Figuring out the mass of carbon in the atmosphere with some back-of-the-envelope number crunching
Following on from a previous post, where I looked at how to estimate the mass of Earth’s atmosphere, there’s a question that inevitably suggests itself in this era of climate crisis and the most recent IPCC assessment report: how much carbon is there in the atmosphere?
Again this is an estimation exercise: what general knowledge might we know — and what specific knowledge would it help to know — and, given the back of an envelope, what can we do with it?
Take 1: Atmospheric mass and the proportion of CO₂
What you’ll need to know or to find out for this estimate, which draws on the least knowledge and is the most approximate:
- The mass of the atmosphere.
- The proportion of the atmosphere that is carbon dioxide.
- How many carbon atoms there are per molecule of carbon dioxide.
We’ve already prepared one ingredient in advance — the mass of the atmosphere — so in keeping with the back-of-the-envelope spirit, let’s use that result rather than one from a text book: 5×10¹⁸ kg, which is within 3% of more precisely determined results.
What we need now is a starting point for figuring out how much carbon there is. The main store of carbon in the atmosphere is carbon dioxide. Focusing only on CO₂ and ignoring all details of other gases — oxygen, nitrogen, etc. — what do we know? If we take this as a question of general knowledge rather than one of what we can find on the web, if you follow the news you may recall CO₂ content in the atmosphere is measured in parts per million (ppm). The easiest way to understand this measure is to compare it with percentages: a percentage figure is parts per hundred, so clearly ppm is a few orders of magnitude scarcer.
One newsworthy figure that may have stuck in your mind is 400 ppm, which was reached as a peak value in 2013 and a global mean in 2015. We’re now peaking well past 410 ppm, but let’s stick to 400 ppm as it is a sufficiently accurate and arithmetic friendly figure — the difference between the current mean and 400 ppm is only a couple of percent and is only slightly more than seasonal variation.
Other numbers that you may recall in connection with this are 350 ppm, which sometimes appears in the media and campaigning as what is considered the safest maximum level. Or you may recall 300 ppm, which is a figure quoted (as 0.03%) in some older books, although the world has not seen 300 ppm since before the Titanic left dry dock. Or perhaps you recall a value between 270 ppm and 280 ppm as the pre-industrial level. Although we’ll stick with 400 ppm, any of these values is in the right ballpark for an estimation-based calculation.
What does it mean to say carbon dioxide content in the atmosphere is 400 parts per million? For every million molecules of gas we expect 400 to be CO₂. This means that carbon dioxide makes up 0.04% of the molecules, of the volume (hence ppmv is often used, which is parts per million by volume) and of the pressure. This figure does not, however, describe a mass relationship, i.e., 0.04% by volume does not imply 0.04% by mass.
For our first estimate, however, let’s assume the difference in density between CO₂ and other atmospheric gases is small enough that it can be ignored, and therefore we can also use 0.04% as a plausible approximation of mass.
This means that, using our atmospheric mass estimate from earlier, our estimate of the mass of CO₂ in the atmosphere is 400×10⁻⁶ × 5×10¹⁸ kg = 2×10¹⁵ kg. To get the mass of carbon, we know from its name and its formula that one molecule of carbon dioxide contains one carbon atom and two oxygen atoms.
As we are doing this first estimate without taking account any chemical knowledge beyond the formula of CO₂, let us assume that the mass of a molecule is divided equally across its atoms, meaning that for a given mass of carbon dioxide, approximately one third of that mass is carbon. This gives us ⅔×10¹⁵ kg as the estimated mass of carbon in the atmosphere, which we can round and write more conveniently as 7×10¹⁴ kg (seven hundred billion tonnes).
Take 2: Masses and proportions of carbon, nitrogen and oxygen
Let’s assume a little more knowledge and see how that affects our estimate:
- The mass of the atmosphere.
- The proportion of the atmosphere that is carbon dioxide, oxygen and nitrogen.
- The atomic and molecular masses of carbon, carbon dioxide, oxygen and nitrogen.
We’ll use the same atmospheric mass figure as before (5×10¹⁸ kg) and the same proportions CO₂ concentration (400 ppm), but this time we’ll also take on board some knowledge of (1) other atmospheric gases and (2) atomic and molecular masses.
The Earth’s atmosphere is a mix of many gases, but the two dominant ones are nitrogen (~80%) and oxygen (~20%), whose proportions are fraction- and decimal-friendly fifths. Both nitrogen and oxygen occur in the atmosphere in their molecular forms: dinitrogen, N₂, and dioxygen, O₂. In the first estimate we ignored variation in mass across different molecules, but to give proper weighting to each gas we need to know a little more.
All the elements we need to know are in the second period (i.e., second row) of the periodic table. Depending on how far you took science, this may be wholly etched or partly sketched in your memory from school, university, work, music, etc., or you may happen to recall the masses or atomic numbers without context for other reasons. Either way, carbon, nitrogen and oxygen are adjacent elements, with atomic numbers 6, 7 and 8 respectively. The atomic numbers indicate the number of protons in the nucleus. The number of protons determines the element, i.e., an atom with 6 protons in its nucleus is by definition a carbon atom; there is no such thing as a carbon atom with 5 or 7 protons in its nucleus.
With the exception of hydrogen, all nuclei also contain neutrons. Neutrons are almost identical in mass to protons (they are about 0.1% heavier). The number of neutrons does not dictate which element an atom is considered to be, but it does determine the isotope. For example, carbon most commonly occurs with 6 neutrons, giving it an atomic mass of 12. This isotope is known as carbon-12 or ¹²C. One of its isotopes has 8 neutrons, so it is known as carbon-14 or ¹⁴C. This isotope is radioactive and its decay is the basis for carbon dating. We don’t need to worry about the less common isotopes of carbon, oxygen or nitrogen in our calculations.
At the lighter end of the periodic table, the number of neutrons is often the same as the number of protons in the most commonly occurring isotopes. In other words, the atomic mass is double that of the atomic number, giving us ¹²C, ¹⁴N and ¹⁶O as the atomic masses. The molecular mass, therefore, of N₂ is 28, of O₂ is 32 and of CO₂ is 44. Looking back to our first estimate, we can see that ignoring mass differences was not an unreasonable approximation.
Because nitrogen and oxygen dominate atmospheric content, we can calculate the mean molecular mass of the atmosphere by ignoring other gases. Based on 80% and 20% proportions, which are arithmetically convenient, this gives us 0.8×28+0.2×32 = 28.8. By mass, the proportion of O₂ in the atmosphere is 0.2×32÷28.8 = 0.222… (two ninths) and of N₂ is 0.8×28÷28.8 = 0.777… (seven ninths).
But we’re here for the CO₂, whose proportional contribution to the atmosphere’s mass is 0.0004×44÷28.8 ≈ 0.0006 (0.06%). This gives our estimate for mass of carbon dioxide in the atmosphere as 6×10⁻⁴×5×10¹⁸ kg = 3×10¹⁵ kg, which is 50% higher than our first estimate.
To determine the mass of carbon, we use the proportion of carbon dioxide’s mass that is carbon, which is 12÷44 = 27÷99 ≈ 0.27, which gives us a total mass of around 8×10¹⁴ kg (eight hundred billion tonnes), which is a hundred billion tonnes higher than our previous estimate.
Take 3: Look it up
We could try to nudge a little more precision out of our estimate by using slightly more accurate figures for atmospheric content (e.g., nitrogen 78%, oxygen 21% and argon 1%), but the percentage differences introduced by these figures is lower than the precision we’re already operating at (e.g., the actual mass of the atmosphere and proportion of carbon dioxide in the atmosphere are slightly higher than the figures we’re using). More precision won’t buy better accuracy unless we revisit all the values we’re using.
It is better to be vaguely right than exactly wrong.
Similarly, accounting for methane (CH₄), which also contains carbon, makes no meaningful difference to the outcome because it is orders of magnitude less present in the atmosphere than carbon dioxide (methane concentration is measured in parts per billion).
So, let’s call time on the estimation and look up the values that will get us an answer we can compare our estimates to. Using a figure of 410 ppm as the global mean for 2019 and a mass of 2.13×10¹² kg as the mass of carbon per 1 ppm of carbon dioxide, that gives us a figure of 8.73×10¹⁴ kg, which is within 10% of our second estimate and 25% of our first estimate.
Although the estimation of atmospheric carbon demanded less knowledge of maths and physics than the previous estimation atmospheric mass, the second estimate did require more specific knowledge of chemistry and the atmosphere, which may be less likely to stick in the mind as general knowledge — there may be more blanks to fill on the back of the envelope.
That said, our first estimate, which assumed less knowledge and approximated off the back of more assumptions, was not unreasonably far from the actual figure, illustrating that estimation is a sliding scale that lets us use almost anything we have to hand to get something usable and useful.
We live in a world where information and misinformation flow as freely as one another and with fire-hydrant abundance, a world that is defined by numbers — economics, climate, health, etc. Although we cannot attain or claim expertise in everything that matters, we may surprise ourselves with how much we can grasp if we take stock of what we know. More importantly, we are not in fact constrained to the back of an envelope and our personal recollection of facts. The constraint, however, is useful both for encouraging us to pare down the flood of information and for highlighting what we might need to find out.
Returning to our second estimate of 8×10¹⁴ kg, there is one last calculation that is within easy arithmetic reach: how much carbon is that per person? The global population is approaching eight billion (8×10⁹), which means 10⁵ kg or 100 tonnes per person — a figure that only looks set to rise in coming decades.