Consider the following two half-cell reactions along with the standard reduction potential given:
\( \text{CO}_{2} + 6\text{H}^{+} + 6e^{-} \rightarrow \text{CH}_{3}\text{OH} + \text{H}_{2}\text{O} \quad E^{0}_{\text{red}} = 0.02 \, \text{V} \)
\( \dfrac{1}{2}\text{O}_{2} + 2\text{H}^{+} + 2e^{-} \rightarrow \text{H}_{2}\text{O} \quad E^{0}_{\text{red}} = 1.23 \, \text{V} \)
A fuel cell was set up using the above two reactions such that the cell operates under the standard condition of \( 1 \, \text{bar} \) pressure and \( 298 \, \text{K} \) temperature. The fuel cell works with \( 80\% \) efficiency. If the work derived from the cell using \( 1 \, \text{mol} \) of \( \text{CH}_{3}\text{OH} \) is used to compress an ideal gas isothermally against a constant pressure of \( 1 \, \text{kPa} \), then the change in the volume of the gas, \( \Delta V = \) _________ \( \text{m}^{3} \). (nearest integer)
Given : \( F = 96500 \, \text{C mol}^{-1} \)
