Half-life calculations describe how unstable radioactive isotopes decay over time. This cheat sheet helps students connect time, number of half-lives, remaining sample amount, and percent remaining. It is useful for chemistry problems involving nuclear decay, medical tracers, radioactive waste, and radiometric dating.
The goal is to make each calculation method clear and easy to choose.
Key Facts
- One half-life means half of the radioactive nuclei remain, so after half-lives the fraction remaining is .
- The number of half-lives is found with , where is elapsed time and is the half-life.
- The remaining amount after half-lives is , where is the starting amount.
- The percent remaining is .
- The percent decayed is .
- Radioactive decay can also be modeled by , where is the decay constant.
- The decay constant and half-life are related by and .
- For dating an object, solve for time using or after finding .
Vocabulary
- Half-life
- The time required for half of the radioactive nuclei in a sample to decay.
- Parent isotope
- The original radioactive isotope that undergoes nuclear decay.
- Daughter product
- The new atom or isotope formed when a parent isotope decays.
- Decay constant
- The value that measures the probability of decay per unit time in the model .
- Activity
- The rate of radioactive decay, often measured in becquerels, where .
- Carbon dating
- A radiometric dating method that estimates the age of once-living material using the decay of carbon-.
Common Mistakes to Avoid
- Using the full original amount after each half-life is wrong because each half-life halves the current amount, not the starting amount.
- Dividing the half-life by time instead of time by half-life is wrong because the number of half-lives is .
- Treating percent decayed as percent remaining is wrong because a sample that is remaining is decayed.
- Forgetting units in is wrong because the decay constant must use the inverse of the time unit, such as .
- Rounding too early is wrong because half-life and dating problems can change noticeably when intermediate values like or are rounded.
Practice Questions
- 1 A radioactive sample starts at and has a half-life of . How much remains after ?
- 2 An isotope has . What percent of the original sample remains after ?
- 3 A fossil contains of its original carbon-. If carbon- has a half-life of , estimate the fossil's age.
- 4 Why does radioactive decay usually follow an exponential model instead of a linear model?
Understanding Half-Life Calculations Reference
Radioactive decay happens inside the atomic nucleus. An unstable nucleus can release a particle or energy and become a different nucleus. No one can predict the exact moment when one particular atom will decay.
This randomness often confuses students. A sample still follows a very reliable pattern because it contains an enormous number of atoms. During each equal time interval, the same fraction tends to decay.
The fraction is not a fixed number of grams or atoms. A large sample loses more material than a small sample during the same interval, yet both can have the same half-life.
Half-life describes the isotope, not the starting sample. Carbon fourteen has one characteristic half-life, whether the sample is a tiny bone fragment or a whole tree. Temperature, pressure, crushing, dissolving, and ordinary chemical reactions usually do not change nuclear decay rates.
This differs from reactions studied earlier in chemistry, where heat or concentration can alter reaction speed. Very unusual physical conditions can affect a few special isotopes, but this is not important for standard school calculations.
Keep the units consistent. If a half-life is given in days, convert the elapsed time to days before calculating.
A useful first step is to decide whether the elapsed time contains a whole number of half-lives. If it does, a repeated halving table is often safer than a calculator. Write the starting amount, then halve it once for each interval.
This method makes it easier to notice mistakes. Students commonly subtract half of the original amount each time. That is incorrect.
The second half-life removes half of what remained after the first one. Another frequent error is mixing up amount remaining with amount decayed. If one quarter remains, three quarters has decayed.
A quick estimate helps. After several half-lives, the remaining amount should be small but not exactly zero.
The exponential form is useful when the time is not an exact multiple of the half-life. The decay constant measures how quickly an isotope decays on a continuous time scale. A larger decay constant means a shorter half-life.
When using logarithms to find an unknown age, check that the starting amount is larger than the measured amount. The resulting time should be positive. Carbon dating needs extra care because living organisms exchange carbon with their surroundings while alive.
After death, that exchange stops, so the carbon fourteen amount can decrease. Scientists compare the remaining carbon fourteen with an expected original level and use calibration data because atmospheric carbon fourteen has changed over time.
Decay is often measured through activity, meaning the number of decays occurring each second. Activity falls with the same overall pattern as the number of undecayed nuclei. This matters in medical imaging, where a tracer must remain detectable long enough for a scan but should not expose the patient for longer than necessary.
It matters for radioactive waste because a material can become far less active after many half-lives, though a tiny remaining fraction may still require careful handling. In problems, identify what is being tracked.
It may be mass, number of atoms, percentage, activity, or a ratio. They all decrease by the same fraction over the same time, so the calculation structure stays consistent.