine electrolysis, the oxidation of ${\text{Cl}}^{-}$ to ${\text{Cl}}_2$ ($E^{\circ }=+1.36\text{V}$) is thermodynamically less favourable than the oxidation of water to ${\text{O}}_2$ ($E^{\circ }=+1.23\text{V}$), but ${\text{Cl}}^{-}$ is oxidised preferentially because the oxygen evolution reaction has a much larger overpotential on most electrodes.

Key result Intermediate+

Faraday's laws combined: the universal charge-to-mass relation

The combined form of Faraday's laws gives a single equation that predicts the mass of any substance deposited or dissolved:

$$\boxed{m=\frac{M\cdot I\cdot t}{n\cdot F}}$$

This equation is exact (within the assumption that all charge goes to the desired reaction) and applies universally to all electrode processes. The current efficiency (or Faradaic efficiency) accounts for side reactions:

$$\text{Current efficiency}=\frac{\text{actual mass deposited}}{\text{theoretical mass from Faraday’s law}}\times 100\%$$

Current efficiencies above 95% are common for well-controlled processes like copper refining and silver plating. Chlor-alkali electrolysis achieves about 95--97% efficiency for both chlorine and hydrogen production.

The Tafel equation: quantifying activation overpotential

At high overpotentials, the Butler-Volmer equation (14.11.01) reduces to the Tafel equation:

$$\eta =a+b{\log }_{10}j$$

where $j$ is the current density and $b=2.303RT/(\alpha nF)$ is the Tafel slope. At $25{}^{\circ }\text{C}$ with $\alpha =0.5$ and $n=1$: $b\approx 0.118\text{V/decade}$. A Tafel plot of $\eta$ vs $\log j$ gives a straight line whose slope yields $\alpha$ and whose intercept yields the exchange current density $j_0$.

Typical activation overpotentials for the hydrogen evolution reaction at $j=10{\text{mA/cm}}^2$: Pt ($\sim 0.02\text{V}$), Ni ($\sim 0.3\text{V}$), Hg ($\sim 1.0\text{V}$). The nine-order-of-magnitude range in exchange current density between Pt and Hg for the same reaction is one of the largest kinetic variations known for any chemical reaction.

Exercises Intermediate+

Electroplating, electrodeposition, and the Hall-Heroult process Master

Electroplating fundamentals

Electroplating deposits a thin metal coating onto a substrate by making the substrate the cathode in an electrolytic cell. The key parameters are:

Current density ($j$): Controls the deposition rate and the quality of the deposit. Too high a current density causes rough, porous, or powdery deposits. Too low wastes time. Typical plating current densities range from 1 to $500{\text{mA/cm}}^2$.

Throwing power: The ability of a plating bath to deposit metal uniformly over a complex-shaped object. Good throwing power requires a bath formulation that equalises the current distribution, often achieved by adding complexing agents or conducting salts. The Wagner number $Wa=\kappa (d\eta /dj)/L$, where $\kappa$ is solution conductivity, $d\eta /dj$ is the slope of the overpotential-current curve, and $L$ is a characteristic length, quantifies throwing power: high $Wa$ means uniform deposition.

Adhesion and nucleation: Metal deposition begins with nucleation of crystallites on the substrate surface. The nucleation density depends on the substrate preparation (cleaning, etching, striking) and the bath chemistry. A strike bath is a dilute, low-current-density plating solution used to deposit a thin, adherent initial layer that improves bonding of the main plating layer.

Common plating systems: Copper (from CuSO4/H2SO4 for printed circuit boards), nickel (from Watts bath: NiSO4/NiCl2/boric acid for corrosion resistance), chromium (from chromic acid/H2SO4 for decorative and hard chrome), gold (from cyanide or sulfite baths for electronics), and zinc (from cyanide or acid baths for galvanised steel).

The Nernst diffusion layer in electrodeposition

During electrodeposition, metal ions are consumed at the cathode surface faster than they can be replenished by diffusion from the bulk solution. A concentration gradient develops in the Nernst diffusion layer of thickness $\delta$ (typically $0.1$--$0.5\text{mm}$ in unstirred solutions).

The limiting current density is reached when the surface concentration drops to zero:

$$j_L=\frac{nFD{c}_0}{\delta }$$

where $D$ is the diffusion coefficient and $c_0$ is the bulk concentration. Operating above $j_L$ is impossible for a single-metal system; in alloy plating, exceeding $j_L$ for one metal shifts the deposit composition as the other metal continues to plate. Agitation, ultrasonic vibration, or pulse-reverse plating reduces $\delta$ and increases $j_L$, enabling higher deposition rates.

Pulse plating alternates between deposition (on-time) and rest (off-time) to allow diffusion to replenish the surface concentration. This produces denser, smoother deposits than DC plating at the same average current density because the peak current can exceed the DC limiting current during the on-time while the off-time allows recovery.

The Hall-Heroult process for aluminium

The Hall-Heroult process is the world's most energy-intensive electrolytic process and the only industrial method for primary aluminium production. It was independently patented by Charles Hall (US, 1886) and Paul Heroult (France, 1886) within two months of each other.

Aluminium oxide (${\text{Al}}_2{\text{O}}_3$, alumina) is dissolved in molten cryolite (${\text{Na}}_3{\text{AlF}}_6$) at about $960{}^{\circ }\text{C}$. The cell operates at 3.5--4.5 V and 100--400 kA. The electrodes are carbon: the cathode is the carbon-lined cell bottom, and the anode is a consumable carbon block.

Cathode: ${\text{Al}}^{3+}+3e^{-}\to \text{Al}(l)$. The molten aluminium (denser than the electrolyte) pools at the cell bottom and is tapped periodically.

Anode: $2{\text{O}}^{2-}+\text{C}(s)\to {\text{CO}}_2(g)+4e^{-}$. The carbon anode is consumed (about 0.4 kg C per kg Al) and must be continuously lowered into the bath as it burns away.

The theoretical decomposition voltage is about $1.2\text{V}$ (reduced from the pure ${\text{Al}}_2{\text{O}}_3$ value of $2.2\text{V}$ by the thermodynamic activity of alumina dissolved in cryolite). The operating voltage of 3.5--4.5 V includes overpotentials and $IR$ drops. The energy consumption is about 13--15 kWh per kg of aluminium, which accounts for roughly 2% of global electricity consumption.

The current efficiency is about 92--96%. The main loss is the "back reaction" where some dissolved aluminium re-oxidises at the anode, reducing the net metal production without reducing the measured current.

Industrial electrolysis: the chlor-alkali and Downs processes

Three major industrial electrolysis processes illustrate the principles:

Chlor-alkali (membrane cell): Electrolyses concentrated NaCl solution. Products: ${\text{Cl}}_2$ (anode), ${\text{H}}_2$ (cathode), NaOH (cathode compartment). A cation-exchange membrane separates the compartments, allowing ${\text{Na}}^{+}$ to pass but blocking ${\text{Cl}}^{-}$ and ${\text{OH}}^{-}$, preventing the formation of hypochlorite. Operating at about 3.0--3.5 V, 2--5 kA/m${}^2$, producing about 2.2--2.5 kWh/kg ${\text{Cl}}_2$.

Downs process (sodium production): Electrolyses a molten NaCl/${\text{CaCl}}_2$ mixture at about $600{}^{\circ }\text{C}$. The calcium chloride lowers the melting point from $801{}^{\circ }\text{C}$ (pure NaCl) to about $580{}^{\circ }\text{C}$, saving energy. Products: Na(l) at the cathode, ${\text{Cl}}_2(g)$ at the anode. A steel mesh diaphragm prevents recombination. Operating at about 7 V.

Electrorefining of copper: An impure copper anode is dissolved and pure copper is deposited on a cathode in a ${\text{CuSO}}_4/{\text{H}}_2{\text{SO}}_4$ electrolyte. The impure anode oxidises: $\text{Cu}(s)\to {\text{Cu}}^{2+}+2e^{-}$. Pure copper deposits: ${\text{Cu}}^{2+}+2e^{-}\to \text{Cu}(s)$. The overall cell potential is nearly zero (same reaction forwards and backwards), so only a small voltage ($0.2$--$0.4\text{V}$) is needed to overcome resistance and overpotentials. Less noble impurities (Zn, Fe, Ni) dissolve but do not plate at the cathode potential. More noble impurities (Ag, Au, Pt) do not dissolve and collect as "anode slime" -- a valuable byproduct that is processed separately for precious metal recovery.

Overpotential in detail: the three contributions

Activation overpotential dominates at low current densities and is described by the Butler-Volmer equation. For the hydrogen evolution reaction (HER), the activation overpotential on platinum is less than $0.05\text{V}$ at moderate currents, while on mercury it exceeds $1.0\text{V}$. This difference is exploited in electroanalysis: mercury electrodes allow wide negative potential windows because the large HER overpotential prevents hydrogen bubbles from interfering with the measurement.

Concentration overpotential arises from reactant depletion at the electrode surface. For a species with bulk concentration $c_0$ and surface concentration $c_s$:

$${\eta }_{\text{conc}}=\frac{RT}{nF}\ln \frac{c_0}{c_s}$$

At the limiting current, $c_s\to 0$ and ${\eta }_{\text{conc}}\to \infty$ (in theory; in practice, a new reaction takes over). Concentration overpotential is minimised by stirring, rotation (RDE), or operating at current densities well below $j_L$.

Resistance overpotential is the simplest to quantify: ${\eta }_{\text{res}}=j\cdot d/\kappa$, where $d$ is the electrode separation and $\kappa$ is the electrolyte conductivity. Using concentrated electrolytes, minimising electrode separation, and adding supporting electrolytes all reduce resistance overpotential.

Quantitative overpotential analysis: the cell voltage budget

For a water electrolysis cell at $100{\text{mA/cm}}^2$ with Pt electrodes in 1 M H2SO4:

Contribution	Voltage (V)
Thermodynamic $E^{\circ }$	1.23
Anode activation overpotential (OER)	0.35
Cathode activation overpotential (HER)	0.03
Concentration overpotential	0.05
Solution $IR$ drop	0.15
Total applied voltage	1.81

The oxygen evolution reaction (OER) accounts for most of the overpotential. Reducing the OER overpotential is the central challenge in water electrolysis for hydrogen production. State-of-the-art iridium oxide and ruthenium oxide anodes achieve OER overpotentials of 0.2--0.3 V, while nickel-iron oxyhydroxide catalysts in alkaline electrolysis achieve about 0.3--0.4 V at comparable current densities.

Connections Master

• Galvanic cells 14.11.02 are the thermodynamic reverse of electrolytic cells. The half-cell potential framework, the electrochemical series, and the cell potential equation are shared. The key difference is the sign convention and the source of energy.

• Electrochemistry fundamentals 14.11.01 provides the Butler-Volmer equation and the Nernst equation that underpin the overpotential analysis. The Tafel equation derived here is the high-overpotential limit of the Butler-Volmer equation.

• Batteries 14.11.04 are galvanic cells during discharge and electrolytic cells during recharge. The overpotential considerations developed here directly determine charging efficiency and battery degradation.

• Chemical kinetics 14.08.01 provides the transition-state theory framework that the Butler-Volmer equation applies to electrode processes. The activation overpotential is the electrochemical analogue of the Arrhenius activation energy.

• Thermodynamics 14.06.01 gives the $\Delta G=-nFE$ relationship that determines the theoretical decomposition potential. The energy efficiency of electrolysis is the ratio of $\Delta G$ to the actual electrical energy input ($nF{E}_{\text{applied}}$).

• Industrial chemistry connects through the Hall-Heroult process (aluminium), the chlor-alkali process (chlorine and NaOH), electrorefining (copper), and electroplating (corrosion protection, decorative finishes, electronics). These processes consume several percent of global electricity production.

Historical notes Master

Michael Faraday's investigation of electrolysis began in 1832 and culminated in the publication of his quantitative laws in 1834 ^{[Faraday 1834]}. Faraday showed that the same quantity of electricity decomposes chemical equivalents of different substances, establishing the intimate connection between atomic mass, valence, and electric charge. His apparatus -- a voltaic pile connected to solutions of various salts with collected and weighed products -- was simple but the conclusion was profound: electric charge is quantised at the atomic level. Faraday coined the terms electrode, anode, cathode, anion, cation, and electrolyte in consultation with William Whewell.

Julius Tafel published his empirical equation relating overpotential to current density in 1905 ^{[Tafel 1905]}, decades before the Butler-Volmer equation provided its theoretical justification. Tafel's measurements on hydrogen evolution at various metals revealed the logarithmic relationship between overpotential and current and the enormous variation in kinetic facility between electrode materials.

The Hall-Heroult process was discovered independently by Charles Martin Hall (Oberlin, Ohio, February 1886) and Paul Louis Toussaint Heroult (France, April 1886), both 23 years old. Before their invention, aluminium was more expensive than gold and was produced by chemical reduction with sodium. The electrolytic process reduced the price from about $1000/kginthe1850stoabout$1/kg by the 1930s. Hall went on to co-found Alcoa (Aluminium Company of America).

The chlor-alkali industry evolved through three cell designs: the diaphragm cell (1890s), the mercury cell (1890s, now being phased out due to mercury pollution), and the membrane cell (1970s, now dominant). The membrane cell uses a perfluorinated sulfonic acid membrane (Nafion, developed by DuPont) that achieves high selectivity for ${\text{Na}}^{+}$ transport while blocking ${\text{Cl}}^{-}$ and ${\text{OH}}^{-}$, enabling the production of high-purity NaOH without mercury contamination.

Modern developments include proton-exchange membrane (PEM) water electrolysers that achieve current densities of 1--2 A/cm${}^2$ at 1.8--2.0 V, an order of magnitude higher than traditional alkaline electrolysers, and solid-oxide electrolysers that operate at 700--900${}^{\circ }$C, reducing the electrical energy requirement by using thermal energy to supply part of the thermodynamic work.

Bibliography Master

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}

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@patent{Hall1886,
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  title = {Process of Reducing Aluminium by Electrolysis},
  number = {US Patent 400,664},
  year = {1886},
}

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