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Formula
C₁ × V₁ = C₂ × V₂
- C₁
- concentration of the stock solution you start from
- V₁
- volume of stock solution you take
- C₂
- concentration of the diluted (final) solution
- V₂
- total volume of the diluted solution, stock plus diluent
What a dilution is
Diluting a solution means adding more solvent, usually water or a buffer, so the same amount of dissolved substance is spread through a larger volume. The amount of solute stays fixed. That single fact gives the dilution equation:
C₁ × V₁ = C₂ × V₂
The left side is the amount of solute in the stock you take, and the right side is the amount in the finished solution. Because the two are equal, knowing any three of the values gives you the fourth. This dilution calculator does that rearranging for you and adds the two numbers you actually need at the bench: how much diluent to add and the dilution factor.
How to use the dilution calculator
- Choose what to solve for. Most people want the stock volume (V₁): how much concentrate to pipette.
- Enter the other three values. Concentrations can be molar (M, mM, µM, nM) or mass-based (g/L, mg/mL, µg/mL, % w/v), and volumes can be in mL or L.
- C₁ and C₂ may use different units of the same kind, such as a 10 mM stock diluted to 50 µM. The calculator converts them exactly.
- Read the result, the diluent to add (V₂ − V₁), the dilution factor and a step-by-step line showing the formula with your numbers.
If the answer would make the solution stronger than the stock, the calculator stops and asks you to check which value is C₁ and which is C₂. Swapping them is the most common entry mistake.
How to calculate a dilution by hand
Say you need 1 L of 0.3 M hydrochloric acid and you have a 6 M stock. Solve for V₁:
V₁ = C₂ × V₂ ÷ C₁ = 0.3 M × 1,000 mL ÷ 6 M = 50 mL
Measure 50 mL of the stock and make it up to 1 L, which means adding about 950 mL of water. The dilution factor is 6 ÷ 0.3 = 20, so this is a 1:20 dilution. (With acids, always add the acid to the water, never the other way round.)
Finding the final concentration
You take 15 mL of a 20% w/v solution and make it up to 200 mL:
C₂ = C₁ × V₁ ÷ V₂ = 20% × 15 mL ÷ 200 mL = 1.5% w/v
The units of C₂ come out the same as C₁, so there is nothing to convert. This is the dilution calculator percent case, and it works the same way for mg/mL or µg/mL.
How to calculate the dilution factor
The dilution factor (DF) tells you how many times weaker the final solution is:
DF = C₁ ÷ C₂ = V₂ ÷ V₁
A factor of 20 means 1 part stock in 20 parts total. Many protocols write that as 1:20. Cleaning-product labels sometimes write 1:20 to mean 1 part concentrate plus 20 parts water, which is a factor of 21, so read the label’s convention before you mix.
Dilution factor chart for 100 mL
Volumes of stock and diluent for 100 mL of final solution at common dilution factors. Scale everything in proportion for other final volumes.
| Dilution factor | Ratio | Stock (V₁) | Diluent to add |
|---|---|---|---|
| 2× | 1:2 | 50 mL | 50 mL |
| 5× | 1:5 | 20 mL | 80 mL |
| 10× | 1:10 | 10 mL | 90 mL |
| 20× | 1:20 | 5 mL | 95 mL |
| 50× | 1:50 | 2 mL | 98 mL |
| 100× | 1:100 | 1 mL | 99 mL |
| 1,000× | 1:1,000 | 0.1 mL | 99.9 mL |
Above about 100×, a single step means pipetting a very small volume, which magnifies measuring error. A serial dilution in two or three steps is usually more accurate.
Serial dilutions
In a serial dilution, each step uses the previous solution as its stock. The overall factor is the product of the step factors: three 10× steps give 1,000×, and two 20× steps give 400×. To plan one, run the calculator once per step and feed each C₂ in as the next C₁. Serial dilutions are standard for standard curves, antibody titrations, cell counts and bacterial plating.
Where dilutions come up
- Teaching and research labs: making working solutions of acids, bases, buffers and reagents from concentrated stocks.
- Biology and biochemistry: diluting DNA, proteins or drugs from mg/mL or mM stocks to µg/mL or µM working concentrations.
- Pharmacy and peptides: reconstituting a vial and drawing up a dose. The peptide calculator handles the syringe units for that case.
- Cleaning and detailing products: turning a concentrate into a ready-to-use spray at the ratio on the label.
When you only know a mass of solid rather than a stock concentration, start with the molarity calculator to make the stock, then dilute it here. For converting between mL, liters, fluid ounces and gallons, use the volume converter.
Common mistakes
- Using the diluent volume as V₂. V₂ is the total final volume. Adding 100 mL of water to 10 mL of stock gives 110 mL, not 100 mL.
- Mixing molar and mass units. C₁V₁ = C₂V₂ cannot turn mol/L into mg/mL. You need the molar mass for that.
- Mixing volume units by hand. 25 µL and 25 mL differ by a factor of 1,000. Let the unit selectors do the conversion.
- Reading 1:10 two ways. Decide whether the ratio means parts of total or parts of water before you measure.
- Expecting a dilution to concentrate. If your numbers say C₂ is higher than C₁, two values have been swapped.
Glossary
- Stock solution: a concentrated solution kept on hand and diluted before use.
- Diluent: the liquid added to dilute, usually water, saline or buffer.
- Working solution: the diluted solution actually used in the experiment.
- Dilution factor: final volume ÷ stock volume, equal to stock concentration ÷ final concentration.
- Molar (M): moles of solute per liter of solution. NIST prefers writing this as mol/L.
Worked examples
Stock volume for a working solution
Make 250 mL of 0.2 M solution from a 5 M stock.
Inputs: Solve for = Stock volume (V₁); Stock concentration (C₁) = 5; C₁ unit = Molar (mol/L); Final concentration (C₂) = 0.2; C₂ unit = Molar (mol/L); Final volume (V₂) = 250; Final volume (V₂) unit = Milliliters
C₁V₁ = C₂V₂ V₁ = C₂ × V₂ ÷ C₁ = 0.2 M × 250 mL ÷ 5 M = 10 mL Diluent = V₂ − V₁ = 250 mL − 10 mL = 240 mL
Answer: Take 10 mL of 5 M stock and add 240 mL of diluent to make 250 mL at 0.2 M (V₁ solved).
Final concentration after diluting
3 mL of a 12 mg/mL stock is made up to 45 mL.
Inputs: Solve for = Final concentration (C₂); Stock concentration (C₁) = 12; C₁ unit = Milligrams per mL (mg/mL); Stock volume (V₁) = 3; Stock volume (V₁) unit = Milliliters; C₂ unit = Milligrams per mL (mg/mL); Final volume (V₂) = 45; Final volume (V₂) unit = Milliliters
C₁V₁ = C₂V₂ C₂ = C₁ × V₁ ÷ V₂ = 12 mg/mL × 3 mL ÷ 45 mL = 0.8 mg/mL Diluent = V₂ − V₁ = 45 mL − 3 mL = 42 mL
Answer: Take 3 mL of 12 mg/mL stock and add 42 mL of diluent to make 45 mL at 0.8 mg/mL (C₂ solved).
Mixed units: millimolar stock to micromolar
50 mL of a 10 mM stock diluted to 200 µM. What final volume can you make?
Inputs: Solve for = Final volume (V₂); Stock concentration (C₁) = 10; C₁ unit = Millimolar (mmol/L); Stock volume (V₁) = 50; Stock volume (V₁) unit = Milliliters; Final concentration (C₂) = 200; C₂ unit = Micromolar (µmol/L)
C₁V₁ = C₂V₂ V₂ = C₁ × V₁ ÷ C₂ = 10 mM × 50 mL ÷ 200 µM = 2.5 L Diluent = V₂ − V₁ = 2.5 L − 50 mL = 2.45 L
Answer: Take 50 mL of 10 mM stock and add 2.45 L of diluent to make 2.5 L at 200 µM (V₂ solved).
Edge case: stock concentration from a known dilution
0.5 mL of an unknown stock made up to 20 mL gives a 1% w/v solution. How strong was the stock?
Inputs: Solve for = Stock concentration (C₁); C₁ unit = Percent (w/v, g/100 mL); Stock volume (V₁) = 0.5; Stock volume (V₁) unit = Milliliters; Final concentration (C₂) = 1; C₂ unit = Percent (w/v, g/100 mL); Final volume (V₂) = 20; Final volume (V₂) unit = Milliliters
C₁V₁ = C₂V₂ C₁ = C₂ × V₂ ÷ V₁ = 1 % w/v × 20 mL ÷ 0.5 mL = 40 % w/v Diluent = V₂ − V₁ = 20 mL − 0.5 mL = 19.5 mL
Answer: Take 0.5 mL of 40 % w/v stock and add 19.5 mL of diluent to make 20 mL at 1 % w/v (C₁ solved).
Assumptions and limitations
- The amount of solute does not change when you dilute, so C₁ × V₁ = C₂ × V₂. Nothing reacts, precipitates or evaporates.
- V₂ is the total final volume. The diluent to add is V₂ − V₁, which assumes volumes add. For water-based solutions that is close enough for routine work; concentrated acids, ethanol and other mixtures that shrink on mixing should be made up to the mark in a volumetric flask instead.
- C₁ and C₂ must be the same kind of unit, either molar (M, mM, µM, nM) or mass-based (g/L, mg/mL, µg/mL, % w/v). Within a kind, units are converted exactly; 1 mg/mL = 1 g/L and 1% w/v = 10 g/L.
- Converting between molar and mass units needs a molar mass, so mixed kinds are rejected. Use the molarity calculator for that step.
- A dilution can only lower the concentration, so results where C₂ would exceed C₁ or V₁ would exceed V₂ are flagged as entry errors.
- Results are shown to 6 significant figures; volumes switch between L, mL and µL automatically.
Frequently asked questions
What is the formula for dilution?
The dilution equation is C₁V₁ = C₂V₂, where C₁ and V₁ are the stock concentration and the volume you take, and C₂ and V₂ are the final concentration and total volume. It works because diluting adds solvent but no solute.
How do you calculate the dilution factor?
Divide the stock concentration by the final concentration, or the final volume by the stock volume. Taking 4 mL of stock up to 60 mL is a dilution factor of 60 ÷ 4 = 15, so the final solution is one fifteenth as strong.
How much water do I add to dilute a solution?
Subtract the stock volume from the final volume. To make 300 mL from 12 mL of stock, add 288 mL of water. For accurate work, add the stock to a volumetric flask and fill to the mark rather than measuring the water separately.
Does C1V1 = C2V2 work with percent and mg/mL?
Yes. The equation only needs C₁ and C₂ in the same units, so it works with molarity, mg/mL, µg/mL or percent. If one concentration is in mol/L and the other in mg/mL, convert first using the molar mass.
What does a 1:10 dilution mean?
In most lab and clinical usage, 1:10 means 1 part stock in 10 parts total, for example 1 mL of stock plus 9 mL of diluent. Some fields, such as cleaning products, use 1:10 to mean 1 part concentrate plus 10 parts water, so check which convention a label uses.
How do I calculate a serial dilution?
Multiply the dilution factors of each step. Three 1:10 steps give a total factor of 10 × 10 × 10 = 1,000, so a 5 mM stock ends at 5 µM. Run this calculator once per step, using each result as the next stock.
Sources
- Chemistry 2e, 3.3 Molarity (Dilution of Solutions) — OpenStax (Rice University). Checked October 10, 2026.
- Chemistry 2e, 3.4 Other Units for Solution Concentrations — OpenStax (Rice University). Checked October 10, 2026.
- NIST Guide to the SI, Chapter 8: Comments on Some Quantities and Their Units — National Institute of Standards and Technology. Checked October 10, 2026.