Lecture on Exponential Decay and Half-Lives

Overview

• Half-lives: Useful for determining how much of a compound remains after set periods: one, two, or three half-lives.
• Limitation: Not useful for determining remaining compound amount in non-half-life periods (e.g., half a half-life, one day, etc.).

Exponential Decay Formula

• Describes the amount of decaying element over time: [ N(t) = N_0 \times e^{-kt} ]
  • (N(t)): Amount at time (t).
  • (N_0): Initial amount.
  • (k): Decay constant.
  • (t): Time elapsed.
• Derivation: Involves calculus; can be skipped for intro math classes.

Application to Carbon-14

• Half-life of Carbon-14: 5,730 years.
• Determining (k):
  • Given half-life, after 5,730 years: ( N(5730) = \frac{N_0}{2} ).
  • Setup equation: ( N_0 \times e^{-5730k} = \frac{N_0}{2} ).
  • Solve for (k) using natural logs: [ k = \frac{\ln(0.5)}{-5730} \approx 1.2 \times 10^{-4} ]

General Formula for Carbon-14

• [ N(t) = N_0 \times e^{-1.2 \times 10^{-4}t} ]
• Use this formula to calculate remaining carbon-14 over any time period.

Example Problems

Problem 1: Determine Remaining Amount

• Given: 300 grams of C-14, find amount after 2,000 years.
• Solution:
  • ( N(2000) = 300 \times e^{-1.2 \times 10^{-4} \times 2000} )
  • Result: Approximately 236 grams.

Problem 2: Determine Time to Reach Specific Amount

• Given: Start with 400 grams of C-14, find time to reduce to 350 grams.
• Solution:
  • Setup equation: ( 350 = 400 \times e^{-1.2 \times 10^{-4}t} )
  • Solve for (t):
    • ( \ln(0.875) = -1.2 \times 10^{-4}t )
    • ( t = \frac{\ln(0.875)}{-1.2 \times 10^{-4}} )
  • Result: Approximately 1,112 years.

Conclusion

• Key Takeaway: Remember the exponential decay formula and how to apply it using the decay constant (k) specific to the element's half-life.

Note

• Calculation of (k) and solving exponential decay problems are essential for understanding decay processes over arbitrary periods.