Faraday's laws of electrolysis



Faraday's laws of electrolysis are quantitative relationships based on the electrochemical research published by Michael Faraday in 1833.

First law
Michael Faraday reported that the mass ($m$) of a substance deposited or liberated at an electrode is directly proportional to the charge ($Q$; SI units are ampere seconds or coulombs).

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

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 or liberated per unit charge.

Second law
Faraday discovered that when the same amount of electric current is passed through different electrolytes connected in series, the masses of the substances deposited or liberated at the electrodes are directly proportional to their respective chemical equivalent/equivalent weight ($E$). This turns out to be the molar mass ($M$) divided by the valence ($v$)


 * $$\begin{align}

& 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{align}$$

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

Thus, the mass $m$ discharged is $$ m = \frac{x M}{v N_{\rm A}} = \frac{Q M}{e N_{\rm A} v} = \frac{Q M}{vF}$$ where
 * $NA$ is the Avogadro constant;
 * $Q = xe$ is the total charge, equal to the number of electrons ($x$) times the elementary charge $e$;
 * $F$ 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 (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 $$\tfrac{M}{v}$$ (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: $$n = \tfrac m M$$
 * $t$ is the total time the constant current was applied.

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

$$m = \frac{It}{F \times \sum_{i} \frac{w_i v_i}{M_i}}$$

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.