8.2 Nuclear reactions and half-life

Nuclear reactions and half-life are key concepts in understanding radioactive decay. They explain how unstable atoms break down over time, releasing energy and transforming into more stable elements.

These processes are crucial in nuclear chemistry, with applications ranging from energy production to medical treatments. Understanding half-life helps scientists date ancient artifacts and predict the behavior of radioactive materials.

Half-life and Radioactive Decay

Definition and Significance

• Half-life is the time required for half of the atoms in a radioactive sample to decay into a more stable form
• The half-life is a characteristic property of each radioactive isotope and remains constant regardless of the initial amount of the substance (carbon-14, uranium-238)
• After one half-life, the amount of radioactive substance remaining is reduced by half, and this process continues exponentially with each subsequent half-life
  • Example: If a sample starts with 100 atoms, after one half-life, 50 atoms remain; after two half-lives, 25 atoms remain, and so on
• The half-life of a radioactive substance determines the rate at which it decays and the time it takes for the substance to reach a safe level of radioactivity
  • Substances with shorter half-lives decay more quickly, while those with longer half-lives decay more slowly

Calculating Half-life

Decay Constant and Activity

• The decay constant (λ) is the probability of a radioactive atom decaying per unit time and is related to the half-life (t₁/₂) by the equation: $t₁/₂ = ln(2) / λ$
• The activity (A) of a radioactive sample is the number of decays per unit time and is related to the number of atoms (N) and the decay constant (λ) by the equation: $A = λN$
• Given the decay constant or activity, the half-life can be calculated using the appropriate equation
  • Example: If the decay constant of a substance is 0.01 per day, the half-life can be calculated as $t₁/₂ = ln(2) / 0.01 ≈ 69.3$ days

Radioactive Decay and Remaining Substance

Calculating Remaining Amount

• The amount of a radioactive substance remaining after a given number of half-lives (n) can be calculated using the equation: $N(t) = N₀ × (1/2)^n$, where N(t) is the amount remaining at time t, and N₀ is the initial amount
  • Example: If a sample starts with 1000 atoms and has a half-life of 10 days, after 30 days (3 half-lives), the remaining amount is $N(30) = 1000 × (1/2)^3 = 125$ atoms
• The fraction of a radioactive substance remaining after a given number of half-lives (n) is equal to $(1/2)^n$
  • Example: After 4 half-lives, the fraction of the substance remaining is $(1/2)^4 = 1/16$
• The time elapsed (t) can be calculated using the equation: $t = n × t₁/₂$, where n is the number of half-lives and t₁/₂ is the half-life of the substance
  • Example: If a substance has a half-life of 5 days and 20 days have passed, the number of half-lives elapsed is $n = 20 / 5 = 4$

Radioactive Dating and its Applications

Radioactive Dating Methods

• Radioactive dating is a method used to determine the age of materials by measuring the amount of a specific radioactive isotope remaining in the sample and comparing it to the initial amount
• Carbon-14 dating is commonly used for organic materials, as living organisms incorporate carbon-14 from the atmosphere, and the ratio of carbon-14 to carbon-12 begins to decrease after the organism dies
  • Carbon-14 has a half-life of approximately 5,730 years, making it suitable for dating materials up to about 50,000 years old
• Uranium-lead dating is used for inorganic materials, such as rocks and minerals, based on the decay of uranium-238 and uranium-235 into stable lead isotopes
  • Uranium-238 has a half-life of about 4.5 billion years, while uranium-235 has a half-life of about 704 million years, making them suitable for dating very old materials

Applications and Limitations

• Radioactive dating has applications in archaeology (dating artifacts), geology (dating rocks and minerals), and paleontology (dating fossils)
• The accuracy of radioactive dating depends on factors such as:
  • The half-life of the isotope: Longer half-lives provide a wider dating range but may be less precise for younger samples
  • The initial amount of the isotope in the sample: Higher initial amounts make dating more accurate
  • The assumption that the sample has remained in a closed system: Contamination or loss of the isotope can lead to inaccurate results
• Other limitations include the need for a sufficient amount of the isotope to be present in the sample and the potential for different parts of a sample to yield different ages due to varying exposure to the environment