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Lots of people talking about air capture of CO2. Even Prez candidates. It's a great idea, but it takes a lot of energy. To understand this, let's work out the thermodynamics. [note: nerd twitter thread] 1/

You have an insulated volume containing a total of 1 mole of gas; it contains two compartments separated by an impermeable wall. One compartment contains pure CO2 at 293C and 1 atm, the other contains 80% N2 and 20% O2, also at 293C and 1 atm. 2/

The initial volumes are such that, after the partition is removed and they mix, the CO2 will have a mixing ratio of 400 ppmv. So let's remove the partition and let the gases mix. 3/

First, there's no change in temperature b/c no work is done. So there's no change in enthalpy of the system. But there is an increase in entropy — it's the sum of the free expansion of each compartment into a vacuum. 4/

the change in entropy is equal to n R Log[Vf/Vi] for both sides (n = number of moles, R = gas constant, Vf and Vi are the final and initial volumes). For the CO2 expanding into the other space, ∆S = 0.026 J/K; for O2/N2 expanding, ∆S = 0.0033 J/K. Total ∆S = 0.029 J/K. 5/

Let's calculate the Gibb's free energy for this process. ∆H = 0, so ∆G = -T∆S = -8.59 J. The negative sign means this mixing is spontaneous. You can think of separating CO2 from air as the reverse of this mixing process. 6/

This means that separating the CO2 from 1 mole of air takes +8.6 J of energy. From this, you can estimate that it would require about 500 kJ to separate one kg of CO2 from the air. 7/

If you want to remove 35 billion tons of CO2 from the air (about one year's emissions), that would require 500 GW of power. Humans consume about 15 TW of power, so this corresponds to a few percent of the power we're now generating. That seems pretty reasonable. 8/

But ... this is the thermodynamic limit. In reality, you won't be able to do nearly this well. In addition, you need to do something with the CO2. If you want to store it underground, for example, then you have to compress it, which takes more energy. 9/

So I'm guessing that it will actually take 10x as much to pull CO2 out of the air. This would mean that we need about 40% of the energy generated to capture the carbon emitted by generating the energy. 10/

Could we do that? Certainly! But before you advocate for going down this road, you have to identify where the energy comes from. 11/

Gas coming out of the smoke stack might be 30% CO2 instead of 0.04% in ambient air. Is it better to capture CO2 at the stack? The answer is slightly — ∆S goes as the Log[Vf/Vi], so capturing C at the smoke stack might decrease required energy by a factor of 2, give or take. 12/

For tweet 8, 500 GW is the power required to remove that much carbon from the atmosphere every year.

Also, 293 C in the problem set up should be 293 K.

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