2.950 mg sample of technetium-99 m is used for a diagnostic test. technetium- 99 m has a half-life of 6.0 hr , how much of the technetium-99 sample remains 24.0 hr after the test? Answer Answer:

**Step 1: Write the decay formula** The decay of a radioactive substance is given by \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \] where: - \( N(t) \) is the remaining amount after time \( t \), - \( N_0 \) is the initial amount, - \( T_{1/2} \) is the half-life. **Step 2: Identify the given values** - Initial mass \( N_0 = 2.950 \) mg - Half-life \( T_{1/2} = 6.0 \) hr - Time elapsed \( t = 24.0 \) hr **Step 3: Determine the number of half-lives elapsed** \[ \frac{t}{T_{1/2}} = \frac{24.0}{6.0} = 4 \] **Step 4: Substitute the values into the decay formula** \[ N(24.0\ \text{hr}) = 2.950 \left(\frac{1}{2}\right)^4 \] **Step 5: Calculate the remaining amount** Since \(\left(\frac{1}{2}\right)^4 = \frac{1}{16}\), we have: \[ N(24.0\ \text{hr}) = 2.950 \times \frac{1}{16} = \frac{2.950}{16} \text{ mg} \] **Step 6: Perform the division** \[ \frac{2.950}{16} \approx 0.1844 \text{ mg} \] **Final Answer:** Approximately \(0.1844\) mg of technetium-99m remains after 24.0 hours.