How is equilibrium constant calculated from the Nernst's equation ? Derive the relation using suitable example.

Explanation

Equilibrium constant can be calculated from the Nernst.s equation in the state of equilibrium.
Consider the following reaction: `Zn(s)+Cu^(2+)(aq)iffZn^(2+)(aq)+Cu(s)`
For the above reaction, equilibrium constant `K_(c )` is given as :
`K_(c ) =([Zn^(2+)(aq)][Cu(s)])/([Zn(s)][Cu^(2+)(aq)])=([Zn^(2+)(aq)])/([Cu^(2+)(aq)]){:[(because[Cu(s)]=1),(" "[Zn(s)]=1)]:}`
The Nernst.s equation for above reaction is written below:
`E_("cell")=E_("cell")^(@)-(2.303RT)/(nF)log""([Zn^(2+)(aq)])/([Cu^(2+)(aq)])`
At equilibrium, `E_("cell")=0`
`0=E_("cell")^(@)-(2.303RT)/(nF)log""([Zn^(2+)(aq)])/([Cu^(2+)(aq)]),E_("cell")^(@)=(2.303RT)/(nF)logK_(c)`
In general, we have
`E_("cell")^(@)=(2.303RT)/(nF)logK_(c)`
Using equation (iii), the value of `K_(c)` can be determined if the values of `E_("cell")^(@)`, T and n are known.