Notes on Radioactive Decay

Radioactive Decay

• Definition: Radioactive decay is a random process by which an unstable atomic nucleus loses energy by emitting radiation. This can involve the emission of particles or electromagnetic waves.

• Mathematical Representation:

  • The rate of change of mass (m) or the number of radioactive atoms present in a sample is proportional to the amount currently present at that time.
  • This relationship can be expressed mathematically as:
    \frac{dm}{dt} = -km
  • Where:
    • dm : represents the change in mass or quantity over time.
    • dt : represents the change in time.
    • k : is the decay constant, which is a positive number that is unique to each radioactive isotope. It signifies the probability per unit time that a nucleus will decay.
    • The negative sign indicates that as time increases, the mass or number of radioactive atoms present decreases, consistent with the nature of radioactive decay.

• Implications of the Decay Constant:

  • The decay constant is essential for calculating the half-life of a radioactive substance.
  • The half-life (T) is the time required for half the quantity of a radioactive substance to decay. It can be calculated using:
    T_{1/2} = \frac{\ln(2)}{k}
  • Where \ln(2) is the natural logarithm of 2 (approximately 0.693).

• Example Situations:

  • If a sample originally contains 100 grams of a radioactive isotope with a decay constant ( k ) of 0.1 per year, the mass remaining after a certain time can be calculated using the above equations.
  • After one half-life (10 years for this example), 50 grams would remain, as half of the original mass has decayed.

• Applications of Radioactive Decay:

  • Used in radiometric dating to determine the age of materials based on the concentrations of radioactive isotopes.
  • Essential in fields like nuclear medicine for diagnosing and treating diseases, as well as in nuclear power generation and safety evaluations.