The discussion presented below is not a part of the Environmental Chemistry syllabus, but it is interesting and should be well within the grasp of an A level maths student. Click here to skip ahead.
The rate of input of pollutant to the reservoir is denoted by Rin and is measured in units of concentration per unit time, for example, mol dm-3 s-1 (moles per cubic decimetre per second) or more likely for an atmospheric pollutant ppm day-1 (parts per million per day).
We are assuming for this model that the rate of removal (Rout) depends on the concentration of the pollutant in the reservoir. The more X there is in the reservoir, the faster X will be removed. This is a reasonable assumption for some atmospheric pollutants but not for all of them.
Rout = k[X]
The rate of change of the concentration of X is denoted by d[X]/dt. Since [X] is being changed by the input of more pollutant at rate Rin and the removal of more pollutant at rate Rout, it follows that
d[X]/dt = Rin - Rout = Rin - k[X]
After some time the system will usually reach a steady state where the rate of removal is the same as the rate of input and the rate of change in the concentration of X is zero.
If d[X]/dt = 0
then Rin = Rout and Rin = k[X]ss
where [X]ss is the steady state concentration of X
It follows that
[X]ss = Rin/k = Rout/k
That is, the steady state concentration of X can be calculated if we know the rate of input (or the rate of removal) and the rate constant for the removal process (k).
A variation on this equation can be used to estimate the global rate of input of a given pollutant using experimental measurements of its atmospheric concentration and laboratory measurements of the rate constant for its removal.
Rin = k[X]ss