Half-life Notes

Although it is impossible to predict when an individual atom of parent isotope will decay into the daughter isotope, the rate at which a sample of a given isotope decays, the half-life of the isotope, is quite predictable.

The half-life of an isotope is the amount of time required for half of the atoms of the

parent isotope in the sample to decay into atoms of the daughter isotope.

The half-life of an isotope is measured in units of time (years, days, seconds, etc.).

Example: is an isotope with a half-life of 4 days. This means that every 4 days the amount

of remaining parent isotope is cut in half.

If you started with 120 g of parent isotope, after 4 days only ½ the parent isotope is

still present (60 g)

After 8 days (two half-lives) only ¼ of the parent isotope remains (30 g)

After 12 days (three half-lives) only 1/8 of the parent isotope remains (15 g)

Note that every half-life (4 days in this case) the remaining parent isotope is reduced by ½

The graph below shows the decay of in the example above

Look at the graph below. What is the half-life of the parent isotope?

Answer: 2 years (the amount of time needed to reduce the initial amount by half, from 60g to 30g and from 30g to 15g)

% parent isotope number of

remaining half-lives

100% 0

50% 1

25% 2

12.5% 3

6.25% 4

3.13% 5

1.56% 6

0.78% 7

0.39% 8

0.20% 9

0.10% 10

Unstable isotopes are most dangerous during the first few half-lives when they emit most of their decay particles. If you look at the graph above you can see that 75% of the radiation will be emitted in the first two half-lives. Of course we must keep in mind that the daughter isotope, although more stable than the parent, may also be radioactive.

As an isotope sample ages, less of the unstable parent isotope remains, so it gives off less radiation.

Isotopes with short half-lives are more radioactive (give off more particles per day) than those with longer half-lives.

We can use radioactive isotopes to determine the age of a specimen if we know the following: how much parent isotope there was initially, how much parent isotope remains, and the half-life of the isotope.

To determine the age of the sample, first determine the percentage of remaining parent isotope by dividing the amount parent isotope remaining by the initial amount of parent isotope.

Next, determine how many half-lives have gone by. The first half-life reduces the original percentage to 50%, the second to 25%, the third to 12.5%, the fourth to 6.25%, etc.

Finally multiply the number of half-lives by the half-life of the isotope.

Example: as a sample of Radium, which has a half-life of 3.8 years, originally had a mass of 70 g

but now 61.25 g of it have been converted into Radon. How old is the sample?

Percentage remaining isotope = 8.75 ¸ 70 = 0.125 = 12.5% which is equivalent to 3 half-lives

Since the half-life is 3.8 years, we get 3 ´ 3.8 = 11.4 years