Half-Life Notes

Half-Life

• Definition: The time it takes for half of the atoms in a radioactive sample to decay.

Visual Representation of Decay

• Imagine radioactive elements as black dots that decay into red dots.
• Beginning: 100% black dots (radioactive element).
• After One Half-Life:
  • 50% black dots, 50% red dots. If you started with 20 black dots, you'd have 10 black and 10 red.
• After Another Half-Life:
  • 25% black dots, 75% red dots. Starting from the previous state (10 black, 10 red), you'd have 5 black and 15 red.

Factors Independent of Half-Life

• Half-life is independent of:
  • Temperature
  • Pressure
  • Concentration

Half-Life Uniqueness

• Each radioisotope has its own unique half-life.
  • Some are very long (billions of years).
  • Some are very short (microseconds).

Example: Uranium-238

• Half-life of Uranium-238 is approximately 4.47 \text{ billion years}. This is useful for approximating the age of the Earth.

Formulas for Half-Life Calculations

• n = Number of half-lives
• Fraction Remaining: \frac{1}{2}^n
• Mass Remaining = (Original Mass) * (Fraction Remaining)

Sample Problem: Radium-222

• Problem: How long will it take for 30 grams of Rn (Radium) -222 to decay to 7.5 grams?

• Set up a T-chart to track mass vs time.

• Identify what you are looking for (time elapsed) and what is given (initial and final masses).

• Solution:

  • Original mass: 30 grams at time 0.
  • After 1 half-life: 15 grams.
  • After 2 half-lives: 7.5 grams.

• Find the half-life of Rn-222 from Table N: 3.823 \text{ days}.

• Time elapsed to reach 7.5 grams = 2 * (half-life) = 2 * 3.823 = 7.646 \text{ days}.

Sample Problem: N-16

• Problem: How many grams of N-16 will be left on a 60 gram sample after 21.6 seconds?
• Set up a T-chart to track mass vs time
• Initial time: 0 seconds. Initial mass: 60 grams
• From Table N, the half-life of N-16 is 7.13 seconds.
• After one half-life (7.13 seconds): 30 grams
• After two half-lives (14.26 seconds): 15 grams
• After three half-lives (21.39 seconds): 7.5 grams
• The presenters solve it a second time for verification
• The final answer is 7.5 grams

Sample problem: Determining Fraction Remaining and Time

• Problem: What fraction of a sample will be left after 42.9 days, and how long is three half lives?
• Identify the knowns and unknowns.
• After one half life is 14.28 days.
• After two half lives 28.56 days.
• After three half lives is 42.84 days.
• Fraction remaining is calculated as the third power of one half (1/2), or (1/2)^3.
• Find the half life for O^{15}, which is 1.599 s
• Multiple 1.599 by three for the final answer (4.07).