Radioactive Decay
Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration by emitting radiation in the form of particles or electromagnetic waves. This process occurs because the nucleus has too many protons, too many neutrons, or excess energy, making it thermodynamically unstable. It is the foundation of nuclear medicine, radiometric dating, and nuclear power generation.
Key Formula
N(t) = N₀ × e^(−λt)
LaTeX: N(t) = N_0 \, e^{-\lambda t}
| Symbol | Meaning | Unit |
|---|---|---|
| N(t) | Number of undecayed nuclei at time t | dimensionless |
| N₀ | Initial number of undecayed nuclei | dimensionless |
| λ | Decay constant | s⁻¹ |
| t | Elapsed time | s |
Worked Example
Problem
A radioactive sample initially contains 8.0 × 10¹² atoms of an isotope with a decay constant λ = 1.39 × 10⁻⁴ s⁻¹. How many atoms remain after 1.0 × 10⁴ s?
Solution
Step 1: Identify values. N₀ = 8.0 × 10¹² atoms, λ = 1.39 × 10⁻⁴ s⁻¹, t = 1.0 × 10⁴ s. Step 2: Apply the decay equation: N(t) = N₀ × e^(−λt). Step 3: Calculate the exponent: λt = 1.39 × 10⁻⁴ × 1.0 × 10⁴ = 1.39. Step 4: Evaluate: N(t) = 8.0 × 10¹² × e^(−1.39) = 8.0 × 10¹² × 0.2491.
Answer
N(t) ≈ 1.99 × 10¹² atoms
Types of Radioactive Decay and Their Properties
| Decay Type | Particle Emitted | Charge Change | Mass Change | Example |
|---|---|---|---|---|
| Alpha (α) | Helium-4 nucleus | −2 | −4 u | U-238 → Th-234 |
| Beta-minus (β⁻) | Electron + antineutrino | +1 | ≈0 | C-14 → N-14 |
| Beta-plus (β⁺) | Positron + neutrino | −1 | ≈0 | Na-22 → Ne-22 |
| Gamma (γ) | Photon | 0 | 0 | Co-60* → Co-60 |
| Electron Capture | Neutrino emitted | −1 | ≈0 | Fe-55 → Mn-55 |
Interactive Tools
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Related Terms
Half-Life
Half-life (t₁/₂) is the time required for exactly half of the radioactive atoms in a sample to undergo decay, reducing the number of undecayed nuclei to 50% of the original count. It is a constant property of each radioactive isotope, independent of temperature, pressure, or chemical state, ranging from microseconds for highly unstable nuclei to billions of years for stable isotopes. Half-life is essential in radiometric dating, nuclear medicine dosage, and radioactive waste management.
Alpha Decay
Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle — a helium-4 nucleus consisting of two protons and two neutrons — thereby reducing its atomic number by 2 and its mass number by 4. This process is common in heavy nuclei (Z > 82) such as uranium and radium, where the nuclear repulsion between protons becomes too great to maintain stability. Alpha particles have low penetrating power and can be stopped by a sheet of paper, but are highly ionising and dangerous if ingested or inhaled.
Beta Decay
Beta decay is a type of radioactive decay mediated by the weak nuclear force, in which a neutron converts to a proton (beta-minus decay, emitting an electron and an antineutrino) or a proton converts to a neutron (beta-plus decay, emitting a positron and a neutrino). Unlike alpha decay, the mass number of the nucleus remains unchanged, but the atomic number increases or decreases by one. Beta decay is responsible for the natural transmutation of elements and is exploited in positron emission tomography (PET scanning) and food irradiation.
From Latin "radius" (ray) + "activus" (active). The term "radioactivity" was coined by Marie Curie in 1898 to describe the spontaneous emission of rays she observed from uranium and thorium compounds.