## First law

Michael Faraday reported that the mass ($m$ ) of elements deposited at an electrode is directly proportional to the charge ($Q$ in 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 in g 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 (in grams per mol) 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 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($\tau$ ) integrated over time $\tau$ :

$Q=\int _{0}^{t}I(\tau )\,d\tau$ Here t is the total electrolysis time.