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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