Iodine-131 has a short half-life of 8.14 days. Suppose the initial amount of Iodine-131 is 50 grams. How much of the initial amount of 50 grams will be radioactive after 30 days?
The correct answer and explanation is:
To find the amount of Iodine-131 remaining after 30 days, we use the radioactive decay formula:
A(t) = A₀ × (1/2)^(t / T)
Where:
- A(t) is the amount remaining after time t
- A₀ is the initial amount
- T is the half-life
- t is the elapsed time
Given:
A₀ = 50 grams
T = 8.14 days
t = 30 days
Step 1: Plug the values into the formula
A(30) = 50 × (1/2)^(30 / 8.14)
Step 2: Calculate the exponent
30 ÷ 8.14 ≈ 3.686
Step 3: Calculate (1/2)^3.686
(1/2)^3.686 ≈ 0.0786
Step 4: Multiply by the initial amount
A(30) = 50 × 0.0786 ≈ 3.93 grams
Correct Answer:
Approximately 3.93 grams of the original 50 grams of Iodine-131 will remain radioactive after 30 days.
Explanation:
Radioactive isotopes like Iodine-131 decay over time following an exponential pattern. The rate of decay is characterized by the half-life, which is the time it takes for half of a substance to decay. In this case, the half-life is 8.14 days, which means that every 8.14 days, the amount of Iodine-131 is halved.
After one half-life, 25 grams would remain. After two, 12.5 grams. However, 30 days is not a perfect multiple of the half-life, so the decay must be calculated using the exponential formula. This approach gives a more accurate result than repeated halving, especially for times that are not exact multiples of the half-life.
This exponential decay ensures that the substance never truly reaches zero but becomes negligible after several half-lives. Iodine-131’s rapid decay is useful in medical applications, especially in treatments where short-term radiation is beneficial.