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This is a really good question, and a bit deeper than it first appears. So here is some semi-educated spitballing (I'm a chemist, but thermodynamics was a while ago):

1. Immediately after ignition, you have a low-volume, high-pressure, high-temperature amount of gas. Sequestration does not aim to turn CO2 back to this exact same state, but only a high-ish, average-temperature state.

2. Combustion often evolves more molecules of gas (look at the formula for the combustion of octane, and remember that water after combustion will be a gas). This increases the pressure, but is not something that needs to be reversed during sequestration.

3. Carbon dioxide isn't bad, but having too much in the atmosphere is. Sequestration doesn't aim to completely reverse the reaction in the first place, it just aims to remove it from the atmosphere so that it can't act as a greenhouse gas.

Enthalpy of combustion for CH4 is 802 kJ and for an equivalent amount of gaseous hydrogen it's 286 kJ so most of the energy does come from the carbon.

That's not how chemistry works. Burning 1 CH4 is not the same as burning 1 C and 2 H2. The formation of carbon dioxide (aka the combustion of carbon) contributes 393.5 kJ per mole of CH4 combusted, and the formation of water (aka the combustion of hydrogen) contributes 483.6 kJ per mole of CH4, the activation energy to split the CH4 molecule is 74.8 kJ per mole of CH4 combusted. So 55% of the heat released per mole is from the hydrogen.

The goal is not to re-create fuel, but to clean up the waste. This is more like sweeping out the ashtray

> he heat energy increases the pressure of the newly created CO2. This higher pressure is placed on one side of a turbine or a piston, and we extract useful work by moving it from a high density state to a low density state, causing it to cool in the process.

You run the hot, high-pressure gas through a turbine to give you less-hot, lower-pressure gas. You then extract as much waste heat out of that stream as you can via a heat exchanger process, to pre-heat incoming fuel/air and to recover more energy by boiling water to run through another turbine.

At the end of the process, you have a medium-temperature stream of combustion product that has high concentrations of CO2. You capture the carbon from this stream, before releasing the last bits of gas to the atmosphere.

You gain usable energy out of the process because all of the heat movement happens through a turbine (to directly generate energy) or through a heat exchanger (to recycle the heat to other more useful parts of the process).

Expanding on what others have noted:

Heat engines take advantage of the fact that expanding a hot fluid releases more energy than it takes to compress a cold fluid - the difference being that combustion energy. Indeed the pressure difference is only there to make the process more efficient - so long as there is a temperature gradient you can extract energy even with no pressure gradient.

You are mostly right, but the missing part is that you don't need to turn the CO2 back into a hydrocarbon fuel, you just need to turn it into something that isn't gaseous CO2.

So carbon capture hinges on the idea that we can find a low-energy route that involves a chemical reaction with CO2 that produces something that isn't a fuel but isn't gaseous CO2 either. And that we can find a LOT of it.