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.

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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

  1. 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.

  2. 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:
    $$ \text{Number of half-lives} = \frac{72 \text{ hours}}{6 \text{ hours}} = 12 $$

  3. 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): $$ A_0 = A \times 2^{n} $$

  1. 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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