Faraday's laws of electrolysis

Michael Faraday.

Faraday's laws of electrolysis are quantitative relationships based on the electrochemical research published by Michael Faraday in 1833.

First law

Michael Faraday reported that the mass ($m$) of elements deposited at an electrode is directly proportional to the charge ($Q$; SI units are ampere seconds or coulombs).

$${\begin{aligned}m&\propto Q\\\implies {\frac {m}{Q}}&=Z\end{aligned}}$$

Here, the constant of proportionality, $Z$, is called the electro-chemical equivalent (e.c.e) of the substance. Thus, the e.c.e. can be defined as the mass of the substance deposited/liberated per unit charge.

Second law

Faraday discovered that when the same amount of electric current is passed through different electrolytes/elements connected in series, the mass of the substance liberated/deposited at the electrodes is directly proportional to their chemical equivalent/equivalent weight ($E$). This turns out to be the molar mass ($M$) divided by the valence ($v$)

$m\propto E$

$E={\frac {\text{Molar mass}}{\text{Valence}}}$

$\implies m_{1}:m_{2}:m_{3}:...=E_{1}:E_{2}:E_{3}:...$

$\implies Z_{1}Q:Z_{2}Q:Z_{3}Q:...=E_{1}:E_{2}:E_{3}:...$ (From 1st Law)

$\implies Z_{1}:Z_{2}:Z_{3}:...=E_{1}:E_{2}:E_{3}:...$

Derivation

A monovalent ion requires 1 electron for discharge, a divalent ion requires 2 electrons for discharge and so on. Thus, if $x$ electrons flow, ${\frac {x}{v}}$ atoms are discharged.

So the mass discharged

$${\begin{aligned}m&={\frac {xM}{vN_{\rm {A}}}}\\&={\frac {QM}{eN_{\rm {A}}v}}\\&={\frac {QM}{vF}}\end{aligned}}$$

where $N_{\rm {A}}$ is the Avogadro constant, Q = xe, and $F$ is the Faraday constant.

Mathematical form

Faraday's laws can be summarized by

$Z={\frac {m}{Q}}={\frac {1}{F}}\left({\frac {M}{v}}\right)={\frac {E}{F}}$

where $M$ is the molar mass of the substance (usually given in SI units of grams per mole) and $v$ is the valency of the ions .

For Faraday's first law, $M$, $F$, and $v$ are constants, so that the larger the value of $Q$ the larger $m$ will be.

For Faraday's second law, $Q$, $F$, and $v$ are constants, so that the larger the value of ${\frac {M}{v}}$ (equivalent weight) the larger $m$ will be.

In the simple case of constant-current electrolysis, $Q=It$, leading to

$m={\frac {ItM}{Fv}}$

and then to

$n={\frac {It}{Fv}}$

where:

• n is the amount of substance ("number of moles") liberated: n = m/M
• t is the total time the constant current was applied.

For the case of an alloy whose constituents have different valencies, we have

$$m={\frac {It}{F\times \sum _{i}{\frac {w_{i}\times v_{i}}{M_{i}}}}}$$

where w_i represents the mass fraction of the i-th element.

In the more complicated case of a variable electric current, the total charge Q is the electric current I($\tau$) integrated over time $\tau$:

$Q=\int _{0}^{t}I(\tau )\,d\tau$

Here t is the total electrolysis time.

See also

• Electrolysis
• Faraday's law of induction
• Tafel equation

Further reading

• Serway, Moses, and Moyer, Modern Physics, third edition (2005), principles of physics.
• Experiment with Faraday's laws