How do you Find exponential decay rate?

framtalshg 2022-01-29 Answered
How do you Find exponential decay rate?
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Expert Answer

Rylee Marshall
Answered 2022-01-30 Author has 6 answers
Explanation:
Exponential decays typically start with a differential equation of the form:
dNdtN(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=αd(t)
{{dNN={{αdtln(N)=αt+C
N(t)=Aeαt 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 {ex[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=1024A=1024
Now at 1 million years: 106years
N(106)=1024eα(106)=2.865×1011
Rearrange to get:
α=1106ln(2.865×10111024)2.888×105yr{1}
So N(t)=1024e2.888×105t
For the next part:
N(t)=5×108=1024e2.888×105t
Rearrange to get t:
t=12.888×105ln(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=2.888×105(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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