In short: to make 1,000 mL of a 0.5% solution from a 10% stock, you need 50 mL of stock — add 950 mL of diluent to reach the full 1,000 mL. Pick a different value to solve for above, or change any of the three known values for your own dilution.
This is a planning calculation. Verify against your specific reagent's documentation before preparing standards for critical use.
How the dilution equation works
The amount of solute doesn't change when you dilute, only the volume it's spread through does. That conservation gives the dilution equation, which rearranges to solve for whichever value you don't know:
C₁V₁ = C₂V₂
C₁ and V₁ describe the stock (before diluting); C₂ and V₂ describe the final solution (after diluting). Diluent to add = V₂ − V₁.
Worked example. To make 1,000 mL of a 0.5% working solution from a 10% stock: V₁ = (C₂ × V₂) ÷ C₁ = (0.5 × 1,000) ÷ 10 = 50 mL of stock, topped up with 950 mL of diluent.
Solving for each variable
The same equation, rearranged for whichever value you're missing.
| Solving for | Formula |
|---|---|
| Stock concentration (C₁) | C₁ = (C₂ × V₂) ÷ V₁ |
| Stock volume (V₁) | V₁ = (C₂ × V₂) ÷ C₁ |
| Final concentration (C₂) | C₂ = (C₁ × V₁) ÷ V₂ |
| Final volume (V₂) | V₂ = (C₁ × V₁) ÷ C₂ |
Why the units you pick don't change the math
C1V1 = C2V2 is a ratio, so it works with any concentration unit, percent, mg/L, ppm, molarity, as long as C₁ and C₂ use the same unit as each other. The same goes for volume: mL, L or gallons all work, as long as V₁ and V₂ match. What breaks the calculation isn't which unit you choose, it's mixing units between the two sides.
This is also why the equation can't tell you anything about a solid solute dissolved from powder or crystal form. It only tracks a solute that's already in solution and being spread through more or less liquid — dissolving a measured mass of solid is a different calculation entirely.
Worked scenarios
Solving for final concentration. Adding 20 mL of a 5 M stock into a flask filled to 250 mL total: C₂ = (5 × 20) ÷ 250 = 0.4 M.
Solving for stock concentration. You have 15 mL of stock available and need 500 mL of a 2% solution: C₁ = (2 × 500) ÷ 15 = 66.7% — the stock needs to be roughly two-thirds strength.
Frequently asked questions
What does C1V1 = C2V2 mean?
It's the conservation of solute during a dilution: the concentration times the volume before diluting equals the concentration times the volume after, as long as nothing but diluent is added. C₁ and V₁ describe the stock, C₂ and V₂ describe the final diluted solution.
How do I know how much water (diluent) to add?
Diluent volume equals the final volume minus the stock volume, V₂ minus V₁. Add that much diluent to your measured amount of stock to reach the target total volume.
Does the concentration unit matter for the calculation?
No, as long as C₁ and C₂ use the same unit. The ratio works regardless of whether you're using percent, mg/L, ppm or molarity, but mixing units between C₁ and C₂ will give a wrong answer.
Can I use this to find out how strong my stock solution needs to be?
Yes. Choose "Stock concentration (C₁)" as what you're solving for, then enter the stock volume you have available along with your target final concentration and volume.
How is this different from the stock solution calculator?
The stock solution calculator is built around a single common workflow: given a stock and a target, how much of each to combine. This tool solves any of the four C₁V₁ = C₂V₂ variables, which is useful when you know a different combination of values.
Does this work for solid solutes, not just liquid stock solutions?
This volume-based formula is for liquid-to-liquid dilutions. Dissolving a solid into solution is a mass-based calculation, concentration times volume equals mass, which this tool doesn't solve.
Sources & references
- APHA, AWWA, WEF. Standard Methods for the Examination of Water and Wastewater (reagent and standard solution preparation).
Sarah Bennett, MS
Water Quality & Sustainability Specialist · Portland, Oregon
Environmental scientist (MS, Water Resources) with 9+ years in residential water quality, treatment, and usage efficiency, and 150+ completed assessments. Sarah writes and reviews the treatment, usage, and cost calculators on watercalculator.us. Read full bio
MS Environmental Science, Water Resources (2018) · BS Environmental Engineering (2016)