A Level Physics - How to Solve: Nuclear Physics (Binding Energy, Fusion/Fission, Exponential Decay Law)

How to Solve: Nuclear Physics (Binding Energy, Fusion/Fission, Exponential Decay Law)

Complete Guide for GCSE/A-Level Students & Teachers

Introduction

"Mastering binding energy, fusion/fission, and exponential decay doesn’t just explain how stars shine or how nuclear power works—it’s worth up to 15% of your GCSE/A-Level Physics exam and appears in every major paper (AQA, Edexcel, OCR). One wrong unit or misread graph, and you lose 3-4 marks instantly. This guide gives you the exact steps to solve any question—fast and error-free."

WHAT YOU NEED TO KNOW FIRST

Before diving in, ensure you understand:
1. Atomic structure – Protons, neutrons, nucleons, and mass number (A) vs. atomic number (Z).
2. Energy-mass equivalence – Einstein’s E = mc² (energy = mass × speed of light²).
3. Graph interpretation – Reading binding energy per nucleon curves and decay graphs.

(If any of these are shaky, pause and review them first—this topic builds on them!)

KEY TERMS & FORMULAS

Key Terms

Formulas

1. Binding Energy & Mass Defect

Formula: BE = Δm × c² - BE = Binding energy (J) - Δm = Mass defect (kg) = (Mass of nucleons) – (Mass of nucleus) - c = Speed of light (3 × 10⁸ m/s) → MEMORISE THIS

Alternative (for MeV): BE (MeV) = Δm (u) × 931.5 MeV/u - 1 atomic mass unit (u) = 1.66 × 10⁻²⁷ kg → MEMORISE THIS - 931.5 MeV/u → Given on exam sheet

2. Exponential Decay Law

Formula: N = N₀ e⁻ᶫᵗ - N = Number of undecayed nuclei at time t - N₀ = Initial number of nuclei - λ = Decay constant (s⁻¹) - t = Time (s) - e = Euler’s number (~2.718) → Given on exam sheet

Alternative (for half-life): λ = ln(2) / t₁/₂ - ln(2) ≈ 0.693 → MEMORISE THIS

Activity (A): A = λN - A = Activity (Bq) - λ = Decay constant (s⁻¹) - N = Number of undecayed nuclei

3. Energy Released in Fusion/Fission

Formula: Energy released = BE of products – BE of reactants (If positive, energy is released; if negative, energy is absorbed.)

STEP-BY-STEP METHOD

Part 1: Binding Energy & Mass Defect

Step 1: Identify the nucleus and its nucleons. - Write down the mass number (A) and atomic number (Z). - Calculate total nucleons: A = protons + neutrons.

Step 2: Find the mass defect (Δm). - Look up the actual mass of the nucleus (given in the question or data sheet). - Calculate the total mass of individual nucleons: - Mass of protons = Z × mass of proton (1.00728 u) - Mass of neutrons = (A – Z) × mass of neutron (1.00867 u) - Δm = (Mass of nucleons) – (Mass of nucleus)

Step 3: Convert mass defect to energy. - If Δm is in kg, use BE = Δm × c². - If Δm is in u, use BE (MeV) = Δm × 931.5.

Step 4: Calculate binding energy per nucleon (if asked). - BE per nucleon = BE / A

Part 2: Fusion & Fission

Step 1: Write the nuclear equation. - Balance mass numbers (A) and atomic numbers (Z) on both sides.

Step 2: Find the binding energy of reactants and products. - Use the binding energy per nucleon curve (given in exams) or calculate from mass defect.

Step 3: Calculate energy released. - Energy released = BE of products – BE of reactants - If positive, energy is released (exothermic). - If negative, energy is absorbed (endothermic).

Step 4: Check units. - If BE is in MeV, answer in MeV. - If BE is in J, answer in J.

Part 3: Exponential Decay Law

Step 1: Identify what’s given and what’s asked. - Given: N₀, t, λ, t₁/₂, or A - Asked: N, A, t, or λ

Step 2: Choose the right formula. - If half-life (t₁/₂) is given, use λ = ln(2) / t₁/₂. - If decay constant (λ) is given, use N = N₀ e⁻ᶫᵗ. - If activity (A) is asked, use A = λN.

Step 3: Plug in the numbers. - Use base-10 logs if e is not on your calculator (some exams allow 2⁻ᵗ/ᵗ¹/² instead). - For N = N₀ e⁻ᶫᵗ, take the natural log (ln) of both sides if solving for t or λ.

Step 4: Check units. - λ must be in s⁻¹ (if time is in seconds). - t must match the units of λ (e.g., if λ is in years⁻¹, t must be in years).

WORKED EXAMPLES

Example 1 – Basic: Binding Energy of Helium-4

Question: Calculate the binding energy of a helium-4 nucleus (²⁴He) in MeV. Given: - Mass of helium-4 nucleus = 4.00150 u - Mass of proton = 1.00728 u - Mass of neutron = 1.00867 u

Step 1: Identify nucleons. - A = 4, Z = 2 - Protons = 2, Neutrons = 2

Step 2: Calculate mass defect (Δm). - Mass of nucleons = (2 × 1.00728) + (2 × 1.00867) = 4.03190 u - Δm = 4.03190 u – 4.00150 u = 0.03040 u

Step 3: Convert to energy. - BE = 0.03040 u × 931.5 MeV/u = 28.3 MeV

Step 4: Binding energy per nucleon. - BE per nucleon = 28.3 MeV / 4 = 7.08 MeV/nucleon

What we did and why: We found the mass defect (difference between nucleon mass and nucleus mass), then converted it to energy using E = mc² (via 931.5 MeV/u). This tells us how much energy holds the nucleus together.

Example 2 – Medium: Energy Released in Fusion

Question: Calculate the energy released when two deuterium nuclei (²¹H) fuse to form helium-3 (³²He) and a neutron. Given: - Mass of deuterium (²¹H) = 2.01355 u - Mass of helium-3 (³²He) = 3.01493 u - Mass of neutron = 1.00867 u

Step 1: Write the nuclear equation. ²¹H + ²¹H → ³²He + ¹⁰n

Step 2: Calculate mass defect (Δm). - Mass of reactants = 2 × 2.01355 u = 4.02710 u - Mass of products = 3.01493 u + 1.00867 u = 4.02360 u - Δm = 4.02710 u – 4.02360 u = 0.00350 u

Step 3: Convert to energy. - Energy released = 0.00350 u × 931.5 MeV/u = 3.26 MeV

What we did and why: We compared the total mass before and after fusion. The mass lost (Δm) was converted to energy using E = mc². This is how stars (like the Sun) release energy!

Example 3 – Exam-Style: Half-Life & Activity

Question: A sample of iodine-131 has an initial activity of 800 Bq. Its half-life is 8 days. a) Calculate the decay constant (λ). b) What will the activity be after 24 days?

Part a) Step 1: Use half-life to find λ. - λ = ln(2) / t₁/₂ - λ = 0.693 / (8 × 24 × 3600 s) = 1.00 × 10⁻⁶ s⁻¹

Part b) Step 1: Use exponential decay formula. - A = A₀ e⁻ᶫᵗ - t = 24 days = 24 × 24 × 3600 s = 2,073,600 s - A = 800 e⁻(1.00×10⁻⁶ × 2,073,600) - A = 800 e⁻².⁰⁷³⁶ ≈ 800 × 0.126 ≈ 101 Bq

Alternative (using half-lives): - 24 days = 3 half-lives (24 / 8 = 3) - A = A₀ × (1/2)³ = 800 × 1/8 = 100 Bq (slight rounding difference)

What we did and why: We used half-life to find λ, then applied the exponential decay formula to find activity after a given time. The alternative method (using half-lives directly) is faster in exams—always check if time is a multiple of half-life!

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

"Listen up—this is your last-minute nuclear physics cheat sheet.

1. Binding energy? Find the mass defect (nucleon mass – nucleus mass), then use BE = Δm × 931.5 MeV/u.
2. Fusion/fission? Energy released = BE of products – BE of reactants. If positive, energy is released.
3. Half-life? Use λ = ln(2)/t₁/₂ to find the decay constant, then N = N₀ e⁻ᶫᵗ or A = A₀ (1/2)ᵗ/ᵗ¹/₂.
4. Units matter! Convert everything to kg for E=mc², seconds for λ, and MeV for energy.
5. Exam tricks? Watch for time unit mismatches, missing masses (use the BE curve), and rounding errors.

Now go crush that exam—you’ve got this!"