Ch7: Half Life
Practical Importance of Half-Life
- Dating artifacts
- Carbon-14 half-life enables archaeologists to estimate the age of historical items in households or excavation sites.
- Radioactive-waste management
- Knowing an isotope’s half-life tells engineers how long waste must be stored before it is safe.
- Rule of thumb discussed: “A radioactive sample is considered safe after 10 half-lives.”
- Example: Iodine-131, t_{1/2}=8\text{ days} ⇒ safe after 10\times8=80\text{ days}.
- Example: Plutonium-239, t_{1/2}=24\,000\text{ yr} ⇒ must be stored for \approx 240\,000\text{ yr} (≈ millions when rounded for policy discussions).
- Medical applications
- Physicians select radio-tracers whose half-lives strike a balance between diagnostic usefulness and minimal long-term exposure.
Definition of Half-Life
- “Time required for the amount (or activity) of a radioactive sample to drop to one-half of its initial value.”
- Can be framed in terms of mass or radiation intensity.
Visualizing Decay (Plutonium-239 Example)
- t_{1/2}=24\,000\text{ yr}
- After 1 half-life (24,000 yr): 50 % remains.
- After 2 half-lives (48,000 yr): 50\%\times\tfrac12=25\% remains.
- After 3 half-lives (72,000 yr): 25\%\times\tfrac12=12.5\% remains.
- Continues to 6.25 %, 3.13 %, etc.
- Safety implication: Waiting for plutonium to reach trace safety thresholds requires geologic time scales.
Generic Half-Life Table (Decay vs. Remaining)
- Provided logic (can be rebuilt without memorizing):
- 1 half-life → 50 % decay / 50 % remain.
- 2 half-lives → 75 % decay / 25 % remain.
- 3 half-lives → 87.5 % decay / 12.5 % remain.
- 4 half-lives → 93.75 % decay / 6.25 % remain.
- 5 half-lives → 96.875 % decay / 3.125 % remain.
Extreme Half-Life Spectrum (Selected Isotopes)
- Uranium-238: 4.5\times10^9\text{ yr} (age of Earth scale)
- Potassium-40: 1.3\times10^9\text{ yr}
- Uranium-235: 7\times10^8\text{ yr}
- Plutonium-239: 2.4\times10^4\text{ yr}
- Iodine-131: 8\text{ days}
- Polonium (specific isotope unspecified): 1.6\times10^{-4}\text{ s} (fractions of a millisecond)
Worked Example 1 – Remaining Percentage
Question: “Element X has t_{1/2}=10\text{ days}. After 30 days, what % of the original amount remains?”
- Strategy: Assume an initial 100 g (any convenient mass).
- Number of half-lives elapsed: \frac{30\text{ d}}{10\text{ d}} = 3.
- Sequential halving:
- After 1st half-life: 100\text{ g}\to50\text{ g}
- After 2nd: 50\text{ g}\to25\text{ g}
- After 3rd: 25\text{ g}\to12.5\text{ g}
- Remaining %: \frac{12.5}{100}\times100=12.5\%
- Complement (decayed %): 100\%-12.5\%=87.5\% (if asked).
Worked Example 2 – Determining Half-Life
Question: “After 42 days only 25 % of Element Y remains. What is t_{1/2}?”
- Observed reduction: 100\%\to25\% corresponds to two halvings (100 %→50 %→25 %).
- Number of half-lives = 2.
- Total time = 42 days.
- Therefore t_{1/2}=\frac{42\text{ d}}{2}=21\text{ days}.
Ethical & Policy Connections
- Long-lived isotopes (e.g., Pu-239, U-238) raise inter-generational storage obligations—ethical debates around nuclear energy center on this timeframe.
- Short-lived medical isotopes minimize patient exposure but require rapid synthesis and logistics.
- Decision frameworks for nuclear vs. alternative energy sources often invoke half-life tables to communicate risk.
Links to Previous Lecture
- Previous session introduced categories of nuclear waste; current half-life discussion provides quantitative tool for predicting when each category transitions from high- to low-level waste.
- Reinforces the importance of isotope selection in nuclear medicine, first mentioned earlier.
Key Equations & Concepts Recap
- Exponential decay formula (not explicitly given in transcript but implied): N(t)=N0\left(\tfrac12\right)^{t/ t{1/2}} where:
- N_0 = initial amount/activity
- t = elapsed time
- t_{1/2} = half-life
- Safety criterion: t{\text{safety}} \approx 10\,t{1/2}
- “Percent remaining” after n half-lives: \left(\tfrac12\right)^n\times100\%