In radioactive decay, A = λ N. If the activity is halved after a time equal to the half-life t1/2, what is λ in terms of t1/2?

In radioactive decay, A = λ N. If the activity is halved after a time equal to the half-life t1/2, what is λ in terms of t1/2?

• A. λ = 2 / t1/2
• B. λ = 1 / t1/2
• C. λ = ln 2 / t1/2
• D. λ = t1/2 / ln 2

Answer: (C)

The key idea is that radioactive decay is exponential and the decay constant λ links directly to how long it takes for the quantity to drop by half. The activity A is proportional to the number of undecayed nuclei N, and N(t) = N0 e^{-λ t}. By definition of the half-life, after a time t1/2 the quantity is halved: N(t1/2) = N0/2. Setting N0/2 = N0 e^{-λ t1/2} and dividing by N0 gives 1/2 = e^{-λ t1/2}. Taking natural logs: ln(1/2) = -λ t1/2, which leads to λ = (ln 2) / t1/2. Since A = λ N, the activity also halves after one half-life, consistent with the same relationship. The ln 2 factor appears because the decay is exponential; simple reciprocals like 1/t1/2 or 2/t1/2 don’t satisfy the halving condition, and t1/2 divided by ln 2 has the wrong units for λ.

The key idea is that radioactive decay is exponential and the decay constant λ links directly to how long it takes for the quantity to drop by half. The activity A is proportional to the number of undecayed nuclei N, and N(t) = N0 e^{-λ t}. By definition of the half-life, after a time t1/2 the quantity is halved: N(t1/2) = N0/2.

Setting N0/2 = N0 e^{-λ t1/2} and dividing by N0 gives 1/2 = e^{-λ t1/2}. Taking natural logs: ln(1/2) = -λ t1/2, which leads to λ = (ln 2) / t1/2.

Since A = λ N, the activity also halves after one half-life, consistent with the same relationship. The ln 2 factor appears because the decay is exponential; simple reciprocals like 1/t1/2 or 2/t1/2 don’t satisfy the halving condition, and t1/2 divided by ln 2 has the wrong units for λ.