Dilution Calculator | C₁V₁=C₂V₂
Solve any variable in C₁V₁ = C₂V₂ and generate serial dilution tables. Supports M, mM, µM, mg/mL, ppm and more.
Quick Presets
Solve For
Solving for V₁, volume to transfer
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What Is the Dilution Calculator | C₁V₁=C₂V₂?
The dilution formula C₁V₁ = C₂V₂ expresses conservation of solute. When you add pure solvent to a solution, no solute is created or destroyed, it simply occupies a larger volume. Because moles = concentration × volume, and moles are constant, the product C × V remains equal before and after dilution.
This calculator solves for any one of the four variables given the other three. It also handles serial dilutions, chains of equal dilution steps used to create concentration series spanning several orders of magnitude, which is a standard technique in microbiology, immunology, and analytical chemistry.
Units are flexible: any consistent concentration unit (M, mM, µM, g/L, mg/mL, % w/v, ppm…) and any consistent volume unit (µL, mL, L) can be used, as long as both C values share one unit and both V values share one unit.
Formula
| Symbol | Name | Description |
|---|---|---|
| C₁ | Stock concentration | Concentration of the original solution before dilution |
| V₁ | Aliquot volume | Volume pipetted from the stock into a new vessel |
| C₂ | Working concentration | Desired concentration of the final diluted solution |
| V₂ | Final volume | Total volume of the working solution (V₁ + solvent added) |
| DF | Dilution factor | C₁ / C₂ or V₂ / V₁, how many times more dilute C₂ is |
| f | Step factor (serial) | The factor each tube is diluted relative to the previous one |
How to Use
- 1Choose a mode: Select "C₁V₁ = C₂V₂" for a single dilution step, or "Serial Dilution" to generate a full step-by-step concentration table.
- 2Select what to solve for: In simple mode, click the variable you want, C₁, V₁, C₂, or V₂. The corresponding field locks and shows "solving". Try a preset to get started quickly.
- 3Set your units: Pick a concentration unit (M, mM, µM, g/L, mg/mL, ppm, etc.) from the dropdown, it applies to both C₁ and C₂. Pick a volume unit (µL, mL, L), it applies to both V₁ and V₂.
- 4Enter three known values: Fill in the three unlocked fields. If you leave one blank accidentally, the calculator will tell you exactly which field is missing.
- 5Press Calculate or Enter: The result appears instantly: the solved variable, all four values, the dilution factor, and the volume of solvent to add (V₂ − V₁).
- 6For serial dilution: Enter starting concentration, dilution factor (e.g. 10 for 1:10 steps), number of steps (up to 12), and total volume per tube. The table shows transfer volume and diluent for every tube.
- 7Copy or reset: Use "Copy result" to copy all values for a lab notebook. Press Reset or Esc to clear all fields and start fresh.
Example Calculation
Example 1: Preparing 1 M HCl from a 12 M stock
You need 500 mL of 1 M HCl. Your stock is 12 M concentrated HCl. Solve for V₁:
Lab note: always add acid to water, never water to acid, to avoid a dangerous exothermic splash.
Example 2: Antibody dilution from 1 mg/mL to 10 µg/mL
Stock antibody is 1 mg/mL. Working concentration needed is 10 µg/mL = 0.01 mg/mL in 1 mL total:
Example 3: Serial 1:10 dilutions from 100 mM, 6 steps, 1000 µL tubes
Understanding Dilution | C₁V₁=C₂V₂
Dilution is one of the most common operations in any wet lab, yet small errors in dilution calculations can cascade into failed experiments, inaccurate assays, or wasted reagents. Understanding the maths behind C₁V₁ = C₂V₂, rather than just plugging in numbers, helps you catch mistakes before they happen.
Dilution Factor and Fold-Change
The dilution factor (DF) is the ratio C₁ / C₂, or equivalently V₂ / V₁. A DF of 100 means the working solution is 100 times less concentrated than the stock.
Watch out for the difference between "1 in 10" and "1 to 10":
- ›"1 in 10", 1 part stock in 10 parts total → DF = 10 (common in UK clinical convention)
- ›"1 to 10" (1:10), 1 part stock + 10 parts diluent → 11 parts total → DF = 11 (common in microbiology)
This calculator uses the 1-in-N convention: V₁ is the stock volume, V₂ is the final total. If your protocol says "add 1 part stock to 9 parts buffer," set V₁ = 1 and V₂ = 10.
When to Use Serial Dilution
A single-step dilution by a factor of 10,000 would require pipetting just 1 µL of stock into 9,999 µL of diluent, the 1 µL measurement alone carries 5–10% uncertainty from typical micropipette error. Four consecutive 1:10 steps (each transferring 100 µL into 900 µL) give the same overall dilution with much better precision.
- ›Use serial dilutions for dose–response curves, antibody titrations, and standard curves.
- ›The geometric concentration series (log scale) matches the way biological responses typically behave.
- ›Mix each tube thoroughly before transferring to the next, incomplete mixing compounds the error at every step.
- ›Discard the last transfer volume from the final tube so each tube has the same total volume.
Supported Concentration Units
| Unit | Full name | Typical use |
|---|---|---|
| mol/L (M) | Molar | Buffer and reagent preparation in chemistry labs |
| mmol/L (mM) | Millimolar | Enzyme assays, cell signalling reagents |
| µmol/L (µM) | Micromolar | Drug concentrations, fluorescent dyes |
| nmol/L (nM) | Nanomolar | Potent ligands, receptor binding studies |
| g/L | Grams per litre | Protein solutions, saline, food science |
| mg/mL | Milligrams per mL | Protein and antibody stocks (= g/L numerically) |
| µg/mL | Micrograms per mL | Antibody dilutions, cytokines, ELISA standards |
| ng/mL | Nanograms per mL | Hormones, trace analytes, ELISA assays |
| % w/v | Percent weight/volume | Gel preparation, common chemical reagents |
| ppm | Parts per million | Environmental water quality, trace analysis |
| ppb | Parts per billion | Trace contaminants, drinking water standards |
Where Dilution Calculations Are Used
- ›Biochemistry & molecular biology: preparing enzyme substrates, running Western blot antibody incubations at the right dilution, diluting SYBR Green for qPCR master mixes.
- ›Cell culture: diluting 100× concentrated antibiotic stocks to 1× in complete media; adjusting cell density before seeding experiments.
- ›Clinical & pharmaceutical: calculating drug dilutions for IV preparations, reconstituting lyophilised antibiotics, preparing immunoassay standard curves.
- ›Microbiology: diluting dense bacterial cultures to a countable range (30–300 CFU/plate) using serial 1:10 steps before plating.
- ›Environmental testing: diluting water and soil extracts to within the linear range of an instrument (ICP-MS, UV-Vis, ion chromatography) before analysis.
- ›Food & beverage: preparing flavour concentrates, adjusting ethanol content, standardising titration solutions in quality control labs.
Practical Tips for Accurate Dilutions
- ›Always add concentrated solution to solvent, mixing generates heat, especially with strong acids and bases.
- ›For very small volumes (< 10 µL), use a calibrated micropipette and consider an intermediate dilution step to reduce pipetting error.
- ›Use volumetric flasks and graduated pipettes rather than measuring cylinders for best accuracy.
- ›For calibration curves, prepare serial dilutions from a single verified stock, independently weighing each standard introduces unnecessary variability.
- ›Store concentrated stocks in tightly sealed containers: evaporation or hygroscopic absorption changes the true concentration over time.
Frequently Asked Questions
Why is C₁V₁ = C₂V₂ valid?
The equation expresses conservation of solute: when you add pure solvent, no solute is created or destroyed. Because moles = concentration × volume, and moles are constant:
n (before) = C₁ × V₁ = C₂ × V₂ = n (after)
This holds for any homogeneous solution where the diluent contains none of the solute. For mixing two solutions of the same solute at different concentrations, use the weighted average: C_mix = (C₁V₁ + C₂V₂) / (V₁ + V₂).
Can I use different concentration units for C₁ and C₂?
No, both concentrations must be in the same unit for the formula to cancel correctly. The calculator enforces this by applying one concentration unit to both C₁ and C₂.
- ›1 g/L = 1 mg/mL numerically, so these can be used interchangeably.
- ›1 mol/L (M) ≠ 1 mmol/L (mM), always convert before entering values.
- ›ppm and ppb are mass/volume units (mg/L and µg/L respectively in water), consistent within themselves.
What is a serial dilution and when should I use it?
A serial dilution makes a large overall dilution achievable with manageable pipette volumes.
Use serial dilutions when:
- ›The single-step dilution would require < 1–2 µL of stock (below reliable pipette accuracy).
- ›You need a log-scale concentration series for dose–response curves or standard curves.
- ›You are counting bacteria and need to dilute a dense culture (10⁸ CFU/mL) to a countable range (30–300 CFU/plate).
- ›You need multiple working concentrations from one stock without re-calculating each one individually.
What does the dilution factor mean?
Dilution factor (DF) = C₁ / C₂ = V₂ / V₁. A DF of 100 means the solution is 100 times less concentrated.
- ›DF = 10: 100 mM → 10 mM, or 1 mL stock into 10 mL total.
- ›Serial dilution: 4 steps of 1:10 = DF of 10⁴ = 10,000.
- ›Serial dilution: 6 steps of 1:2 = DF of 2⁶ = 64.
- ›Overall concentration = C₀ / DF.
What is the solvent to add value?
Solvent to add = V₂ − V₁. This is the volume of water, buffer, or diluent you pipette into the vessel (in addition to the V₁ of stock).
Best practice in the lab:
- ›Add about half the solvent to the flask first.
- ›Transfer V₁ of stock solution and mix.
- ›Let the solution reach room temperature if heat was generated.
- ›Top up to the V₂ volume line in a volumetric flask for best accuracy.
Can I use this calculator for percent solutions?
Yes, select "% w/v" as the concentration unit. The formula works identically.
Example: 70% ethanol → 30% in 200 mL total:
V₁ = 0.30 × 200 / 0.70 = 85.7 mL (70% ethanol) Solvent to add = 200 − 85.7 = 114.3 mL water
Note: % v/v dilutions are strictly valid only when volumes are additive, a good approximation for dilute aqueous solutions but less accurate for high-concentration alcohol mixtures.
Why does the calculator give an error when C₂ is greater than C₁?
Dilution can only reduce concentration. If C₂ > C₁, you are asking the formula to give V₁ > V₂, which is physically impossible, you cannot transfer more stock than the total volume of the final solution.
- ›To get a higher concentration: find a more concentrated stock.
- ›Or concentrate the existing solution by evaporation, lyophilisation, or ultrafiltration.
- ›Or use a different starting material with the needed concentration.
Does the formula work for gases and other states of matter?
The formula applies to any homogeneous mixture where the diluent contains none of the analyte.
- ›Gas dilutions: mixing 1000 ppm calibration gas with zero-air to prepare 50 ppm working standard, same equation, concentration in ppm, volumes in L.
- ›Liquid solutions: all aqueous and non-aqueous systems where volume additivity is a good approximation.
- ›Dissolved solids: if you start from a known mass (m) and volume (V₁), then C₁ = m/V₁, and C₁V₁ = C₂V₂ gives the final volume for your target concentration.