$16$ Consider a pure $H_2-O_2$ fuel cell at $T=80C$ and $P_{\text{c}\text{a}\text{t}\text{h}\text{o}\text{d}\text{e}}=P_{\text{a}\text{n}\text{o}\text{d}\text{e}}=1$atm :

$($a$)$ Calculate the ideal thermodynamic voltage for this fuel cell given $E^0=1\mathrm{.}23V$ and $S_{rxn}=-163\frac{J}{K}*mol{H}_2($remember $E^0$ is given for STP conditions; assume liquid water product$).$

$($b$)$ At $j=1\frac{A}{c}{m}^2,$ calculate ${}_{\text{a}\text{c}\text{t}}$ for the cathode given $=0\mathrm{.}3,n=4,$ and $j_0={10}^{-3}\frac{A}{c}{m}^2.$ Check any assumptions$/$simplifications made.

$($c$)$ Calculate $j_L$ at the cathode given $D_{\text{e}\text{f}\text{f}}={10}^{-2}c\frac{m^2}{s}$ and $=150m.$

$($d$)$ At $j=1\frac{A}{c}{m}^2,$ calculate ${}_{\text{c}\text{o}\text{n}\text{c}}$ at the cathode. $($Assume $c=0\mathrm{.}10V.)$

$($c$)$ We now pressurize the fuel cell cathode to $10$atm $($but the anode pressure remains $1$atm. Calculate the new thermodynamic voltage for this situation.

$($f$)$ Atj$=1\frac{A}{c}{m}^2,$ calculate ${}_{\text{a}\text{c}\text{t}}$ for the pressurized cathode given $=0\mathrm{.}3,n=4,$ and $j_0={10}^{-3}\frac{A}{c}{m}^2.$ Keep in mind that $j_0$ is given for $1$atm pressure conditions and thus ${}_{\text{a}\text{c}\text{t}}$ will need to be corrected for the new cathode pressure. Check any assumptions$/$simplifications made.

$($g$)$ Calculate $j_L$ for the pressurized fuel cell cathode, again assuming $D_{\text{e}\text{f}\text{f}}={10}^{-2}c\frac{m^2}{s}$ and $=150m.$

$($b$)$ At $j=1\frac{A}{c}{m}^2,$ calculate ${}_{\text{c}\text{o}\text{n}\text{c}}$ at the pressurized fuel cell cathode $($again$,$ assume $c=0\mathrm{.}10V).$ How much total voltage boost is gained when operating at $j=1\frac{A}{c}{m}^2$ by pressurizing the fuel cell cathode to $10$atm $?$