Topic 15.7 Notes – Fission, Fusion, and Nuclear Decay

1. What Governs Nuclear Reactions

The strong nuclear force

Inside the nucleus, protons and neutrons (nucleons) are held together by the strong nuclear force.

• Acts over an extremely short range, about $1\text{-}3$ femtometers.
• At that distance, it is stronger than electric repulsion between protons.
• Beyond that range, it drops off quickly.

This short range explains why:

• Small and medium nuclei are stable.
• Very large nuclei can become unstable because distant protons don’t “feel” enough strong force to stay bound.
• The force shows saturation. Each nucleon mainly interacts with nearby neighbors, not the entire nucleus.

If a free-response question asks why large nuclei are unstable, talk about the short range of the strong force plus increasing proton-proton repulsion.

Conservation laws in nuclear reactions

Every nuclear reaction must obey:

• Conservation of nucleon number
  Total mass number $A$ (protons + neutrons) stays the same.
• Conservation of charge
  Total atomic number $Z$ stays the same.
• Conservation of energy
  Includes rest mass energy and kinetic energy.
• Conservation of momentum
  Products recoil to balance momentum.

Mass and energy are connected by:

$$E = mc^{2}$$

If total mass decreases, that missing mass appears as energy. Often as:

• Kinetic energy of fragments
• Gamma photons

On tests, balancing $A$ and $Z$ is usually the first step before identifying an unknown particle.

2. Binding Energy and Why Energy Is Released

Mass defect and binding energy

A bound nucleus has less mass than its separated protons and neutrons.

That difference is the mass defect $\Delta m$.

Binding energy:

$$E_{b} = \Delta m c^{2}$$

This is the energy required to pull the nucleus apart. It is also the energy released when the nucleus forms.

Binding energy per nucleon

What matters for stability is binding energy per nucleon.

The graph below shows binding energy per nucleon as a function of mass number $A$.

Binding energy per nucleon vs. mass number

• It peaks near iron-56 (A ≈ 56), labeled as the most stable nucleus.
• Nuclei tend to move toward that peak.
• Reactions that move nuclei toward higher binding energy per nucleon release energy.

That single graph explains both fusion and fission.

3. Fusion and Fission

Nuclear fusion

Fusion combines light nuclei into a heavier nucleus.

• Requires very high temperature.
• Needed to overcome Coulomb repulsion between positive nuclei.
• Occurs in stars.

Why energy is released:

• Light nuclei (left side of the graph) combine.
• Product has higher binding energy per nucleon.
• Total mass decreases → energy released via $E = mc^{2}$.

Nuclear fission

Fission splits a heavy nucleus into smaller nuclei plus neutrons.

• Common for very heavy elements.
• Products are closer to iron on the graph.

Why energy is released:

• Heavy nucleus has relatively low binding energy per nucleon.
• Products are more tightly bound.
• Mass decreases → energy released.

Spontaneous vs induced fission

• Spontaneous fission happens naturally in very unstable heavy nuclei.
• Induced fission happens after absorbing a neutron.
  • Absorption makes nucleus unstable.
  • Often releases 2-3 neutrons.

Those emitted neutrons can trigger more fissions.

If each event causes at least one more, you get a chain reaction:

• Controlled → nuclear reactor.
• Uncontrolled → explosion.

When explaining this in writing, mention neutron multiplication and conservation of energy.

4. Radioactive Decay and Half-Life

What radioactive decay is

Radioactive decay is a spontaneous transformation of an unstable nucleus.

• Exact decay time of one nucleus is unpredictable.
• Large samples follow predictable statistics.
• This is fundamentally probabilistic.

Half-life

The half-life $t_{1/2}$ is the time for half the nuclei to decay.

After:

• 1 half-life → 50% remains
• 2 half-lives → 25%
• 3 half-lives → 12.5%

Equation:

$$N = N_{0} \left(\tfrac{1}{2}\right)^{t/t_{1/2}}$$

Half-lives vary wildly, from fractions of a second to billions of years.

Decay constant and exponential form

The decay constant $\lambda$ is the probability per unit time that a nucleus decays.

Relationship:

$$\ln 2 = \lambda t_{1/2}$$

Exponential form:

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

Activity:

$$A = \lambda N$$

Larger $\lambda$ means shorter half-life and faster decay.

On AP problems, you’ll either:

• Count half-lives, or
• Use the exponential form with $\lambda = \frac{\ln 2}{t_{1/2}}$.