## First law

Michael Faraday reported that the mass (${\displaystyle m}$) of elements deposited at an electrode is directly proportional to the charge (${\displaystyle Q}$ in ampere seconds or coulombs).[3]

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

Here, the constant of proportionality ${\displaystyle Z}$ 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 in g is directly proportional to their chemical equivalent/equivalent weight (${\displaystyle E}$).[3] This turns out to be the molar mass (${\displaystyle M}$) divided by the valence (${\displaystyle v}$)

${\displaystyle m\propto E}$
${\displaystyle E={\frac {\text{Molar mass}}{\text{Valence}}}}$
${\displaystyle \implies m_{1}:m_{2}:m_{3}:...=E_{1}:E_{2}:E_{3}:...}$
${\displaystyle \implies Z_{1}Q:Z_{2}Q:Z_{3}Q:...=E_{1}:E_{2}:E_{3}:...}$ (From 1st Law)
${\displaystyle \implies Z_{1}:Z_{2}:Z_{3}:...=E_{1}:E_{2}:E_{3}:...}$

## Derivation

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

So the mass discharged

{\displaystyle {\begin{aligned}m&={\frac {xM}{vN_{\rm {A}}}}\\&={\frac {QM}{eN_{\rm {A}}v}}\\&={\frac {QM}{vF}}\end{aligned}}}
where ${\displaystyle N_{\rm {A}}}$ is the Avogadro constant, Q = xe, and ${\displaystyle F}$ is the Faraday constant.

## Mathematical form

Faraday's laws can be summarized by

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

where ${\displaystyle M}$ is the molar mass of the substance (in grams per mol) and ${\displaystyle v}$ is the valency of the ions .

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

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

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

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

and then to

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

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

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

where wi 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(${\displaystyle \tau }$) integrated over time ${\displaystyle \tau }$:

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

Here t is the total electrolysis time.[4]