Dilution & concentration (Q₁C₁ = Q₂C₂)

What is this for?

When a preparation is diluted or concentrated, the absolute amount of active ingredient does not change; only the total quantity does. That conservation is expressed as Q₁C₁ = Q₂C₂. Quantities may be in any consistent unit, and concentrations in any consistent expression, as long as you do not mix them.

How to use it

1. Identify which of the four quantities you do not know and leave that field blank.
2. Enter the other three. Keep Q₁ and Q₂ in the same unit, and C₁ and C₂ in the same expression of strength.
3. If you solved for Q₂, the "diluent to add" line tells you how much vehicle to add to your original quantity.

Worked example

You have 70% w/v stock solution and need 500 mL of a 20% w/v solution. How much stock is required, and how much diluent?

Answer: Q₁ = (500 × 20) ÷ 70 = 142.9 mL of stock, made up to 500 mL with 357.1 mL of diluent.

Clinical pearls & pitfalls

• Q₂ is the final total quantity, not the amount of diluent. Adding 500 mL of water to 142.9 mL of stock gives 642.9 mL, not 500 mL — a classic error. Make up to the final volume.
• The equation only works when the quantity and concentration units are internally consistent. Mixing % w/v with mg/mL, or mL with grams, produces a plausible-looking but wrong answer.
• If your calculated Q₁ exceeds Q₂, check your inputs: you cannot make a stronger preparation by dilution.

Assumptions & limitations

• Assumes volumes are additive. For alcohol-water mixtures this is not true — mixing 50 mL of alcohol with 50 mL of water yields slightly less than 100 mL — so always make up to volume rather than adding volumes.
• Does not account for the displacement volume of a solid solute, which matters when reconstituting powders for injection.

References

• United States Pharmacopeia. General Chapter <795> Pharmaceutical Compounding — Nonsterile Preparations.
• Allen LV. The Art, Science, and Technology of Pharmaceutical Compounding. American Pharmacists Association.

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