How do you Find exponential decay rate?

1 Answer
Andrew I.
Jan 12, 2018

See below.

Explanation:

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

dNdt∝−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:

dNdt=−αN(t)

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

→dNN=−αdt

→∫dNN=∫−αdt→ln(N)=−αt+C

→N(t)=Ae−αt 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 1024 atoms. After one million years have elapsed years the plutonium now has 2.865×1011 atoms left. Work out, A and α. When will the plutonium have only 5×108 atoms left and what is the decay rate here?

We are told the lump has 1024 atoms at t=0 so:

N(0)=Ae0=1024→A=1024

Now at 1 million years: 106 years:

N(106)=1024e−α(106)=2.865×1011

Rearrange to get:

α=−1106ln(2.865×10111024)≈2.888×10−5yr−1

So N(t)=1024e−2.888×10−5t

For the next part:

N(t)=5×108=1024e−2.888×10−5t

Rearrange to get t:

t=−12.888×10−5ln(5×1081024)≈1.22×106yr

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:

dNdt=−αt=−2.888×10−5(1.22×106)

=−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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