A Sample of radioactive material A, that has an activity of 10 mCi (1 Ci $$= 3.7 \times 10^{10}$$ decays s$$^{-1}$$), has twice the number of nuclei as another sample of a different radioactive material B which has an activity of 20 mCi. The correct choices for half-lives of A and B would then be, respectively:

We have two radioactive samples, call them A and B.

First recall the basic relation that connects the activity $$R$$ (also called decay rate) of a sample to the number of undecayed nuclei $$N$$ present in it: $$ R \;=\; \lambda N, $$ where $$\lambda$$ is the decay constant.

The decay constant itself is related to the half-life $$T_{1/2}$$ by the well-known formula $$ \lambda \;=\; \frac{\ln 2}{T_{1/2}}. $$

Now translate the data given in the question into symbols:

• Activity of A: $$R_A \;=\; 10\ \text{mCi}.$$

• Activity of B: $$R_B \;=\; 20\ \text{mCi}.$$

• Number of nuclei: “sample A has twice the number of nuclei as sample B”, so $$ N_A \;=\; 2N_B. $$

Use the activity relation $$R=\lambda N$$ for each sample.

For sample A we can write $$ R_A \;=\; \lambda_A\,N_A. $$ Substituting the numerical activity, $$ 10 \;=\; \lambda_A\,N_A. \quad -(1) $$

For sample B we have similarly $$ R_B \;=\; \lambda_B\,N_B, $$ so with its given activity, $$ 20 \;=\; \lambda_B\,N_B. \quad -(2) $$

Use the relation between the numbers of nuclei. From $$N_A=2N_B$$, equation (1) becomes $$ 10 \;=\; \lambda_A\,(2N_B), $$ which is $$ \lambda_A\,N_B \;=\; 5. \quad -(3) $$

Equation (2) remains $$ \lambda_B\,N_B \;=\; 20. \quad -(4) $$

Divide (4) by (3) to compare the two decay constants:

The common factor $$N_B$$ cancels, giving

Thus $$ \lambda_B \;=\; 4\lambda_A. $$

Convert this statement about decay constants into one about half-lives. Since $$\lambda=\dfrac{\ln 2}{T_{1/2}}$$, we can write

From relation (5) we have $$\dfrac{\lambda_B}{\lambda_A}=4,$$ so

Equation (6) tells us that the half-life of sample A is four times the half-life of sample B.

Now consult the options and pick the pair that satisfies $$T_{1/2,A}=4\,T_{1/2,B}$$:

A. 5 days and 10 days  → 5 is not 4 × 10.

B. 10 days and 40 days  → 10 is not 4 × 40.

C. 20 days and 10 days  → 20 is 2 × 10, not 4 × 10.

D. 20 days and 5 days  → 20 is exactly 4 × 5. ✔️

Hence, the correct answer is Option D.