Let's break down and solve each part step by step. --- ## (b) **Calculate the Standard Cell Voltage at 80°C** ###Given:** - **Equation:** \[ E^(T) = E^(298) - \frac{\Delta S}{nF}(T - 298) \] - \(\Delta S = -163.2 \text{ J/mol·K}\) - Standard temperature = \(298\,\text{K}\) - \(T = 80^\circ \text{C} = 353\,\text{K}\) - Number of electrons, \(n = 2\) (for PEMFC: \(2H_2 + O_2 \to 2H_2O\)) - Faraday's constant, \(F = 96485\,\text{C/mol}\) - \(E^(298) = 1.229\,\text{V}\) (Standard cell potential for PEMFC at \(25^\circ\text{C}\)) --- ### **Step 1: Substitute Values into the Equation** \[ E^(353) = 1.229 - \frac{-163.2}{2 \times 96485}(353 - 298) \] --- ### **Step 2: Calculate the Fraction** - Numerator: \(-163.2\) - Denominator: \(2 \times 96485 = 192970\) - \((T-298) = 353 - 298 = 55\) \[ \frac{-163.2}{192970} = -.000846\,\text{V/K} \] --- ### **Step 3: Multiply by \((T-298)\)** \[ -.000846 \times 55 = -.04653 \] --- ### **Step 4: Plug Back into the Equation** \[ E^(353) = 1.229 - (-.04653) \] \[ E^(353) = 1.229 + .04653 = 1.2755\,\text{V} \] --- ### **Final Answer for (b):** \[ \boxed{E^(80^\circ\text{C}) = 1.28\,\text{V} \quad (\text{rounded to two decimals})} \] --- ## (c) **Determine the Actual Cell Potential (with liquid water)** ### **Approach:** - Use the Nernst equation: \[ E = E^(T) + \frac{RT}{nF} \ln Q \] - For **PEMFC**: \(2H_2(g) + O_2(g) \to 2H_2O(l)\) - For liquid water, \(a_{H_2O} = 1\) - Assume gases at 1 atm: \(P_{H_2} = P_{O_2} = 1\,\text{atm}\) (standard conditions, so \(Q = 1\)) - \(R = 8.314\,\text{J/mol·K}\) - \(T = 353\,\text{K}\) - \(n = 2\) - \(F = 96485\,\text{C/mol}\) --- ### **Step 1: Reaction Quotient, \(Q\)** \[ Q = \frac{a_{H_2O}^2}{a_{H_2}^2 \cdot a_{O_2}} = \frac{1^2}{1^2 \cdot 1} = 1 \] \[ \ln Q = \ln 1 = \] --- ### **Step 2: Nernst Equation** \[ E = E^(T) + \frac{RT}{nF} \ln Q \] \[ E = 1.28 + \implies E = 1.28\,\text{V} \] --- ### **Final Answer for (c):** \[ \boxed{E_{\text{cell, actual}} = 1.28\,\text{V} \quad (\text{for liquid water, gases at 1 atm)}} \] --- ## **Summary Table** | Part | Step | Value | |------|--------------------------------------------|-----------------| | (b) | Standard cell voltage at 80°C | \(1.28\,\text{V}\) | | (c) | Actual cell potential (liquid water, std) | \(1.28\,\text{V}\) | --- **Note:** If partial pressures for H\(_2\) or O\(_2\) are not 1 atm, you would need to calculate \(Q\) accordingly, but with standard conditions and liquid water, \(E_{cell}\) equals the standard cell voltage at the given temperature.