For 1000k atoms I will have a 500k after 1 half life, and 250k after 2 half lives. But when I try to do the same with fractions something doesn’t add up. Because after “quarter life” i will have 750k atoms, but then I’ll have 562500 atoms, because of 750k multiplied by 0.75. So quarter life should be equal to 0.5 of half life, but half life doesn’t work that way. I am confused.
In: Physics
Radioactive decay is just
N(t) = N0 e^-λt
N(t) is the number of atoms remaining
N0 is the number of atoms you started with
One half life will be when N(t)/N0=1/2
N(t)/N0 = e^-λt
1/2 = e^-λt
ln(1/2) = -λt
ln(2)/λ = t
This gives use one half life, or τ
The half life is usually measured directly, and we use ln(2)/τ to calculate lambda
We can apply similar math to get the quarter life, which would be ln(4/3)/λ
Let’s plug that in for
N(t)/N0 = e^-λln(4/3)/λ
When we simplify, we get N(t)/N0 = 3/4
If you Google the equation to get λ, you’ll often see .693/τ, that’s just a decimal approximation of ln(2)/τ
Two quarter lives isn’t a half life, rather you’re left with 9/16 of the original (note, that’s (3/4)^(2))
Just like how two half lives isn’t a whole life, you’re left with 1/4 of the original (that is (1/2)^(2))
To get rid of all of the material, you would need to wait an infinite amount of time (although in reality there is a last atom to decay, but once we get below a few micrograms of atoms it becomes impossible to measure or predict.
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