Nuclear Half Life: Calculations
Alpha Decay of Radium to Radon
Process
Radium undergoes alpha decay to form radon.
Half-life of radium is 11 days.
Question: Starting with a 120 g sample of radium, how much remains after 44 days?
Calculation of Half-lives
44 days ÷ 11 days/half-life = 4 half-lives.
Decay Chart:
Initial: 120 g
After 1 half-life: 60 g (120 g / 2)
After 2 half-lives: 30 g (60 g / 2)
After 3 half-lives: 15 g (30 g / 2)
After 4 half-lives: 7.5 g (15 g / 2)
Conclusion: After 44 days, 7.5 g of radium remains.
Percentage Calculation:
Start at 100%.
Percent after each half-life:
After 1 half-life: 50% (100% / 2)
After 2 half-lives: 25% (50% / 2)
After 3 half-lives: 12.5% (25% / 2)
After 4 half-lives: 6.25% (12.5% / 2)
Conclusion: After 44 days, 6.25% remains.
Fraction Calculation:
Start with whole amount: 1 (or 120 g)
Fraction after each half-life:
After 1 half-life: 1/2
After 2 half-lives: 1/4
After 3 half-lives: 1/8
After 4 half-lives: 1/16
Conclusion: After 44 days, 1/16 of the original amount remains (equivalent to 6.25%).
Beta Decay of Tritium to Helium-3
Process: Hydrogen-3 (tritium) undergoes beta decay to make helium-3.
Half-life Duration: 12.3 years.
Question:
Starting with an 80 g sample of tritium that decays to 2.5 g. How long does this take?
Decay Chart:
Initial: 80 g
After 1 half-life: 40 g (80 g / 2)
After 2 half-lives: 20 g (40 g / 2)
After 3 half-lives: 10 g (20 g / 2)
After 4 half-lives: 5 g (10 g / 2)
After 5 half-lives: 2.5 g (5 g / 2)
Total Half-lives: 5 half-lives.
Total Time: 12.3 years × 5 = 61.5 years.
Percentage Calculation:
Start at 100%.
Percent after each half-life:
After 1 half-life: 50%
After 2 half-lives: 25%
After 3 half-lives: 12.5%
After 4 half-lives: 6.25%
After 5 half-lives: 3.125%
Conclusion: To reach approximately 3.1%, still takes 61.5 years.
Fraction Calculation:
Start: 1
Fraction after each half-life:
After 1 half-life: 1/2
After 2 half-lives: 1/4
After 3 half-lives: 1/8
After 4 half-lives: 1/16
After 5 half-lives: 1/32
Total Time: 5 half-lives × 12.3 years = 61.5 years.
Beta Decay of Thallium to Lead
Process: Thallium-207 decays to lead.
Question:
Starting with 200 g of thallium, decays to 12.5 g in 20 minutes. What is the half-life?
Decay Chart:
Initial: 200 g
After 1 half-life: 100 g
After 2 half-lives: 50 g
After 3 half-lives: 25 g
After 4 half-lives: 12.5 g
Total Half-lives: 4 half-lives in 20 minutes.
Calculation: 20 minutes ÷ 4 = 5 minutes per half-life.
Summary of Methods
Most decay problems can use simple calculations (halves), but complex problems may require exponents and logarithms for resolution.