The half-life of an element is 5.8 x 10^11. How long does it take a sample of the element to decay to 2/5 its original mass?

1 Answer
Chuck W.
Dec 1, 2016

The expression for the first-order decay of a population is
A/A_0=e^(-kt)
where the rate constant k is related to the half-life by
k=ln2/t_(1/2)

Explanation:

In the question, the half-life should have units of time. Let's assume that the half-life is 5.8 times 10^11 s

In this case, the value of the rate constant is
k=ln2/t_(1/2)=ln2/(5.8times10^11 s)=1.20times 10^-12 s^-1

Using the first equation, we can find the time, t at which the fraction of remaining atoms is 2/5.

2/5 = e^(-(1.20times10^-12 s^-1)(t))

Solve for t by first taking the natural logarithm of both sides:

-0.92=-(1.2 times 10^-12 s^-1) t

t=0.92/(1.2times10^-12 s^-1)=7.67times10^11 s

(or about 24,300 years)

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