half life of plutonium-239 is 24,300 years. if a nuclear bomb released 8kg of this isotope, how many years would pass before the amt is reduced to 1kg
The correct answer and explanation is:
Correct Answer: 72,900 years
To determine how long it will take for 8 kg of plutonium-239 to decay to 1 kg, the law of radioactive decay is used. This law is based on the principle that the amount of a radioactive substance decreases by half every half-life.
The formula used is:
N = N₀ × (1/2)^(t/T)
Where:
- N is the final amount (1 kg)
- N₀ is the initial amount (8 kg)
- t is the time in years (what we are solving for)
- T is the half-life (24,300 years)
Divide both sides of the equation by N₀:
(1/8) = (1/2)^(t/24,300)
Now rewrite 1/8 as (1/2)^3:
(1/2)^3 = (1/2)^(t/24,300)
Since the bases are the same, equate the exponents:
3 = t / 24,300
Solve for t:
t = 3 × 24,300 = 72,900 years
This means that it will take 72,900 years for the 8 kilograms of plutonium-239 to decay to 1 kilogram.
Radioactive decay is exponential, meaning the substance decays by half every fixed period (in this case, every 24,300 years). After the first half-life, 8 kg becomes 4 kg. After the second, it becomes 2 kg. After the third half-life, it becomes 1 kg. Each half-life takes the same amount of time, so the total time for three half-lives is 3 × 24,300 years.
Understanding radioactive decay is important for nuclear safety, environmental science, and calculating the long-term impact of nuclear materials. The persistence of radioactive substances like plutonium-239 demonstrates why handling and storage of nuclear waste must be managed carefully over extremely long timescales.