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Half-Life worksheet

Grades 10–12 · CCSS HSF.LE.A.1.c, HSF.LE.A.4

Solve each problem and show your work. Round answers to 2 decimal places where needed.

1.
Problem 1.
- Solve for: Half-life
- Initial amount (N₀): 140
- Amount remaining (N): 28
- Time elapsed (t): 20
- Time unit (for both times): Seconds
Find: Half-life (T½)
Answer
2.
Problem 2.
- Solve for: Amount remaining
- Initial amount (N₀): 80
- Time elapsed (t): 16
- Half-life (T½): 8.5
- Time unit (for both times): Seconds
Find: Amount remaining (N)
Answer
3.
Problem 3.
- Solve for: Half-life
- Initial amount (N₀): 110
- Amount remaining (N): 30
- Time elapsed (t): 19
- Time unit (for both times): Hours
Find: Half-life (T½)
Answer
4.
Problem 4.
- Solve for: Amount remaining
- Initial amount (N₀): 82
- Time elapsed (t): 26
- Half-life (T½): 8
- Time unit (for both times): Days
Find: Amount remaining (N)
Answer
5.
Problem 5.
- Solve for: Initial amount
- Amount remaining (N): 34
- Time elapsed (t): 20
- Half-life (T½): 11
- Time unit (for both times): Years
Find: Initial amount (N₀)
Answer
6.
Problem 6.
- Solve for: Amount remaining
- Initial amount (N₀): 120
- Time elapsed (t): 20
- Half-life (T½): 8.75
- Time unit (for both times): Days
Find: Amount remaining (N)
Answer
7.
Problem 7.
- Solve for: Initial amount
- Amount remaining (N): 18
- Time elapsed (t): 20
- Half-life (T½): 5.6
- Time unit (for both times): Years
Find: Initial amount (N₀)
Answer
8.
Problem 8.
- Solve for: Time elapsed
- Initial amount (N₀): 90
- Amount remaining (N): 22
- Half-life (T½): 9
- Time unit (for both times): Days
Find: Time elapsed (t)
Answer
9.
Problem 9.
- Solve for: Amount remaining
- Initial amount (N₀): 116
- Time elapsed (t): 28
- Half-life (T½): 5.25
- Time unit (for both times): Years
Find: Amount remaining (N)
Answer
10.
Problem 10.
- Solve for: Half-life
- Initial amount (N₀): 120
- Amount remaining (N): 26
- Time elapsed (t): 18
- Time unit (for both times): Minutes
Find: Half-life (T½)
Answer

Answer key: Half-Life worksheet

1. Half-life (T½): 8.61 s
1. Find how many half-lives fit into the decay: n = ln(N₀ ÷ N) ÷ ln 2
  $n$ $= \frac{ln (\frac{140}{28})}{ln 2}$ $= 2.321928$
2. Divide the elapsed time by that count: T½ = t ÷ n
  $T½$ $= \frac{20 s}{2.321928}$ $= 8.613531 s$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{8.613531 s}$ $= 0.0804719 per s$
2. Amount remaining (N): 21.7
1. Count the half-lives that have passed: n = t ÷ T½
  $n$ $= \frac{16 s}{8.5 s}$ $= 1.882353$
2. Halve the starting amount once per half-life: N = N₀ × (1/2)ⁿ
  $N$ $= 80 \times {(\frac{1}{2})}^{1.882353}$ $= 21.699278$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{8.5 s}$ $= 0.08154673 per s$
3. Half-life (T½): 10.14 h
1. Find how many half-lives fit into the decay: n = ln(N₀ ÷ N) ÷ ln 2
  $n$ $= \frac{ln (\frac{110}{30})}{ln 2}$ $= 1.874469$
2. Divide the elapsed time by that count: T½ = t ÷ n
  $T½$ $= \frac{19 h}{1.874469}$ $= 10.136203 h$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{10.136203 h}$ $= 0.06838331 per h$
4. Amount remaining (N): 8.62
1. Count the half-lives that have passed: n = t ÷ T½
  $n$ $= \frac{26 d}{8 d}$ $= 3.25$
2. Halve the starting amount once per half-life: N = N₀ × (1/2)ⁿ
  $N$ $= 82 \times {(\frac{1}{2})}^{3.25}$ $= 8.619188$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{8 d}$ $= 0.0866434 per d$
5. Initial amount (N₀): 119.9
1. Count the half-lives that have passed: n = t ÷ T½
  $n$ $= \frac{20 yr}{11 yr}$ $= 1.818182$
2. Double the remaining amount once per half-life to go back in time: N₀ = N × 2ⁿ
  $N₀$ $= 34 \times 2^{1.818182}$ $= 119.896411$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{11 yr}$ $= 0.06301338 per yr$
6. Amount remaining (N): 24.61
1. Count the half-lives that have passed: n = t ÷ T½
  $n$ $= \frac{20 d}{8.75 d}$ $= 2.285714$
2. Halve the starting amount once per half-life: N = N₀ × (1/2)ⁿ
  $N$ $= 120 \times {(\frac{1}{2})}^{2.285714}$ $= 24.610061$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{8.75 d}$ $= 0.07921682 per d$
7. Initial amount (N₀): 213.98
1. Count the half-lives that have passed: n = t ÷ T½
  $n$ $= \frac{20 yr}{5.6 yr}$ $= 3.571429$
2. Double the remaining amount once per half-life to go back in time: N₀ = N × 2ⁿ
  $N₀$ $= 18 \times 2^{3.571429}$ $= 213.983178$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{5.6 yr}$ $= 0.12377628 per yr$
8. Time elapsed (t): 18.29 d
1. Find how many halvings turn N₀ into N: n = ln(N₀ ÷ N) ÷ ln 2
  $n$ $= \frac{ln (\frac{90}{22})}{ln 2}$ $= 2.032421$
2. Multiply by the half-life to get the elapsed time: t = n × T½
  $t$ $= 2.032421 \times 9 d$ $= 18.291793 d$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{9 d}$ $= 0.07701635 per d$
9. Amount remaining (N): 2.88
1. Count the half-lives that have passed: n = t ÷ T½
  $n$ $= \frac{28 yr}{5.25 yr}$ $= 5.333333$
2. Halve the starting amount once per half-life: N = N₀ × (1/2)ⁿ
  $N$ $= 116 \times {(\frac{1}{2})}^{5.333333}$ $= 2.877164$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{5.25 yr}$ $= 0.13202803 per yr$
10. Half-life (T½): 8.16 min
1. Find how many half-lives fit into the decay: n = ln(N₀ ÷ N) ÷ ln 2
  $n$ $= \frac{ln (\frac{120}{26})}{ln 2}$ $= 2.206451$
2. Divide the elapsed time by that count: T½ = t ÷ n
  $T½$ $= \frac{18 min}{2.206451}$ $= 8.157897 min$
3. The decay constant is the fraction decaying per unit time: λ = ln 2 ÷ T½
  $λ$ $= \frac{0.693147}{8.157897 min}$ $= 0.0849664 per min$