5.2.3 & 5.2.4: Decay Equations & Half-Life
From unstable nuclei to the mathematics of radioactive decay
🎯 The Hook: Nature's Random Timer
Imagine you have a bag of 1000 popcorn kernels. Every minute, exactly half of the remaining unpopped kernels pop. After 1 minute: 500 left. After 2 minutes: 250 left. After 3 minutes: 125 left. This predictable halving pattern is exactly how radioactive atoms decay, except instead of popping, they transform into different elements by firing out particles. Today, we'll crack the code of nuclear transformations.
Part 1: Understanding Radioactive Decay
📘 Core Concept
Radioactive decay is when an unstable nucleus changes by emitting radiation (α, β, or γ). This process is:
- Spontaneous: Happens on its own, no external trigger needed
- Random: You cannot predict exactly WHEN a specific nucleus will decay
- Irreversible: Once changed, the nucleus becomes a different element
Baby Steps: Why Do Nuclei Decay?
Step 1: A stable nucleus has a balanced ratio of protons and neutrons. Think of it like a perfect Jenga tower.
Step 2: Some nuclei have too many neutrons or are too heavy (too many nucleons total). This makes them unstable, like a wobbly tower.
Step 3: To become stable, the nucleus "fixes" itself by ejecting particles. This releases energy and creates a more stable nucleus.
Part 2: The Three Types of Decay (Baby Steps Edition)
🔴 Alpha (α) Decay: The Heavy Ejection
What leaves: 2 protons + 2 neutrons (helium nucleus)
Effect on nucleus:
- Mass number (A) decreases by 4
- Proton number (Z) decreases by 2
- Result: Different element (2 places back in periodic table)
AZX → A-4Z-2Y + 42α
🔵 Beta (β⁻) Decay: The Neutron Transform
What happens: A neutron transforms into a proton + electron (the electron shoots out)
Effect on nucleus:
- Mass number (A) stays the same (no mass lost)
- Proton number (Z) increases by 1 (gained a proton)
- Result: Different element (1 place forward in periodic table)
AZX → AZ+1Y + 0-1β
Inside nucleus: n → p + e⁻
🟢 Gamma (γ) Decay: The Energy Release
What leaves: High-energy electromagnetic wave (no particle)
Effect on nucleus:
- Mass number (A) no change
- Proton number (Z) no change
- Result: Same element, just lower energy (more stable)
AZX* → AZX + γ
(*excited state)
⚠️ Examiner's Trap: Element Changes
Common mistake: "The element stays the same in all decay types."
Truth: Only γ-decay keeps the same element. α and β decay ALWAYS produce a different element because they change the proton number (Z). Remember: Proton number = Element identity.
🧪 Interactive Lab 1: Decay Equation Builder
Balance these nuclear equations by filling in the missing numbers. Use what you learned about how A and Z change!
Scenario 1: Alpha Decay of Uranium-238
Scenario 2: Beta Decay of Carbon-14
💡 Hint: Total mass (A) and total charge (Z) must be equal on both sides of the equation!
Part 3: Half-Life (The Predictable Pattern)
📘 Core Definition
Half-life is the time taken for half the nuclei in a sample to decay.
Alternatively: The time for the activity (count rate) to drop to half its original value.
Baby Steps: Understanding Half-Life
Step 1: You start with 1000 radioactive atoms at time = 0.
Step 2: After 1 half-life: 500 atoms remain (half decayed).
Step 3: After 2 half-lives: 250 atoms remain (half of 500).
Step 4: After 3 half-lives: 125 atoms remain (half of 250).
Key insight: The half-life is CONSTANT. It doesn't matter if you have 1000 atoms or 10 atoms—the time to halve is always the same for that isotope.
⚠️ Examiner's Trap: Half-Life is NOT Linear
Common mistake: "If the half-life is 2 days, then after 4 days all atoms are gone (2 half-lives = 100% decay)."
Truth: You never reach zero! Each half-life only removes HALF of what's left:
• After 1 half-life: 50% remain
• After 2 half-lives: 25% remain
• After 3 half-lives: 12.5% remain
• After 4 half-lives: 6.25% remain
📊 Interactive Lab 2: The Half-Life Graph
This graph shows activity (count rate) dropping over time. Find the half-life by identifying when the activity drops to 50% of its value.
💡 Strategy: Find where the curve crosses 400 (half of 800). Note the time. Then find where it crosses 200 (half of 400). The time difference should be the same!
⏰ Interactive Lab 3: The Nuclear Time Machine
Watch atoms decay step by step. Each click advances time by one half-life.
Half-Life Calculations: The Formula Approach
🧮 The Half-Life Formula
Remaining = Initial × (½)n
where n = number of half-lives elapsed
Alternative form: n = Total Time ÷ Half-Life
For example: If half-life = 2 days, and total time = 6 days, then n = 6 ÷ 2 = 3 half-lives
📝 Worked Example 1: Simple Calculation
Question: A sample has 800 atoms. The half-life is 5 hours. How many atoms remain after 15 hours?
Step 1: Find number of half-lives: n = 15 ÷ 5 = 3 half-lives
Step 2: Apply formula: Remaining = 800 × (½)³
Step 3: Calculate: 800 × 0.5 × 0.5 × 0.5 = 800 × 0.125 = 100 atoms
Or think: 800 → 400 → 200 → 100
📝 Worked Example 2: Activity (Count Rate)
Question: A source has activity 1200 counts/s. Half-life is 3 minutes. What is the activity after 9 minutes?
Step 1: Find number of half-lives: n = 9 ÷ 3 = 3 half-lives
Step 2: Apply formula: Activity = 1200 × (½)³
Step 3: Calculate: 1200 × 0.125 = 150 counts/s
Or think: 1200 → 600 → 300 → 150
✅ Syllabus Checklist: Can You Do This?
Core Requirements
Supplement Requirements
🎯 Final Examiner's Wisdom
1. When balancing equations: Check BOTH mass (A) and charge (Z) are equal on both sides.
2. Beta decay: Remember the neutron transforms INTO a proton (Z goes up, not down).
3. Half-life graphs: Always look for when count drops to exactly HALF the previous value.
4. Calculations: Show your working! n = time ÷ half-life, then apply (½)ⁿ
5. Units: Time can be in seconds, minutes, hours, days, or years—always match units in division!
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