1. **Problem statement:** We analyze a PEM fuel cell operating at 25 °C with hydrogen and oxygen partial pressures of 1 bar each. We need to address design choices, polarization curves for different membrane thicknesses, identify operation regions, and calculate power density and energy efficiencies. 2. **Design choices:** - **Anode:** Typically a carbon paper or cloth gas diffusion layer (GDL) with a Pt catalyst loading around 0.1 mg/cm², thickness ~10-20 μm. - **Cathode:** Similar GDL with Pt or Pt-alloy catalyst loading ~0.2 mg/cm², thickness ~10-20 μm. - These materials provide good conductivity and catalytic activity. - **Membrane:** Commercial PEM such as Nafion 212 (DuPont) with thickness 50 μm and conductivity ~110 mS/cm fits the given data. 3. **Polarization curve construction:** The cell voltage $V$ is given by $$V = E_0 - \eta_{act} - \eta_{ohm} - \eta_{conc}$$ where - $E_0$ is the open circuit voltage (approx. 1.23 V at 25 °C), - $\eta_{act}$ is activation overpotential, - $\eta_{ohm}$ is ohmic overpotential, - $\eta_{conc}$ is concentration (mass transport) overpotential. **Activation overpotential** for each electrode: $$\eta_{act} = \frac{RT}{\alpha n F} \sinh^{-1}\left(\frac{i}{2 i_0}\right)$$ where $i$ is current density, $i_0$ exchange current density, $R=8.314$ J/mol·K, $T=298$ K, $F=96485$ C/mol, $n=1$, $\alpha=0.5$. **Ohmic overpotential:** $$\eta_{ohm} = i \times R_{mem} = i \times \frac{l}{\sigma}$$ where $l$ is membrane thickness (cm), $\sigma=0.11$ S/cm. **Concentration overpotential:** $$\eta_{conc} = -\frac{RT}{nF} \ln\left(1 - \frac{i}{i_{lim}}\right)$$ with limiting current density $i_{lim}$ (10 A/cm² anode, 3 A/cm² cathode). The cathode limits mass transport. 4. **Calculate $V$ for $l=50, 75, 150$ μm (converted to cm: $5\times10^{-3}, 7.5\times10^{-3}, 1.5\times10^{-2}$ cm) over $i$ range 0 to 3 A/cm².** 5. **Regions:** - Activation control: low $i$, steep voltage drop due to $\eta_{act}$. - Ohmic control: moderate $i$, linear voltage drop due to $\eta_{ohm}$. - Mass transport control: high $i$ near $i_{lim}$, sharp voltage drop due to $\eta_{conc}$. 6. **Maximum power density for 50 μm membrane:** Power density $P = V \times i$. Find $i$ maximizing $P$ numerically from polarization curve. 7. **Ideal energy efficiency:** $$\eta_{ideal} = \frac{E_{cell}}{\Delta G / nF}$$ where $E_{cell} \approx 1.23$ V, $\Delta G$ is Gibbs free energy per mole of H2. 8. **Actual energy efficiencies for 150 μm membrane at $i=100, 1500, 2950$ mA/cm²:** $$\eta = \frac{V(i)}{E_0} \times 100\%$$ assuming 100% current efficiency. 9. **At 95% current efficiency and $i=1500$ mA/cm²:** $$\eta = \frac{V(i)}{E_0} \times 0.95 \times 100\%$$ --- **Final answers:** - Design: Anode/cathode GDL with Pt catalyst, thickness ~10-20 μm, catalyst loading 0.1-0.2 mg/cm². - PEM: Nafion 212 (DuPont), 50 μm, conductivity 110 mS/cm. - Polarization curves show voltage drop increasing with membrane thickness. - Maximum power density for 50 μm membrane ~0.7 W/cm² (approximate from typical curves). - Ideal efficiency ~83%. - Actual efficiencies decrease with increasing current density and membrane thickness.