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.