Calculate `K_(sp)` if `(PbSO_(4))E_(cell)` at 298 K of this electrode is 0.236 V
`Pb(s)|PbSO_(4)(s)|underset(0.01M)(Na_(2)SO_(4)(aq))||underset(0.1M)(Pb(NO_(3))|Pb(s))`
`E_(cell)=E_(cell)^(@)-(0.059)/(2)log[(0.01)/(0.1)]`
`0.236=E_(cell)^(@)+(0.059)/(2)`
`E_(cell)^(@)=0.236-0.03=0.206-0.059log[K_(sp)]`

Explanation

Let it be conc. Cell
Anode `Pb(s)toPb_((a))^(2+)+2e^(-)`
cothode `Pb_((as))^(2+)+2e^(-)toPb(s)`
`Pb_((c))^(2+)hArrPb_((a))^(2+)`
`E_(cell)=E_(cell)^(0)-(0.059)/(2)log[((pb^(2+))_(a))/((Pb^(2+))_(c))]`
`0.236=(0.059)/(2)log[((Pb^(2+))_(a))/((Pb^(2+))_(c))]`
`(Pb^(2+))_(a)=10^(-9)M`
`K_(sp)=(Pb^(2+))_(a)(SO_(4)^(2-))_(a)=10^(-11)`
Let if it is not a conc cell
Anode: `Pb(s)+SO_(4)^(2-)(aq)toPbSO_(4)(s)+2e^(-)`
cathode: `Pb^(2+)(aq)+2e^(-)toPb(s)`
`Pb^(2+)(c)+SO_(4)^(2-)(a)toPbSO_(4)(s)`
`E_(cell)^(0)=E_(Pb^(2+)//Pb)^(0)-E_(SO_(4)^(2-)|PbSO_(4)|Pb)^(0)=E_(Pb^(2+)|Pb)^(0)-[E_(Pb^(2+)|Pb^(+))^(0)+(0.059)/(2)logK_(sp)]`
`E_(cell)^(0)=-(0.059)/(2)logK_(sp)`
`0.236=(0.059)/(2)logK_(sp)logK_(sp)-(0.059)/(2)log[(1)/((Pb^(2+))_(c)(SO_(4)^(2-))_(a))]=log[(K_(SP))/(0.1xx0.01)]=log[(K_(SP))/(10^(-3))]log10^(-8)`