Thermodynamics of air capture of carbon dioxide
If you like this post, you are a nerd
About 4 years ago I did a twitter thread on the thermodynamics of air capture, which got a surprising amount of positive attention. It had a few minor errors in it and a few places that needed to be improved, so I’ve spruced it up and converted it into this substack post.
Thermodynamics is like your mom: it may not tell you what you can do, but it’s iron fisted about what you cannot do. Thermodynamics gives you the best-case estimate of the energetics of a process, which can be useful as a way to evaluate how effective a process can be. After all, if the best case isn’t very good, then any real-world attempts will be even worse.
Thermodynamics can tell us a lot about air capture of carbon dioxide (CO2). Air capture of carbon dioxide, also known as Direct Air Capture (DAC), is a process that involves machines extracting CO2 directly from the ambient atmosphere.
Going a step beyond conventional emissions reduction strategies, DAC is a negative emissions technology provides the potential to decrease atmospheric CO2 concentrations at a rate significantly exceeding the natural carbon cycle.
The potential significance of this technology is profound. Given the current levels of atmospheric CO2, merely reducing emissions to zero might not be sufficient to prevent dangerous levels of global warming. We may need DAC to avoid temperature increases that many would consider dangerous.
To work out the thermodynamics of DAC, we usually start with an idealized setup — remember, we’re calculating the best-case energetics here. So let’s start with an insulated container containing one mole of gas.
It is divided into two compartments by an impermeable wall. One compartment contains pure CO2 at 1 atm, the other contains 80% N2 and 20% O2, also at 1 atm.
The initial volumes of the two compartments are such that, after the partition is removed and they mix, the carbon dioxide will have a mixing ratio of 400 ppm in the combined volume. Throughout this calculation, we’ll assume the gases are at a temperature of 293 K, about room temperature.
So let's remove the partition and let the gases mix. What happens next? Because the mixing does no work, the temperature doesn’t change, and hence, the enthalpy of the system stays constant (∆H = 0).
But, as the two gases mix, the overall entropy of the system increases. The increase in entropy can be calculated by adding up the entropy changes that occur when the two compartments freely expand into a vacuum. The entropy change (∆S) for a reversible free expansion of n moles of gas is:
Calculating the entropy change for the two free expansions yields 0.026 J/K for CO2 and 0.0033 J/K for O2+N2, resulting in a total ΔS for mixing of 0.029 J/K.
Let's calculate the change in Gibb's free energy for this process. As discussed above, ∆H = 0, so ∆G = -T∆S = -8.59 J. The negative value tells us that this mixing is a spontaneous event.
You can think of the separation of carbon dioxide from air as the inverse of this mixing process, so it would require about +8.6 J of energy to separate 400 ppm of carbon dioxide from from 1 mole of air.
Because Gibbs free energy is a state function, it doesn’t matter exactly how you get from mixed to unmixed. The value we calculate here is the best you can do for the unmixing problem.
From this, we estimate that about 500 kJ1 would be needed to separate one kg of CO2 from the air. Now imagine that we want to offset present day emissions by removing 40 billion tons of CO2 from the air each year. This would require about 2x10^19 J each year, which corresponds to an average power of 630 GW.
This is a lot of power — equal to about 300 Hoover Dams or 30 Three Gorges Dams. And this isn't the end of the energy story.
What about the subsequent storage of the carbon dioxide? If we decide to sequester it underground, it needs to be compressed, which requires additional energy.
Being optimistic, let's assume isothermal compression:
Compressing 40 billion tons of CO2 from 1 atm to 100 atm requires 10^19 J of work be done on the gas. Spread over a year, this equates to 320 GW, about half the power needed for extracting carbon dioxide from the air.
So, in total, we're looking at roughly 1 TW of power — about 6% of human society’s total power demand.
[update June 7, 2023: I have been pointed to an estimate of energy required for DAC of 2000 kWh/ton of CO2 by one of the companies trying to commercialize this. That corresponds to 7200 kJ/kg, or 10 times more energy than the thermodynamic limit. In that case, the power required to pull all of our emissions out of the atmosphere each year is close to how much power humans consume.]
This estimate is the thermodynamic limit. In reality, it will be more than this. Exactly how much more will be determined by the engineering going into the process, but it certainly could be a several times this limit. Thus, carbon capture will require an enormous amount of energy.
And this energy better be climate-safe. If you’re generating the energy by burning coal, then you’re better off shutting down the fossil fuel power plants rather than using it for air capture.
This post is not intended to advocate for or against DAC. The benefits of developing the ability to do this is profound, but so are the technical, economic, and industrial hurdles. Whether it makes economic or political sense is still (IMHO) to be determined.
My view is that our priority must be decarbonizing our economy — this is and always will be the single most important step in our pursuit of a sustainable future.
Once we successfully achieve this, however, the appeal of DAC technologies is likely to increase. I think it’s very unlikely that we will restrict global warming to 1.5C above preindustrial without employing DAC. And development of DAC will make it a lot easier to keep global warming below 2°C.
This value is similar to other calculations.
This is a great article in so many ways. You cannot imagine how many social gatherings I go to with my wife, and no one wants to discuss the thermodynamics of DAC with me! ;)
But this article has a weakness. This statement needs to be reconsidered. "Because Gibbs free energy is a state function, it doesn’t matter exactly how you get from mixed to unmixed."
You have analyzed DAC as being one endpoint and one strategy and attribute the results to all endpoints and strategies. For this article, the endpoint is compressed, pure CO2 sequestered underground and under great pressure. The energy required to accomplish this is high because the choices are poor.
Now, I understand that this is the leading DAC solution being pursued by the US DOE. The energy required is actually higher as your analysis has not included losses attributed to transportation. That captured and compressed CO2 is going into a pipeline and transported to somewhere (Wyoming?) to be sequestered. There will be energy considerations associated with the friction of transport.
The point that needs to be considered is that different endpoints can have different energy requirements. And different strategies for an endpoint can influence the necessary energy as well. For an understanding of how strategy influences energy considerations, review Lackner's analysis of Sherwood's rule, Passive Direct Air Capture and his support for mechanical trees.
The major problem with traditional DAC is that it is a process that concentrates very dilute CO2 into a pure gas stream, compresses and transports that stream and then stores all that CO2 and all that energy in a geological formation. This process requires a great deal of energy because it is designed by chemical engineers who optimize processes for mass transport and deal with the energy consequences later.
DAC needs to be defined as any process that removes CO2 from the air and renders that CO2 inert to the atmosphere. This broader definition of DAC will provide a greater variety of solutions to be considered.
All around us, living organisms capture CO2 directly from the air while using much less energy. They don't ever create a pure concentration of CO2. And they never compress that captured CO2. Why? Too much damn energy.
I am not a nature-based CO2 kinda guy. I am firmly in the engineered CO2 capture/sequester camp. I see nature providing us with insights as to how to better design engineered CO2 capture and engineered CO2 sequestration solutions.
And the key to all this is to optimize for energy rather than mass transport. That is a whole 'nuther discussion.
Hi Andrew -- I share your skepticism about direct air capture. In 2019 I spent a summer writing a commissioned report reviewing the literature on the many social and political challenges in developing, deploying, and scaling direct air capture. The link below is to a column I wrote for Issues in Science and Technology summarizing my analysis and conclusions. Among the key takeaways related to your analysis is the immense land footprint / land use required to scale direct air capture considering that each plant needs its own power source either in the form of renewables, nuclear, or in the nearer term natural gas. And that you have to build a massive pipeline system to transport the captured CO2 to geographic locations where it is possible to be buried. https://issues.org/sciences-publics-politics-carbon-removal/