How do you Find exponential decay rate?

Exponential decays typically start with a differential equation of the form:

`(dN)/dt prop -N(t)`

That is, the rate at which a population of something decays is directly proportional to the negative of the current population at time `t`. So we can introduce a proportionality constant:

`(dN)/dt=-alphaN(t)`

We will now solve the equation to find a function of `N(t)`:

`->(dN)/N=-alphadt`

`->int(dN)/N=int-alphadt-> ln(N)=-alphat+C`

`->N(t)=Ae^(-alphat)` where `A` is a constant.

This is the general form of the exponential decay formula and will typically have graphs that look like this:

graph{e^-x [-1.465, 3.9, -0.902, 1.782]}

Perhaps an example might help?

Consider a lump of plutonium 239 which initially has `10^24` atoms. After one million years have elapsed years the plutonium now has `2.865times10^11` atoms left. Work out, `A` and `alpha.` When will the plutonium have only `5times10^8` atoms left and what is the decay rate here?

We are told the lump has `10^24` atoms at `t=0` so:

`N(0)=Ae^(0)=10^24-> A=10^24`

Now at 1 million years: `10^6` years:

`N(10^6) = 10^24e^(-alpha(10^6))=2.865times10^11`

Rearrange to get:

`alpha=-1/(10^6)ln((2.865times10^11)/10^24)~~2.888times10^(-5)yr^-1`

So `N(t)=10^(24)e^(-2.888times10^(-5)t)`

For the next part:

`N(t) =5times10^8=10^(24)e^(-2.888times10^(-5)t)`

Rearrange to get `t`:

`t=-1/(2.888times10^(-5))ln((5times10^8)/(10^24))~~1.22times10^6yr`

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:

`(dN)/dt=-alphat=-2.888times10^(-5)(1.22times10^6)`

`=-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.