What is the standard EMF of the cell? — Electrochemistry Chemistry Question

💡 Solution & Explanation

Step 1 - Relate Standard Gibbs Free Energy Change and Standard Cell EMF The thermodynamic relationship between the standard Gibbs free energy change ($\Delta G^\circ$) and the standard electromotive force ($E^\circ_{\text{cell}}$) of an electrochemical cell is given by the formula: $$\Delta G^\circ = -nFE^\circ_{\text{cell}}$$ Where: * $\Delta G^\circ$ is the standard Gibbs free energy change in $\text{J mol}^{-1}$. * $n$ is the number of moles of electrons transferred per mole of reaction. * $F$ is Faraday's constant ($F \approx 96,500\text{ C mol}^{-1}$). * $E^\circ_{\text{cell}}$ is the standard electromotive force of the cell in volts ($\text{V}$). Rearranging this formula to solve for the standard cell potential ($E^\circ_{\text{cell}}$): $$E^\circ_{\text{cell}} = -\frac{\Delta G^\circ}{nF}$$ Step 2 - Determine the Moles of Electrons Transferred ($n$) The given thermodynamic values are specified per mole of standard water formation: $$\ce{H2(g) + 1/2 O2(g) -> H2O(l)}$$ For the reduction of half a mole of oxygen gas ($\ce{1/2 O2}$) or the oxidation of one mole of hydrogen gas ($\ce{H2}$), the total number of moles of electrons transferred ($n$) is: $$n = 2$$ Step 3 - Substitute the Given Values and Calculate the Standard EMF We are given: * $\Delta G^\circ = -237.39\text{ kJ mol}^{-1} = -237.39 \times 10^3\text{ J mol}^{-1} = -237,390\text{ J mol}^{-1}$ * $F = 96,500\text{ C mol}^{-1}$ * $n = 2$ Substituting these values with their respective units into our rearranged formula: $$E^\circ_{\text{cell}} = -\frac{-237,390\text{ J mol}^{-1}}{2 \times 96,500\text{ C mol}^{-1}}$$ $$E^\circ_{\text{cell}} = \frac{237,390\text{ J}}{193,000\text{ C}}$$ $$E^\circ_{\text{cell}} = \boxed{1.23\text{ V}}$$ Step 4 - Evaluate the Options * **Option (A) is incorrect:** This potential ($0.615\text{ V}$) is exactly half of the actual standard cell EMF, which is obtained if one incorrectly uses $n = 4$ for the single-mole free energy change value of water. * **Option (B) is correct:** As mathematically proven above, the standard EMF of the hydrogen-oxygen fuel cell is $+1.23\text{ V}$. * **Option (C) is incorrect:** This value ($2.46\text{ V}$) is double the correct potential, which is thermodynamically incorrect. * **Option (D) is incorrect:** This value ($0.74\text{ V}$) does not correspond to correct thermodynamic calculations. $$\text{Correct Option: } \boxed{B}$$