\(\mathrm{H}_{2}\) gas is produced by reforming methane via the following reaction:

\[ \mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O}=3 \mathrm{H}_{2}+\mathrm{CO} \]

The reaction is endothermic and carried out at \(1200 \mathrm{~K}\). Methane and steam, each at \(800 \mathrm{~K}\) and \(1 \mathrm{~atm}\), are fed to the reactor in a \(1: 1\) ratio. The reaction essentially goes to completion, and the hydrogen and CO leave at \(1200 \mathrm{~K}\). How much heat must be supplied per mole of methane reacted?