Radioactive decay is the spontaneous transformation of unstable atomic nuclei into more stable forms, accompanied by the emission of particles or electromagnetic radiation. The process follows an exponential decay law, meaning the rate of decay is proportional to the number of unstable nuclei present.
Radioactive decay is a quantum mechanical process that occurs spontaneously due to nuclear instability. The exponential nature arises because each nucleus has the same probability of decaying per unit time, regardless of the nucleus's age. This reflects the probabilistic nature of quantum mechanics.
Radioactive decay is a first-order kinetic process, meaning the rate of decay is directly proportional to the number of undecayed nuclei. Its properties are statistical and are best described for a large population of atoms.
| Property | Details |
|---|---|
| Nature | The number of nuclei (N), decay constant (λ), and time (t) are all scalar quantities. The decay of any single nucleus is a random, probabilistic event. |
| SI Units | The decay constant (λ) is measured in inverse seconds (s⁻¹). The activity (A), or rate of decay, is measured in Becquerels (Bq), where 1 Bq = 1 decay per second. |
| Key Quantities | The process is characterized by the <strong>decay constant (λ)</strong>, the probability per unit time that a nucleus will decay, and the <strong>half-life (t₁/₂)</strong>, the time required for half of the sample to decay. |
| Governing Law | The number of nuclei N(t) remaining at time t follows an exponential decay law: N(t) = N₀e⁻ˡᵗ, where N₀ is the initial number of nuclei. |
| Conservation Laws | All radioactive decay processes must obey fundamental conservation laws, including the conservation of energy, linear momentum, angular momentum, electric charge, and nucleon number. |
| Dimensional Formula | The dimensional formula for the decay constant (λ) is [T⁻¹]. The number of nuclei (N) is a dimensionless quantity. |
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| \(N(t)\) | Number of nuclei | unitless | Number of radioactive nuclei remaining at time t. |
| \(N_0\) | Initial number of nuclei | unitless | Number of radioactive nuclei present at t = 0. |
| \(\lambda\) | Decay constant | s⁻¹ | Probability per unit time that a nucleus will decay. |
| \(t\) | Time | s | Elapsed time. |
| \(T_{1/2}\) | Half-life | s | Time required for half the nuclei in a sample to decay. |
| \(A(t)\) | Activity | Bq (s⁻¹) | Rate of radioactive decay at time t (decays per second). |
| \(A_0\) | Initial activity | Bq (s⁻¹) | Activity of the sample at t = 0. |
| \(\tau\) | Mean lifetime | s | Average time a nucleus exists before decaying. |
| \(m(t)\) | Mass | kg | Mass of the radioactive material at time t. |
| \(N_A\) | Avogadro's number | mol⁻¹ | 6.022 × 10²³ particles per mole. |
| \(M\) | Molar mass | kg/mol | Mass of one mole of the radioactive isotope. |
The law of radioactive decay is derived from the observation that the rate of decay of a sample of radioactive nuclei is directly proportional to the number of nuclei present. The rate of change of the number of nuclei, \(dN/dt\), is negative because the number of nuclei decreases over time.
Here, \(\lambda\) is the decay constant. This is a first-order separable differential equation. We can rearrange it to separate the variables \(N\) and \(t\):
Next, we integrate both sides. We integrate from the initial number of nuclei \(N_0\) at time \(t=0\) to the number of nuclei \(N(t)\) at a later time \(t\).
Evaluating the integrals gives:
Using the properties of logarithms, we can combine the terms on the left side:
Finally, to solve for \(N(t)\), we exponentiate both sides with base \(e\):
Radioactive decay occurs through several primary modes, distinguished by the type of particle or radiation emitted as the unstable nucleus transforms into a more stable configuration.
| Type / Case | Description | When to Use |
|---|---|---|
| Alpha (α) Decay | Emission of an alpha particle (a helium nucleus, ⁴He). The parent nucleus's mass number decreases by 4 and atomic number decreases by 2. | Typically occurs in very heavy nuclei (e.g., Uranium, Thorium) where the nuclear force cannot hold the large nucleus together against the electrostatic repulsion of the protons. |
| Beta-Minus (β⁻) Decay | Emission of an electron and an electron antineutrino. A neutron in the nucleus is converted into a proton. | Occurs in nuclei that have an excess of neutrons compared to protons (i.e., a high neutron-to-proton ratio). |
| Beta-Plus (β⁺) Decay / Positron Emission | Emission of a positron (the antiparticle of an electron) and an electron neutrino. A proton in the nucleus is converted into a neutron. | Occurs in nuclei that have an excess of protons compared to neutrons (i.e., a low neutron-to-proton ratio). |
| Gamma (γ) Decay | Emission of a high-energy photon (a gamma ray). This process does not change the number of protons or neutrons in the nucleus. | Occurs when a nucleus is in a high-energy, excited state. It transitions to a lower energy state by emitting a gamma ray. This often follows an alpha or beta decay event. |
Carbon-14 dating determines the age of organic materials up to 50,000 years old by measuring the ratio of C-14 to C-12, revolutionizing archaeology and paleontology.
Radioisotopes like Iodine-131 (thyroid treatment), Technetium-99m (imaging), and Cobalt-60 (cancer therapy) are crucial for nuclear medicine diagnostics, treatment, and equipment sterilization.
The decay of heavy elements like Uranium-235 is the basis for nuclear power generation. Additionally, radioisotope thermoelectric generators (RTGs) use the heat from decay of isotopes like Plutonium-238 to power spacecraft and remote stations.
Radioactive tracers are used to study complex environmental processes, such as tracking pollution pathways in groundwater, measuring ocean current circulation, and analyzing sediment deposition rates.
Ionization Smoke Detectors. Many household smoke detectors contain a tiny amount of Americium-241. This isotope undergoes alpha decay, ionizing the air inside a small chamber and allowing a small electric current to flow. When smoke particles enter the chamber, they disrupt this current, which triggers the alarm.
Earth's Internal Heat. A significant portion of the Earth's internal heat, which drives geological processes like plate tectonics and volcanism, is generated by the radioactive decay of long-lived isotopes within the mantle and crust. Elements like Uranium-238, Thorium-232, and Potassium-40 have half-lives of billions of years, acting as a slow-release natural nuclear furnace.
Food Irradiation. To extend shelf life and kill harmful bacteria, some foods are exposed to controlled doses of gamma radiation, often from a Cobalt-60 source. The gamma rays pass through the food, destroying pathogens and insects without making the food itself radioactive. This process is used for spices, poultry, and some fruits.
| Quantity | Symbol | SI Unit | Dimensional Formula |
|---|---|---|---|
| Number of nuclei | \(N\) | unitless | [1] |
| Decay constant | \(\lambda\) | s⁻¹ | [T⁻¹] |
| Time / Half-life | \(t, T_{1/2}\) | second (s) | [T] |
| Activity | \(A\) | Becquerel (Bq) | [T⁻¹] |
| Mass | \(m\) | kilogram (kg) | [M] |
| Energy (Q-value) | \(Q\) | Joule (J) | [M L² T⁻²] |
The formula is N(t) = N₀e^(-λt). It calculates the number of radioactive nuclei, N(t), remaining in a sample after a specific amount of time, t, has passed. This is based on the initial number of nuclei, N₀, and the substance's unique decay constant, λ.
N₀ is the initial number of radioactive nuclei (a dimensionless count) at time t=0. N(t) is the number of nuclei remaining at time t. The decay constant, λ, is the probability of decay per nucleus per unit time, with units of inverse time (e.g., s⁻¹ or yr⁻¹). Time, t, is the elapsed time, measured in units consistent with λ.
In carbon-14 dating, scientists measure the current amount of C-14, N(t), in an organic sample. They compare this to the initial amount, N₀, estimated from atmospheric ratios. Using the known decay constant (λ) for C-14, they solve the decay equation for time (t) to determine how long ago the organism died.
A frequent error is substituting the half-life (T₁/₂) directly for the decay constant (λ) or time (t) in the formula N(t) = N₀e^(-λt). The decay constant must first be calculated from the half-life using the relationship λ = ln(2) / T₁/₂. Using the half-life incorrectly will produce a mathematically incorrect result.
Radioactive decay is fundamental to nuclear medicine, particularly in PET scans and radiation therapy. For instance, Technetium-99m is used as a tracer for imaging because its decay characteristics and short half-life can be calculated to provide a safe, effective dose that minimizes patient exposure while being detectable by scanners.
The exponential decay model is not unique to atomic physics. It describes any process where the rate of change is proportional to the current amount. This same mathematical form is used in electromagnetism to model the voltage decay in a discharging capacitor (V(t) = V₀e^(-t/RC)) and in thermodynamics for Newton's law of cooling.