Half - Life
We have already learned the types of decay that a radioactive isotope can undergo
(alpha, beta, gamma), but we have not talked about how long it takes for decay to
happen.
Different isotopes have different half-lives.
Half-Life: the time required for a substance to decay onehalf of its initial value.
Here is a table that lists half-lives of selected radioisotopes:
This rate of decay is constant for each isotope and different
from any other. The rate is not affected by factors like
temperature or pressure.
We can’t predict when an individual atom will decay, but we
can know when half of the sample will, just like predicting
the probability that a coin will land heads-up.
Nuclear decay is
an exponential
function:
Each time a half-life passes half of the sample decays and turns into something else
(depending on the decay mode). For example, every 5715 years, half of the nuclei in a
sample of C-14 will undergo beta decay and turn into N-14:
14
6C
 0-1e + 147N
The shorter a half-life is, the less time an unstable isotope is emitting radiation before it
decays into something more stable and less dangerous. For this reason, elements with
shorter half-lives are used for medical reasons. They can decay before they cause
harm to the body.
Elements with longer half-lives can be used to trace the age of substances, since their
rate of decay is constant. As long as a substance contains a radioactive element its
approximate age can be traced.
For example, all living things are made of carbon, such as people, plants and animals.
C-14 can be used to trace the age of any of these things, such as a Viking ship, or
papyrus used in ancient Egypt. (Watch C-14 dating video clip)
Elements like uranium, with an extremely long half-life, can be used to trace the age of
the Earth, since uranium is a natural part of some rocks.
Half-Life Calculations
For example, if a substance has a half-life of 14 days, half of its atoms will have
decayed within 14 days. In 14 more days, half of the remaining half will have decayed.
If we start with 10g, after 14 days 5g will be undecayed and after another 14 days, 2.5g
will be undecayed. This continues.
Initial
After 14 Days: Half-Life 1
After 28 Days: Half-Life 2
Decayed
10g
5g
Decayed
5g
2.5g
7.5g
Each time a half-life passes, half of the sample decays.
We can also visualize this using arrows to represent each half-life that passes:
10g
 5g
 2.5g
14d
14d
 1.25g
14d
If one half-life is 14 days, 3 half-lives takes 42 days (14 x 3).
We can also consider this as fractions of a whole:
1
 1/2
 1/4
 1/8
Example 1:
Consider the radioactive isotope (or radioisotope) iodine-131. If we started with 20g of
this isotope, how much would we have left after 32.08 days?
First, you can find on the table that the half-life of iodine-131 is 8.021 days.
We have a half-life of 8.021 days and the time the problem gives us is 32.08 days.
How many half-lives have passed?
# of Half-Lives = Total Time = 32.08d = 4 half-lives have gone by
Half-Life
8.021d
Each time a half-life passes, half of the sample decays. Start with 20g and divide that
in half 4 times:
Draw 4 arrows, which represent each of the 4 Half Lives



 [then start with 20g]
20 


 [then proceed to halve each of the end results]
20  10  5
 2.5 
1.25
At the end of 32,08 days, 4 half-lives have gone by and 1.25g of the original 20g
of I-131 remains undecayed.
Example 2:
What total mass of a 32g sample of 220Fr will remain unchanged after 82.2 seconds?
Look up the half-life: Half-Life = 27.4 seconds
# of Half-Lives = Total Time = 82.2s/27.4s = 3 half-lives
Half-Life
32g  16g
 8g
 4g
At the end of 82,2 seconds, 3 half-lives have gone by and 4g of the original 32g
of Fr-220 remains undecayed.