 Key Concepts

• Unstable isotopes undergo nuclear decay (also known as radioactive decay).
• An unstable isotope is also called a radioactive isotope, or, a radioisotope.
• The half-life of a radioisotope is the time required for half the atoms in a given sample to undergo radioactive decay (nuclear decay).
• Half-life is given the symbol t½
• Different radioisotopes have different half-lives.
• The amount of radioactive isotope remaining in a sample after a given time interval can be calculated:

Nt = No × (0.5)number of half-lives

Where:
Nt = amount of radioisotope remaining after time t
No = original amount of radioisotope
number of half-lives = time elapsed ÷ half-life

• The amount of radioisotope that has decayed in a sample after a given time interval can be calculated:

Nd = No - Nt

Where:
Nd = amount of radioisotope that has decayed during time t
No = original amount of radioisotope in the sample at time 0 (t=0)
Nt = amount of radioisotope remaining in the sample after time t

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Defining the Half-life of a Radioisotope

The half-life 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 half-lives, some have very short half-lives.
The half-life of some radioisotopes is given in the table below:

Name Symbol Half-life (t½)
fluorine-20
 20 F 9
11 seconds
magnesium-27
 27 Mg 12
9.5 minutes
sodium-24
 24 Na 11
15 hours
iodine-131
 131 I 53
8 days
cobalt-60
 60 Co 27
5.3 years
tritium
(hydrogen-3)
 3 H 1
12.3 years
strontium-90
 90 Sr 38
28 years
carbon-14
 14 C 6
5,700 years
 226 Ra 88
1,600 years
plutonium-239
 239 Pu 94
24,000 years
uranium-238
 238 U 92
4,500,000,000 years

The earth is about 4.5 × 109 years old.
The half-life of uranium-238 is also about 4.5 × 109 years.
This means that if a rock contained 100 g of uranium-238 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 uranium-238.
If you wait another 4.5 × 109 years and measure the mass of uranium-238 in that rock you will find there will be only ½ × 50 = 25 g left.

We could tabulate the mass of uranium-238 remaining after an interval of time measured in numbers of half-lives as shown below:

Time as a number of half-lives Time elapsed in years Mass of Uranium-238 (g)
0 0 100.0 g
1 4.50 × 109 50.00 g
2 9.00 × 109 25.00 g
3 1.35 × 1010 12.50 g
4 1.80 × 1010 6.250 g
5 2.25 × 1010 3.125 g

And we could plot the mass of uranium-238 in the rock against time on a graph as shown below:

 mass (g) Mass of 238U in rock time (× 1010 years)

We could then use this graph to find the mass of uranium-238 remaining in the sample of rock at any time.

For example, if we want to know how much uranium-238 is in the rock after 1 × 1010 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 uranium-238 left in the rock.

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Determining Half-life of a Radioisotope from Data Tables and Graphs

In an experiment, the mass of strontium-90 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 strontium-90 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 half-life of strontium-90 by inspecting the mass of strontium-90 remaining in the bone.
Remember, half-life of a radioisotope is defined as the time it takes for half the isotope to undergo nuclear decay.

At time 0, the mass of strontium-90 in the bone is 36.00 mg.
After one half-life, only half this amount of strontium-90 will remain, that is, mass of strontium-90 will be ½ × 36.00 mg = 18.00 mg
From the table we see that it takes 28 years for the mass of strontium-90 in the bone to be 18.00 mg so the half-life of strontium-90 is 28 years (t½ = 28 years)

We can see that we can take any time interval of 28 years and find that the mass of strontium-90 in the bone will be halved:

For example, at time = 7 years the mass of strontium-90 is 30.27 mg, then after 1 half-life (7 + 28 = 35 years), the mass of strontium-90 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 half-life of strontium-90 in the same way:

 mass (mg) Mass of 90Sr in bone time (years)

Choose a point on the graph, for example, at time 21 years the mass of strontium-90 is 21.41 mg
After one half-life, the mass of strontium-90 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 half-life is the time it takes for 21.41 mg to be halved to 10.70 g, that is,
the half-life of strontrium-90 is 49 - 21 = 28 years.

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Calculating the Amount of Radioisotope Remaining in a Sample

If you know:

• how much radioisotope was present in a sample at a given time, No
• half-life of that radioisotope, t½
• how much time has elapsed, t

then you calculate the mass of radioisotope remaining in the sample, Nt:

Nt = No × (0.5)t/t½

For example, iodine-131 has a half-life of 8 days.
If we start our experiment with a mass of 1.50 g of iodine-131, how much iodine-131 will be present in 14 days time?

No = 1.50 g
t½ = 8 days
t = 14 days

Nt = No × (0.5)t/t½
Nt = 1.50 × (0.5)14/8 = 0.446 g

Note the t ÷ t½ is actually the time that has elapsed in terms of the number of half-lives.
In the example above, t ÷ t½ = 14 ÷ 8 = 1.75 half-lives

So we could re-write our equation in terms of the number of half-lives that has elapsed:

Nt = No × (0.5)number of half-lives

For our 1.50 g sample of iodine-131, if we wait 5 half-lives (5 × 8 = 40 days), then the amount of iodine-131 remaining in the sample will be:

Nt = No × (0.5)number of half-lives
Nt = 1.50 × (0.5)5 = 0.0469 g

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Calculating How Much Radioisotope has Decayed

If we know:

• how much radioisotope was present in the original sample, No
• how much radioisotope is present in the sample after time t, Nt

we can calculate how much of the radioisotope has undergone nuclear decay (Nd):

Nd = No - Nt

For example, the mass of iodine-131 remaining in a sample of iodine-131 after 40 days is 0.0469 g.
If the sample originally contained 1.50 g of iodine-131, what mass of iodine-131 has undergone nuclear decay?

No = 1.50 g
Nt = 0.0469 g

Nd = No - Nt = 1.50 - 0.0469 = 1.45 g

You could also express this amount as a percentage.
What percentage of the iodine-131 has undergone radioactive decay?

 %Nd = Nd No × 100 = 1.45 1.50 × 100 = 96.7 %

And you could use this to calculate the percentage of iodine-131 that still remains in the sample:

 %Nt = 100 - %Nd = 100 - 96.7 = 3.30 %

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