Radioactive Half Life Activity
Assumed Knowledge
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Simulating the Half Life of a Radioisotope
Introduction
An unstable isotope, or radioisotope, of an element is an isotope that undergoes radioactive decay, or nuclear decay. At any moment in time, an atom of this radioisotope has either decayed or it has not, there are only two possibilities. This is very similar to flipping a coin since, when it lands, it could land with either with its "head" up or its "tail" up, just two possibilities. Therefore, we can use a collection of identical coins to represent a collection of atoms of the same radioisotope. The "head" side represents an atom of the original, undecayed radioisotope, while the "tail" side represents an atom of the new species created after nuclear decay, the daughter.
- Head-up coin = an atom of the radioisotope that has NOT decayed
- Tail-up coin = an atom of the radioisotope that HAS decayed
One coin is a model of one radioisotope.
Many other things could be used instead of coins as models for radioisotopes in this activity. You could use dice; an even number face-up represents the original radioisotope while an odd number face-up represents the daughter. In the video demonstration below we use scrabble letters; letter face-up is an undecayed radioisotope, letter face-down is a daughter.
In this activity, we simulate radioactive half life using our model radioisotopes by effectively flipping a large number of coins simultaneously.
Procedure
- Step 1:
- Place 80 identical coins in a box. Use your fingers to sweep through the coins several times, or shake the box vigorously, to mix the coins up.
- Step 2:
- Wait 3 minutes, then up-end the box, allowing the coins to fall on level ground.
- Step 3:
- Pick up the "tail-up" coins and pile them up, one on top of the other, to make a tower.
- Step 4:
- Count the number of "head-up" coins (atoms that have NOT decayed), and record this.
- Step 5:
- Place the coins that were "head-up" (undecayed) back into the box and mix with your fingers or shake the box.
- Step 6:
- Repeat steps 2 to 5, placing each new tower of coins to the right of the one before. Stop the activity when just a few coins remain "head-up" (undecayed).
Results
At the end of the activity, you should have a table of results similar to that below:
| Time (min) |
No. head-up coins (undecayed atoms) |
|---|---|
| 0 | 80 |
| 3 | 40 |
| 6 | 20 |
| 9 | 10 |
| 12 | 5 |
Note that your experimental results will be a little bit different. One way to improve the results is to do this activity in pairs, pool everyone's results, and average the number of head-up coins for each time interval.
Can you think of any other ways to improve the results?
You should also have 4 towers of coins that will look a lot like the diagram below:
Discussion
First we will consider the tabulated data for the radioisotopes that have NOT undergone nuclear decay, these were the "head-up" coins.
Can you see any patterns in the data?
| Time (min) |
No. head-up coins (undecayed atoms) |
Pattern |
|---|---|---|
| 0 | 80 | |
| 3 | 40 | 80 ÷ 2 = 40 |
| 6 | 20 | 40 ÷ 2 = 20 |
| 9 | 10 | 20 ÷ 2 = 10 |
| 12 | 5 | 10 ÷ 2 = 5 |
The number of undecayed radioisotopes remaining is halved every 3 minutes.
We say that the half life of this radioisotope is 3 minutes, and it is represented as t½. For this radioisotope, t½ = 3 minutes
The half life of a radioisotope is defined as the time taken for half the atoms of the radioisotope to undergo nuclear decay (radioactive decay).
This enables us to write 0 half-lives at the start of the experiment, at the 3 minute mark we have reached 1 half life, at the 6 minute mark 2 half lives have past, etc.
We could also change the number of "head-up" coins to a fraction or a percentage of the total number of coins we had at the start of the experiment as shown below:
(a) 0 half lives: % heads = (80/80) × 100 = 100%
(b) after 1 half life: % heads = (40/80) × 100 = 50%
(c) after 2 half lives: % heads = (20/80) × 100 = 25%
This enables us to make two alterations to our table to make it more general:
- Replace the heading "Time (min)" with "Number of half lives", and write the numbers 0, 1, 2, 3 etc underneath.
- Convert the "No. of head-up coins" to a fraction or a percentage of the original number of coins.
- Extend the table for as many half lives as we need.
The new table is shown below:
| No. half lives | % undecayed radioisotope remaining |
|---|---|
| 0 | 100 |
| 1 | 50 |
| 2 | 25 |
| 3 | 12.5 |
| 4 | 6.25 |
| 5 | 3.125 |
| 6 | 1.04167 |
| 7 | 0.520833 |
| 8 | 0.2604167 |
| 9 | 0.13020833 |
| 10 | 0.065104167 |
We can use this table to tell us:
- how much of a radioisotope is left after a given number of half lives
- how much time has passed given the original and final quantity of radioisotope
What we can't do very well using the table is get an estimate for values that are not included in the table.
For example, how much undecayed radioisotope remains after 1.2 half lives have passed? The table tells us that between 25% and 50% will remain. We might guess that about 45% remains after 1.2 half lives, 40% remains after 1.4 half lives, 35% remains after 1.6 half lives, and 30% remains after 1.8 half lives. But how good are these guesses?
A graph would be so much better than a table!
So, let's compare the data in this new table to our "tower of coins". What do you think the tower of coins represents?
Did you notice that the height of each column is about half, or 50%, of the height of the column to its left? This suggests that the height of each tower represents the percentage of radioisotope remaining. If this were a graph, the vertical axis, or y-axis, would be labelled "% radioisotope remaining".
Why are there 4 towers? Because we waited 3 minutes, or 1 half life, between coin tosses, and we did this 4 times. Once again, if this were a graph the horizontal axis, the x-axis, could be labelled either "time" or "Number of half lives".
So, let's turn our tower of coins into a graph by marking the height of each tower, the right hand side of the base of each tower, and drawing in each axis, as shown below:
Note that we have included a tower of coins that was not present, 100% of the radioisotope was present at time 0, that is, at the start of the experiment no radioisotope had decayed yet, but we did not build a tower of coins at this point.
We can use this graph(3) to determine:
- how much of a radioisotope is left after a given number of half lives
- how much time has passed given the original and final quantity of radioisotope
but, unlike the table, we can use this curve to get quite accurate estimates of values that are NOT included in the table, that is, fractions of half life time or other percentages of isotopes remaining.
Video: Simulating Half Life Demonstration
Comparing Half Lives
Different radioisotopes have different half lives: some radioisotopes have short lives, others have long half lives.
Let's compare two different radioisotopes of the same element: gold-199 and gold-196. Gold-196 has a longer half, t½ ≈ 6 days. Gold-199 has a shorter half life, t½ ≈ 3 days. This means that after 3 days, 50% of the original amount of gold-199 remains, but it takes twice as long for gold-169 to decay so it takes 6 days for only 50% of the original gold-169 to remain.
We can construct a table to compare the length of time for each radioisotope to acheive certain percentages remaining as shown below:
| % Gold-199 remaining | Time (minutes) | % Gold-196 remaining |
|---|---|---|
| 100 | 0 | 100 |
| 50 | 3 | |
| 25 | 6 | 50 |
| 12.5 | 9 | |
| 6.25 | 12 | 25 |
We can plot this data on a graph as shown below:
Key:
Black = gold-199 (t½ = 3 days)
Red = gold-196 (t½ = 6 days)
The graph shows us that a radioisotope with a shorter half life has a steeper curve than a radioisotope with a longer half life.
Worked Examples: Half Life
Using a Table
You may refer to the following table to answer the questions below:
| No. half lives | % radioisotope remaining |
|---|---|
| 0 | 100 |
| 1 | 50 |
| 2 | 25 |
| 3 | 12.5 |
| 4 | 6.25 |
| 5 | 3.125 |
| 6 | 1.04167 |
| 7 | 0.520833 |
| 8 | 0.2604167 |
| 9 | 0.13020833 |
| 10 | 0.065104167 |
Example 1: How much radioisotope remains?
Question: What percentage of the original radioisotope remains after a length of time equal to 4 half lives has passed?
How to find the answer: Find "No. half lives" heading, go down the column to "4", go across to the column on the right titled "% undecayed atoms" and read the value "6.25".
How to write your answer: write 6.25 and then the % sign to the right of this number (the % sign is the "units" of measurement in this table).
Answer the question: 6.25% of the original radioisotope remains after 4 half lives has passed.
Example 2: What mass of radioisotope remains?
Question: The mass of carbon-14 in a wooden park bench was determined to be 0.120 g. Predict the mass of carbon-14 remaining in this park bench when a period of time equal to 5 half lives has passed.
How to find the answer: Use the table to determine the % carbon-14 remaining when the number of half-lives is 5, that is 3.125% remains.
Calculate the mass: mass of carbon-14 remaining will be 3.125% of the original 0.120 g:
| mass | = | 3.125 100 |
× 0.120 |
| = | 0.00375 g |
Answer the question: After 5 half lives, 0.00375 g of carbon-14 remain in the park bench.
Example 3: How much time has passed?
Question: If 25% of the original quantity of radioisotope remains in a sealed jar, how much time has passed since the quantity of radioisotope in this jar was last measured?
How to find the answer: Find the "% undecayed atoms" column, go down the column to 25, then go across to the left hand column labelled "No. half lives" and read the value "2".
How to write your answer: write 2 half lives (half lives are the "units" of measurement here).
Answer the question: 2 half lives have passed since the quantity of radioisotope in this jar was last measured.
Example 4: How many years have elapsed?
Question: A 6.24 g sample of radioactive lead-210 is produced and sealed in jar. After some time, the jar was opened and the mass of lead-210 in this same sample was determined to be 0.78 g. If the half life of lead-210 is 22.3 years, how many years elapsed between the production of the sample and when it was opened?
How to find the answer: Determine the percentage of lead-210 that remains in the sample:
| % remaining | = | 0.78 6.24 |
× 0.100 |
| = | 12.5% |
Use the table to find how many half lives have passed in order for 12.5% of the radioisotope to remain: 3 half lives.
Since 1 half life = 22.3 years, and we know that 3 half lives have passed, the length of time must be equal to 3 times the length of a half life:
| time elapsed | = | no. half lives × length of a half life |
| = | 3 × 22.3 | |
| = | 66.9 |
How to write your answer: write 66.9 with units years, that is, 66.9 years
Answer the question: 66.9 years have elapsed since this sample of lead-210 was produced.
Using a Graph
You may refer to the following graph to answer the questions below:
Example 1: How much radioisotope remains?
Question: What percentage of the original radioisotope remains after a length of time equal to 1.2 half lives has passed?
How to find the answer: Find 1.2 on the x-axis, "No. half lives" axis (this will be the first mark after the number 1). Move vertically up this line until you reach the curve, then go along horizontally to the left to read of the "% remaining" axis (the y-axis). The value should be 43 (although a value of between 42 and 44 would be acceptable).
How to write your answer: write 43 and then the % sign to the right of this number (the % sign is the "units" of measurement in this table).
Answer the question: 43% of the original radioisotope remains after 1.2 half lives has passed.
How to write your answer: 0.52 g (the units of measurement is grams)
Answer the question: After 3.5 half lives, 0.52 g of the radioisotope remains.
Find 93% remaining on the vertical axis of the graph, then go horizontally until you hit the curve, then go down vertically to read the value off the x-axis, and the value should be 0.1 half lives.
Use this half life value to calculate the number of years that have passed, that is, 1 half life = 1600 years so
How to write your answer: 160 years (the units are years)
Answer the question: 160 years will have passed between the extraction of 0.0010 g and the time when the mass of the radioisotope is 0.00093 g
Question: What percentage of the original radioisotope remains after a length of time equal to 1.2 half lives has passed?
How to find the answer: Find 1.2 on the x-axis, "No. half lives" axis (this will be the first mark after the number 1). Move vertically up this line until you reach the curve, then go along horizontally to the left to read of the "% remaining" axis (the y-axis). The value should be 43 (although a value of between 42 and 44 would be acceptable).
How to write your answer: write 43 and then the % sign to the right of this number (the % sign is the "units" of measurement in this table).
Answer the question: 43% of the original radioisotope remains after 1.2 half lives has passed.
Example 2: What mass of radioisotope remains?
Question: 5.86 g of a radioisotope is sealed in a jar. After a time equal to 3.5 half lives, the jar is opened. What mass of the radioisotope remains in the jar?
How to find the answer: Find 3.5 on the "No. half lives" axis (half way between 3 and 4), go up vertically until you hit the curve, then go across horizontally to the left and read the value off the "% remaining" axis which should be 8.9%
Use this value to calculate the mass of radioisotope remaining:
| mass | = | % remaining 100 |
× original mass |
| = | 8.9 100 |
× 5.86 | |
| = | 0.52 g |
How to write your answer: 0.52 g (the units of measurement is grams)
Answer the question: After 3.5 half lives, 0.52 g of the radioisotope remains.
Example 3: How much time has passed?
Question: If 35% of the original quantity of radioisotope remains in a sealed jar, how much time has passed since the quantity of radioisotope in this jar was last measured?
How to find the answer: Find 35 on the y-axis, "% remaining" axis. Follow this line horizontally to the right until it hits the curve then go down vertically on this line until it reaches the x-axis, "No. half lives" axis. The value should be 1.5 (for this graph, any value between 1.4 and 1.6 would acceptable).
How to write your answer: write 1.5 half lives (half lives are the "units" of measurement here).
Answer the question: 1.5 half lives have passed since the quantity of radioisotope in this jar was last measured.
Example 4: How many years have elapsed?
Question: Marie and Pierre Curie extracted 0.0010 g of radium-226 from pitchblende in 1898. If the half life of this radioisotope is 1600 years, how much time must elapse so that the mass of radium-226 in this sample is only 0.00093 g?
How to find the answer: Calculate the percentage of radium-226 remaining in the sample at the end of the experiment:
| % remaining | = | 0.00093 0.0010 |
× 100 |
| = | 93% |
Find 93% remaining on the vertical axis of the graph, then go horizontally until you hit the curve, then go down vertically to read the value off the x-axis, and the value should be 0.1 half lives.
Use this half life value to calculate the number of years that have passed, that is, 1 half life = 1600 years so
| time elapsed | = | no. half lives × length of 1 half life |
| = | 0.1 × 1600 | |
| = | 160 years |
How to write your answer: 160 years (the units are years)
Answer the question: 160 years will have passed between the extraction of 0.0010 g and the time when the mass of the radioisotope is 0.00093 g
Problem Solving: Radioactive Half Life
The Problem:
The level of radioactivity of a radioisotope can be measured in units of gigabecquerel (GBq)(4). The level of radioactivity of an unknown sample is measured at 24 hour intervals. The results are given in Table 1 below.
| Table 1 | |
|---|---|
| Days | Level of radioactivity (GBq) |
| 0 | 1.1100 |
| 1 | 1.01565 |
| 2 | 0.9324 |
| 3 | 0.8547 |
| 4 | 0.7881 |
| 5 | 0.7215 |
| 6 | 0.66045 |
| 7 | 0.60495 |
| 8 | 0.555 |
| 9 | 0.5106 |
| 10 | 0.4662 |
| 11 | 0.42735 |
| 12 | 0.39405 |
| 13 | 0.36075 |
| 14 | 0.333 |
It is believed that this sample could be one of the radioisotopes listed below in Table 2:
| Table 2 | |
|---|---|
| Radioisotope | t½ (approximate) |
| chromium-51 | 28 days |
| gold-198 | 3 days |
| iodine-131 | 8 days |
| phosphorus-32 | 14 days |
| technetium-99m | 6 hours |
Identify the radioisotope in the sample.
Solving the Problem using the StoPGoPS model for problem solving.
STOP! |
State the question. | What is the question asking you to do?
Identify the radioisotope in the sample. |
PAUSE! |
Pause to Plan. | What information have we been given?
At time 0, radioactivity = 1.1100 GBq What do we know?Definition of half life: time taken for half the atoms of the radioisotope to decay What do we need to find out?(1) Half life of this radioisotope : Use Table 1. (2) The identity of the radioisotope with this value for half life: Use Table 2. |
GO! |
Go with the Plan. | (1) Find the half life of this radioisotope:
At time 0, radioactivity = 1.1100 GBq After 1 half life, radioactivity = ½ × 1.1100 = 0.555 GBq From Table 1: 0.555 GBq occurs at time = 8 days. The half life of this radioisotope is 8 days. (2) Identify the radioisotope Find the radioisotope in Table 2 that has a half life of 8 days. The radioisotope is iodine-131 |
PAUSE! |
Ponder Plausability. | Have you answered the question that was asked?
Yes, we have identified (named) the radioisotope. Is your solution to the question reasonable? Consider the other possibilities: chromium-51, t½ = 28 days, therefore, 0.555 g would remain at 28 days, not 8 days. gold-198, t½ = 3 days, therefore, 0.555 g would remain at 3 days, not 8 days. phosphorus-32, t½ = 14 days, therefore, 0.555 g would remain at 14 days, not 8 days. technetium-99m, t½ = 6 hours = 6/24 = 0.25 days, therefore, 0.555 g would remain at 0.25 days, not 8 days. |
STOP! |
State the solution. | The unknown sample is iodine-131. |
Resources for Students and Teachers
Footnotes
(1) The terms radioisotope and radionuclide are often used interchangeably, but there is a subtle difference. Radioisotope refers specifically to an unstable isotope of an element that emits radiation. On the other hand, radionuclide refers to any atom that emits radiation. Hence, isotopes can be defined as nuclides having the same atomic number but different mass numbers, and radioisotopes can be defined as radioactive nuclides having the same atomic number but different mass numbers.
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(2) You may also see half life written as half-life or halflife.
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(3) You might recognise the shape of this curve, it is a graph of an exponential decay function: y=a(1-b)x where b is the decay factor and a is the starting amount. You can learn more about this in the Radioisotope Half Life Calculations tutorial.
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(4) The Becquerel, abbreviation Bq, is the SI unit of measurement of radioactivity. 1 Becquerel is a rate of one radioisotope decaying in 1 second (also referred to as 1 disintegration per second).
1 gigabecquerel = 109 becquerel
1 GBq = 1 × 109 Bq