Radioactive Decay

Introduction

students, every atom in nature is made of a tiny nucleus surrounded by electrons, but some nuclei are unstable. When an unstable nucleus changes into a more stable one by emitting particles or electromagnetic radiation, it is called radioactive decay ☢️. This lesson explains the main ideas and terms, how to use the half-life model, and how radioactive decay connects to the bigger picture of nuclear and quantum physics.

Learning objectives:

Explain the key ideas and terminology behind radioactive decay.
Use IB Physics HL reasoning to solve radioactive decay problems.
Connect radioactive decay to nuclear structure, quantum ideas, fission, and fusion.
Describe evidence and examples that show how radioactive decay is observed in real life.

Radioactive decay is important because it is random for any single nucleus, but predictable for a large sample. That mix of chance and pattern is one of the most important ideas in nuclear physics.

What radioactive decay is

An unstable nucleus changes because its current arrangement of protons and neutrons does not give enough binding stability. The nucleus may emit a particle or energy to become more stable. This process happens on its own, without being started by temperature, pressure, or chemical reactions.

In chemistry, atoms can form new substances by rearranging electrons. In radioactive decay, the nucleus itself changes, so the element may change into a different element or into a different isotope of the same element.

The main types of radioactive decay you should know are:

Alpha decay: the nucleus emits an alpha particle, which is a helium nucleus, $\,^{4}_{2}\text{He}\,$.
Beta minus decay: a neutron changes into a proton, and the nucleus emits an electron, $\beta^{-}$, and an antineutrino.
Gamma decay: the nucleus emits a high-energy photon, $\gamma$, without changing its proton or neutron numbers.

A useful idea is the nuclear equation. For example, in alpha decay:

$$\,^{238}_{92}\text{U} \rightarrow \,^{234}_{90}\text{Th} + \,^{4}_{2}\text{He}$$

The mass number and atomic number must balance on both sides. In every nuclear equation, total nucleon number and charge are conserved.

Alpha decay

Alpha particles contain $2$ protons and $2$ neutrons. Because they are relatively heavy and carry charge, they interact strongly with matter and ionize atoms efficiently. This means alpha radiation is highly ionizing but weakly penetrating. A sheet of paper or a few centimeters of air can stop most alpha particles.

Alpha decay often happens in very heavy nuclei, such as uranium or radium, because the nucleus is too large to remain stable. By losing an alpha particle, the nucleus becomes smaller and often more stable.

Example:

$$\,^{226}_{88}\text{Ra} \rightarrow \,^{222}_{86}\text{Rn} + \,^{4}_{2}\text{He}$$

Here, the mass number decreases by $4$ and the atomic number decreases by $2$.

Beta minus decay

In beta minus decay, a neutron changes into a proton inside the nucleus:

$$n \rightarrow p + e^{-} + \bar{\nu}_{e}$$

The nucleus gains one proton, so the atomic number increases by $1$, while the mass number stays the same. Beta particles are lighter than alpha particles, so they are less ionizing but more penetrating. They can usually pass through paper but are stopped by a thin sheet of aluminum.

Example:

$$\,^{14}_{6}\text{C} \rightarrow \,^{14}_{7}\text{N} + e^{-} + \bar{\nu}_{e}$$

This is the basis of carbon dating, where the amount of carbon-14 in an old sample is compared with the amount expected in living material.

Gamma decay

After alpha or beta decay, the nucleus may still be in an excited state. It can release extra energy as a gamma photon:

$$\,^{*}\text{X} \rightarrow \text{X} + \gamma$$

Gamma radiation has no mass and no charge, so it is weakly ionizing but highly penetrating. Thick lead or concrete is needed to reduce it significantly. Gamma decay does not change the mass number or atomic number; it only lowers the energy of the nucleus.

Randomness, decay constant, and half-life

Radioactive decay is random for one nucleus, meaning you cannot predict the exact moment a particular nucleus will decay. However, for a large number of nuclei, the overall decay rate follows a reliable pattern.

The number of undecayed nuclei $N$ decreases with time according to:

$$N = N_{0}e^{-\lambda t}$$

where:

$N_{0}$ is the initial number of undecayed nuclei,
$\lambda$ is the decay constant,
$t$ is time.

The decay constant $\lambda$ is the probability per unit time that a nucleus will decay. A larger $\lambda$ means a substance decays faster.

The activity $A$ is the rate of decay, or the number of decays per second:

$$A = \lambda N$$

The SI unit of activity is the becquerel, $\text{Bq}$, where $1\,\text{Bq} = 1\,\text{decay s}^{-1}$.

Half-life

The half-life $T_{1/2}$ is the time taken for half of the radioactive nuclei in a sample to decay. It is related to the decay constant by:

$$T_{1/2} = \frac{\ln 2}{\lambda}$$

Half-life is a very useful concept because it is constant for a given isotope and does not depend on how much of the isotope you start with.

If a sample starts with $N_{0}$ nuclei, after one half-life the number remaining is:

$$N = \frac{N_{0}}{2}$$

After two half-lives:

$$N = \frac{N_{0}}{4}$$

After $n$ half-lives:

$$N = N_{0}\left(\frac{1}{2}\right)^{n}$$

Example: If a sample of iodine-131 has an initial activity of $800\,\text{Bq}$ and a half-life of $8$ days, then after $16$ days the activity will be:

$$A = 800\left(\frac{1}{2}\right)^{2} = 200\,\text{Bq}$$

This works because activity is proportional to the number of undecayed nuclei.

Applications and evidence in the real world

Radioactive decay is measured using detectors such as Geiger-Müller tubes, scintillation detectors, and cloud chambers. These devices show the presence of radiation by counting events or displaying tracks. In a Geiger counter, each detected decay produces a pulse, which helps measure activity.

Real-world uses include:

Medical tracers: small amounts of radioactive isotopes help image organs and track body processes.
Cancer treatment: radiation can damage the DNA of cancer cells.
Carbon dating: uses the known half-life of carbon-14 to estimate the age of once-living material.
Industrial inspection: radiation can detect cracks or flaws in metal parts.

A key IB idea is that radioactive decay is not affected by normal chemical changes. Heating, cooling, pressure changes, or mixing with other substances does not significantly alter the decay rate of a nucleus. This is because decay is a nuclear process, not an electronic one.

Example reasoning question: If a sealed radioactive source is placed in different temperatures, its half-life stays the same. The atoms in the source may move faster or slower, but the nucleus and the forces inside it are not changed by ordinary temperature changes.

Radioactive decay in the larger nuclear and quantum picture

Radioactive decay is part of nuclear physics because it deals with the structure and stability of the nucleus. It is also linked to quantum physics because decay is fundamentally probabilistic. The exact time when one nucleus decays cannot be predicted, only the probability that it will decay in a given time interval.

This probabilistic nature is related to quantum theory, where some processes happen by chance rather than by deterministic motion. In nuclear decay, the nucleus can be thought of as a system with a certain probability of escaping its unstable state.

Radioactive decay also connects to nuclear fission and fusion:

In fission, a heavy nucleus splits into smaller nuclei, releasing energy and often more neutrons.
In fusion, light nuclei combine to form a heavier nucleus, also releasing energy.

Both processes depend on nuclear binding energy and stability, just like radioactive decay. A nucleus becomes more stable when it moves toward a lower-energy arrangement.

Solving IB-style problems

When answering IB Physics HL questions on radioactive decay, students, use these steps:

Identify the type of decay from the change in atomic number and mass number.
Write a balanced nuclear equation.
Use the half-life or decay equation if the problem involves time.
Check units carefully, especially for activity in $\text{Bq}$ and time in seconds when needed.
Explain your reasoning clearly using nuclear terminology.

Example problem: A radioactive sample has $N_{0} = 1200$ nuclei and a half-life of $6\,\text{h}$. How many nuclei remain after $18\,\text{h}$?

Since $18\,\text{h}$ is $3$ half-lives:

$$N = 1200\left(\frac{1}{2}\right)^{3} = 150$$

So $150$ nuclei remain.

Another common task is reading a decay curve. A graph of $N$ versus $t$ falls steeply at first and then levels off. The curve is not linear because each nucleus has the same probability of decay at any moment, so fewer nuclei remain to decay later.

Conclusion

Radioactive decay is the spontaneous change of an unstable nucleus into a more stable one, often by emitting alpha, beta, or gamma radiation. It is random for individual nuclei but follows clear mathematical laws for large samples. The ideas of decay constant, activity, and half-life make it possible to predict how a radioactive sample changes over time.

This topic is central to Nuclear and Quantum Physics because it shows how nuclei behave, how energy is released, and how quantum probability helps explain natural processes. Understanding radioactive decay gives students a strong foundation for fission, fusion, atomic structure, and practical applications in medicine, dating, and industry.

Study Notes

Radioactive decay is a spontaneous change in an unstable nucleus.
It is a nuclear process, so it is not affected by ordinary chemical changes, temperature, or pressure.
Alpha decay emits $\,^{4}_{2}\text{He}\,$ and reduces mass number by $4$ and atomic number by $2$.
Beta minus decay emits $e^{-}$ and $\bar{\nu}_{e}$, increasing atomic number by $1$ with no change in mass number.
Gamma decay emits a photon $\gamma$ and does not change mass number or atomic number.
Nuclear equations must conserve mass number and charge.
Radioactive decay is random for one nucleus but predictable for large numbers of nuclei.
The decay law is $N = N_{0}e^{-\lambda t}$.
Activity is $A = \lambda N$ and is measured in $\text{Bq}$.
Half-life is $T_{1/2} = \frac{\ln 2}{\lambda}$.
After $n$ half-lives, $N = N_{0}\left(\frac{1}{2}\right)^{n}$.
Radioactive decay is used in medical tracers, cancer treatment, carbon dating, and industrial testing.
The topic connects nuclear stability with quantum probability and helps explain fission and fusion.