Quick answer: Radioactive decay is the random change of an unstable nucleus, and a sample decays exponentially with a fixed half-life. Nuclei are lighter than their separate nucleons, and E = mc² turns this mass defect into binding energy. Fission and fusion release energy because the products are more tightly bound per nucleon.
Level: Physics grades 11-12 (A Level · IB · Class 11-12) · Curriculum: IB E.3-E.5
What happens in radioactive decay?
An unstable nucleus changes into a more stable one by emitting radiation. The process is random: nobody can say when one particular nucleus will decay, yet a large sample behaves very predictably. Temperature, pressure and chemical bonding have no effect, because the change happens deep inside the nucleus.
Radioactive decay is the spontaneous and random change of an unstable nucleus into another nucleus (the daughter) with emission of alpha, beta or gamma radiation. A nuclide is written with its nucleon number A (protons plus neutrons) and its proton number Z.
| Decay | Emitted | Change in A | Change in Z | Example |
|---|---|---|---|---|
| alpha (α) | helium-4 nucleus | −4 | −2 | uranium-238 → thorium-234 + α |
| beta-minus (β⁻) | fast electron and antineutrino | 0 | +1 | carbon-14 → nitrogen-14 + e⁻ + antineutrino |
| gamma (γ) | high-energy photon | 0 | 0 | excited daughter nucleus settles down |
In every nuclear equation the total nucleon number A and the total charge (proton number Z) are the same on both sides. Check the top numbers add up, then check the bottom numbers.
How does the decay law work?
Every unstable nucleus has the same small chance of decaying in each second, called the decay constant λ. The number of decays per second, the activity, is therefore proportional to the number of undecayed nuclei, and the sample decays exponentially.
N = N₀ e−λt, A = A₀ e−λt, A = λN, T½ = ln 2 / λ ≈ 0.693 / λ
N = undecayed nuclei, N₀ = initial number, A = activity (Bq, decays per second), λ = decay constant (s⁻¹), t = time (s), T½ = half-life (s).
The half-life is the time for the activity, or the number of undecayed nuclei, to fall to half. After n half-lives a fraction (½)ⁿ remains.
Strontium-90 has a half-life of 28.8 years. What fraction of a sample remains after 100 years?
λ = 0.693 / 28.8 = 0.0241 y⁻¹. Then N / N₀ = e−λt = e−2.41 = 0.0901.
About 9.0 % remains.
After two half-lives the sample has not gone: one quarter remains, not zero. The half-life tells you how fast the sample shrinks, never when it ends.
What is binding energy and how does E = mc² explain it?
A nucleus is lighter than the separate protons and neutrons that make it. The difference is the mass defect Δm, and the energy equivalent of that missing mass was released when the nucleus formed.
The binding energy of a nucleus is the minimum energy needed to separate it completely into free protons and neutrons. It equals the mass defect times c².
E = Δm c², and with 1 u = 931.5 MeV/c², E (MeV) = Δm (u) × 931.5
Δm = mass of separate nucleons − mass of the nucleus (u or kg), c = 3.00 × 10⁸ m/s, 1 MeV = 1.60 × 10⁻¹³ J.
Find the binding energy per nucleon of helium-4, using atomic masses: hydrogen atom 1.007825 u, neutron 1.008665 u, helium-4 atom 4.002603 u.
Mass of separate parts = 2 × 1.007825 + 2 × 1.008665 = 4.032980 u, so Δm = 0.030377 u.
E = 0.030377 × 931.5 = 28.3 MeV, and per nucleon 28.3 / 4 = 7.07 MeV.
Why do fission and fusion release energy?
Energy is released whenever nuclei change into others with a higher binding energy per nucleon. Light nuclei can climb the curve by joining together (fusion). Very heavy nuclei can climb it by splitting (fission).
| Feature | Fission | Fusion |
|---|---|---|
| Process | heavy nucleus splits into two medium ones | light nuclei join into a heavier one |
| Typical reaction | uranium-235 + n → xenon-140 + strontium-94 + 2n | deuterium + tritium → helium-4 + n |
| Energy released | about 200 MeV per nucleus | 17.6 MeV per reaction |
| Condition | slow neutrons trigger a chain reaction | temperatures of 10⁷ to 10⁸ K to beat repulsion |
For any nuclear equation, write A on top and Z at the bottom, then add each row separately. If a row does not balance, a particle is missing. In decay, ask which row changes: only Z changes in beta decay.
- N = N₀ e^(−λt), A = λN and T½ = ln 2 / λ
- After n half-lives a fraction (½)ⁿ remains
- Binding energy E = Δm c², and 1 u = 931.5 MeV/c²
- Iron-56 sits near the peak of binding energy per nucleon, about 8.8 MeV
- Fission and fusion both move nuclei toward higher binding energy per nucleon
Radioactive decay and nuclear binding energy: frequently asked questions
What is half-life?
It is the time taken for the activity of a sample, or the number of undecayed nuclei, to fall to half its value. It is constant for a given nuclide and is not changed by heating, pressure or chemical state.
How are decay constant and half-life related?
T½ = ln 2 / λ, which is about 0.693 divided by λ. A large λ means rapid decay and a short half-life.
What is the mass defect?
It is the amount by which a nucleus is lighter than its separate protons and neutrons. Multiplying it by c² gives the binding energy released when the nucleus formed.
Why is iron-56 the most stable nucleus?
It has about the highest binding energy per nucleon, near 8.8 MeV. Nuclei lighter or heavier than iron can release energy by moving toward it, by fusion or by fission.
How is nuclear fusion different from fission?
Fusion joins light nuclei into a heavier one, while fission splits a heavy nucleus into lighter ones. Both release energy because the products have a higher binding energy per nucleon.
