Radioactive decay describes how unstable nuclei transform into more stable nuclei while emitting radiation. Half-life problems are common in physics, chemistry, earth science, and medicine because they model how quickly radioactive materials change. This cheat sheet helps students organize the main equations and apply them to worked example style questions.
It is useful for calculating remaining nuclei, activity, elapsed time, and the age of samples.
The core idea is that radioactive decay is exponential, not linear, so equal time intervals remove equal fractions, not equal amounts. The half-life equation can be written with powers of or with the decay constant . Activity is proportional to the number of undecayed nuclei, so it follows the same exponential pattern.
Dating problems work by comparing the amount or activity left in a sample to the original amount.
Key Facts
- After half-lives, the remaining fraction is .
- The number of nuclei remaining is .
- The activity remaining is .
- The exponential decay model is , where is the decay constant.
- The decay constant and half-life are related by .
- Activity is the decay rate and is calculated by .
- Elapsed time can be found from the remaining fraction using .
- If a sample has of its original nuclei left, then and half-lives have passed.
Vocabulary
- Radioactive decay
- Radioactive decay is the spontaneous change of an unstable nucleus into a more stable nucleus with the release of radiation.
- Half-life
- Half-life is the time required for half of the radioactive nuclei in a sample to decay.
- Parent isotope
- A parent isotope is the original radioactive isotope that decays into another isotope or particle.
- Daughter product
- A daughter product is the nucleus or isotope formed after radioactive decay.
- Activity
- Activity is the number of decays per second and is measured in becquerels, where .
- Decay constant
- The decay constant is the probability per unit time that a nucleus will decay.
Common Mistakes to Avoid
- Treating decay as subtraction of the same amount each half-life is wrong because radioactive decay removes the same fraction, not the same number of nuclei.
- Using instead of in is wrong because the exponent must be the number of half-lives.
- Confusing remaining amount with decayed amount is wrong because if remains, then has decayed, and those values answer different questions.
- Forgetting consistent time units is wrong because , , and must use matching units such as seconds, years, or days.
- Assuming activity and number of nuclei are unrelated is wrong because , so activity decreases in the same ratio as the number of undecayed nuclei.
Practice Questions
- 1 A radioactive isotope has a half-life of . If the sample starts with , how much remains after ?
- 2 A sample has an initial activity of and a half-life of . What is its activity after ?
- 3 A fossil contains of its original carbon-14. If the half-life of carbon-14 is , estimate the fossil's age.
- 4 Explain why a radioactive sample never reaches exactly nuclei in the ideal exponential decay model, even after many half-lives.
Understanding Radioactive Decay and Half Life Worked Examples
Radioactive decay is random for each individual nucleus. No one can predict the exact moment when one particular nucleus will decay. A large sample contains an enormous number of nuclei, however, so its overall behaviour is very predictable.
This is why a graph of the count from a detector may jump around slightly during short measurements, yet follow a smooth decay curve over longer periods. The half life is not a countdown timer built into every atom. It is a statistical property of a particular isotope.
Worked examples become easier when the information is sorted before any calculation. Identify the starting quantity, the quantity left, the half life, and the elapsed time. Check whether the question gives nuclei, mass, activity, or a fraction.
A mass can be used in place of a number of nuclei when the isotope is pure, because the same fraction of the sample remains. Activity needs extra care because it measures decays per second. Its SI unit is the becquerel, meaning one decay each second.
A detector may report counts per second instead. This can be lower than the real activity because some radiation misses the detector or is absorbed before it arrives.
Half life is especially useful when the elapsed time is a whole number of half lives. A sample that begins with eight hundred grams has four hundred grams after one half life, two hundred grams after two, and one hundred grams after three. Notice that the amount lost gets smaller each interval.
In the first interval, four hundred grams disappear. In the third interval, only one hundred grams disappear. Students often subtract a fixed amount repeatedly by mistake.
That produces a straight line and does not describe nuclear decay. For times that fall between whole half lives, a calculator, graph, or logarithms are needed.
Radioactive dating relies on a justified starting value. Carbon dating works for once living material because a living organism exchanges carbon with its environment. After death, that exchange stops and the carbon fourteen amount falls.
Other dating methods suit rocks or minerals, where scientists must consider how the material formed and whether radioactive atoms or daughter atoms entered or escaped later. A calculated age is reliable only when the sample remained a closed system.
In medicine, short half life isotopes are chosen for scans so they give useful signals without remaining active for long. Always include units, keep time units consistent, and state any assumptions about the original sample and detector readings.