## Thursday, December 17, 2015

### Carbon Cycle: Rate of Transfer

Our study of the radioactive isotope carbon-14, which we began in Carbon-14: Origins and Reservoir, led us to the conclusion that roughly one in every eighteen atmospheric carbon atoms are absorbed by a vast oceanic reservoir each year. When the atmosphere and the reservoir are in equilibrium, the same amount of carbon flows out of the reservoir as into it, so that the mass of carbon in the atmosphere remains constant. But the oceanic reservoir contains over one hundred times as much carbon as the atmosphere.

We now embark upon a series of posts in which we calculate the effect of mankind's carbon emissions upon the atmospheric CO2 concentration. Our starting point for these calculations will be the natural, equilibrium atmosphere and oceans of the late nineteenth century. This atmosphere contains 300 ppmv CO2, which implies a total atmospheric carbon mass of 650 Pg (see here). The oceanic reservoir contains 77,000 Pg, and each year 37 Pg of carbon is exchanged between the atmosphere and the reservoir (see here).

As we showed in our previous post, the probability of an atmospheric carbon atom being transferred into the oceanic reservoir is independent of the number of carbon atoms in the atmosphere. The number of carbon atoms moving into the reservoir is equal to the total number of carbon atoms in the atmosphere divided by eighteen. If we have twice as many carbon atoms, the rate at which they move into the oceanic reservoir will double. If mA is the mass of carbon moving into the reservoir every year, and MA is the mass of carbon in the atmosphere, we must have:

mA = kAMA, where kA = 37 Pg/yr ÷ 650 Pg = 0.057 Pg/yr/Pg.

The probability of a carbon atom in the oceanic reservoir being released into the atmosphere is likewise independent of the number of carbon atoms in the reservoir. If mR is the mass of carbon leaving the reservoir every year and MR is the mass of carbon in the reservoir, we must have:

mR = kRMR, where kR = 37 Pg/yr ÷ 77,000 Pg = 0.00048 Pg/yr/Pg.

If we were to double suddenly the mass of carbon in the atmosphere, carbon would start to move into the oceanic reservoir at double the rate. The graph of atmospheric carbon dioxide concentration would look almost exactly like the graph of carbon-14 concentration after the nuclear bomb tests, which we present here. The movement of carbon between the atmosphere and the oceanic reservoir is governed by almost exactly the same equations as the movement of carbon-14. The only difference is that carbon-14 decays, while carbon-12 lasts forever.

UPDATE: The above calculations are consistent with Henry's Law of gasses dissolving in liquids. Henry's Law applies when the concentration of gas in the liquid has reached equilibrium at a particular temperature with the concentration of gas above the liquid. At equilibrium, the rate at which the liquid emits the gas is equal to the rate at which the liquid absorbs the gas. Henry's Law states that the equilibrium concentration of the gas in the liquid at a particular temperature is proportional to the partial pressure of the gas above the liquid. According to Boyle's Law, the partial pressure of a gas at a particular temperature is proportional to the number of gas molecules per unit volume. In our rate of transfer equations, the rate at which CO2 is absorbed by the ocean is proportional to the concentration of CO2 in the atmosphere, and the rate at which CO2 is emitted is proportional to the concentration in the liquid. If we double the concentration of CO2 in the atmosphere, our equations tell us that the rate of emission by the oceans will equal the rate of absorption only if the concentration of CO2 in the oceans also doubles, which is precisely what Henry's Law requires. Of course, we have not yet considered how changing the temperature of the atmosphere and ocean will affect the rates of transfer, but we will get to that later.