explain the relevance of binding energy per nucleon to nuclear reactions, including nuclear fusion and nuclear fission

Mass Defect, Nuclear Binding Energy & Their Relevance to Nuclear Reactions

1. Nuclear Notation \(^{A}_{Z}\!X\)

• A – mass number (total nucleons \(A = Z+N\)).
• Z – atomic number (number of protons).
• X – chemical symbol of the element.

Example: ²³⁸₉₂U represents a uranium nucleus with A = 238, Z = 92 and N = 146 neutrons.

2. Mass Defect (Δm)

The “missing” mass when free nucleons combine to form a nucleus is called the mass defect:

\[

\Delta m = Z\,m{p}+N\,m{n}-m_{\text{nucleus}}

\]

• \(m_{p}=1.007276\;\text{u}\) (proton mass)
• \(m_{n}=1.008665\;\text{u}\) (neutron mass)
• \(m_{\text{nucleus}}\) – measured atomic mass (including the electrons; the electron‑mass contribution is negligible for A‑level work).

2.1 Converting Δm to Energy

Einstein’s relation \(E=mc^{2}\) together with the atomic‑mass unit conversion gives:

\[

1\;\text{u}=931.5\;\text{MeV}\,c^{-2}\quad\Longrightarrow\quad

E_{b}= \Delta m\,c^{2}= \Delta m\;(\text{u})\times 931.5\;\text{MeV}

\]

2.2 Worked Example (from atomic‑mass data)

The calculation shows how the “missing” mass of 0.030 u becomes the 28.3 MeV that holds the \(^4_2\)He nucleus together.

3. Nuclear Binding Energy (\(E_{b}\))

Binding energy is the energy required to separate a nucleus into its constituent protons and neutrons. It is numerically equal to the mass defect converted to energy (see §2).

4. Binding Energy per Nucleon (\(E_{b}/A\))

To compare nuclei of different size we use the average binding energy per nucleon:

\[

\frac{E{b}}{A}= \frac{E{b}}{Z+N}

\]

This quantity tells, on average, how tightly each nucleon is bound.

4.1 Sample Data

4.2 Binding‑Energy‑per‑Nucleon Curve

Why the curve has its shape

• Rising region (A ≲ 56): Adding nucleons increases the short‑range strong nuclear force faster than the electrostatic repulsion, so each nucleon becomes more tightly bound.
• Falling region (A ≳ 56): In heavy nuclei the long‑range Coulomb repulsion between many protons outweighs the additional strong‑force attraction, reducing the average binding per nucleon.

5. Relevance to Nuclear Reactions

The curve tells us which reactions can release energy. A reaction is energetically favourable when the total binding energy of the products exceeds that of the reactants.

5.1 Q‑value of a reaction

\[

Q = \sum{\text{products}}E{b} - \sum{\text{reactants}}E{b}

\]

Because \(E{b}= (E{b}/A)\times A\), the same result can be obtained using per‑nucleon values.

5.2 Fusion (light nuclei, \(A\lesssim56\))

Two or more light nuclei combine to form a heavier nucleus that lies higher on the curve → \(E_{b}/A\) increases → excess energy is liberated.

Example: Deuterium–Tritium Fusion

\[

^{2}{1}\text{H}+^{3}{1}\text{H}\;\rightarrow\;^{4}{2}\text{He}+^{1}{0}\text{n}+Q

\]

• \(E_{b}(^{2}\text{H})\approx2.2\) MeV
• \(E_{b}(^{3}\text{H})\approx8.5\) MeV
• \(E_{b}(^{4}\text{He})=28.3\) MeV
• Neutron has \(E_{b}=0\).

Binding before = 2.2 + 8.5 = 10.7 MeV

Binding after = 28.3 MeV

\(Q = 28.3-10.7 \approx 17.6\) MeV (energy released).

5.3 Fission (heavy nuclei, \(A\gtrsim56\))

A heavy nucleus splits into lighter fragments that lie higher on the curve → \(E_{b}/A\) increases → excess energy is liberated.

Example: Neutron‑induced Fission of \(^{235}_{92}\)U

\[

^{235}{92}\text{U}+^{1}{0}\text{n}\;\rightarrow\;^{92}{36}\text{Kr}+^{141}{56}\text{Ba}+3\,^{1}_{0}\text{n}+Q

\]

• Average \(E_{b}/A\) for \(^{235}\)U ≈ 7.6 MeV.
• Average \(E_{b}/A\) for the fission fragments (Kr, Ba) ≈ 8.5 MeV.

Total binding before = 236 × 7.6 ≈ 1793 MeV

Total binding after = (92 + 141 + 3) × 8.5 ≈ 1996 MeV

\(Q \approx 1996-1793 \approx 200\) MeV released.

6. Radioactive Decay (Cambridge AS & A‑Level 11.1)

6.1 Types of Nuclear Radiation

• α‑radiation: emission of a helium‑4 nucleus \((^{4}_{2}\text{He})\). Mass number ↓ 4, atomic number ↓ 2.
• β⁻‑radiation: emission of an electron \((^{0}_{-1}\text{e})\) when a neutron converts to a proton. Mass number unchanged, atomic number ↑ 1.
• β⁺‑radiation (positron emission): emission of a positron \((^{0}_{+1}\text{e})\) when a proton converts to a neutron. Mass number unchanged, atomic number ↓ 1.
• γ‑radiation: high‑energy photon; no change in A or Z, but removes excess nuclear excitation energy.

All three particle radiations obey conservation of nucleon number (A) and charge (Z). γ‑radiation conserves both automatically.

6.2 Activity, Decay Constant and Half‑life

• Activity (A) – number of decays per second: \(A = \lambda N\).
• Decay constant (\(\lambda\)) – probability per unit time that a nucleus will decay.
• Half‑life (\(t_{1/2}\)) – time for half the original nuclei to decay: \[

  t_{1/2}= \frac{\ln 2}{\lambda}= \frac{0.693}{\lambda}

  \]

• Exponential decay law: \[

  N(t)=N{0}\,e^{-\lambda t}=N{0}\left(\frac{1}{2}\right)^{t/t_{1/2}}

  \]

6.3 Example Calculation – Carbon‑14

Given \(t{1/2}=5730\) yr for \(^{14}{6}\)C, find the activity of a 1 g sample (≈\(5.0\times10^{22}\) atoms).

\[

\lambda = \frac{0.693}{5730\ \text{yr}} = 3.82\times10^{-12}\ \text{s}^{-1}

\]

\[

A = \lambda N = 3.82\times10^{-12}\times5.0\times10^{22}\approx 1.9\times10^{11}\ \text{decays s}^{-1}

\]

6.4 Random Nature of Decay

• Each nucleus decays independently; the exact instant of a particular decay cannot be predicted.
• Counting statistics follow a Poisson distribution; the relative fluctuation in the count rate is \(\sigma/N \approx 1/\sqrt{N}\).

7. Summary of Key Points

• Mass defect \(\Delta m\) quantifies the “missing” mass; \(1\;\text{u}=931.5\;\text{MeV}\) gives the conversion to binding energy.
• Binding energy \(E_{b}= \Delta m\,c^{2}\) is the energy that holds a nucleus together.
• The binding‑energy‑per‑nucleon curve peaks at iron‑56; nuclei far from this peak have the greatest potential to release energy.
• Fusion (light nuclei) and fission (heavy nuclei) both move nuclei toward the peak, increasing \(E_{b}/A\) and releasing the difference as the Q‑value.
• Q‑value = total binding energy of products − total binding energy of reactants.
• Radioactive decay is characterised by activity \(A=\lambda N\), half‑life \(t{1/2}=0.693/\lambda\) and the exponential law \(N=N{0}e^{-\lambda t}\). Decay is random, leading to statistical fluctuations in measured count rates.
• Three main types of radiation – α, β⁻, β⁺ (and γ) – obey conservation of nucleon number and charge.
• Notation \(^{A}_{Z}X\) is used throughout the syllabus to identify a nuclide and to keep track of A and Z in reactions.