Climate change. It’s bad and it’s getting worse. The main cause is burning fossil fuels, which spews CO₂ into the atmosphere. Carbon dioxide, as we all know too well by now, is a greenhouse gas, meaning it absorbs heat radiation from the Earth, preventing it from escaping out into space.

A certain amount of this is good—without CO₂ the Earth would be so cold, the oceans would freeze. But in preindustrial times the concentration of CO₂ in the atmosphere was about 280 parts per million (ppm). Now it’s 420 ppm, or 50 percent higher. (You might be surprised to know that CO₂ is still only .04 percent of the air we breathe, but that’s enough to ruin everything.)

What if we could remove carbon dioxide from the atmosphere? This is an idea some startups are already experimenting with—it’s called direct air capture. The only problem is that removing the tiny fraction of C0₂ from the air, which is 99 percent nitrogen and oxygen, takes a lot of energy, and our hunger for energy is what got us into this mess in the first place.

How much energy would it take? I’m glad you asked. We can estimate that using some fundamental ideas in thermodynamics.

Free Expansion of Gas

Let’s start with a simple example. Imagine we have a box with a volume of 1 cubic meter, and it has a divider that splits it into two equal halves. On one side it contains nitrogen at atmospheric pressure and temperature, and the other side is completely empty. Here’s a diagram:

We can model this gas as a bunch of tiny balls (molecules of nitrogen) bouncing around. When a nitrogen ball collides with a wall of the container, it gives it a tiny little push. All these pushes are what cause the gas to have a pressure. In this case, it’s a pressure of 1 atmosphere, or about 100,000 newtons per square meter. (One N/m² is also called a pascal).

Suppose we let the partition wall in the middle move? Since there is a pressure from the gas, there is a force on the wall as it slides to the end. According to the work-energy principle, a force applied over a distance means that the motion of the wall can do work, and that work can change the energy of something. By letting this gas expand, we can get energy from this system.

In fact, any time you increase the entropy of a system you can get some extra energy. Wait, what? What the heck is entropy? We should talk about that. In short, entropy is a measure of the amount of disorder in a system. In thermodynamics, systems tend to go from a state of low entropy to higher entropy—meaning they get more disordered over time, like a teenager’s room.

So in our box, when the gas expands to fill the whole container (we call this free expansion), the system is “more disordered,” since it is no longer separated into two compartments, and entropy is increased.

But why do we care? Well, it’s possible to calculate the change in entropy of an expanding gas and use that to find the change in energy. Since this isn’t a thermodynamics 401 course, I’ll just show you the equation and give a brief explanation:

In this expression, ΔS is the change in entropy (in joules per kelvin). V₁ and V₂ are the starting and ending volumes, N is the number of particles. Finally, k is the Boltzmann constant with a value of 1.38 x 10^–23 joules per kelvin. We can calculate the change in energy (ΔE) of the gas as:

Here T is the temperature of the gas in kelvins. OK, let’s just do a quick calculation. Suppose I have 0.5 m³ of air that I let expand into 1 m³. If the temperature stays constant, then this change in entropy will result in an energy decrease of 37,000 joules. That means that this gas could lend that energy to some productive use—like toasting a Pop-Tart.

Separating Mixed Gases

Here’s the cool thing: We can use the same idea about changes in entropy when separating gases to estimate the energy requirement of removing CO₂ from the air. Instead of letting a gas expand into a larger volume, we want to take a mixture of two gases and move them to opposite sides of the container.

Just to show you what this would look like, suppose we have a red-ball gas mixed with a blue-ball gas. We could then use two magical moving walls. One wall lets only blue balls through, the other lets only red balls through. After moving these walls, we would have a gas separation.

Basically, we’re compressing both of these gases from their original space (the whole container) into separate, orderly compartments, like socks and shorts in different drawers. Let’s say we start with a volume V; now there are N₁ red balls in a new volume V₁, and N₂ blue balls in the volume V₂.

Both V₁ and V₂ are smaller than the original volume V (duh), so V₁/V and V₂/V are less than 1—and when we take the log of a fraction, we get a negative number. This means the change in entropy, ΔS, will be negative. With a decline in entropy, we get a positive change in energy, so we must add energy to the system to make this happen. (Yes, just like it takes energy to clean up your room.)

Carbon Dioxide Capture

OK, now we have the physics of separating gases, so let’s use it. How much energy would it take to pluck carbon dioxide molecules out of the air? This is a tough calculation, but let’s start small with that 1 cubic meter of air. (For simplicity, let’s again say it’s mainly nitrogen.) At a room temperature of 293 K (68 Fahrenheit) and air pressure of 1 atmosphere, this would be 2.47 x 10²⁵ molecules. At 400 ppm, 9.89 x 10²¹ of these will be carbon dioxide. We want to take the CO₂ molecules and move them into their own compartment.

If we assume both the nitrogen and CO₂ molecules behave as ideal gases, then we actually deal with the gas pressure instead of the volume. We would have to use the pressure to find the volume anyway—but in this case we can consider the nitrogen to be at close to 1 atmosphere of pressure and the partial pressure of the CO₂ would be 400 millionths (0.0004) of the atmospheric pressure. The energy required to separate the CO₂ would then be:

Crunching the numbers, it would take 313 joules of energy to capture the carbon dioxide in this box. If we look at it in terms of mass, the total gas mass would be 1.2 kilograms, of which 0.48 grams are CO₂. That means it’s going take 652,000 joules of energy for every kilogram of carbon dioxide we get from the air. Or I should say, it’s going to take at least that much energy. No process is ever perfectly efficient, so that’s a theoretical minimum.

But how much carbon dioxide is in the atmosphere? That’s not an easy question. Yes, the proportion is around 400 ppm, but we need to know how many particles are in the atmosphere. Well, that’s fine. Let’s just make a rough approximation (it’s what physicists do).

Suppose we just take the air up to 5 kilometers above the surface of the Earth. I can assume this has a constant density of 1.2 kilograms per cubic meter and 400 out of 1 million of these molecules are CO₂, giving us a total CO₂ mass of 1.2 x 10¹⁵ kilograms, or 1.2 trillion metric tons. Yes, that’s a LOT, and it’s probably a lower-end estimation.

If we wanted to remove enough CO² to get back to the pre-industrial level of 280 ppm, it would take  2.39 x 10²⁰ joules of energy. For a reality check, that’s almost as much as the world’s total annual energy consumption (5.8 x 10²¹ joules every year).

But never mind that. The more urgent problem is that we’re still putting more carbon dioxide in the atmosphere all the time. In 2023, total CO₂ emissions were about 37 billion metric tons. If we wanted to remove just this amount each year, to keep the carbon dioxide level from rising, it would require 764 gigawatts of power.

Just to be clear, a nuclear power plant produces around 1 GW. If you include the inefficiencies of the whole direct air capture process, it would probably take more than 1,000 nuclear power plants.

Now, this really is a rough estimate—things get much more complicated when we account for the interaction of atmospheric CO₂ with rocks and oceans and things. But I think the takeaway is pretty clear: Any idea that we can maintain our current lifestyle and just suck the CO₂ out of the air afterwards is a fantasy. We’d be a lot better off devoting that effort and expense to eliminating emissions.