Radioactive decay is a spontaneous process in which an unstable atomic nucleus emits radiation and transforms into another nucleus. This phenomenon underlies archaeological dating, nuclear medicine, and nuclear power. The key concept describing the pace of this process is half-life - the time after which half of the initial amount of a radioactive substance has decayed. Understanding the math behind radioactive decay opens the door to many practical applications.
What is radioactive decay?
Some atomic nuclei are unstable - they have too many protons, too many neutrons, or simply too much energy. To reach a more stable state, they emit particles or electromagnetic radiation. There are three main types of decay:
- Alpha decay - the nucleus emits an alpha particle (2 protons + 2 neutrons), losing 4 in mass number and 2 in atomic number.
- Beta decay - a neutron converts into a proton (or vice versa) with emission of an electron or positron and a neutrino.
- Gamma radiation - the nucleus releases excess energy in the form of a gamma photon, without any change in the number of protons or neutrons.
The half-life law - the formula
Radioactive decay is a random process at the level of a single atom, but for a huge number of atoms it behaves in a statistically predictable way. The amount of substance decreases exponentially:
- N(t) = N0 x (1/2)^(t / T½)
- N(t) - the amount of substance (or activity) after time t
- N0 - the initial amount of substance
- t - the time elapsed
- T½ - the half-life (characteristic of each isotope)
The formula can also be written using the decay constant λ: N(t) = N0 x e^(-λt), where λ = ln(2) / T½.
Examples of isotopes and their half-lives
Half-lives vary astronomically depending on the isotope:
- Polonium-214 - T½ = 0.000164 s (near-instant decay)
- Iodine-131 - T½ = 8 days (used in thyroid therapy)
- Cobalt-60 - T½ = 5.27 years (a source of gamma radiation in radiotherapy)
- Carbon-14 - T½ = 5,730 years (radiocarbon dating)
- Plutonium-239 - T½ = 24,110 years (nuclear fuel, radioactive waste)
- Uranium-238 - T½ = 4.47 billion years (comparable to the age of the Earth)
Worked example
We have an iodine-131 sample with an activity of 800 MBq. After how many days will the activity drop to 100 MBq?
We're looking for t, at which N(t)/N0 = 100/800 = 1/8 = (1/2)^3. So we need three halvings: t = 3 x T½ = 3 x 8 = 24 days.
Checking with the formula: N(24) = 800 x (1/2)^(24/8) = 800 x (1/2)^3 = 800 x 0.125 = 100 MBq. The result checks out.
Radiocarbon dating - a practical example
Carbon-14 is created in the atmosphere by cosmic radiation and enters the biological cycle. Living organisms constantly exchange carbon with the environment, so the ratio of C-14 to C-12 in a living organism is constant. After death, the exchange stops and C-14 begins to decay. By measuring the ratio of C-14 to C-12 in a sample, we can calculate when the organism was alive. The method is effective for objects up to 50,000 years old - older samples have too little C-14 to measure accurately.
Applications of radioactive decay
Nuclear medicine
Isotopes with a short half-life, such as technetium-99m (T½ = 6 h) or fluorine-18 (T½ = 110 min), are used in imaging diagnostics (scintigraphy, PET). A short half-life means the radioactivity fades quickly, minimizing patient exposure.
Nuclear power
In nuclear reactors, radioactive decay generates heat. Managing radioactive waste requires accounting for half-lives - some fission products decay within seconds, others remain hazardous for thousands of years.
FAQ
1. What is radioactive activity? Activity is the number of decays per second, measured in becquerels (Bq) or curies (Ci).
2. Does half-life depend on temperature or pressure? No, it's a nuclear constant independent of external conditions.
3. What is the decay constant? The decay constant λ = ln(2) / T½ and expresses the probability of one atom decaying per unit of time.
4. After how many half-lives is a substance safe? After 10 x T½, less than 0.1% of the initial amount remains - a practical safety threshold.
5. Can decay be accelerated or slowed? Not under normal conditions. Extremely strong electromagnetic fields can slightly change the pace, but the effect is negligible.
6. What is a decay chain? A decay product is often itself radioactive and undergoes further transformations - forming a decay series.
7. What's the difference between an isotope and an element? Isotopes of the same element have the same number of protons, but a different number of neutrons.
8. How does an atomic bomb work compared to a reactor? In a bomb, fission proceeds uncontrolled (a chain reaction), while in a reactor it's controlled with moderators and control rods.
9. What is background radiation? Natural radiation from rocks, space, and our own bodies, present everywhere on Earth.
10. Can I calculate half-life from activity measurements? Yes, by measuring activity at two points in time and applying a logarithmic formula.
To calculate the remaining amount of a substance or its activity after any period of time, use the radioactive decay calculator on Liczbnik.pl.