Usually, exponential decays begin with a differential equation of the following form:
In other words, the rate of population decay is directly inversely proportional to the current population at a given time. . So we can introduce a proportionality constant:
We will now solve the equation to find a function of
where is a constant
This is the general form of the exponential decay formula and will typically have graphs that look like this:
Perhaps an example might help?
Consider a lump of plutonium 239 which initially has atoms. After one million years have elapsed years the plutonium now has atoms left. Work out, and . When will the plutonium have only atoms left and what is the decay rate here?
We are told the lump has atoms at t = 0 so:
Now at 1 million years:
Rearrange to get:
For the next part:
Rearrange to get t:
Now for the last part, the decay rate is already defined a way back at the very start, simply evaluate it at the given time:
= -35.23 atoms per year.
The idea is to start with differential equation above, which gives the decay rate, and solve it to get the population at any given time.
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