Radioactive decay is the process in which an unstable atomic nucleus changes into a more stable nucleus by emitting particles or energy. The time pattern of this decay is predictable for a large sample, even though no one can predict exactly when a single atom will decay. Half-life is the key idea because it tells how long it takes for half of a radioactive sample to decay.
This makes half-life useful in medicine, archaeology, geology, and nuclear physics.
Radiometric dating uses the fraction of a radioactive parent isotope remaining in a sample to estimate how much time has passed. Carbon-14 dating is used for once-living materials because living organisms keep exchanging carbon with the environment, but after death the carbon-14 begins to decrease. The decay follows an exponential curve, so equal time intervals remove equal fractions, not equal amounts.
By measuring parent and daughter isotopes, scientists can calculate ages from thousands to billions of years depending on the isotope used.
Understanding Physics: Half-Life and Radiometric Dating
Inside an unstable nucleus, the balance of forces is not quite right. The strong nuclear force holds protons and neutrons together at very short distances. Electric repulsion pushes protons apart.
Some arrangements have excess energy or an unhelpful mix of particles, so the nucleus can change into a lower-energy arrangement. It may release an alpha particle, a beta particle, or a gamma ray. These are different processes, with different effects on atomic number and mass.
An alpha emission changes the element because the nucleus loses two protons. Beta decay can turn a neutron into a proton, or a proton into a neutron. Gamma emission releases energy without changing the numbers of protons or neutrons.
The decay of one nucleus is random, but a large collection behaves in a very regular way. Each undecayed nucleus has the same chance of decaying during a short time interval. That chance does not grow because the nucleus has existed for a long time.
This is unlike a battery or a candle, which changes as it is used. As the sample gets smaller, fewer nuclei are available to decay each second. Its activity therefore falls over time.
A graph of activity against time curves downward. It drops quickly at first, then more slowly. This pattern is important in hospitals, where radioactive tracers must give a useful signal while keeping radiation exposure as low as practical.
Radiometric dating works best when scientists understand the history of a sample. They need a suitable parent isotope, a measurable daughter product, and evidence that the material has stayed mostly closed since it formed. A rock that gains or loses atoms through heating, weathering, or groundwater can give a misleading age.
Scientists may choose minerals whose crystal structures trap certain atoms well. They may date several minerals from the same rock, compare different isotope systems, and check the result against the rock layers around it.
The calculated age usually describes a particular event, such as when a mineral crystal cooled enough to lock its atoms in place. It does not always mean the age of the whole rock material.
Carbon dating has its own limits. Plants take in carbon while alive, and animals obtain carbon by eating. Once an organism dies, this exchange stops.
The remaining carbon-14 can then act as a clock, but contamination from soil, glue, preservatives, or newer plant roots can alter the measurement. Very old samples contain so little carbon-14 that the uncertainty becomes large. Students should pay close attention to units, powers of two, and the difference between amount remaining and amount decayed.
After one half-life, half remains. After two half-lives, one quarter remains, not zero. A small remaining fraction can represent many half-lives, which is why the curve never reaches zero in the simple model.
Key Facts
- Half-life t1/2 is the time required for half of the radioactive nuclei in a sample to decay.
- Number remaining after time t: N = N0(1/2)^(t/t1/2).
- Activity also decreases exponentially: A = A0(1/2)^(t/t1/2).
- Decay constant relation: t1/2 = ln(2)/lambda.
- Carbon-14 has a half-life of about 5730 years and is useful for dating once-living materials.
- After n half-lives, the fraction remaining is (1/2)^n, so 3 half-lives leaves 1/8 of the original parent isotope.
Vocabulary
- Half-life
- The time it takes for half of the radioactive nuclei in a sample to decay.
- Parent isotope
- The original radioactive isotope that decays into a different nucleus.
- Daughter isotope
- The product nucleus formed when a parent isotope undergoes radioactive decay.
- Decay constant
- A probability rate, usually represented by lambda, that describes how likely each nucleus is to decay per unit time.
- Radiometric dating
- A method for estimating the age of a sample by measuring radioactive parent isotopes and their decay products.
Common Mistakes to Avoid
- Treating half-life as a countdown for every atom is wrong because individual decay events are random and half-life only predicts the behavior of large samples.
- Subtracting the same amount each half-life is wrong because exponential decay removes the same fraction each half-life, not the same number of nuclei.
- Using carbon-14 to date very old rocks is wrong because carbon-14 is best for once-living materials and its half-life is too short for most geological ages.
- Forgetting to convert time units is wrong because t and t1/2 must be in the same units before using N = N0(1/2)^(t/t1/2).
Practice Questions
- 1 A 80.0 g sample of a radioactive isotope has a half-life of 5.0 days. How many grams remain after 15.0 days?
- 2 A fossil has 25.0 percent of its original carbon-14 remaining. Using a carbon-14 half-life of 5730 years, estimate the age of the fossil.
- 3 Two radioactive isotopes have the same starting mass, but isotope A has a shorter half-life than isotope B. Explain which sample has the greater activity at the beginning and which sample will have more parent isotope remaining after several half-lives of isotope A.