Carbon-14: Removal from the Atmosphere

We are going to estimate the mass of carbon-14 in the Earth's atmosphere prior to our burning significant quantities of fossil fuels, and prior to our detonating atomic bombs. We will use the Earth's atmosphere at the end of the nineteenth century as an approximation to the atmosphere in its natural, equilibrium state, undisturbed by human activity.

The concentration of carbon dioxide in our equilibrium atmosphere is 300 ppmv (parts per million by volume, as estimated for the turn of the nineteenth century here). The total mass of the atmosphere is 5.3×10¹⁸ kg. (Atmospheric pressure is generated by the weight of the atmosphere per square meter, so divide sea-level atmospheric pressure by gravitational acceleration and multiply by the surface area of the Earth to obtain atmospheric mass.) In accordance with the gas law, the density of CO2 is 1.5 times higher than the density of air (the molar mass of CO2 is 44 g, and of air is 29 g). Thus 300 ppmv of CO2 in the atmosphere is the same as 450 ppm (parts per million by mass). The mass of CO2 in our equilibrium atmosphere is 2.4×10¹⁵ kg (450 ppm of 5.3×10¹⁸ kg). The molar mass of CO2 is 44 g, and that of carbon is 12 g, so the mass of carbon in the atmosphere is 6.5×10¹⁴ kg (2.4×10¹⁵ kg × 12 g ÷ 44 g). We will use petagrams (Pg) to represent large masses, where 1 Pg = 10¹² kg = 10¹⁵ g. Our equilibrium atmosphere contains 650 Pg of carbon.

The concentration of carbon-14 in our equilibrium atmosphere is 1.0 ppt (parts per trillion by mass). Almost all carbon in the atmosphere is contained in CO2, so the mass of carbon-14 in our equilibrium atmosphere is 650 kg (650 Pg of CO2 × 1ppt). As we already showed, the equilibrium reservoir of carbon-14 on Earth is 62 Mg (7.5 kg/yr production by cosmic rays ÷ 0.00012 kg/kg/yr decay rate = 62,500 kg = 62 Mg). Of this reservoir, only 1% is to be found in the atmosphere. From now on, when we refer to the Earth's carbon-14 reservoir we will be referring to the 62 Mg that is not in the atmosphere.

The 650 kg of carbon-14 in our equilibrium atmosphere decays at 0.078 kg/yr (650 kg × 0.00012 kg/kg/yr) and is added to by cosmic ray production of 7.5 kg/yr. In order for the carbon-14 content of the atmosphere to remain constant, carbon-14 must pass out of the atmosphere at 7.4 kg/yr. Let us suppose, for the sake of argument, that this 7.4 kg/yr does not pass into the carbon-14 reservoir. In that case, the 7.4 kg/yr goes somewhere else, and a new reservoir starts to build up, while the existing reservoir decays, which would mean that our carbon-14 reservoir would not be in equilibrium, which contradicts our observation that the reservoir had millions of years to reach equilibrium before the nineteenth century. Thus 7.4 kg/yr of carbon-14 must pass directly from the atmosphere into the reservoir. It could be that the reservoir contains many sub-divisions communicating with one another in complex ways, but this does not alter the fact that 7.4 kg of carbon-14 is passing out of the atmosphere and into the reservoir every year.

The figure below illustrates the origin and fate of Carbon-14 on Earth. We use M for mass, and m for mass flow. We use subscript A for atmosphere, R for reservoir, D for decay, T for transfer, and P for production. Superscript C14 means carbon-14.


Where is the Earth's carbon-14 reservoir? How does it acquire 7.4 kg of carbon-14 from the atmosphere every year?

Carbon-14: Origins and Reservoir

This is the first of a series of posts in which we use our knowledge of carbon-14 concentrations to arrive at firm conclusions about the way in which carbon dioxide (CO2) cycles between the atmosphere and the oceans. The implications of the atmosphere's carbon-14 concentration were studied thoroughly and objectively prior to 1960, in papers such as Arnold et al. But these authors did not have available to them the results of the nuclear bomb tests of the 1960s, so their conclusions could not be as firm as ours. The same implications have been studied more recently in work such as Mearns and Pettersson, but these authors did not attend to the rate of production of carbon-14 by cosmic rays, and so did not appreciate the necessary size of the global CO2 reservoir. Modern models of the CO2 cycle are presented in papers such as Archer et al., but these models are contradicted by carbon-14 observations, so they cannot be correct.

Carbon-14 has been produced in our atmosphere by cosmic rays for billions of years. A cosmic ray is an energetic particle arriving from space. Most are protons. Some have energy 1×10²⁰ eV. (The Large Hadronic Collider, for comparison, produces protons with energy 7×10¹³ eV.) Cosmic rays collide with atmospheric nuclei and produce showers of photons and particles. Among the particles produced are neutrons, and these neutrons can react with nitrogen-14 nuclei to produce carbon-14.

A nitrogen-14 nucleus has seven protons and seven neutrons. During its reaction with a neutron, it ejects a proton but retains the neutron. The result is a nucleus with six protons and eight neutrons, which is carbon-14. The carbon-14 nucleus is unstable. Eventually, one of its neutrons will emit an electron and turn into a proton. The nucleus is once again nitrogen-14. The electron shoots out of the nucleus with energy up to 156 keV. It is called a beta particle, and the decay of carbon-14 is called a beta decay. The decay happens at random, but the probability that any given carbon-14 nucleus will decay each year is 0.012%. If we have one kilogram of carbon-14, there will be only half a kilogram left after 5700 years.

The electrons emitted by carbon-14 decay have sufficient energy to penetrate 50 mm of air. With care, we can measure the concentration of carbon-14 in a sample of air, or in a sample of wood, cloth, or animal tissue, by counting the electrons it produces, and weighing its carbon content. We find that one in a trillion carbon atoms in the atmosphere is a carbon-14 atom. The rest is carbon-12, with one part in a thousand carbon-13.

Almost all the carbon-14 in our atmosphere ends up in CO2 molecules. One in every trillion atmospheric CO2 molecules contains carbon-14. The rate at which cosmic rays produce carbon-14 is of order two atoms per square centimeter of the Earth's surface per second (see Lingenfelter for measurement 2.5 atoms/cm²/s and Kovaltsov et al. for 1.7 atoms/cm²/s). The creation rate varies as the Earth moves through the galaxy, and with cycles of solar activity, but to the best of our knowledge, the creation rate has been constant to within ±25% over the past ten million years.

Because we know carbon-14's rate of decay and its rate of production, which has been stable for at least a million years, we can calculate the equilibrium quantity of carbon-14 on our planet. Cosmic rays produce 2 atoms/cm²/s of carbon-14, so they produce 7.5 kg of carbon-14 every year. (Multiply 2 by the Earth's surface area in square centimeters, the number of seconds in a year, the atomic weight of carbon-14, and divide by Avogadro's number to get the number of grams produced per year.) In the past million years, cosmic rays produced 7.5 million kilograms of carbon-14. But each carbon-14 nucleus has a 0.012% chance of decaying each year, so only a small fraction of this 7.5 million kilograms still exists. Suppose 75,000 kg remained. In the coming year 9.0 kg would decay (0.012% of 75,000 kg) and only 7.5 kg would be created. The Earth's reservoir of carbon-14 would be decreasing at 1.5 kg/yr. Suppose only 50,000 kg remained. In the coming year, only 6.0 kg would decay (0.012% of 50,000 kg) and 7.5 kg would be created. The Earth's reservoir would be increasing at 1.5 kg/yr. The equilibrium size for Earth's carbon-14 reservoir is 62,000 kg (7.5 kg ÷ 0.012%). At this size, the rate at which carbon-14 in the reservoir decays is equal to the rate at which new carbon-14 is added to the reservoir by cosmic rays.

Historically, carbon-14 atoms have been produced exclusively by cosmic rays. But in the 1960s, nuclear bomb tests doubled the concentration of carbon-14 in the atmosphere. Since then, the concentration has relaxed to its historical value. For ethical and practical reasons, it is hard to perform experiments upon the Earth's atmosphere and climate. But the doubling of the carbon-14 concentration by bomb tests amounts to a gigantic experiment upon the atmosphere, and this experiment turns out to be profoundly revealing when it comes to estimating the effect of human CO2 emissions upon the climate.