If the entire atmosphere is taken as the reservoir, then under steady-state conditions
As an illustration of the concept of mean lifetime, consider all sulfur-containing compounds in the troposphere. If the average mixing ratio of these compounds is 1 part per billion by mass (ppbm) and a steady state is assumed to exist, then with the mass of the troposphere about 4 x 1021 g, the total mass of sulfur-containing compounds in the troposphere is Q = 4 x 1012 g. If natural and anthropogenic sources of sulfur contribute to give a total P of about 200 x 1012 g yr 1, the mean lifetime of sulfur compounds in the troposphere is estimated to be
Calculations of lifetimes can be useful in estimating how far from its source a species is likely to remain airborne before it is removed from the atmosphere.
If we consider a particular region of the atmosphere, say, the volume of air over a city or the volume of air in the Northern Hemisphere or the entire stratosphere, we can define a characteristic mixing time for that volume as the time needed to thoroughly mix a chemical in that volume of air. Call the characteristic mixing time xM. A reservoir is poorly mixed for a particular species if the characteristic mixing time, xM, is not small compared with the species residence time, x. Note that this means that a particular reservoir can be well mixed for some species and poorly mixed for others, depending on the residence time of each species. Furthermore, as we have seen, the mixing times in the atmosphere are different for different directions. For example, as we have noted, the characteristic vertical mixing time in the troposphere, the time required to mix a species uniformly from the ground up to the tropopause, is about one week; whereas the troposphere's horizontal mixing time, the time required to mix a constituent thoroughly around the globe in the troposphere, is about one year. Thus the troposphere can be considered well mixed for 85Kr, which has a residence time of 10 years; but for sulfur compounds, which are estimated to have a residence time of about one week, the troposphere is not even well mixed vertically.
The stratosphere can be considered well mixed vertically only for atmospheric species with lifetimes substantially exceeding 50 years. In fact, one of the only examples of such a long-lived species is He, which has its source at the Earth's surface and its sink as escape through the very top of the atmosphere into space. Thus the stratosphere is poorly mixed vertically for essentially all atmospheric trace constituents.
Frequently the rate at which a chemical is removed from the atmosphere is proportional to its concentration (first-order loss)—the more that is present, the faster its rate of removal. This is generally the case for both dry deposition at the Earth's surface and scavenging by cloud droplets. Consider a species for which steady-state conditions hold and which is removed at a rate proportional to its concentration with a proportionality constant X. Such a species is 85Kr, the only significant removal process for which is radioactive decay. For 85Kr, then
Thus to estimate the lifetime of 85Kr does not even require knowledge of its atmospheric abundance Q, but only of the radioactive decay constant. Consequently, it does not matter whether 85Kr is uniformly mixed throughout the entire atmosphere or not to estimate its lifetime. In a case where the removal process is first-order, then even for a poorly mixed species a simple and accurate estimate for its lifetime can be obtained provided that its removal rate constant can be accurately estimated.
Now consider a species with mass Q in the atmosphere that is removed by two independent processes, the first at a rate k\ Q and the second at a rate k2Q, where k\ and k2 are the first-order removal coefficients. Its overall lifetime is given by
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