## 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).[3]

${\displaystyle m\propto Q\quad \implies \quad {\frac {m}{Q}}=Z}$

Here, the constant of proportionality, Z, is called the electro-chemical equivalent (ECE) of the substance. Thus, the ECE 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).[3] This turns out to be the molar mass (M) divided by the valence (v)

{\displaystyle {\begin{aligned}&m\propto E;\quad E={\frac {\text{molar mass}}{\text{valence}}}={\frac {M}{v}}\\&\implies m_{1}:m_{2}:m_{3}:\ldots =E_{1}:E_{2}:E_{3}:\ldots \\&\implies Z_{1}Q:Z_{2}Q:Z_{3}Q:\ldots =E_{1}:E_{2}:E_{3}:\ldots \\&\implies Z_{1}:Z_{2}:Z_{3}:\ldots =E_{1}:E_{2}:E_{3}:\ldots \end{aligned}}}

## 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, ${\displaystyle {\tfrac {x}{v}}}$  atoms are discharged.

So the mass m discharged is

${\displaystyle m={\frac {xM}{vN_{\rm {A}}}}={\frac {QM}{eN_{\rm {A}}v}}={\frac {QM}{vF}}}$

where

## 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 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, v are constants; thus, the larger the value of Q, the larger m will be.

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

In the simple case of constant-current electrolysis, 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: ${\displaystyle n={\tfrac {m}{M}}}$
• t is the total time the constant current was applied.

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

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

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(τ) integrated over time τ:

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

Here t is the total electrolysis time.[4]