🧠 The theory: the powder takes up space
The final concentration depends on the total final volume, which is the diluent you add PLUS the volume the dissolved powder occupies. If you added a volume of diluent equal to the final volume you wanted, the powder would push the total higher and the solution would end up too dilute. So you subtract the displacement volume from the target final volume to decide how much diluent to add. It is simple bookkeeping of volumes: diluent plus powder equals final volume.
final volume = dose of drug / desired concentration
diluent to add = final volume minus displacement volume
Worked example
A vial contains 1 g of antibiotic with a displacement volume of 0.8 mL. What volume of water gives a final concentration of 100 mg/mL?
- Step 1 — Work out the final volume you need. You have 1000 mg and want 100 mg in each mL, so final volume = 1000 / 100 = 10 mL.
- Step 2 — The dissolved powder already provides 0.8 mL, so subtract it to find the water to add: 10 minus 0.8 = 9.2.
Answer: 9.2 mL of water for injection
Worked example
A 600 mg vial has a displacement volume of 0.5 mL. What volume of diluent gives 200 mg/mL?
- Step 1 — Final volume needed: 600 / 200 = 3 mL.
- Step 2 — Subtract the displacement volume: 3 minus 0.5 = 2.5.
Answer: 2.5 mL
Worked example
A 1.2 g vial has a displacement volume of 0.7 mL. What volume of diluent gives 200 mg/mL?
- Step 1 — Final volume: 1200 / 200 = 6 mL.
- Step 2 — Subtract: 6 minus 0.7 = 5.3.
Answer: 5.3 mL
Harder: working backwards
If the question tells you how much diluent was added and asks for the concentration, add the displacement volume back to get the true final volume first.
Worked example
If 4 mL of water is added to a 500 mg vial with a displacement volume of 0.5 mL, what is the final concentration?
- Step 1 — Find the true final volume by adding the powder's volume to the water: 4 + 0.5 = 4.5 mL.
- Step 2 — Concentration is the amount divided by that final volume: 500 / 4.5 = 111.1.
Answer: about 111 mg/mL
Tip: Ignoring the displacement volume here would wrongly give 125 mg/mL, an 11% error.
⚠️ Always subtract
For injections you add less diluent than the final volume. If your diluent volume is larger than the final volume, you have added rather than subtracted the displacement volume. If the question says the displacement is negligible, the diluent simply equals the final volume.
Test yourself
A 2 g vial has a displacement volume of 1.5 mL. What diluent volume gives 200 mg/mL?
Answer: Final volume = 2000 / 200 = 10 mL; diluent = 10 - 1.5 = 8.5 mL
A 1 g vial has a displacement volume of 0.8 mL. What diluent gives 200 mg/mL?
Answer: Final volume = 1000 / 200 = 5 mL; diluent = 5 - 0.8 = 4.2 mL
A 750 mg vial has a displacement volume of 0.6 mL. What diluent gives 150 mg/mL?
Answer: Final volume = 750 / 150 = 5 mL; diluent = 5 - 0.6 = 4.4 mL
3 mL is added to a 300 mg vial, displacement volume 0.3 mL. What is the concentration?
Answer: Final volume = 3 + 0.3 = 3.3 mL; 300 / 3.3 = about 91 mg/mL