Exponential decay half life problem. ?
Every chemical element goes through natural exponential decay, which means that over time its atoms fall apart. The speed of each element's decay is described by its half-life, which is the amount of time it takes for the number of radioactive atoms of this element to be reduced by half.
The half-life of the isotope dubnium-263 is 29 seconds. A sample of dubnium-263 was first measured to have 1024 atoms. After t seconds, there were only 32 atoms of this isotope remaining.
Write an equation in terms of t that models the situation.
Here is my answer:
32 = 1024(1/2)^(t/29)
Please verify my answer..Provide correction if I have something wrong..
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Verified answer
After t seconds, number of half-lives = t/29
Fraction of dubnium-263 remains:
32/1024 = (1/2)^(t/29)
Hence, 32 = 1024 × (1/2)^(t/29)
I don't like the wording of the problem
1.. "every chemical element goes through natural exponential decay"
.. "it's atoms fall apart"
that entire first sentence should be stricken
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remember this equation
.. At = Ao * (1/2)^(t / half life)
and for this problem
.. 32 = 1025 * (1/2)^(t / 29sec)
Yup, that looks OK. Since 32 = 1024 * (1/32),
apparent t = 5*29 = 145 seconds.
Radioactive decay ...
Simple solution ....
1024 --> 512 --> 256 --> 128 --> 64 --> 32 .......... number of atoms
..........1......... 2.......... 3 ........ 4 ....... 5 ................. number of half-lives
5 half-lives x (29 s / 1 half-life) = 145 s
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Radioactive decay follows first order kinetics. The integrated rate equation is:
N = No e^(-kt) ...... where k (or λ) is the decay constant, t is the elapsed time
k is related to the half-life by: k = ln2 / t½
Solve for t
t = ln(N/No) / -(ln2 / t½)
t = ln(32/1024) / -(ln2 / 29 s)
t = 145 s