More details on the simulation

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 R_{in}
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 (R_{out})
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.

R_{out} = 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 R_{in}
and the removal of more pollutant at rate R_{out}, it follows that

d[X]/dt = R_{in} - R_{out} = R_{in} - 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 R_{in} = R_{out} and R_{in} = k[X]_{ss}

where [X]_{ss} is the steady state concentration of X

It follows that

[X]_{ss} = R_{in}/k = R_{out}/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.

R_{in} = k[X]_{ss}

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