In preparing capsules of a very potent drug, 500 mg of the drug was weighed out and diluted to 10 g with Lactose; 1.000 g of this mix was further diluted to 15 g with Lactose; 1.000 g of this second mix was diluted to 15 g with Lactose; 1.00 g of this third mix was diluted to 15 g with Lactose, and 500 mg of the final mix was filled into a No.1 empty gelatin capsule. What is the amount of drug in each capsule ?

Question

BlackTom AI · Accepted Answer

Dilution and proportion are the key concepts: track the drug fraction (drug mass / total mass) through each dilution, then multiply by the mass placed in the capsule.

1) Relevant knowledge points
- Mass fraction = (drug mass) / (total mass).
- When you take a portion of a mixture and dilute it, the drug fraction is multiplied by (mass taken)/(mass after dilution).

2) Step-by-step reasoning with intermediate calculations
- Initial mixture: 500 mg drug diluted to 10 g (10 g = 10,000 mg).
  Drug fraction f0 = 500 mg / 10,000 mg = 1/20.

- First further dilution: 1.000 g (1000 mg) of the first mix is diluted to 15 g (15,000 mg).
  Amount of drug in that 1.000 g = 1000 mg × f0 = 1000 × (1/20) = 50 mg.
  New fraction f1 = 50 mg / 15,000 mg = 50/15000 = 1/300.
  (Equivalently f1 = f0 × (1000/15000) = (1/20) × (1/15) = 1/300.)

- Second further dilution: 1.000 g of second mix → diluted to 15 g.
  Amount drug in 1.000 g = 1000 mg × f1 = 1000 × (1/300) = 10/3 mg ≈ 3.333... mg.
  New fraction f2 = (10/3) / 15,000 = 10/(45,000) = 1/4500.
  (f2 = f1 × (1/15) = 1/300 × 1/15 = 1/4500.)

- Third further dilution: 1.00 g of third mix → diluted to 15 g.
  Amount drug in 1.00 g = 1000 mg × f2 = 1000 × (1/4500) = 1000/4500 = 2/9 mg ≈ 0.222222 mg.
  New fraction f3 = (2/9) / 15,000 = 2/(135,000) = 1/67,500.
  (f3 = f2 × 1/15 = 1/4500 × 1/15 = 1/67,500.)

- Final fill: 500 mg of the final mix is put into a capsule.
  Drug in capsule = 500 mg × f3 = 500 × (1/67,500) = 500/67,500 = 1/135 mg.
  Convert to micrograms: (1/135) mg × 1000 μg/mg = 1000/135 μg ≈ 7.407407 μg ≈ 7.4 μg.

3) Why the correct answer is correct
The repeated successive dilutions multiply the original drug fraction (1/20) by (1/15) three times, giving an overall fraction 1/20 × (1/15)^3 = 1/67,500. Multiplying that by the 500 mg finally filled yields 1/135 mg = 7.4 μg.

4) Why other options are incorrect (brief)
- 740 ng = 0.74 μg (10× too small).
- 7.4 ng = 0.0074 μg (1000× too small).
- 74 μg = 10× too large.
- 74 ng = 0.074 μg (100× too small).

Conclusion: the computed drug amount per capsule is approximately 7.4 μg, matching answer a.

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