$\Delta G =\Delta H - T \Delta S$
$ \Delta G =\Delta H + T \left(\frac{ d \Delta G }{ dT }\right)_{ p } $
$- nF \left(\frac{ dE _{\text {cell }}}{ dT }\right)=-\Delta S $
$ \left.\frac{ dE _{\text {cell }}}{ dT }=\frac{\Delta S }{ nF }=\frac{ R }{ F } \text { (given }\right)$
$ \Rightarrow \Delta S = nR$
For the reaction, $M ( g )+2 H ^{\oplus}( aq ) \longrightarrow H _2( g )+ M ^{2 \oplus}( aq )$
$ n =2$
$\Rightarrow \Delta S =2 R$
Hence, option (A) is incorrect
For the reaction, $Pt _{( s )} \mid H _{2( g )}, 1$ bar $\left| H ^{\oplus}{ }_{ aq }(0.01 M )\right|\left| H ^{\oplus}( aq , 0.1 M )\right| H _2( g , 1 bar ) \mid Pt _{( s )}$
$E _{\text {cell }}= E _{\text {cell }}^{\circ}-\frac{0.0591}{1} \log \frac{0.01}{0.1}=0.0591 V$
$E_{cell}$ is positive $\Rightarrow \Delta G <0$ and $\Delta S >0(\Delta H =0$ for concentration cells $)$
Hence, option (B) is correct
Racemization of an optically active compound is a spontaneous process.
Here, $\Delta H =0$ (similar type of bonds are present in enantiomers)
$\Rightarrow \Delta S >0$
Hence, option (C) is correct.
$\left[ Ni \left( H _2 O \right)_6\right]^{2+}+3 en \rightarrow\left[ Ni ( en )_3\right]^{2+}+6 H _2 O$ is a spontaneous process
more stable complex is formed
$\Rightarrow \Delta S >0$