Calculate the standard voltage that can be obtained from an ethane oxygen fuel cell at `25^(@)C`. `C_2H_6 (g) + 7/2 O_2 (g) to 2 CO_2 (g) + 3 H_2O (l)` `ΔG^(@) = -1467 kJ/mol.`

Options

A. `+0.91`
B. `+0.54`
C. `+0.72`
D. `+1.08`

Answer

To calculate the standard voltage that can be obtained from an ethane-oxygen fuel cell at \(25^{\circ}C\), we will follow these steps:

### Step 1: Write the balanced chemical equation
The balanced chemical equation for the reaction is:
\[ C_2H_6 (g) + \frac{7}{2} O_2 (g) \rightarrow 2 CO_2 (g) + 3 H_2O (l) \]

### Step 2: Determine the change in oxidation states
- For carbon in ethane (\(C_2H_6\)), the oxidation state is \(-3\).
- In carbon dioxide (\(CO_2\)), the oxidation state of carbon is \(+4\).
- The change in oxidation state for each carbon atom is \(4 - (-3) = 7\).
- Since there are 2 carbon atoms, the total change in oxidation state for carbon is \(2 \times 7 = 14\).

### Step 3: Determine the number of electrons transferred (n)
The number of electrons transferred in the reaction is \(n = 14\).

### Step 4: Use the given standard Gibbs free energy change
We are given that:
\[ \Delta G^{\circ} = -1467 \, \text{kJ/mol} \]
We need to convert this to joules:
\[ \Delta G^{\circ} = -1467 \times 10^3 \, \text{J/mol} = -1467000 \, \text{J/mol} \]

### Step 5: Use the relationship between Gibbs free energy and cell potential
The relationship between Gibbs free energy change and the standard electrode potential is given by:
\[ \Delta G^{\circ} = -nFE^{\circ}_{cell} \]
Where:
- \(F\) is Faraday's constant, approximately \(96500 \, \text{C/mol}\).

### Step 6: Rearranging the equation to find \(E^{\circ}_{cell}\)
Rearranging the equation gives:
\[ E^{\circ}_{cell} = -\frac{\Delta G^{\circ}}{nF} \]

### Step 7: Substitute the values into the equation
Substituting the values we have:
\[ E^{\circ}_{cell} = -\frac{-1467000 \, \text{J/mol}}{14 \times 96500 \, \text{C/mol}} \]

### Step 8: Calculate \(E^{\circ}_{cell}\)
Calculating this gives:
\[ E^{\circ}_{cell} = \frac{1467000}{14 \times 96500} \approx \frac{1467000}{1351000} \approx 1.08 \, \text{V} \]

### Final Answer
The standard voltage that can be obtained from the ethane-oxygen fuel cell at \(25^{\circ}C\) is approximately:
\[ \boxed{1.08 \, \text{V}} \]