Dilution Calculator
Dilute a stock solution with C₁V₁ = C₂V₂ - stock volume, diluent, dilution factor & serial dilutions
Last updated September 5, 2026
Method: Uses the exact conservation-of-solute equation C₁V₁ = C₂V₂, rearranged to solve for whichever of the four values you leave blank. Molar prefixes (M, mM, µM, nM) and volume units (µL, mL, L) are converted with fixed SI factors; percent, mg/mL and X-fold stocks are handled as plain ratios.
Included: Solving for V₁, C₁, C₂ or V₂; diluent volume to add; dilution factor and 1:N ratio; a mixing recipe; a serial dilution table with 1:2, 1:5, 1:10 or 1:100 steps and per-tube transfer volumes.
Not included: Molar-mass conversions between percent and molarity, density or volume-contraction corrections for concentrated acids and alcohols, and temperature effects. Volumes are assumed to be additive.
🧫 Stock volume needed (V₁)
🧪 Mixing recipe
- Measure 5 mL of the 1 M stock.
- Add 95 mL of diluent (water, buffer or solvent).
- Mix well. You now have 100 mL at 50 mM.
Ratio: 1 part stock to 19 parts diluent (a 1:20 dilution). Diluent = V₂ − V₁.
📊 Serial dilution from the stock
Starting at 1 M, each tube is diluted by the same factor from the previous one.
Per tube: transfer 100 µL from the previous tube into 900 µL of diluent (total 1 mL).
| Tube | Dilution | Cumulative factor | Concentration |
|---|---|---|---|
| 1 | 1:10 | 10× | 100 mM |
| 2 | 1:100 | 100× | 10 mM |
| 3 | 1:1,000 | 1,000× | 1 mM |
| 4 | 1:10,000 | 10,000× | 100 µM |
| 5 | 1:100,000 | 100,000× | 10 µM |
| 6 | 1:1,000,000 | 1,000,000× | 1 µM |
Pure math, no chemistry corrections. C₁V₁ = C₂V₂ assumes volumes are additive and the solute does not react with the diluent. For concentrated acids and alcohols, mix to the final volume in a volumetric flask rather than adding a calculated diluent volume.
Dilution calculator: how C₁V₁ = C₂V₂ works and how to use it
A dilution calculator tells you how much concentrated stock to measure and how much solvent to add to hit a target concentration. It solves the C₁V₁ = C₂V₂ equation for whichever value is missing: to make 100 mL of 50 mM from a 1 M stock, you need 5 mL of stock and 95 mL of water, a 20-fold (1:20) dilution.
Dilution is a ratio problem, so it sits next to our other ratio tools. Use this page when a concentration and a volume are involved; for plain proportions such as 3:5 = x:20 use the Ratio Calculator, for mass-per-volume of a material the Density Calculator, for a share of a whole the Percentage Calculator, and to switch between µL, mL, liters, fluid ounces and gallons the Unit Converter.
The dilution formula
When you dilute, you add solvent but no new solute. The number of moles (or grams) of dissolved substance is the same before and after, and moles = concentration × volume, so:
C₁ × V₁ = C₂ × V₂ Here C₁ is the stock (starting) concentration, V₁ is the volume of stock you take, C₂ is the final (target) concentration, and V₂ is the final total volume after mixing. Chemistry classes often write the same equation with M for molarity, M₁V₁ = M₂V₂, but it works for any concentration unit as long as both sides use the same one. Rearranged, one equation gives four answers:
V₁ = C₂V₂ ÷ C₁ C₁ = C₂V₂ ÷ V₁ C₂ = C₁V₁ ÷ V₂ V₂ = C₁V₁ ÷ C₂ Two derived numbers matter just as much in the lab. The diluent volume is V₂ − V₁, the amount of water or buffer you actually pour in. The dilution factor is C₁ ÷ C₂, which by the same equation equals V₂ ÷ V₁; a 20-fold dilution is written 1:20 and means one part stock in twenty parts total.
Worked example 1: 50 mM from a 1 M stock
You have a 1 M stock of sodium chloride and need 100 mL of a 50 mM working solution. First put both concentrations in the same unit: 1 M = 1,000 mM. Then solve for V₁:
V₁ = 50 mM × 100 mL ÷ 1,000 mM = 5 mL Measure 5 mL of stock, add 100 − 5 = 95 mL of water and mix. The dilution factor is 1,000 ÷ 50 = 20, so this is a 1:20 dilution: 1 part stock to 19 parts diluent. The calculator does the mM-to-M conversion for you; these are its default inputs, so you can check the numbers on load.
Worked example 2: 70% ethanol from 95%
Disinfectant-strength ethanol is 70%. Starting from a 95% bottle and wanting 500 mL, switch the calculator to the percent family:
V₁ = 70% × 500 mL ÷ 95% = 368.42 mL Measure 368.42 mL of 95% ethanol and add 131.58 mL of water. The dilution factor is 95 ÷ 70 = 1.357, a mild dilution. Because ethanol and water contract slightly on mixing, exact work adds the ethanol first and tops up to the 500 mL mark rather than adding exactly 131.58 mL; the difference is small but real.
Worked example 3: solving for the other three values
- Final concentration (C₂): you pipette 2 mL of a 5 mg/mL protein stock and bring it to 25 mL. C₂ = 5 × 2 ÷ 25 = 0.4 mg/mL, a 12.5-fold dilution.
- Stock concentration (C₁): an unlabelled stock, diluted 1 mL into 50 mL total, measures 0.2 mM. C₁ = 0.2 × 50 ÷ 1 = 10 mM, so the stock was 50 times stronger than the sample.
- Final volume (V₂): you have 10 mL of a 10X buffer and want 1X. V₂ = 10 × 10 ÷ 1 = 100 mL, so the 10 mL of stock plus 90 mL of water makes 100 mL of working buffer.
Stock volume needed by final volume and dilution factor
For any dilution factor, the stock volume is simply V₁ = V₂ ÷ DF, and the diluent is the rest. This table shows the stock volume in mL for common final volumes; a 1:20 dilution to 100 mL takes 5 mL of stock, exactly the default example above.
| Final volume (V₂) | 1:2 | 1:5 | 1:10 | 1:20 | 1:50 | 1:100 |
|---|---|---|---|---|---|---|
| 10 mL | 5 mL | 2 mL | 1 mL | 0.5 mL | 0.2 mL | 0.1 mL |
| 50 mL | 25 mL | 10 mL | 5 mL | 2.5 mL | 1 mL | 0.5 mL |
| 100 mL | 50 mL | 20 mL | 10 mL | 5 mL | 2 mL | 1 mL |
| 250 mL | 125 mL | 50 mL | 25 mL | 12.5 mL | 5 mL | 2.5 mL |
| 1 L | 500 mL | 200 mL | 100 mL | 50 mL | 20 mL | 10 mL |
Notice that the columns scale exactly with the final volume. If your target is not listed, take any row and multiply: 1:50 to 200 mL needs 2 × 2 mL = 4 mL of stock.
Serial dilution: a 1:10 series from a 1 M stock
A serial dilution chains identical steps so you can reach very low concentrations without pipetting an impractically tiny volume of stock. With 1 mL per tube and a 1:10 step, each tube receives 100 µL from the previous tube plus 900 µL of diluent. Starting from 1 M:
| Tube | Transfer in | Diluent | Cumulative dilution | Concentration |
|---|---|---|---|---|
| 1 | 100 µL of stock | 900 µL | 1:10 | 100 mM |
| 2 | 100 µL of tube 1 | 900 µL | 1:100 | 10 mM |
| 3 | 100 µL of tube 2 | 900 µL | 1:1,000 | 1 mM |
| 4 | 100 µL of tube 3 | 900 µL | 1:10,000 | 100 µM |
| 5 | 100 µL of tube 4 | 900 µL | 1:100,000 | 10 µM |
| 6 | 100 µL of tube 5 | 900 µL | 1:1,000,000 | 1 µM |
The cumulative factor multiplies: after n steps of 1:f the total dilution is 1:fn. Six 1:10 steps give a million-fold dilution, something you could never pipette in one go (1 µL into 1 L). A 1:2 series is common for assay standard curves: from 100 µg/mL, eight halving steps give 50, 25, 12.5, 6.25, 3.125, 1.5625, 0.78125 and 0.390625 µg/mL.
Common lab dilutions at a glance
Each row is a full C₁V₁ = C₂V₂ solution: the stock volume to take, the diluent to add and the resulting dilution factor.
| Dilution | Final volume | Stock (V₁) | Diluent | Factor |
|---|---|---|---|---|
| 10X buffer to 1X | 1 L | 100 mL | 900 mL | 10 |
| 95% ethanol to 70% | 500 mL | 368.42 mL | 131.58 mL | 1.357 |
| 1 M NaCl to 150 mM | 250 mL | 37.5 mL | 212.5 mL | 6.667 |
| 5 mg/mL to 0.4 mg/mL | 25 mL | 2 mL | 23 mL | 12.5 |
| 100 mM to 2 mM | 50 mL | 1 mL | 49 mL | 50 |
| 37% HCl to 10% | 100 mL | 27.03 mL | 72.97 mL | 3.7 |
The last row is a reminder that the arithmetic is only half the job: concentrated acid is always added to water, never the other way round, and the heat released means the final volume should be set after the mixture cools.
How to use this dilution calculator
- Choose what to solve for. The four cards at the top select V₁ (how much stock to take, the usual question), C₁, C₂ or V₂. The selected field turns into a read-only result box.
- Pick the concentration family. Molar lets you mix M, mM, µM and nM freely between the stock and the target. Percent, mg/mL and X-fold are ratio units, so both concentrations share that unit.
- Enter the three known values. Type the numbers and tap the unit toggles under each field - for instance 1 M stock, 50 mM target, 100 mL final volume.
- Read the result card. The headline shows the solved value in the unit you chose. The four tiles list stock to measure, diluent to add, final volume and dilution factor, and the mixing recipe spells out the three steps.
- Build a serial dilution if needed. Set the step factor (1:2, 1:5, 1:10 or 1:100), the number of tubes and the volume per tube; the table shows every tube's cumulative dilution and concentration, and the line above it gives the transfer and diluent volumes.
Everything updates as you type and runs in your browser, so nothing is uploaded.
Who this calculator is for
- Chemistry and biology students solving M₁V₁ = M₂V₂ homework or preparing solutions in a teaching lab.
- Research and clinical lab staff making working solutions from concentrated stocks, buffers from 10X or 20X concentrates, and standard curves by serial dilution.
- Pharmacy and nursing learners practising concentration-and-volume problems, where the same equation appears with mg/mL and percent strengths.
- Home brewers, aquarists, gardeners and cleaners diluting sanitizer, fertilizer, dosing solutions or bleach to a labelled strength.
- Anyone with a concentrate that says "dilute 1:20" or "use at 1X" and wants to know how much water that actually means.
Key terms
- Stock solution: the concentrated solution you start with (C₁). Stocks are made strong so they take up less space and keep longer.
- Working solution: the diluted solution you actually use (C₂), made fresh from the stock.
- Diluent (solvent): the liquid you add - water, buffer, saline or an organic solvent. Its volume is V₂ − V₁.
- Dilution factor (DF): C₁ ÷ C₂ = V₂ ÷ V₁. A DF of 20 is written 1:20 or "20-fold". The final concentration is C₁ ÷ DF.
- Molarity (M): moles of solute per liter of solution. 1 M = 1,000 mM = 1,000,000 µM. Chemistry problems usually state C₁ and C₂ in molarity.
- Percent concentration: % w/v is grams of solute per 100 mL of solution; % v/v is milliliters of solute per 100 mL. Both dilute with the same equation.
- X-fold stock: a concentrate labelled 10X or 50X is 10 or 50 times the working (1X) strength; the dilution factor equals the X number.
- Serial dilution: a chain of identical dilutions where each tube is made from the previous one; cumulative dilution is the product of the steps.
What changes the result the most
- The concentration ratio: V₁ scales with C₂ ÷ C₁. Halving the target concentration halves the stock volume; doubling the stock strength does the same.
- The final volume: V₁ is directly proportional to V₂. Making 1 L instead of 100 mL takes exactly ten times as much stock.
- Unit prefixes: confusing mM with M, or µL with mL, changes the answer by a factor of 1,000. Always match prefixes before you trust a number.
- Pipetting precision: a 5 µL error on a 50 µL stock volume is a 10% error in concentration; on a 5 mL stock volume it is 0.1%. Serial dilutions exist so that every transfer stays in a comfortable pipette range.
- Stock purity and age: the equation assumes C₁ is exactly what the label says. An evaporated or degraded stock gives a working solution that is off by the same percentage.
Tips for accurate dilutions
- Make to volume, not by addition, when accuracy matters: put the stock in a volumetric flask and fill to the mark, so V₂ is exact even if volumes do not add perfectly.
- Keep V₁ pipettable. If the calculator asks for less than about 2 µL, make an intermediate 1:10 or 1:100 dilution first and dilute again from that.
- Mix every tube in a serial dilution before transferring to the next; an unmixed tube carries the error down the whole series.
- Label with the final concentration and date. A tube marked "1:20" is useless a week later if nobody remembers the stock strength.
- Check the dilution factor against the volumes: C₁ ÷ C₂ must equal V₂ ÷ V₁. If the two do not match, one of the four numbers is wrong.
Limitations and assumptions
- Volumes are treated as additive. Mixtures such as ethanol and water or concentrated sulfuric acid and water contract slightly, so V₂ can be a little less than V₁ plus the added diluent.
- The calculator does not convert between families. Turning a 5% (w/v) stock into a molar target requires the solute's molar mass and density, which are not part of C₁V₁ = C₂V₂.
- Percent by mass (w/w) dilutes correctly only when the mass of diluent is used for V; the formula here uses volumes.
- Results are reported to six significant figures. Real pipettes and cylinders are far less precise, so round to what you can actually measure.
- No temperature, pH, reaction or solubility effects are modelled. Diluting a saturated solution or an acid that reacts with the diluent is a chemistry question, not a ratio question.
How it compares to related calculators
This page answers "how do I dilute a solution to a target concentration?" For nearby questions a sibling tool fits better: use the Ratio Calculator when the problem is a bare proportion with no concentrations in it, the Density Calculator when you need mass from a volume (or the reverse) of a single material, the Percentage Calculator when you want a percent of a number rather than a percent-strength solution, the Significant Figures Calculator to round a result to the precision of your glassware, and the Unit Converter when a protocol is written in fluid ounces or gallons instead of milliliters.
Sources and method
C₁V₁ = C₂V₂ is a direct consequence of conservation of solute and needs no external data; every result on this page is that one equation rearranged. The unit factors and definitions come from official references:
- National Institute of Standards and Technology (NIST) - SP 811, Guide for the Use of the International System of Units (SI): SI prefixes (milli, micro, nano) and the liter as an accepted unit of volume.
- IUPAC - Compendium of Chemical Terminology (Gold Book): definitions of amount concentration (molarity), mass concentration and volume fraction.
⚠️ Common mistakes & edge cases
Adding V₂ of diluent instead of V₂ − V₁
The final volume already includes the stock. For 100 mL of 50 mM from 1 M you add 95 mL of water to 5 mL of stock, not 100 mL. Adding 100 mL gives 105 mL at 47.6 mM instead of 50 mM.
Reading 1:10 as 1 plus 10
1:10 means one part in ten total (1 mL + 9 mL). Mixing 1 mL with 10 mL is an 11-fold dilution: from a 1 M stock you get 0.0909 M instead of 0.1 M, a 9% error that compounds through every step of a serial dilution.
Mismatched unit prefixes
Plugging 1 (M) and 50 (mM) into the formula as if they were the same unit gives V₁ = 5,000 mL - fifty times the final volume. Convert to a common unit first, or use the per-field toggles so the calculator does it.
Asking for a higher concentration than the stock
If C₂ is bigger than C₁ the equation returns V₁ larger than V₂ and a negative diluent volume. Dilution can only go down; the calculator shows a warning instead of a result.
Pipetting a volume too small to measure
A 1:2,000 dilution of 10 mM into 10 mL calls for 5 µL of stock. Errors on tiny volumes are large; make a 1:20 intermediate first (100 µL into 2 mL total), then dilute that 1:100 (100 µL into 10 mL total), and the smallest transfer becomes 100 µL.
Mixing concentration families
A 5% stock and a 20 mM target cannot go into C₁V₁ = C₂V₂ directly. Convert one to the other with the molar mass first; the ratio equation only works when both concentrations share a unit.
❓ Frequently asked questions
How do I use this dilution calculator?
Pick the value you want to solve for - stock volume (V1), stock concentration (C1), final concentration (C2) or final volume (V2) - then choose the concentration family (molar, percent, mg/mL or X) and enter the three known values with their units. The calculator applies C1V1 = C2V2, shows the missing value in the unit you selected, and lists the stock volume to measure out, the diluent to add, the dilution factor and a serial dilution table. With the default 1 M stock, 50 mM target and 100 mL final volume, it returns 5 mL of stock plus 95 mL of diluent.
What is the C1V1 = C2V2 formula?
C1V1 = C2V2 (also written M1V1 = M2V2) says that the amount of dissolved substance does not change when you dilute: the stock concentration times the stock volume equals the final concentration times the final volume. Rearranged, V1 = C2 x V2 / C1 gives the stock volume you need, C1 = C2 x V2 / V1 gives an unknown stock concentration, C2 = C1 x V1 / V2 gives the concentration after mixing, and V2 = C1 x V1 / C2 gives the final volume a stock can be stretched to.
How do I calculate the dilution factor?
The dilution factor is the stock concentration divided by the final concentration, which is the same as the final volume divided by the stock volume: DF = C1 / C2 = V2 / V1. Diluting a 1 M stock to 50 mM is a 20-fold dilution, written 1:20, and it means 1 part stock in 20 parts total (1 part stock plus 19 parts diluent). The final concentration is always the stock concentration divided by the dilution factor.
How much water do I add to dilute a solution?
Subtract the stock volume from the final volume: diluent = V2 - V1. To make 500 mL of 70% ethanol from a 95% stock, V1 = 70 x 500 / 95 = 368.42 mL of stock, so you add 500 - 368.42 = 131.58 mL of water. The calculator prints this diluent volume for every result, and the mixing recipe lists it as step 2.
What is a serial dilution and how do I calculate it?
A serial dilution repeats the same dilution step tube after tube. In a 1:10 series with 1 mL per tube, you transfer 100 µL from each tube into 900 µL of diluent in the next one. Concentrations fall by a factor of 10 each step, so a 1 M stock becomes 100 mM, 10 mM, 1 mM, 100 µM, 10 µM and 1 µM in tubes 1 through 6, and the cumulative dilution in tube 6 is 1:1,000,000. Choose the factor and number of tubes in the serial dilution panel of the calculator.
Does 1:10 mean 1 part in 10 or 1 part plus 10?
In this calculator, and in most laboratory conventions, 1:10 means 1 part stock in 10 parts total - 1 mL of stock plus 9 mL of diluent, a 10-fold dilution. Adding 1 mL to 10 mL of diluent is an 11-fold dilution: a 1 M stock would give 0.0909 M instead of 0.1 M, a 9% error. Always read a protocol carefully, because some fields write dilutions as stock-to-diluent ratios (1 + 9) rather than stock-to-total.
Can I mix units like a 1 M stock and a 50 mM target?
Yes. In the molar family each concentration has its own M, mM, µM or nM toggle and each volume has its own µL, mL or L toggle. The calculator converts everything to moles per liter and milliliters internally, then shows the answer in the unit chosen for the solved field. Percent, mg/mL and X-fold concentrations are treated as plain ratios, so both concentrations must use the same one of those families.
Why can't the final concentration be higher than the stock?
Diluting adds solvent, which can only lower the concentration. If C2 is greater than C1 the formula would demand a stock volume larger than the final volume, meaning a negative amount of diluent. The calculator flags this instead of showing a nonsensical result. To reach a higher concentration you need a stronger stock, or you must evaporate solvent or add solid solute, neither of which is a dilution.
How do I dilute a 10X buffer to 1X?
Use the X (fold) family and enter C1 = 10, C2 = 1 and your final volume. For 1 L of 1X buffer, V1 = 1 x 1,000 / 10 = 100 mL of 10X stock plus 900 mL of water, a 10-fold dilution. The same rule applies to any concentrated stock labelled in X: the stock volume is the final volume divided by the X number.
How do I make 70% ethanol from 95% or 100%?
Enter C1 = 95 (or 100), C2 = 70 and the final volume in the percent family. For 500 mL from 95% ethanol you need 368.42 mL of stock and 131.58 mL of water; from 100% ethanol you need 350 mL of stock and 150 mL of water. Note that ethanol and water contract slightly when mixed, so for exact percentages bring the ethanol up to the final volume mark in a graduated cylinder rather than adding the calculated water volume.
Does the dilution formula work for percent and mg/mL, not just molarity?
Yes. C1V1 = C2V2 works with any concentration unit as long as both concentrations use the same unit, because the units cancel in the ratio C2 / C1. Percent (w/v or v/v), mg/mL, µg/mL, ppm, X-fold stocks and molarity all work. What you cannot do is mix families - for example a 5% stock and a 20 mM target - without first converting one to the other using the molar mass.
Is this dilution calculator free?
Yes. The dilution calculator is completely free, needs no sign-up and has no limit on how many dilutions you run. All arithmetic happens in your browser, nothing you enter is sent to a server, and you can switch between solving for V1, C1, C2 and V2 and rebuild the serial dilution table as often as you like.
💡 Good to know
The dilution factor is a shortcut
Once you know the factor, you never need the concentrations again: stock volume = final volume ÷ factor. A 1:20 dilution to any volume is always one-twentieth stock and nineteen-twentieths diluent.
Serial dilutions multiply, they do not add
Three 1:10 steps are a 1:1,000 dilution, not 1:30. Six of them reach one part in a million, which is why serial dilution is the standard way to prepare standard curves and count bacteria.
Acid into water, never water into acid
The math is the same either way, but the heat is not. Add concentrated acid slowly to the water so the large water volume absorbs the heat; adding water to acid can boil and spatter.