[muawiyah]

Quick ~ somebody work this out ~ Coal + Air =, then we take the CO2 and convert that to Methane ~ then we burn that again. Remember to account for the additional O from the atmosphere.

[LucianOfSamasota]

You don't need to figure in coal at all.

The chemical reaction for combustion of methane is:

CH4 + 2 O2 → CO2 + 2 H2O (ΔH = −891 kJ/mol (at standard conditions))

Any reaction converting CO2 back into methane will consume 891 kJ/mol if 100% efficient. And no process is 100% efficient. So (and thermodynamics guarantees this result, hence no 'perpetual motion' machines) converting the same chemical reaction back and forth must ALWAYS result in a net energy LOSS. I'm a physicist, not a chemist, but this is pretty basic stuff.