AP Physics 1 Exam Question: Nuclear Physics

A radioactive isotope, Technetium-99m ($^{99m}$Tc), is commonly used in medical imaging. It undergoes beta decay with a half-life of 6.01 hours. Suppose a sample of $^{99m}$Tc is initially found to contain 256 radioactive atoms.

a) How many $^{99m}$Tc atoms will remain after 24.04 hours?

b) Determine the decay constant (λ) for $^{99m}$Tc.

c) Calculate the activity of the $^{99m}$Tc sample after 12.03 hours.

d) What will be the activity of the sample after 36.12 hours?

e) Explain why the activity does not depend on the size of the radioactive sample.

Provide answers with step-by-step detailed explanations.

Answer:

a) The number of remaining atoms after a certain time can be calculated using the decay equation:

N(t) = N₀ × (1/2)^(t/T)

where N₀ is the initial number of atoms, t is the time elapsed, and T is the half-life.

Given: N₀ = 256, t = 24.04 hours, T = 6.01 hours

Plugging in the values:

N(t) = 256 × (1/2)^(24.04/6.01)

N(t) = 256 × (1/2)^(4)

N(t) = 256 × 1/16

N(t) = 16

Therefore, 16 $^{99m}$Tc atoms will remain after 24.04 hours.

b) The decay constant (λ) can be determined using the formula:

λ = ln(2) / T where T is the half-life.

Given: T = 6.01 hours

λ = ln(2) / 6.01

λ ≈ 0.1156 hours⁻¹

Therefore, the decay constant (λ) for $^{99m}$Tc is approximately 0.1156 hours⁻¹.

c) The activity at a certain time is given by:

A(t) = λ × N(t)

where A(t) is the activity and N(t) is the number of remaining atoms.

Given: λ = 0.1156 hours⁻¹, t = 12.03 hours

Using the result from part (a), N(t) = 16

A(t) = 0.1156 × 16

A(t) ≈ 1.85 Bq (becquerels)

Therefore, the activity of the $^{99m}$Tc sample after 12.03 hours is approximately 1.85 Bq.

d) Using the same formula as in part (c):

Given: λ = 0.1156 hours⁻¹, t = 36.12 hours

Using the result from part (a), N(t) = 16

A(t) = 0.1156 × 16

A(t) ≈ 1.85 Bq (becquerels)

Therefore, the activity of the $^{99m}$Tc sample after 36.12 hours is approximately 1.85 Bq.

e) The activity of a radioactive sample does not depend on its size because activity is defined as the number of decay events per unit time (measured in becquerels). It is a property of the radioactive substance itself and is independent of the quantity of the substance present. Therefore, even if the size of the sample changes, the activity remains the same as long as the radioactive substance remains the same. The rate of decay depends on the decay constant, which remains constant for a specific radioactive isotope.