You are ordering a batch of Tc-99m for your hospital. You are to order the Tc-99m for use on Friday of next week at 12 noon. The manufacturer will make the Tc-99m on Monday at 12 n... You are ordering a batch of Tc-99m for your hospital. You are to order the Tc-99m for use on Friday of next week at 12 noon. The manufacturer will make the Tc-99m on Monday at 12 noon of next week. If you want the Tc-99m to be 637 MBq on Friday, what activity will it be at the date of manufacturing? The half-life of Tc-99m is 6 hours.
Understand the Problem
The question is asking for the calculation of the activity of Tc-99m at the time of manufacturing, given its half-life and the required activity at a later date. We will use the half-life decay formula to find the initial activity needed on the manufacturing date.
Answer
The initial activity of Tc-99m at the time of manufacturing is approximately \(2,605,312 \text{ MBq}\).
Answer for screen readers
The initial activity of Tc-99m at the time of manufacturing is approximately (2,605,312 \text{ MBq}).
Steps to Solve
-
Determine the time between manufacturing and use
The Tc-99m will be manufactured on Monday at 12 noon and used on Friday at 12 noon.
This is a period of 3 days, or 72 hours. -
Calculate the number of half-lives
Since the half-life of Tc-99m is 6 hours, we calculate the number of half-lives over 72 hours:
$$ \text{Number of half-lives} = \frac{72 \text{ hours}}{6 \text{ hours}} = 12 $$ -
Use the half-life decay formula
The decay formula is given by: $$ A = A_0 \times \left(\frac{1}{2}\right)^{n} $$ where:
- (A) is the final activity (637 MBq)
- (A_0) is the initial activity (what we want to find)
- (n) is the number of half-lives (12)
Rearranging the equation to solve for (A_0): $$ A_0 = A \times 2^{n} $$
- Calculate initial activity
Substituting the values into the formula: $$ A_0 = 637 \text{ MBq} \times 2^{12} $$
Calculating (2^{12}): $$ 2^{12} = 4096 $$
Now calculate: $$ A_0 = 637 \text{ MBq} \times 4096 $$
Calculating the initial activity: $$ A_0 = 2,605,312 \text{ MBq} $$
The initial activity of Tc-99m at the time of manufacturing is approximately (2,605,312 \text{ MBq}).
More Information
Tc-99m is widely used in medical imaging due to its suitable half-life and emission properties. Its half-life of 6 hours allows for enough time to produce the radiopharmaceutical and use it shortly afterward.
Tips
- Confusing the dates or times which can lead to incorrect calculations of the time between manufacturing and use. Always double-check the timeline.
- Misapplying the decay formula, especially when rearranging for the initial activity.
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