Unit 10: Rates of Decay & Half-Life
Understanding the predictable nature of radioactive decay and the concept of half-life.
10.10 Rate of Radioactive Decay
Radioactive decay is a spontaneous and random process. We cannot predict exactly when a single unstable nucleus will decay. However, for a large sample containing many radioactive atoms, the rate of decay is predictable and follows first-order kinetics.
The rate of decay (also known as the activity of a sample) is directly proportional to the number of radioactive nuclei present in the sample. This means that as the nuclei decay, the number of remaining radioactive nuclei decreases, and therefore the rate of decay also decreases over time.
Unlike chemical reaction rates, the rate of radioactive decay is not affected by external factors such as:
- Temperature
- Pressure
- Surface area
- The presence of a catalyst
- The chemical compound the atom is in
Solved Examples:
- What is the rate of radioactive decay proportional
to?
Solution: The number of radioactive nuclei present in the sample. - What happens to the activity of a radioactive sample over
time?
Solution: It decreases. - Would heating a sample of uranium-238 increase its rate of
decay?
Solution: No, the rate of radioactive decay is not affected by temperature. - A student has two samples of carbon-14. Sample A has twice as many atoms as
Sample B. How does the initial activity of Sample A compare to Sample
B?
Solution: The initial activity of Sample A will be twice that of Sample B. - What does it mean that radioactive decay is a "random"
process?
Solution: It means it is impossible to predict which specific nucleus will decay at any given moment. - Name two factors that do NOT affect the rate of
decay.
Solution: Temperature and pressure. - Why does the rate of decay decrease over
time?
Solution: Because as nuclei decay, there are fewer radioactive nuclei remaining in the sample to decay in the future. - Is the rate of decay a chemical or a nuclear
property?
Solution: A nuclear property, as it is determined by the stability of the nucleus. - How is the activity of a radioactive sample
measured?
Solution: Using a detector like a Geiger-Müller counter, which measures the number of decay events (counts) per unit of time. - If a radioactive isotope is part of a compound (e.g., in uranium oxide),
will it decay at a different rate than the pure
element?
Solution: No, the chemical state of the atom does not affect its nuclear decay rate.
10.11 Half-Life (Definition & Calculation)
Since the rate of decay is constantly changing, it is more convenient to describe it using a constant value called the half-life ($t_{1/2}$).
The half-life of a radioactive isotope is the time taken for half of the radioactive nuclei in any given sample to decay. It is also the time taken for the activity (or count rate) of the sample to fall to half its initial value.
Each radioactive isotope has its own unique and constant half-life, which can range from fractions of a second to billions of years.
Half-Life Calculations:
- After 1 half-life, 50% (1/2) of the original sample remains.
- After 2 half-lives, 25% (1/4) of the original sample remains.
- After 3 half-lives, 12.5% (1/8) of the original sample remains.
- After 'n' half-lives, $(1/2)^n$ of the original sample remains.
Solved Examples:
- Define half-life.
Solution: The time it takes for half of the radioactive nuclei in a sample to decay. - The half-life of iodine-131 is 8 days. If you start with 100g, how much will
be left after 16 days?
Solution: 16 days is two half-lives (16/8 = 2). After 1 half-life: 50g remain. After 2 half-lives: 25g remain. 25g will be left. - A radioactive sample has an initial count rate of 800 counts per minute.
After 30 minutes, the count rate is 100 counts per minute. What is its
half-life?
Solution: The count rate has dropped from 800 → 400 (1 half-life) → 200 (2 half-lives) → 100 (3 half-lives). Three half-lives took 30 minutes, so one half-life is 30 / 3 = 10 minutes. - Cobalt-60 has a half-life of 5.3 years. What fraction of a sample will
remain after 21.2 years?
Solution: Number of half-lives = 21.2 / 5.3 = 4. The fraction remaining is $(1/2)^4 = 1/16$. - A sample of a radioisotope has a half-life of 20 hours. What percentage of
the isotope has decayed after 60 hours?
Solution: 60 hours is three half-lives. The percentage remaining is $(1/2)^3 = 1/8 = 12.5\%$. Therefore, the percentage that has decayed is 100% - 12.5% = 87.5%. - Does the half-life of an isotope depend on the initial amount of the
sample?
Solution: No, the half-life is a constant property of the isotope, regardless of the sample size. - After how many half-lives will less than 1% of a radioactive sample
remain?
Solution: After 6 half-lives, 1.56% remains. After 7 half-lives, 0.78% remains. So, 7 half-lives are needed. - The half-life of uranium-238 is 4.5 billion years. If a rock originally
contained 1g of U-238, how much would be left after 4.5 billion
years?
Solution: 0.5 g. - If 75% of a radioactive sample decays in 24 hours, what is its
half-life?
Solution: If 75% has decayed, 25% (or 1/4) remains. This takes two half-lives. So, the half-life is 24 hours / 2 = 12 hours. - What happens to the atoms that have
decayed?
Solution: They have transformed into a new, more stable element (the daughter isotope).
10.12 Relationship between Stability and Half-Life
The half-life of a radioactive isotope is a direct measure of its nuclear stability.
- A highly unstable nucleus will decay very quickly. It has a short half-life (e.g., Krypton-90 has a half-life of 33 seconds).
- A more stable (but still radioactive) nucleus will decay very slowly. It has a long half-life (e.g., Uranium-238 has a half-life of 4.5 billion years).
A completely stable, non-radioactive nucleus can be thought of as having an infinite half-life, as it will never decay.
Isotopes with very short half-lives are often more of an immediate, intense radiation hazard but disappear quickly. Isotopes with very long half-lives pose a long-term environmental hazard as they remain radioactive for geological timescales, which is a major concern for the storage of nuclear waste.
Solved Examples:
- What is the relationship between the stability of a nucleus and its
half-life?
Solution: The more stable the nucleus, the longer its half-life. The less stable the nucleus, the shorter its half-life. - Isotope A has a half-life of 10 minutes. Isotope B has a half-life of 10
million years. Which isotope is more
stable?
Solution: Isotope B is much more stable. - Which isotope in the previous question would present a greater immediate
radiation risk from a sample containing the same number of
atoms?
Solution: Isotope A, because its short half-life means it has a much higher initial rate of decay (activity). - What does a very short half-life imply about a
nucleus?
Solution: It implies the nucleus is highly unstable and decays rapidly. - Why are isotopes with long half-lives a problem for nuclear waste
disposal?
Solution: Because they will remain radioactive and hazardous for thousands or millions of years, requiring secure long-term storage. - What is the half-life of a non-radioactive isotope like
carbon-12?
Solution: It is considered to be infinite. - An isotope decays very quickly. Is its half-life long or
short?
Solution: Short. - If an isotope is described as having a high activity, what does this suggest
about its half-life?
Solution: It suggests it has a short half-life. - Why are radioisotopes used in medical imaging often chosen to have short
half-lives (e.g., a few hours)?
Solution: So that they provide a detectable signal for the scan but decay away quickly, minimizing the radiation dose to the patient. - Does half-life measure chemical stability or nuclear
stability?
Solution: Nuclear stability.