Lecture Notes: Understanding Half-Life and Radioactive Decay

Key Concepts

• Half-life Definition: The time required for a quantity to reduce to half its initial amount.
• Radioactive Decay: The process by which an unstable atomic nucleus loses energy by radiation.

Application of Half-Life

• Multiples of Half-Life:
  • Time = 0: 100% of substance remains.
  • After 1 half-life: 50% remains.
  • After 2 half-lives: 25% remains.
  • Continuing this pattern allows us to calculate remaining substance over multiple half-lives.
• Example: Carbon's half-life is roughly 15,000 years.

General Function for Radioactive Decay

• Challenge: Calculating remaining substances at arbitrary times (e.g., 1/2 year, 10 minutes).
• Objective: Develop a general function of time that describes the amount of decaying substance.

Mathematical Approach

• Rate of Change: Proportional to the amount of substance present.
  • Dependent on substance type (e.g., carbon vs. uranium).
• Differential Equation: Describes the decay process:
  • ( \frac{dN}{dt} = -\lambda N )
  • ( N ) is the number of particles; ( \lambda ) is the decay constant.

Solving the Differential Equation

• Separation of Variables:
  • ( \frac{1}{N} dN = -\lambda dt )
  • Integrate both sides:
    • ( \ln(N) = -\lambda t + C )
• Solving for ( N ):
  • ( N = e^{-\lambda t + C} )
  • Rewrite as ( N = C_4 e^{-\lambda t} )

Initial Conditions

• ( N_0 ): Initial quantity of the substance.
• Using Initial Conditions to Solve for Constants:
  • At ( t = 0 ), ( N = N_0 )
  • ( C_4 = N_0 )
  • Final function: ( N(t) = N_0 e^{-\lambda t} )

Relating to Half-Life

• Example with Carbon-14:
  • Half-life: 5,700 years.
  • Initial amount: 100 units.
  • After 5,700 years: 50 units.
• **Solving for ( \lambda ):
  • ( \ln(0.5) = -5,700\lambda )
  • ( \lambda \approx 1.21 \times 10^{-4} )

General Equation for Carbon-14

• ( N(t) = N_0 e^{-1.21 \times 10^{-4} t} )
• Usage: Calculate remaining carbon at any time ( t ).

Conclusion

• This formula allows for calculation of remaining substance at any given time.
• More practice problems will follow to reinforce understanding.