In this explainer, we will learn how to explain the half-life of radioactive elements.

The nucleus of an atom contains protons and neutrons. Inside the nucleus, the positively charged protons repel one another strongly; however, there are also many short-range attractive forces that occur between two protons, two neutrons, or a proton and a neutron. The nucleus is stable when the attractive forces outweigh the repulsive forces. When the repulsive forces prevail, the nucleus is unstable.

Unstable nuclei can become more stable by emitting ionizing radiation, such as gamma rays (), alpha particles (), and beta particles (). The spontaneous emission of radiation by an unstable nucleus is called radioactive decay. A material that can undergo radioactive decay is considered to be radioactive, and the amount of ionizing radiation released by the material is its radioactivity.

### Definition: Radioactive Decay

Radioactive decay is the spontaneous emission of radiation from an unstable nucleus.

The radioactivity of a material can be measured using a GeigerβMΓΌller tube connected to a counter, often collectively referred to as a Geiger counter. When ionizing radiation enters the tube, an electrical pulse is sent to the counter. The counter displays the count rate, the number of electrical pulses, either per second (CPS) or per minute (CPM). Geiger counters may also have a device that creates an audible click every time ionizing radiation is detected. The figure below shows a GeigerβMΓΌller tube and counter.

Before using a Geiger counter to measure the radioactivity of a sample, we must first measure the background radiation. Background radiation is radiation emitted from all other radioactive sources in the area. Sources of background radiation include terrestrial (radon in soil), cosmic (ionizing radiation from solar flares), man-made (nuclear power plant fallout), and internal (potassium-40 in the human body) radiation.

Once the background radiation count rate is known, we can use the Geiger counter to measure the count rate of the radioactive sample. However, we must recognize that this measurement is really the combined count rate of the sample and the background radiation. To know the true count rate of the radioactive source, we need to subtract the background radiation count rate from the combined count rate:

Over time, the radioactivity of a material will decrease as the unstable nuclei decay. The average amount of time it takes for one-half of the radioactive nuclei in a sample to undergo radioactive decay is called the half-life and is represented by the symbol .

### Definition: Half-Life (π‘_{1/2})

Half-life is the amount of time required for one-half of the radioactive nuclei in a sample to decay.

Half-life is an intrinsic property. This means that, regardless of the amount of material present, the half-life will be the time required for half of the material to decay.

### Example 1: Recognizing the Definition of Half-Life

Which of the following statements best defines the concept of half-life?

- Half the time taken for all of the unstable nuclei to decay
- The time taken for all of the unstable nuclei to decay
- Half the time taken for half of the unstable nuclei to decay
- The time taken for half of the unstable nuclei to decay

### Answer

Radioactive or unstable nuclei release radiation in the form of energy and particles in a process known as radioactive decay. The length of time required for a nucleus to decay varies from isotope to isotope. One measure of the length of time required is half-life. Half-life, represented by the symbol , is an intrinsic property of a material defined as the amount of time required for one-half of the radioactive nuclei to decay. Therefore, the best statement is answer choice D.

Let us examine iodine-131, a radioactive isotope of iodine that is commonly used to treat hyperthyroidism. Iodine-131 undergoes beta decay to become stable xenon-131. The half-life of this reaction is approximately 8.04 days. This means that if we had a sample of eight iodine-131 atoms, after 8.04 days, half of the atoms will have decayed to form xenon-131 and half of the atoms will still be radioactive iodine-131. After an additional 8.04 days, half of the still radioactive iodine atoms will decay leaving two atoms of iodine-131. This process is represented by the diagram below.

This same idea would apply if we measured our sample in grams instead of atoms. For example, if we started with 25 grams of iodine-131, after one half-life, 12.5 grams would have decayed into xenon-131 and 12.5 grams would still be radioactive. After an additional half-life, 6.25 grams of the original sample would still be iodine-131 while the remaining 18.75 grams would have decayed into xenon-131. This paragraph can be summarized in the following figure

If we used a Geiger counter to measure the count rate per minute of a sample of iodine-131, we would see that the count rate would halve every 8.04 days as shown in the following figure.

### Example 2: Determining Which Sample Has a Half-Life of Two Days from Count Rate Data

Five radioactive samples are measured over three days, using a GeigerβMΓΌller tube, to provide a count rate. Which of the samples has a half-life of 2 days?

Day 1 | Day 2 | Day 3 | |
---|---|---|---|

Source A | 1βββ200 | 600 | 300 |

Source B | 100 | 95 | 85 |

Source C | 600 | 420 | 300 |

Source D | 800 | 690 | 550 |

Source E | 1βββ200 | 300 | 75 |

### Answer

We want to examine the data to determine which source has a half-life of two days. The half-life is the amount of time it takes for half of the radioactive material to decay. The data does not provide information regarding the amount of each source but does give us information regarding their count rates as measured by a GeigerβMΓΌller tube.

Count rate is a measurement of the reactivity of a sample. As the radioactive sample decays, the count rate will decrease as less and less ionizing radiation is released. The count rate directly correlates to the amount of radioactive material remaining. Therefore, we can also define the half-life as the amount of time it takes for the count rate to decrease to half of its original value.

The table below shows the expected count rate for each source after one half-life has passed.

Day 1 (Initial Count Rate) | Count Rate after One Half-Life | |
---|---|---|

Source A | 1βββ200 | 600 |

Source B | 100 | 50 |

Source C | 600 | 300 |

Source D | 800 | 400 |

Source E | 1βββ200 | 600 |

We need to compare the expected count rates after one half-life with the count rate recorded after two days have passed. Recognize that the column labeled day 2 does not represent the count rate after two days but rather the count rate after one day of monitoring. Therefore, the source with a count rate on day 3 that matches the expected count rate after one half-life should have a half-life of two days.

Comparing the day 3 count rates to the expected count rates, we can see that these values match for source C. Source C had a count rate of 600 on day 1 and a count rate of 300 on day 3. The count rate halved over the course of two days. Therefore, source C has a half-life of two days.

The half-life of a radioactive isotope can range anywhere from a fraction of a second to well over a trillion years. Francium-223 has the shortest half-life of all naturally occurring isotopes at 22 minutes while xenon-124 has the longest half-life ever recorded in a laboratory setting at years. Synthetic or man-made isotopes often have very short half-lives; however, this is not the case for all man-made isotopes. Oganesson, first created in 2006, has a half-life of seconds but technetium-98 has a half-life of years.

By looking at a graph of undecayed radioactive material over time, we can estimate the half-life of a substance.

The starting amount of iodine-131 is 40 grams. After one half-life, only 20 grams of radioactive iodine-131 should remain. The amount of time it takes for the sample to decay to 20 grams is the half-life . Looking at the graph, it takes eight days for the sample to decay to one-half of its original amount. Furthermore, after an additional eight days, the sample again decays by half to 10 grams.

### Example 3: Estimating the Half-Life of a Sample from a Decay Curve

Using the decay curve on the graph, find the half-life of the sample.

### Answer

This is a graph of the number of counts per minute over the course of several hours. We want to examine the graph to determine the half-life of the sample. The half-life is the amount of time it takes for half of the radioactive material to decay. The graph does not provide information regarding the amount of the sample but does give us information regarding the count rate.

Count rate is a measurement of the reactivity of a sample that is often collected by using a Geiger counter. As the radioactive sample decays, the count rate will decrease as less and less ionizing radiation is released. The count rate directly relates to the amount of radioactive material remaining. Therefore, we can also define the half-life as the amount of time it takes for the count rate to decrease by half of its original value.

At an initial time of zero days, the count rate was 1βββ100 counts per minute. After one half-life has passed, the count rate should be half as much as this original value, 550 counts per minute.

Looking at the graph above, we can see that the amount of time it took for the count rate to halve was two hours. Thus, the half-life of the sample is two hours.

Mathematical problems involving half-life can be solved in a number of ways, but regardless of the method, three of the following four pieces of information must be known:

- the initial amount of radioactive material,
- the half-life ,
- the total time the sample is allowed to decay,
- the amount of radioactive material remaining after a given time has passed.

For simple problems, a diagram can be used to help us determine any one of these values.

We can increase the diagram by adding more half-lives as necessary. The diagram helps us to recognize the following relationship:

Let us examine how to use the diagram for the following problem. A 208 g sample of sodium-24 decays to 13 g of sodium-24 within 60 hours. What is the half-life of this radioactive isotope?

We can begin by filling in the initial amount of sodium-24, the total time the sample is allowed to decay, and the amount of sodium-24 that remains after 60 hours.

Next, we divide the initial amount of sodium-24 by two to determine the amount of sodium-24 that remains after one half-life has passed.

We continue to divide the amount of sodium-24 that remains by two until we have 13 grams remaining.

Notice that sodium-24 needed to go through four half-lives in order for the initial mass to decay to 13 grams. We can use the relationship between half-life, the total time, and the number of half-lives to determine the half-life of sodium-24:

### Example 4: Determining the Amount of Unstable Nuclei Present after a Given Amount of Time

A radioactive isotope with a half-life of 2 hours contains 100 billion unstable nuclei. How many unstable nuclei would remain after 10 hours?

### Answer

The half-life is the amount of time it takes for half of the radioactive material to decay. We know that there are originally 100 billion unstable nuclei of a radioactive isotope. The half-life of the isotope is two hours. This means that, after two hours have passed, half of the unstable nuclei, 50 billion, will have decayed while the other half remain unstable. We want to determine how many unstable nuclei remain after 10 hours have passed.

First, we need to know how many half-lives will occur over the 10 hours. The following equation can be used to relate the half-life , total time allowed to decay, and the number of half-lives:

We can substitute the half-life and total time into the equation. The variable n will be used to represent the number of half-lives: We then rearrange the equation to solve for :

We have determined that, over the course of 10 hours, the isotope will decay by half five times. Using this information, we can construct the following diagram.

The diagram includes the original amount of unstable nuclei in the uppermost box and five additional boxes for the amount of unstable nuclei that remain after each of the five half-lives. We know that, after one half-life, the amount of unstable nuclei will decrease to half of its original value (50 billion nuclei).

The remaining 50 billion nuclei will have decayed by half by the end of the second half life.

We can continue to fill in the diagram, dividing each value by two, until we have completed all five half-lives.

The value in the lowermost box indicates the number of unstable nuclei still present in the sample after 10 hours, or five half-lives. The amount of unstable nuclei still present is 3.125 billion.

### Example 5: Using the Mass and Half-Life of a Radioactive Substance to Determine the Mass Present Prior to Being Tested

A sample was tested and found to contain 0.32 g of a radioactive substance with a half-life of 4 hours. How much of the substance was present in the sample 20 hours before it was tested?

### Answer

The half-life is the amount of time it takes for half of the radioactive material to decay. The half-life of the substance is four hours. This means that every four hours, half of the radioactive material will have decayed and half will remain undecayed. We want to determine what mass of radioactive material was present 20 hours ago.

First, we need to know how many half-lives occurred over the 20 hours. The following equation can be used to relate the half-life , total time allowed to decay, and the number of half-lives:

We can substitute the half-life and total time into the equation. The variable n will be used to represent the number of half-lives:

We then rearrange the equation to solve for :

We have determined that, over the course of 20 hours, the isotope will decay by half five times. Using this information, we can construct the following diagram.

The diagram includes the final mass of radioactive material in the lowermost box and five additional boxes for the mass of radioactive material present after each half-life. We know that after one half-life the amount of radioactive material will decrease to half of its original value. However, we know the final value, not the original. Working backward, we know that one half-life ago, there must have been twice as much radioactive material.

Four hours prior to there being 0.64 grams of radioactive material, there must have been twice as much present.

We can continue to fill in the diagram, multiplying each value by two, until we have completed all five half-lives.

The value in the uppermost box indicates the mass of radioactive material present 20 hours before the test was completed. The mass of radioactive material present was 10.24 grams.

By understanding half-life, scientists can estimate the age of materials. When cosmic rays bombard nitrogen in the atmosphere, radioactive carbon-14 is produced. While plants and animals are alive, they absorb some carbon-14 through the atmosphere and food sources. They also release carbon-14 back into the environment as part of the carbon cycle. While living, the amount of carbon-14 in a specimen remains constant. However, once the plant or animal dies, the carbon-14 is no longer being replenished. Over time, the carbon-14 will decay. The half-life of carbon-14 is 5βββ730 years. By measuring the amount of carbon-14 in a sample and performing calculations involving the half-life, scientists can estimate the age of the material.

### Key Points

- Unstable nuclei release radiation in the form of particles and energy in a process known as radioactive decay.
- The radioactivity of a sample can be measured in counts per minute or per second using a GeigerβMΓΌller tube.
- The half-life is the amount of time it takes for half of a sample to decay.
- The half-life of an isotope may range from fractions of a second to well over a billion years.
- The half-life of a material can be estimated from a graph of radioactive material remaining versus time.
- A diagram can be used to calculate
- the half-life,
- the total amount of time the sample is allowed to decay,
- the initial amount of a radioactive substance,
- the final amount of a radioactive substance.