Derive an expression for Nernst equation.

Explanation

Nernst equation is the one which relates the cell potential and the concentration of the species involved in an electrochemical reaction.
Let us consider an electrochemical cell for which the overall redox reaction is,
`xA+yB hArr lC+mD`
The reaction quotient Q is, `Q=([C]^(l)[D]^(m))/([A]^(x)[B]^(y))`
We know that,
`DeltaG=DeltaG^(@)=RT" ln "Q " "......(1) `
`therefore DeltaG=-nFe_("cell"), DeltaG^(@)=-nFE_("cell")^(@)`
`therefore` equation (1) becomes
`-nFE_("cell")=-nFE_("cell")^(@)+RT" lnQ "......(2)`
Substitute the Q value in equation (2)
`-nFE_("cell")=-nFE_("cell")^(@)+RT" ln "([C]^(l)[D]^(m))/([A]^(x)[B]^(y))" ".......(3)`
Divide the whole equation (3) by - nF
`E_("cell")=E_("cell")^(@)=(RT)/(nF)"ln"(([C]^(1)[D]^(m))/([A]^(x)[B]^(y)))`
`E_("cell")=E_("cell")^(@)-(2.303RT)/(nF)log(([C]^(1)[D]^(m))/([A]^(x)[B]^(y)))" ".......(4)`
This is called the Nernst equation.
At `25^(@)C(298K)` equation (4) becomes,
`E_("cell")=E_("cell")^(@)-(2.303xx8.314xx298)/(n xx 96500)log(([C]^(l)[D]^(m))/([A]^(x)[B]^(y)))`
`E_("cell")=E_("cell")^(@)-(0.0591)/(n)log(([C]^(l)[D]^(m))/([A]^(x)[B]^(y)))`