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): $N(t)=N<em>0\left(\tfrac12\right)^{t/ t</em>{1/2}}$ where:
  • $N_0$ = initial amount/activity
  • $t$ = elapsed time
  • $t_{1/2}$ = half-life
• Safety criterion: $t<em>{\text{safety}} \approx 10\,t</em>{1/2}$
• “Percent remaining” after $n$ half-lives: $\left(\tfrac12\right)^n\times100\%$