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How do we know the CO2 that is building up in the atmosphere comes mainly from burning fossil fuels? The respiration of plants and animals adds CO2 to the atmosphere, CO2 comes from volcanoes, and CO2 comes out of the ocean when the water temperature rises. Why can't we attribute most of the additional CO2 to these sources?
In preindustrial times, atmospheric CO2 concentrations were fairly stable for thousands of years. On both land and in the oceans, plants take up CO2 for photosynthesis, and animals and plants respire CO2 into the atmosphere in what was a largely balanced relationship - i.e., respiration, which releases CO2 to the atmosphere, requires the metabolism of food materials that originated in plants, which produced it by removing CO2 from the atmosphere. The amount of CO2 dissolved in the ocean was also balanced with the atmospheric concentration, with rates of ocean uptake similar to rates of release.
Since those preindustrial times, however, atmospheric CO2 has risen by about 41% (from 280 parts per million to about 395 ppm) in association with two important human activities. The main one is burning of fossil fuels (coal, oil, natural gas), and the lesser one is deforestation, which reduces the number of trees that can remove CO2. We know from records kept by various nations approximately how much CO2 has been emitted from fossil fuel burning, and it appears that about half of that amount has remained in the atmosphere, with the rest being taken up on land by plants and soil, and in the ocean by plant life and as dissolved CO2 in the form of carbonic acid, bicarbonate and carbonate. The land and ocean have become a "CO2 sink", in that more has been entering it than leaving it.
Additional evidence comes from carbon isotopes. Most carbon has an atomic weight of 12, but about 1% is carbon-13. Plants take up C13 less efficiently than C12, and so the C13/12 ratio is reduced in plant products, including fossils, which were derived from plants or from animals that ate plants. As the atmospheric CO2 has risen, the ratio of C13 to C12 has dropped, indicating that the CO2 came from plants, the source of fossil fuels, and not from non-biologic sources such as volcanoes (which in any case emit only a small fraction annually of the amount of CO2 emitted from fossil fuel burning). We also know that most of the CO2 came from fossil sources rather than plants that have lived recently, based on another carbon isotope found in CO2, C14, which is a tiny fraction of total carbon. C14 is produced in the atmosphere from cosmic rays, and so plants alive today will include some C14. However, C14 is radioactive, and decays with a half life of about 5000 years. Once buried underground, it disappears with time because there is no uptake from the atmosphere. The 41% increase in atmospheric CO2 has been associated with a significant depletion in the C14/C12 ratio, indicating that most of its carbon was underground for a very long time (at least 10 half lives, but from what we know of fossils, probably hundreds of millions of years). An exception is the excess CO2 resulting from deforestation, which accounts for about 20% of the total excess due to human activity. Finally, the rise in atmospheric CO2 has been accompanied by a reduction in atmospheric oxygen, which is what we expect from a combustion process that produces CO2 from carbon and oxygen.
In summary, the relatively stable preindustrial CO2 concentrations, national records of fossil fuel burning, and the evidence from carbon isotopes and atmospheric oxygen tell us that human activities (fossil fuel burning and to a lesser extent deforestation) account for all or almost all the increase we have observed in atmospheric CO2.