Know the relationship between the nucleon number and the relative mass of a nucleus

5.1.2 The Nucleus – Relationship Between Nucleon Number and Relative Mass

Objective

• Define the nucleon (mass) number \(A\) and the relative atomic mass unit \(u\).
• Write and interpret nuclide notation \(\,^{A}_{Z}\!X\).
• Explain why the relative mass of a nucleus is ≈ \(A\) u.
• Calculate mass‑defect and binding energy for a simple nucleus.
• Identify isotopes, determine the number of neutrons and relate \(Z\) to nuclear charge.
• Describe, in brief, the relevance of the mass‑defect concept to nuclear fission and fusion (Supplement).

Syllabus Context – Where This Note Fits

Key Concepts

• Nucleons: protons and neutrons that reside in the nucleus.
• Mass number (A): total number of nucleons (protons + neutrons) in a nucleus.
• Atomic number (Z): number of protons; also the nuclear charge in units of the elementary charge (+1 e per proton).
• Relative atomic mass unit (u): 1 u = \( \frac{1}{12}\) the mass of a carbon‑12 atom (exact by definition).
• Nuclide notation: \(\,^{A}_{Z}\!X\) where \(X\) is the element symbol, \(A\) the mass number and \(Z\) the atomic number.
• Isotopes: atoms of the same element (same \(Z\)) that have different mass numbers \(A\) (different numbers of neutrons).

Nuclide Notation & Isotopes

Examples of nuclide notation required by the Cambridge syllabus:

• Carbon‑12 \(\,^{12}_{6}\!C\)
• Uranium‑235 \(\,^{235}_{92}\!U\)
• Chlorine‑35 \(\,^{35}_{17}\!Cl\)

Number of neutrons:

\[N = A - Z\]

Example: \(\,^{14}_{6}\!C\) → \(N = 14 - 6 = 8\) neutrons.

Relative Masses of Sub‑Atomic Particles

Relationship Between Nucleon Number and Relative Mass

The mass of a neutral atom is the sum of the masses of its protons, neutrons and electrons. Because the electron mass is < 0.05 % of a nucleon’s mass, the atomic (or nuclear) mass can be approximated by the nucleon number:

This “\(A\) u” rule is sufficient for most IGCSE (AO1/AO2) questions. Two refinements are needed for higher‑level (AO2) work:

1. Mass defect – the binding energy that holds the nucleus together reduces the actual mass slightly (Einstein’s \(E=mc^{2}\)).
2. Exact nucleon masses – protons and neutrons are not exactly 1 u; using their precise values gives a more accurate result.

Mass‑Defect and Binding Energy – Worked Example

Calculate the mass defect and binding energy of an \(\,^{4}_{2}\!He\) nucleus (alpha particle).

1. Constituents: 2 p + 2 n.
2. Mass of separate nucleons:
   \[m_{\text{sep}} = 2(1.0073\;\text{u}) + 2(1.0087\;\text{u}) = 4.0320\;\text{u}\]
3. Measured atomic mass of \(\,^{4}_{2}\!He\) = 4.0026 u. Subtract the two electrons (2 × 0.00055 u = 0.0011 u) to obtain the nuclear mass:
   \[m_{\text{nucleus, meas}} = 4.0026\;\text{u} - 0.0011\;\text{u} = 4.0015\;\text{u}\]
4. Mass defect:
   \[\Delta m = m_{\text{sep}} - m_{\text{nucleus}} = 4.0320\;\text{u} - 4.0015\;\text{u} = 0.0305\;\text{u}\]
5. Convert to energy (1 u = 931.5 MeV):
   \[E_{\text{binding}} = \Delta m \times 931.5\;\text{MeV u}^{-1} \approx 28.4\;\text{MeV}\]

Approximation Example

Find the relative mass of a nitrogen‑14 nucleus using the simple rule.

1. \(A = 14\) (7 p + 7 n).
2. Approximation: \(m_{\text{N‑14}} \approx 14\;\text{u}\).
3. Measured mass = 13.999 u → mass defect ≈ 0.001 u, showing the rule is very accurate for IGCSE calculations.

Supplement: Connection to Nuclear Fission & Fusion

Both fission and fusion involve a change in the total binding energy of the system. Because mass and energy are equivalent (E = mc²), a small change in mass (the mass defect) releases or absorbs a large amount of energy.

• Fission: A heavy nucleus (e.g., \(\,^{235}_{92}\!U\)) splits into lighter fragments. The combined binding energy of the products is greater than that of the original nucleus, so the products have a lower total mass. The mass defect appears as the large energy released in a nuclear reactor or bomb.
• Fusion: Light nuclei (e.g., \(\,^{2}_{1}\!H\) and \(\,^{3}_{1}\!H\)) combine to form a heavier nucleus (e.g., \(\,^{4}_{2}\!He\)). The fused nucleus has a higher binding energy per nucleon, so the final mass is slightly less than the sum of the reactants. The missing mass is released as energy (the principle behind the Sun and experimental fusion reactors).

Understanding the mass‑defect concept therefore underpins the quantitative description of energy released in both processes – a requirement for the optional Supplementary material in the Cambridge syllabus.

Practical Activity – Exploring Nuclear Mass with a Mass‑Spectrometer Diagram

Objective: Use a simple magnetic‑deflection diagram to determine the relative masses of three isotopes and discuss the limits of the \(A\approx m\) approximation.

1. Provided diagram shows ion beams for \(\,^{12}_{6}\!C^{+}\), \(\,^{14}_{6}\!C^{+}\) and \(\,^{16}_{8}\!O^{+}\) moving through a uniform magnetic field.
2. Measure the radius of curvature \(r\) for each beam (same charge \(q\) and velocity). Record values.
3. Use \(r \propto \sqrt{m/q}\) → \(\displaystyle \frac{r_{1}}{r_{2}} = \sqrt{\frac{m_{1}}{m_{2}}}\) to calculate the relative masses.
4. Compare the experimental masses with the integer mass numbers (12, 14, 16). Discuss why the measured values are slightly lower (binding‑energy mass defect).
5. Write a brief evaluation (2–3 sentences) of experimental uncertainties and the usefulness of the \(A\approx m\) rule for quick calculations.

This activity develops AO3 skills: planning, data collection, analysis, and evaluation.

Common Misconceptions

• “All nuclei have a mass exactly equal to their nucleon number.” – A small mass defect exists because part of the nucleons’ mass is converted into binding energy.
• “Electrons contribute significantly to atomic mass.” – Their combined mass is < 0.05 % of the total; it can be ignored for most IGCSE problems.
• “Protons and neutrons have identical masses.” – Their masses differ by ≈ 0.0014 u; the “≈ A u” rule is an approximation.
• “Isotopes are different elements.” – Isotopes have the same \(Z\) (same element) but different \(A\) (different numbers of neutrons).
• “Binding energy is only relevant to radioactivity.” – It also explains the energy released in fission and fusion.

Quick‑Check Questions (AO1 & AO2)

1. What is the nucleon number \(A\) of an oxygen‑16 atom?
   Answer: 16.
2. Using the approximation, give the relative mass of a calcium‑40 nucleus.
   Answer: ≈ 40 u.
3. Write the nuclide notation for a chlorine atom that has 18 neutrons.
   Answer: \(\,^{35}_{17}\!Cl\) (since \(Z=17\), \(A=Z+N=35\)).
4. Calculate the number of neutrons in \(\,^{235}_{92}\!U\).
   Answer: \(N = 235 - 92 = 143\) neutrons.
5. Explain why the measured mass of a carbon‑12 nucleus is defined as exactly 12 u, whereas other nuclei show a slight mass defect.
   Answer: By definition, one mole of \(\,^{12}_{6}\!C\) atoms has a mass of exactly 12 g, giving a nuclear mass of 12 u. All other nuclei have a mass defect because part of the nucleons’ mass is converted into binding energy, so their measured masses are a little less than the integer \(A\) value.

Summary – Key Points to Remember

• The nucleon (mass) number \(A\) equals the total number of protons + neutrons.
• For IGCSE calculations, the relative mass of a nucleus can be approximated as \(A\) u.
• Electrons contribute negligibly to atomic mass (< 0.05 %).
• Binding energy causes a small mass defect; the defect can be calculated using the precise masses of protons and neutrons.
• Isotopes are written in nuclide notation \(\,^{A}_{Z}\!X\); the number of neutrons is \(N = A - Z\).
• Mass‑defect concepts also explain the large energy released in nuclear fission and fusion (Supplement).
• Practical investigations (e.g., mass‑spectrometer) reinforce the concepts and develop AO3 experimental skills.

Assessment Objective Alignment