Lecture on Radioactive Decay and Half-Life

Introduction to Radioactive Decay

• Atoms and isotopes can experience radioactive decay, turning into other atoms or releasing particles.
• Types of decay include:
  • Beta decay: Electrons are released, turning neutrons into protons.
  • Positron emission: Protons are turned into neutrons.

Understanding Moles and Atomic Mass

• Moles: A mole of carbon-12 weighs 12 grams and contains $6.02 \times 10^{23}$ atoms.
• This is a huge number, illustrating the large amount of atoms even in small masses.

Probability of Decay

• At any moment, isotopes have a probability of decaying, but the exact timing is unknown.
• Decay is probabilistic due to the unknown state of the nucleus.
• On a macro level, terms like "half-life" help understand decay over large numbers of atoms.

Concept of Half-Life

• Half-Life: Time it takes for half of a sample of a radioactive isotope to decay.
• Example: Carbon-14 decays to nitrogen-14 with a half-life of 5,740 years.
• After one half-life, half of the atoms have decayed, but it's probabilistic which atoms decay.

Example with Carbon-14

• Initial state: Start with 10 grams of carbon-14.
• After one half-life (5,740 years):
  • 5 grams of C-14 remain, 5 grams decay to N-14.
• After two half-lives:
  • 2.5 grams of C-14 remain, 7.5 grams of N-14.

Understanding Decay on a Micro Level

• For a single carbon atom, decay probability over time varies:
  • After 1 second: Likely still carbon.
  • After 1 billion years: Likely decayed to nitrogen-14, but not guaranteed.

Calculating Time Passed Based on Remaining Sample

• Example:
  • Start with 80 grams of a substance with a 2-year half-life.
  • After 2 years: 40 grams left.
  • After 4 years: 20 grams left.
  • After 6 years: 10 grams left.
  • Three half-lives have passed to get from 80 grams to 10 grams.

Key Formula

• Use $\frac{1}{2}^{n}$ to determine remaining fraction of sample after n half-lives.
• Example: After three half-lives, 1/8 of the original sample remains.

Conclusion

• Understanding half-life helps predict decay in large samples, but individual atom decay remains probabilistic.

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Note: Future lectures might address how to calculate decay for non-discrete times, such as exactly 10 days or 2.5 years.