Faraday's laws of electrolysis

Faraday's laws of electrolysis are quantitative relationships based on the electrochemical research published by Michael Faraday in 1833.^[1]^[2] ^[3]

First law

Michael Faraday reported that the mass( $m$ ) of elements deposited at an electrode in g is directly proportional to the charge ( $Q$ ) in coulombs.^[3]

m\propto Q

Q=Amperes*seconds

\implies {\frac {m}{Q}}=Z

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 ( $E$ ).^[3] This turns out to be the molar mass ( $M$ ) divided by the valence ( $v$ )

m\propto E

E={\frac {Molar\ mass}{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 $m$

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

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

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

Where ( $F$ ) 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 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.^[4]

References

^ Faraday, Michael (1834). "On Electrical Decomposition". Philosophical Transactions of the Royal Society. 124: 77–122. doi:10.1098/rstl.1834.0008. S2CID 116224057.
^ 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.
^ ^a ^b ^c "Faraday's laws of electrolysis | chemistry". Encyclopedia Britannica. Retrieved 2020-09-01.
^ 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.