a$)$ In a solid oxide fuel cell we use graphite as fuel. $1)$ Write down the cathodic and anodic reactions $2)$ What is the standard operating potential at $1000\backslash$deg C and $1$ bar if $\backslash$Delta $G_{f^o}$CO$2($g$)=-395\mathrm{.}865$ kJ$/$mol b$)1)$When charging a lead battery, the reactions PbSO$4+2$e Pb $+$ SO$4,$ U$^\backslash$theta $=-0\mathrm{.}356$~V

U$^\backslash$theta $=0$V $($in pure lead$)$ with corresponding exchange current densities io$($Pb$)=100$A$/$m$^2,$ io$($H$2)=6\mathrm{.}6*10^-10$ A$/$m$^2.$ Calculate the current density of each reaction using the Butler$-$Volmer equation and when the electrode potential is held constant at $-0\mathrm{.}44$V $($with respect to SHE$).($Take T$=298$K and coefficients aa$=$ac$=0.5)2)$ In case the lead has antimony $($Sb$)$ impurity the new value of io$($H$2)$Sb$=3\mathrm{.}7*10^-4$ A$/$m$^2)$ Calculate the new current density for H$2$ and in each case find the percentage $(\%)$ consumed for the undesired reaction.