A small company is considering using reversible PEEC$/$PEFCs for electrochemical storage of solar power. F&H Figure $9\mathrm{.}3($page $198)$ shows the polarization curye for a PEFC $($square symbols$).$ Based on these data and other estimates below, please answer the following:

a$)$ Assuming the reactants are $H_2,O_2,$ and liquid water at $80C,$ calculate the thermoneutral potential $(U^{TN})$ of the cell. Likewise, assuming air at the cathode and a $50/50$ mixture of $H_2$ and $H_{2}O$ at the anode, calculate the reversible cell potential $(U^{\text{r}\text{e}\text{v}}).$ Briefly explain why $U^{TN}$ and $U^{\text{r}\text{e}\text{v}}$ are so different.

b$)$ During $H_2$ electrolysis $($not shown on the graph$),$ the cell operates at $1\mathrm{.}0\frac{A}{c}{m}^2$ at a voltage of $1\mathrm{.}70$ volts. Under these conditions, calculate the total cell area $($in $c{m}^2)$ needed to convert $10\mathrm{.}0kW$ of solar power to hydrogen. Calculate the exergetic efficiency of the electrolysis process. Calculate the amount of heat generated or consumed in watts $\frac{?}{c{m}^2}$ of cell area during electrolysis.

c$)$ Assuming the cell operates in electrolysis mode for $8$ hours each day, calculate the total amount of $H_2$ that can be stored $($in moles$),$ and the volume of the tank assuming ambient temperature of $25C$ and $1$atm. If the cells are operated in fuel cell mode for the rest of the day$/$night cycle $(16$ hours per day$),$ what current should be drawn $($as a fuel cell$)$ to balance the production and consumption of fuel?

d$)$ During fuel cell operation, the cell operates at $0\mathrm{.}50\frac{A}{c}{m}^2.$ Assuming the same cell area that you calculated in $($b$),$ calculate the total power $($in $kW)$ that the system can produce, and the exergetic efficiency of the fuel cell. Calculate the amount of heat generated or consumed in watts $\frac{?}{c{m}^2}$ of cell area during fuel cell operation.

e$)$ Calculate the round$-$trip storage efficiency of the entire day$-$night cycle, defined as the electrical work recovered divided by the work input. Of the original $80kW*hr$ of daily energy storage, calculate how much is lost. Compare this loss to the heat produced by the cells during electrolysis and fuel cell operation. Does everything add up$?$

f$)$ In order to increase overall efficiency, the company proposes to operate the cells at much lower current density as a fuel cell, $0\mathrm{.}20\frac{A}{c}{m}^2$ at $1\mathrm{.}60$ volts during electrolvsis$).$ Is this a