I used to think that too, but then I found out it wasn't true. It turns out that by instantly laughing at people who disagree with you, you lose the opportunity to learn the things they know that you don't.

Specifically, in this case, the standard enthalpy of formation of carbon dioxide is -393.5 kilojoules per mole (that's the energy you get when you burn coal). If you burn coal in a sealed 1-bar chamber to drive a heat engine (such as a steam turbine), and pump the exhaust gases through a passively-air-cooled 1-bar heat exchanger to cool them back down to room temperature, you have obtained those 394kJ/mol thermal, about 138kJ/mol electric, for only the cost of blowing the gas around, without releasing any carbon dioxide into the atmosphere. Blowing the gas around with blowers takes typically about three orders of magnitude less energy than the heat energy the gas carries, though this depends on things like gas temperature and duct length, diameter, and smoothness. So physics is no bar (heh!) to getting from the coal-and-atmospheric-oxygen state into the electricity-and-captured-carbon-dioxide state.

There is some necessary energy cost involved in atmospheric carbon dioxide capture, because the air is only 400 ppm carbon dioxide and 99.96% other things, so there is an entropic cost to unmixing it. That entropic cost is not where you get the energy from burning it; the mixing happens above your smokestack where you can't extract any energy released and, as demonstrated above, can be avoided entirely. I don't understand thermodynamics well enough to calculate the thermodynamically necessary energy to reverse that entropy, but my understanding is that even at 400ppm it is orders of magnitude lower than the fuel's heating value.

Recompressing the carbon dioxide to liquid form requires about as much energy as you got in mechanical form out of the turbine; carbon dioxide liquefies at room temperature at 60 bar, liberating its enthalpy of vaporization of 16.7 kilojoules per mole, 4.2% of the enthalpy of formation. 59 bar is 5.9 kJ/l, and at room temperature and 1 bar an ideal gas occupies 24.4 l/mol, so you need to put about 140kJ/mol into compression. But as I understand it (and correct me if I'm wrong) almost all of that energy comes back out as heat. In practice, atmospheric carbon capture systems use absorbents or adsorbents such as calcium oxide, zeolites, or ethanolamine solutions to avoid this complicated and inefficient compression step. They use a substantial amount of energy to regenerate the sorbents, but, as I understand it, much less than burning the fuel generates.

For example, https://cobblab.eas.gatech.edu/energy/Readings/rochelle2009.... (DOI: 10.1126/science.1176731) says that a monoethanolamine scrubber used for flue-gas treatment at a 450-megawatt power plant in the year 02006 used 0.37 megawatt hours per ton of CO₂ removed. Assuming they mean metric tonnes, that's 1.332 gigajoules per 22700 moles of CO₂, which works out to 59kJ/mol, which is 43% of the energy obtained from burning the carbon. (Although amine scrubbers have been used for capturing CO₂ from flue gas at a number of power plants since the 01980s, I'm unclear on whether this 59kJ/mol number represents measurements from an actually deployed system or the projected performance of a proposed design.)

You could argue that it's easier to remove CO₂ from power-plant flue gas where it's fairly concentrated (12% rather than 0.04%), but in fact sorbents like monoethanolamine and triethanolamine are also quite effective at removing CO₂ from breathable air in environments such as submarines and space stations. At room temperature, their vapor pressure of CO₂ is quite low indeed, so it's mostly just a question of needing to expose the sorbent solution to air over a longer period of time. It doesn't result in requiring a proportional increase in the energy consumption of the process.

A recent open-access survey of the direct-air-capture problem is https://pubs.rsc.org/en/content/articlepdf/2022/ee/d1ee03523... (doi 10.1039/d1ee03523a). (However, note that that's still from November 02021, at which point photovoltaic panels still cost three times as much as they do now, so its calculations of energy costs are inflated by at least a factor of three.)

So, no, the laws of physics don't require that removing CO₂ will require even as much energy as what went into adding it to the atmosphere, much less "massively more energy", as you claim.

Even if they did, though, that wouldn't make atmospheric carbon capture impractical or a scam. Burning methane (natural gas) yields 55.6 MJ/kg (https://en.wikipedia.org/wiki/Energy_density#Chemical_reacti...) which works out to 892kJ/mol, but each mole of methane only produces one mole of carbon. The other 892 - 394 = 498kJ/mol (HHV) comes from burning the hydrogen (water's enthalpy of formation is -285.83kJ/mol, and methane's is -74.6kJ/mol, so we get 497.1 kJ/mol extra after the rounding errors). So even if we had to pay back all of that 286kJ/mol from the carbon to stuff it back into a bottle, we'd still have 56% of the energy we got from burning the natural gas left over.

And, of course, photovoltaics and wind provide abundant energy we can use for atmospheric carbon capture without releasing any more carbon dioxide, and nowadays they do it far more cheaply than fossil-fuel power plants ever did.

There are also schemes for atmospheric carbon dioxide capture with biological photosynthesis, which I think are probably not feasible at the required scale due to the required area, and with enhanced weathering by pulverizing olivines, which are probably feasible but strike me as risky. In the olivine case, the energy required to extract the gas from the atmosphere was provided billions of years ago by the heat of the earth driving carbon dioxide and other volatiles out of the mantle, so we don't have to care how big it is.

The reason "every single proposal for CO2 removal has failed" is (1) most of them are pretty energy-intensive, and until five years ago, energy was expensive (and cheap energy still hasn't reached most places); and (2) there's no clear way to make money at CO2 removal at current carbon credit prices, which are in part because the carbon credit market is pretty corrupt but also because at this point there's still a lot of low-hanging fruit in the emissions-reduction sector.