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The formula

For simple dilution with a zero-concentration diluent (water, solvent), solute amount is conserved: C₁ × V₁ = C₂ × V₂. Stock volume is V₁ = C₂V₂ / C₁, diluent is V₂ − V₁, and final strength is C₂ = C₁V₁ / V₂. Mixing two solutions uses a volume-weighted average: C_f = (C_aV_a + C_bV_b) / (V_a + V_b).

C₁V₁ = C₂V₂ · C_f = (C_aV_a + C_bV_b)/(V_a + V_b)

Worked example

  1. Stock 10%, target 2%, make 500 mL
  2. V₁ = C₂ × V₂ ÷ C₁ = 2 × 500 ÷ 10 = 100 mL stock
  3. Diluent = 500 − 100 = 400 mL

Result: 100 mL stock + 400 mL diluent → 2%

How mixture and dilution percentages are calculated

Pick a mode, enter concentrations as percentages of the same kind (% v/v, % w/w, or % w/v), and use matching volume units. The calculator rearranges the dilution identity or the two-solution blend formula and shows stock, diluent, and final values.

The C₁V₁ = C₂V₂ dilution identity

C₁ and V₁ are the stock concentration and volume; C₂ and V₂ are the diluted concentration and total volume. The product equals the amount of solute (in “%-volume” units). Solving for any one unknown gives the classic lab and kitchen dilution recipes.

Diluent volume is simply V₂ − V₁ when the diluent contributes no solute. If you start from stock volume and target %, final volume is V₂ = C₁V₁ / C₂.

Mixing two nonzero solutions

When both parts carry solute, add amounts then divide by total volume. Example: 200 mL at 40% plus 300 mL at 10% → (8000 + 3000) / 500 = 22%.

Same basis for every percentage

Do not mix % by volume with % by weight in one calculation. Alcohol proof, acid normality, molarity, and activity coefficients need different models — this tool is for percentage concentration only.

What we do not model

No density corrections, heat of mixing, volume contraction (important for high-proof ethanol), serial dilution chains beyond one step, or chemical reactions. Treat results as ideal volumetric mixing.

Interesting facts

C₁V₁ = C₂V₂ is solute conservation

The identity holds when the diluent adds no solute and volumes are additive. It is the same bookkeeping used in clinical labs, photography chemistry, and pool chlorination charts.

Proof vs percent for spirits

In the US, proof is twice the alcohol-by-volume percentage (80 proof ≈ 40% ABV). Diluting spirits with water still follows C₁V₁ = C₂V₂ in ABV terms if volume change is ignored — real blending can shrink slightly.

Serial dilutions multiply factors

A 1:10 dilution followed by another 1:10 is a 1:100 overall factor. Each step is still C₁V₁ = C₂V₂; multiply dilution ratios when chaining tubes or flasks.

Weighted average, not arithmetic mean

Equal volumes of 10% and 50% average to 30%, but 100 mL of 10% plus 400 mL of 50% average to 42% — volume weights matter.

Frequently asked questions

Use C₂ = C₁ × V₁ / V₂ when you know stock strength, stock volume, and final volume. Or rearrange for the stock volume you need: V₁ = C₂ × V₂ / C₁. This calculator does both.

It states that concentration times volume is constant when you dilute with a solute-free liquid. Enter three of the four values (in matching units) and solve for the fourth.

Compute the stock volume for your target %, then diluent = final volume − stock volume. Example: 10% → 2% in 500 mL needs 100 mL stock and 400 mL water.

Choose “Mix two solutions”, enter each concentration and volume, and read the blend %. The tool uses C_f = (C_aV_a + C_bV_b) / (V_a + V_b).

Not by adding diluent alone — that only lowers concentration. Raising % requires removing solvent, adding solute, or starting from a stronger stock.

Any volume unit works if every volume in the problem uses the same unit. Pick mL, L, fl oz, or gal in the unit menu so labels match your recipe.

Dilution factor is often written V₂/V₁ or C₁/C₂. For a 10% → 2% dilution the factor is 5 (one part stock plus four parts diluent). The calculator reports volumes; the ratio follows from V₂ ÷ V₁.

Yes for percentage recipes when the diluent is effectively 0% of the active ingredient. Always follow product safety labels — some concentrates have minimum dilution limits this tool does not enforce.

References

  1. NIST Guide to the SI — Volume and concentration — National Institute of Standards and Technology (NIST) SI volume units and coherent quantity algebra for concentration work.
  2. IUPAC Gold Book — Dilution — International Union of Pure and Applied Chemistry Formal definition of dilution in chemical terminology.
  3. CDC — How to Dilute Bleach — Centers for Disease Control and Prevention Practical percentage dilution guidance for household disinfectant prep.

Educational mixing aid for ideal percentage dilutions — not a substitute for lab protocols, product labels, or chemical safety guidance.

Last reviewed: 2026-09-12 — Reviewed by: Editorial Team

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