Question:
A radioactive sample of iodine-131 has an initial activity of 1000 decays per second. The half-life of iodine-131 is 8 days.
a) Calculate the decay constant (λ) of iodine-131.
b) Determine the activity of the iodine-131 sample after 16 days.
c) How many iodine-131 atoms remain in the sample after 24 days?
Answer:
a) The decay constant (λ) can be calculated using the half-life (T1/2) of the radioactive sample using the formula:
λ = ln(2) / T1/2
Given: T1/2 = 8 days
Substituting the values into the formula:
λ = ln(2) / 8 = 0.0866 day^(-1)
b) The activity of the radioactive sample can be calculated using the decay constant (λ) and the time (t) using the formula:
A = A0 * e^(-λt)
where A0 is the initial activity and A is the activity after time t.
Given: A0 = 1000 decays/second, t = 16 days, λ = 0.0866 day^(-1)
Substituting the values into the formula:
A = 1000 * e^(-0.0866 * 16) = 1000 * e^(-1.3856) = 285.9 decays/second
Therefore, the activity of the iodine-131 sample after 16 days is 285.9 decays/second.
c) The remaining number of iodine-131 atoms can be calculated using the equation:
N = N0 * e^(-λt)
where N0 is the initial number of atoms and N is the number of atoms remaining after time t.
Given: N0 = initial number of iodine-131 atoms
Since activity is directly proportional to the number of atoms, we know that:
A = λ * N0
Therefore, N0 = A / λ = 1000 / 0.0866 = 11543 atoms
For t = 24 days:
N = 11543 * e^(-0.0866 * 24) = 2510 atoms
Therefore, after 24 days, there are 2510 iodine-131 atoms remaining in the sample.
Note: Make sure to check your answers with the appropriate significant figures or rounding rules depending on the instructions for the exam.