Solution: Correcting Radioactive Dosage Volume
Step 1: Understand the Dosage Requirement
The dose was prepared to be 1.00 mL at 8:00 AM. This means the doctor calculated the specific activity required for the patient at that exact time. Let the required activity be $A_{req}$.
The dose was prepared to be 1.00 mL at 8:00 AM. This means the doctor calculated the specific activity required for the patient at that exact time. Let the required activity be $A_{req}$.
Step 2: Calculate the Decay Factor
The patient arrives late and takes the dose at 8:55 AM. Delay time $\Delta t = 55$ minutes. Half-life $T_{1/2} = 110$ minutes. Number of half-lives passed: $$ n = \frac{55}{110} = 0.5 $$ The activity of the solution has decreased by a factor of: $$ \frac{A_{new}}{A_{old}} = \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^{0.5} = \frac{1}{\sqrt{2}} $$
The patient arrives late and takes the dose at 8:55 AM. Delay time $\Delta t = 55$ minutes. Half-life $T_{1/2} = 110$ minutes. Number of half-lives passed: $$ n = \frac{55}{110} = 0.5 $$ The activity of the solution has decreased by a factor of: $$ \frac{A_{new}}{A_{old}} = \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^{0.5} = \frac{1}{\sqrt{2}} $$
Step 3: Adjust Volume
Since the concentration of activity (Activity per mL) has dropped by $\frac{1}{\sqrt{2}}$, we must increase the volume administered to deliver the same total activity $A_{req}$. $$ V_{new} \times A_{new\_conc} = V_{old} \times A_{old\_conc} $$ $$ V_{new} = V_{old} \times \frac{A_{old\_conc}}{A_{new\_conc}} = 1.00 \text{ mL} \times \sqrt{2} $$ $$ V_{new} \approx 1.414 \text{ mL} $$
Since the concentration of activity (Activity per mL) has dropped by $\frac{1}{\sqrt{2}}$, we must increase the volume administered to deliver the same total activity $A_{req}$. $$ V_{new} \times A_{new\_conc} = V_{old} \times A_{old\_conc} $$ $$ V_{new} = V_{old} \times \frac{A_{old\_conc}}{A_{new\_conc}} = 1.00 \text{ mL} \times \sqrt{2} $$ $$ V_{new} \approx 1.414 \text{ mL} $$
Correct Option: (a) 1.41 mL