Question

Nernst equation, why is $2.303$ value used in some cases of the equation? Mathematically, where does it come from?

Answer 1

Nernst equation is used in electrochemistry to draw a mathematical relationship between electrode potential of a cell and the concentration of electrolyte present in the electrochemical cell. The Nernst equation is a temperature dependent equation.

Complete answer:
To determine the Nernst equation of any electrode reaction we have to consider any reaction hypothetically. Let any metal ${M_{\left( s \right)}}$ is formed at an electrode by acceptance of $n{e^ - }$ number of electrons by ${M^{n + }}_{\left( {aq} \right)}$. The equation will become-
${M^{n + }}_{\left( {aq} \right)}$ $+$ $n{e^ - }$ $\to$ ${M_{\left( s \right)}}$
So, the Nernst equation for the above reaction is written as –
$E\left( {{M^{n + }}/M} \right)$ $=$ ${E^ \circ }\left( {{M^{n + }}/M} \right) - \dfrac{{RT}}{{nF}}\ln \dfrac{{\left[ {{M_{\left( S \right)}}} \right]}}{{\left[ {{M^{n + }}_{\left( {aq} \right)}} \right]}}$
Where $E\left( {{M^{n + }}/M} \right)$ $=$ Electrode potential of electrochemical cell
${E^ \circ }\left( {{M^{n + }}/M} \right)$ $=$ Standard electrode potential of an electrochemical cell which is calculated for one mole solution of metal ions.
$R$ $=$ Gas constant having value of $8.314$ $J/K/mol$
$T$ $=$ temperature of electrochemical cell in which reaction takes place $298K$
$F$ $=$ faraday constant of electricity with numerical value $96500$ coulombs
$n$ $=$ total number of electrons gained by metal ion during the reaction
$\left[ {{M^{ + n}}_{\left( {aq} \right)}} \right]$ $=$ molar concentration of metal ion used in the reaction
$\left[ {{M_{\left( s \right)}}} \right]$ $=$ molar concentration of metal formed in the reaction
This Nernst equation is expressed in terms of base $e$as $\left( {\ln } \right)$. When we convert the base $e$ value into base $10$.
Formula used to convert $\left( {\ln } \right)$into $\left( {\log } \right)$by-
$\ln x = \dfrac{{\log x}}{{\log e}}$
Value of $\log e$is $0.43$.
So, the equation will become
$\ln x = \dfrac{{\log x}}{{0.45}}$
After solving the above equation, we get
$\ln x = 2.303\log x$
Hence, when we multiply the Nernst equation by $2.303$ the equation will change to $\log$ at base $10$.
$\Rightarrow$ In Nernst equation $2.303$ value is used to convert $\ln$to $\log$.

Note:
Nernst equation is used to determine the potential created by different ions present in body fluids which cross through biological membranes. Nernst equation is used for dilute solution but it does not hold good at higher concentration of electrolyte.