A “mole” is a gram-molecular weight.

The atomic weight of carbon is 12 atomic mass units, so a mole of carbon weighs 12 grams (6 protons + 6 neutrons in the nucleus). So a mole turns out to be 6.02 x 10^23 atoms. (Remember Avogadro’s Number”?) A mole of carbon will combust with a mole of O2 molecules (That’s two moles of oxygen atoms) to release a mole of CO2 plus 410 kJ of energy. The reason for this is the electrons surrounding both the carbon and oxygen nuclei reassemble themselves into lower-energy orbits. The excess energy is unloaded as light and heat. If you want to reverse the process, the exact same amount of energy must be introduced to separate the oxygen from the carbon.

That’s how much energy you get by burning carbon. To take apart CO2 molecules and reconstruct the original electron orbitals will require AT LEAST as much energy.

This is the same reaction that occurs in plants, the molecule that does this is chlorophyll, and it uses sunlight as an energy source. Plants use the carbon to build tissue and make sugar, and the oxygen is released into the atmosphere as a waste product. Animals breath in the oxygen and combine it with carbon derived from plants for energy, releasing CO2 as a waste.

Energy from the sun drives the whole process. Light produced in thermonuclear reactions in the sun (hydrogen fusion) produces the photons that power the chlorophyll. We don’t get this energy for free, either. The gravitational energy of the collapsing star provides the temperature and pressure that overcomes the repulsion of the hydrogen nucleus. The hydrogen fuses into helium (plus a lot of energy, that escapes the star as light). So the original source of all energy is gravitational energy working on the repulsive forces of the atomic nucleus.

The density of carbon is about the same as the density of coal (1 g/cm^3) A mole of carbon (a small lump of about 12 cm ^3) will release a lot of energy when burned in air.