Nuclear Decay Calculator

Understanding Nuclear Decay

Nuclear decay, also called radioactive decay, is the process by which unstable atomic nuclei lose energy by emitting radiation. This radiation can be in the form of alpha particles, beta particles, or gamma rays. The process is spontaneous and governed by probability, meaning we cannot predict when a single nucleus will decay, but we can predict the average behavior of a large collection of nuclei with high precision.

Exponential Nature of Radioactive Decay

The fundamental law of radioactive decay states that the number of undecayed nuclei decreases exponentially with time:

N = N₀ e^-λt

where:

• N = number of undecayed nuclei at time t
• N₀ = initial number of nuclei
• λ = decay constant (probability of decay per unit time)
• t = elapsed time

Half-Life

The half-life (T₁/₂) is the time required for half of the radioactive sample to decay. It is related to the decay constant by:

T₁/₂ = ln(2) / λ

Different isotopes have half-lives ranging from fractions of a second to billions of years. For example, Carbon-14 has a half-life of about 5730 years, making it useful in archaeological dating.

Types of Radioactive Decay

• Alpha Decay: The nucleus emits an alpha particle (2 protons, 2 neutrons), reducing its atomic number by 2 and mass number by 4.
• Beta Decay: A neutron is converted into a proton (β⁻ decay) or a proton into a neutron (β⁺ decay), accompanied by emission of electrons or positrons.
• Gamma Decay: The nucleus releases energy as high-energy photons without changing the number of protons or neutrons.

Applications of Nuclear Decay

• Carbon Dating: Archaeologists use carbon-14 decay to determine the age of fossils and artifacts.
• Medical Imaging: Radioisotopes such as Technetium-99m are used in PET and SPECT scans.
• Energy Production: Controlled radioactive decay in nuclear reactors generates electricity.
• Environmental Science: Tracing isotopes helps track pollutants and study climate history.
• Astrophysics: Decay of heavy elements provides clues about supernova explosions and the formation of the universe.

Worked Example

Suppose we start with 100 grams of a radioactive isotope with a half-life of 10 years. The decay constant is:

λ = ln(2) / 10 ≈ 0.0693 year⁻¹

After 30 years:

N = N₀ e^-λt = 100 × e^{-0.0693 × 30} ≈ 12.5 grams

Thus, only 12.5% of the original sample remains.

Historical Importance

Radioactivity was first discovered by Henri Becquerel in 1896 and later studied extensively by Marie and Pierre Curie. Their work not only revealed new elements like polonium and radium but also paved the way for nuclear physics and quantum theory. The exponential decay law has since become a cornerstone in both theoretical and applied physics.

Philosophical Implications

Radioactive decay highlights the role of probability in the physical world. Unlike classical mechanics, where outcomes can be predicted with certainty, quantum processes like decay are fundamentally probabilistic. This has reshaped our understanding of determinism in nature.

Conclusion

The study of nuclear decay bridges fundamental physics and practical applications. From powering nuclear plants to dating ancient relics, its impact is profound. Tools like this calculator allow students, researchers, and engineers to model decay processes quickly, deepening their appreciation of how atomic instability shapes both technology and the natural world.