HalfLife of Radioisotopes Chemistry Tutorial
Key Concepts
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Defining the Halflife of a Radioisotope
The halflife of a radioisotope is the time it takes for half the original number of atoms of the isotope to undergo nuclear decay (radioactive decay).
Some radioisotopes have very long halflives, some have very short halflives.
The halflife of some radioisotopes is given in the table below:
Name 
Symbol 
Halflife (t_{½}) 
fluorine20 

11 seconds 
magnesium27 

9.5 minutes 
sodium24 

15 hours 
iodine131 

8 days 
cobalt60 

5.3 years 
tritium (hydrogen3) 

12.3 years 
strontium90 

28 years 
carbon14 

5,700 years 
radium226 

1,600 years 
plutonium239 

24,000 years 
uranium238 

4,500,000,000 years 
The earth is about 4.5 × 10^{9} years old.
The halflife of uranium238 is also about 4.5 × 10^{9} years.
This means that if a rock contained 100 g of uranium238 at the time the earth came into being, then at the present time the rock would contain only half that amount, ½ × 100 = 50 g, of uranium238.
If you wait another 4.5 × 10^{9} years and measure the mass of uranium238 in that rock you will find there will be only ½ × 50 = 25 g left.
We could tabulate the mass of uranium238 remaining after an interval of time measured in numbers of halflives as shown below:
Time as a number of halflives 
Time elapsed in years 
Mass of Uranium238 (g) 
0 
0 
100.0 g 
1 
4.50 × 10^{9} 
50.00 g 
2 
9.00 × 10^{9} 
25.00 g 
3 
1.35 × 10^{10} 
12.50 g 
4 
1.80 × 10^{10} 
6.250 g 
5 
2.25 × 10^{10} 
3.125 g 
And we could plot the mass of uranium238 in the rock against time on a graph as shown below:
mass (g)

Mass of ^{238}U in rock
time (× 10^{10} years)

We could then use this graph to find the mass of uranium238 remaining in the sample of rock at any time.
For example, if we want to know how much uranium238 is in the rock after 1 × 10^{10} years, we can read it straight of the graph as a mass of approximately 21.5 grams.
Similarly, we could determine how long we would have to wait in order for there to be only 75 grams of uranium238 left in the rock.
Reading this off the graph, we see that the answer is about 0.18 × 10^{10} years.
Determining Halflife of a Radioisotope from Data Tables and Graphs
In an experiment, the mass of strontium90 in a given sample of bone was measured every 7 years.
The results are shown in the table below:
time elapsed in years 
mass of strontium90 in bone (mg) 
0 
36.00 
7 
30.27 
14 
25.46 
21 
21.41 
28 
18.00 
35 
15.14 
42 
12.73 
49 
10.70 
56 
9.000 
63 
7.570 
We can determine the halflife of strontium90 by inspecting the mass of strontium90 remaining in the bone.
Remember, halflife of a radioisotope is defined as the time it takes for half the isotope to undergo nuclear decay.
At time 0, the mass of strontium90 in the bone is 36.00 mg.
After one halflife, only half this amount of strontium90 will remain, that is, mass of strontium90 will be ½ × 36.00 mg = 18.00 mg
From the table we see that it takes 28 years for the mass of strontium90 in the bone to be 18.00 mg so the halflife of strontium90 is 28 years (t_{½} = 28 years)
We can see that we can take any time interval of 28 years and find that the mass of strontium90 in the bone will be halved:
For example, at time = 7 years the mass of strontium90 is 30.27 mg, then after 1 halflife (7 + 28 = 35 years), the mass of strontium90 is ½ × 30.27 mg = 15.14 mg
If the results of the experiment were presented in a graph we could use the graph to determine the halflife of strontium90 in the same way:
mass (mg)

Mass of ^{90}Sr in bone
time (years)

Choose a point on the graph, for example, at time 21 years the mass of strontium90 is 21.41 mg
After one halflife, the mass of strontium90 will be ½ × 21.41 mg = 10.70 mg
From the graph, read off the value of time when the mass is 10.70 mg
time = 49 years.
The halflife is the time it takes for 21.41 mg to be halved to 10.70 g, that is,
the halflife of strontrium90 is 49  21 = 28 years.
Calculating the Amount of Radioisotope Remaining in a Sample
If you know:
 how much radioisotope was present in a sample at a given time, N_{o}
 halflife of that radioisotope, t_{½}
 how much time has elapsed, t
then you calculate the mass of radioisotope remaining in the sample, N_{t}:
N_{t} = N_{o} × (0.5)^{t/t½}
For example, iodine131 has a halflife of 8 days.
If we start our experiment with a mass of 1.50 g of iodine131, how much iodine131 will be present in 14 days time?
N_{o} = 1.50 g
t_{½} = 8 days
t = 14 days
N_{t} = N_{o} × (0.5)^{t/t½}
N_{t} = 1.50 × (0.5)^{14/8} = 0.446 g
Note the t ÷ t_{½} is actually the time that has elapsed in terms of the number of halflives.
In the example above, t ÷ t_{½} = 14 ÷ 8 = 1.75 halflives
So we could rewrite our equation in terms of the number of halflives that has elapsed:
N_{t} = N_{o} × (0.5)^{number of halflives}
For our 1.50 g sample of iodine131, if we wait 5 halflives (5 × 8 = 40 days), then the amount of iodine131 remaining in the sample will be:
N_{t} = N_{o} × (0.5)^{number of halflives}
N_{t} = 1.50 × (0.5)^{5} = 0.0469 g
Calculating How Much Radioisotope has Decayed
If we know:
 how much radioisotope was present in the original sample, N_{o}
 how much radioisotope is present in the sample after time t, N_{t}
we can calculate how much of the radioisotope has undergone nuclear decay (N_{d}):
N_{d} = N_{o}  N_{t}
For example, the mass of iodine131 remaining in a sample of iodine131 after 40 days is 0.0469 g.
If the sample originally contained 1.50 g of iodine131, what mass of iodine131 has undergone nuclear decay?
N_{o} = 1.50 g
N_{t} = 0.0469 g
N_{d} = N_{o}  N_{t} = 1.50  0.0469 = 1.45 g
You could also express this amount as a percentage.
What percentage of the iodine131 has undergone radioactive decay?
%N_{d} 
= 
N_{d} N_{o} 
× 100 

= 
1.45 1.50 
× 100 

= 
96.7 % 
And you could use this to calculate the percentage of iodine131 that still remains in the sample:
%N_{t} 
= 
100  %N_{d} 

= 
100  96.7 

= 
3.30 % 