C1V1 = C2V2
Last reviewed: May 2026
C1V1 = C2V2 is one of the most-used equations in laboratory science, expressing the conservation of solute during dilution.[1] It applies to any solution where you are adding solvent to reduce concentration without adding or removing solute. This calculator solves for any missing variable and handles unit conversions. For related calculations, see the pH Calculator.
| Application | Stock Conc | Target Conc | Final Volume | Stock Needed |
|---|---|---|---|---|
| Lab buffer | 10× (10 M) | 1× (1 M) | 1,000 mL | 100 mL |
| Bleach cleaner | 8.25% | 0.5% | 1,000 mL | 60.6 mL |
| Fertilizer | 20-20-20 | 1 tbsp/gal | 1 gallon | 1 tbsp |
| Antibody staining | 1 mg/mL | 10 µg/mL | 5 mL | 50 µL |
The dilution equation C1V1 = C2V2 works because the total amount of solute remains constant during dilution — you are only adding solvent, not solute. C1 is the initial concentration, V1 is the volume of concentrated solution you need, C2 is the desired final concentration, and V2 is the desired final volume. The formula works with any consistent concentration units — molarity, percent, mg/mL, ppm, or even arbitrary units — as long as both sides use the same unit. This universality makes it one of the most versatile equations in laboratory science.
| Step | Dilution Ratio | Cumulative Factor | If Starting at 1M |
|---|---|---|---|
| Stock | — | 1× | 1 M |
| 1st | 1:10 | 10× | 0.1 M |
| 2nd | 1:10 | 100× | 0.01 M |
| 3rd | 1:10 | 1,000× | 1 mM |
| 4th | 1:10 | 10,000× | 100 µM |
| 5th | 1:10 | 100,000× | 10 µM |
Stock solutions are concentrated solutions prepared in advance and diluted as needed. Preparing one requires calculating the mass of solute: mass = molarity × volume (liters) × molecular weight. For 1M NaCl (MW 58.44 g/mol), dissolve 58.44g in enough water to make exactly 1 liter — not in 1 liter of water, which would produce slightly more than 1 liter total. Common stocks include 10× PBS buffer, 1000× antibiotic stocks, and 5M NaCl. Always label with concentration, date, solvent, and expiration. For molar mass lookups, use our Molar Mass Calculator.
When diluting concentrated acids (especially sulfuric acid), always add acid to water — never water to acid. Adding water to concentrated sulfuric acid causes an extremely exothermic reaction that can produce explosive boiling and splash concentrated acid. The mnemonic "do what you oughta — add acid to water" is taught in every chemistry safety course. Concentrated sulfuric acid (18M, 98%) releases approximately 880 kJ per mole when diluted. Concentrated hydrochloric acid (12M) releases corrosive HCl gas and should only be handled in a fume hood. Always wear splash goggles, chemical-resistant gloves, and a lab coat when handling concentrated solutions.
Not all concentrations use molarity. Weight/volume (w/v) percent means grams of solute per 100 mL of solution. Volume/volume (v/v) percent means mL of solute per 100 mL of solution. A 10% bleach solution means 10 mL of bleach per 100 mL total. Parts per million (ppm) equals mg per liter for dilute aqueous solutions — used for water quality, environmental monitoring, and trace analysis. Parts per billion (ppb) equals µg per liter. C1V1 = C2V2 works with all of these units as long as both sides match. Swimming pool chemistry targets 1–3 ppm free chlorine, and adjusting levels requires dilution math based on pool volume.
Dilution calculations appear outside the laboratory constantly. Mixing cleaning concentrate, preparing baby formula, diluting essential oils for safe topical use, adjusting cocktail strength, and mixing fertilizer all involve C1V1 = C2V2. A cleaning product labeled "dilute 1:20" means 1 part concentrate to 19 parts water (20 parts total). An essential oil "2% dilution" means 12 drops per 30 mL carrier oil. Understanding dilution prevents waste (too much concentrate) and ineffectiveness (too little).
Solution volume changes with temperature due to thermal expansion. A solution prepared as exactly 1.000 liter at 25°C will measure slightly differently at other temperatures. Volumetric glassware is calibrated at 20°C — using it at significantly different temperatures introduces systematic error. For most routine preparations this effect is negligible (less than 0.5%), but pharmaceutical and forensic laboratories must account for it. Solubility also changes with temperature — some solutes may precipitate when cooled, invalidating the concentration even if the volume is correct.
→ V1 is the volume of concentrate, not solvent. If V1 = 25 mL and V2 = 500 mL, add 25 mL of stock plus 475 mL of solvent to reach 500 mL total.
→ Always add acid to water. For exothermic dilutions, add concentrate slowly to solvent while stirring, never the reverse.
→ Use volumetric flasks for precision. Graduated cylinders and beakers are less precise and should only be used for approximate preparations.
See also: Concentration Calculator · pH Calculator · Molar Mass · Equation Balancer
| Application | Typical Dilution | Purpose |
|---|---|---|
| Blood chemistry panel | 1:10 to 1:100 | Bring analyte into assay range |
| Bacterial plating | 1:10⁶ to 1:10⁸ | Countable colonies (30–300/plate) |
| ELISA standards | 1:2 serial (7–8 steps) | Generate standard curve |
| PCR template | 1:10 to 1:100 | Reduce inhibitor concentration |
| Cleaning solutions | 1:10 to 1:50 | Working strength from concentrate |
The dilution factor (DF) is the ratio of final volume to sample volume: DF = V2/V1. A 1:10 dilution has a DF of 10. To find the original concentration from a diluted measurement, multiply the measured concentration by the dilution factor. If you measure 0.5 mg/mL in a sample diluted 1:20 (DF = 20), the original concentration is 0.5 × 20 = 10 mg/mL. For serial dilutions, the cumulative dilution factor is the product of all individual factors — three consecutive 1:10 dilutions produce a cumulative factor of 1,000. This back-calculation is essential in clinical chemistry, where patient samples are routinely diluted before analysis and reported values must reflect original concentrations.
While molarity (M = moles per liter of solution) is the most common concentration unit, two others appear frequently in chemistry. Molality (m = moles per kilogram of solvent) is independent of temperature because mass does not change with thermal expansion — making it preferred for precise thermodynamic calculations like boiling point elevation and freezing point depression. Normality (N = equivalents per liter) accounts for the reactive capacity of a substance: 1M H₂SO₄ is 2N because each molecule provides two H⁺ ions. Normality simplifies acid-base titration calculations because at the equivalence point, N₁V₁ = N₂V₂ directly without needing to account for stoichiometry. However, normality has fallen out of favor in modern chemistry curricula because the number of equivalents depends on the specific reaction, creating potential confusion.
Mastering these concentration units and their interconversions is fundamental to any quantitative work in chemistry, biology, pharmacology, and environmental science.
Dilution calculations, whether in a research lab or a kitchen, all rely on the same fundamental principle: solute is conserved when solvent is added, and the relationship between concentration and volume is always inversely proportional.
→ Add solute to solvent. Especially with acids: add acid to water, never water to acid.[1]
→ Measure to final volume. After adding stock, bring to the target volume, do not add the volume of solvent.[2]
→ Units must match. Both concentrations in the same unit, both volumes in the same unit.
→ For pool chemicals. Use the Pool Chemical Calculator for specific dosing.
See also: pH Calculator · Pool Chemicals · Density · Equation Balancer