An isotope of a radioactive element has half-life equal to 9 thousand years. Imagine a sample that is so old that most of its radioactive atoms have decayed, leaving just 10 percent of the initial quantity of the isotope remaining. How old is the sample? (Give your answer correct to at least one decimal place.)

For this problem, we use the formula for radioactive decay which is expressed as follows:

An = Aoe^-kt

where An is the amount left after time t, Ao is the initial amount and k is a constant.

We calculate as follows:

An = Aoe^-kt
0.5 = e^-k(9000)
k = 7.7 x 10^-5

An = Aoe^-kt
.10 = e^-7.7 x 10^-5(t)
t = 29903.7 years

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raphealnwobi raphealnwobi · Answer 2 · 2022-03-01T12:15:55+01:00

It would take 29897.4 years for the radioactive atoms to have decayed to 10 percent of the initial quantity

Half life

The half life is the amount of time that it takes for a substance to decay to about half of its initial value.

It is given by:

N(t) = N(1/2)^(t / T)

Where N(t) is the amount of substance after t years, N is the initial value and T is the half life

T= 9000, N(t) = 10% = 0.1N, hence:

0.1N = N(1/2)^(t/9000)

0.1 = (1/2)^(t/9000)

t/9000 = ln(0.1) / ln(1/2)

t = 29897.4 years

It would take 29897.4 years for the radioactive atoms to have decayed to 10 percent of the initial quantity

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