## First law

Michael Faraday reported that the mass(${\displaystyle m}$) of elements deposited at an electrode in g is directly proportional to the Charge(${\displaystyle Q}$) in Coulombs.[3]

${\displaystyle m\propto Q}$
${\displaystyle Q=Amperes*seconds}$
${\displaystyle \implies {\frac {m}{Q}}=Z}$

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[clarification needed] 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[clarification needed] (${\displaystyle E}$).[3] This turns out to be the molar mass(${\displaystyle M}$) divided by the valence(${\displaystyle v}$)

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

${\displaystyle ={\frac {xM}{vN_{A}}}}$ (where ${\displaystyle N_{A}}$ is Avogadros number)

${\displaystyle ={\frac {QM}{eN_{A}v}}}$ (From Q = xe)

${\displaystyle ={\frac {QM}{Fv}}}$

## 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 ith 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]

## References

1. ^ Faraday, Michael (1834). "On Electrical Decomposition". Philosophical Transactions of the Royal Society. 124: 77-122. doi:10.1098/rstl.1834.0008. S2CID 116224057.
2. ^ Ehl, Rosemary Gene; Ihde, Aaron (1954). "Faraday's Electrochemical Laws and the Determination of Equivalent Weights". Journal of Chemical Education. 31 (May): 226-232. Bibcode:1954JChEd..31..226E. doi:10.1021/ed031p226.
3. ^ a b c "Faraday's laws of electrolysis | chemistry". Encyclopedia Britannica. Retrieved .
4. ^ For a similar treatment, see Strong, F. C. (1961). "Faraday's Laws in One Equation". Journal of Chemical Education. 38 (2): 98. Bibcode:1961JChEd..38...98S. doi:10.1021/ed038p98.