Solution Dilution Calculator
Enter any three of C₁, V₁, C₂, and V₂ — the calculator solves the fourth using C₁V₁ = C₂V₂.
Percent, molarity, and ppm aren't the same kind of number
What "C" is actually counting
C₁V₁ = C₂V₂ works because it's an accounting identity: the amount of solute doesn't change when you add solvent, so concentration × volume before must equal concentration × volume after. But "concentration" only means one specific thing once you know what it's a fraction of, and the units in this calculator's dropdown don't all pick the same denominator:
- % w/w (weight-in-weight) — grams of solute per 100 g of solution. This is what a reagent bottle or SDS almost always means by a bare "%," and it's mass on both sides.
- % w/v (weight-in-volume) — grams of solute per 100 mL of solution. Common in labs, pharmacy, and biology; mass over volume.
- % v/v (volume-in-volume) — mL of solute per 100 mL of solution, used for liquid-in-liquid mixes like ethanol or isopropanol in water.
- Molarity (M) — moles of solute per liter of solution, the unit the chemistry itself actually reacts in.
- ppm — parts per million by mass, which for a dilute aqueous solution (density close to 1.00 g/mL) collapses to the convenient shortcut of mg per liter, or µg per mL — the same equivalence this site's bleach dilution calculator and its dilution ratios guide use to turn a bleach ratio into ppm available chlorine.
These aren't a house convention — USP General Notices §8.140 defines percentage concentrations for pharmacopeial solutions using exactly this w/w, w/v, v/v split, and it's the same split analytical chemistry and industrial spec sheets use.
Where C₁V₁ = C₂V₂ holds, and where it quietly breaks
The formula is valid the instant C₁ and C₂ are expressed in the same basis — both % w/w, or both molar, or both ppm. It doesn't care which basis you picked, only that you didn't switch baskets partway through. That's also why this calculator locks the C₁ and C₂ unit dropdowns together instead of letting you convert freely between families (see the FAQ below): going from % w/w to % w/v, or either to molarity, needs information the calculator doesn't collect — a density for the mass↔volume switch, a molar mass for the mole↔mass switch.
The gap between mass-basis and volume-basis percent isn't a rounding issue; it scales with density and gets worse the more concentrated the solution is. 98% w/w sulfuric acid has a density around 1.84 g/mL, so a liter of it holds roughly 1,800 g of H₂SO₄ — expressed as % w/v that's about 180%, a number that only makes sense once you remember it's grams per 100 mL of a liquid much denser than water, not grams per 100 g. Run a supplier's 98% w/w spec straight into a target that expected % w/v and the answer is off by close to a factor of two. The dilution math that survives the shop floor guide walks through this conversion in full, including the general g/L formula for turning a mass-based spec into a volume-based working figure.
At dilute, near-water concentrations the two percentages sit close enough that conflating them mostly goes unnoticed — which is exactly what makes the habit dangerous the day the same person reaches for a concentrated stock instead.
Worked examples
Diluting a 10% stock to a 2% working solution
A lab needs 500 mL of a 2% working solution but only stocks 10% concentrate on the shelf.
- C₁
- 10 %
- C₂
- 2 %
- V₂
- 500 mL
Use 100 mL of stock, add 400 mL water
Diluting a 5 M stock to 0.5 M for an experiment
A researcher needs 250 mL of a 0.5 M working solution from a 5 M concentrated stock.
- C₁
- 5 M
- C₂
- 0.5 M
- V₂
- 250 mL
Use 25 mL of stock, add 225 mL water
Converting a 36% HCl reagent into a 10% w/v working acid
A reagent bottle is labeled 36.23% w/w HCl at 1.18 g/cm³ (read directly off this site's HCl concentration table) — but the SOP calls for 1 L of a 10% w/v working solution. The two percentages aren't the same basis, so the stock has to be converted before C₁V₁ = C₂V₂ can run: % w/v = % w/w × density = 36.23 × 1.18 = 42.75% w/v.
- Stock
- 36.23% w/w, 1.18 g/cm³
- Stock, converted
- 42.75% w/v
- C₂
- 10% w/v
- V₂
- 1000 mL
Use 233.91 mL of stock, add ≈766.09 mL water (acid into water, not the reverse)
Diluting a 1000 ppm reference standard to 50 ppm
A lab has a certified 1000 µg/mL (1000 ppm) reference standard and needs 100 mL of a 50 µg/mL (50 ppm) working standard for a calibration curve. µg/mL and ppm are numerically the same for a dilute aqueous solution, so this runs as a direct C₁V₁ = C₂V₂ step with no density conversion needed.
- C₁
- 1000 µg/mL (1000 ppm)
- C₂
- 50 µg/mL (50 ppm)
- V₂
- 100 mL
Use 5 mL of standard, add 95 mL water
Diluting a 10X buffer stock to 1X
Molecular biology stocks are often labeled in fold (X) rather than percent or molarity. A 10X buffer concentrate needs to become 1 L of 1X working buffer for the bench — the calculator's concentration dropdown has an "X" option built in for exactly this.
- C₁
- 10X
- C₂
- 1X
- V₂
- 1000 mL
Use 100 mL of 10X stock, add 900 mL water
How the formula works
Adding solvent to a solution doesn't change how much solute is dissolved in it — it only spreads that same amount of solute through a larger volume, which lowers the concentration. The amount of solute is concentration times volume, so that quantity — C₁V₁ before dilution, C₂V₂ after — must stay the same on both sides. Setting them equal gives the formula below; rearrange it for whichever of the four values you're solving for, and the difference V₂ − V₁ tells you how much solvent to add.
Dilution ratios: 1:X, percent, and fold (X)
Outside a lab, dilution strength often gets quoted as a ratio — "1:10," "1:50" — instead of a percent or molarity. A ratio and a percent describe the same thing, but only once you know which convention the ratio is using. The more common technical convention, and the one the table below uses, is parts-of-total: "1:10" means 1 part concentrate made up to a total of 10 parts, so it's 10% of the mix — not 1 part added to 10 parts of diluent, which would make it roughly 9.1% of an 11-part total. The two conventions sit close together at tight ratios and stay proportionally apart at loose ones; see bleach dilution ratios: what 1:10 and 1:50 really mean for exactly how much that ambiguity costs, and how CDC guidance resolves it in writing.
| Ratio (1:X) | Concentration | Dilution factor |
|---|---|---|
| 1:2 | 50% | 2× |
| 1:4 | 25% | 4× |
| 1:5 | 20% | 5× |
| 1:10 | 10% | 10× |
| 1:20 | 5% | 20× |
| 1:50 | 2% | 50× |
| 1:100 | 1% | 100× |
"Dilution factor" and "fold" are the same number as the ratio's X — a 1:10 dilution is a 10-fold, or 10×, dilution — and it's exactly what this calculator's own "Dilution factor" chip reports once a volume is solved: V₂ ÷ V₁. The concentration dropdown also carries a direct "X" unit for this notation, so a 10X stock diluted to 1X (see the buffer example above) doesn't need converting to percent at all — enter both sides in X and the equation solves the same way it does for any other shared unit.
Where this calculator's assumptions stop holding
Volumes don't always add up
This calculator treats V₂ as V₁ plus the solvent you add — the beaker diagram above draws it exactly that way. That's a very good approximation for dilute aqueous solutions, but it isn't a law of physics: when two different liquids mix, their molecules can pack together more tightly (or, less often, more loosely) than either did alone, so the combined volume comes out slightly under — or over — the sum of the parts. Ethanol and water are the textbook case: 500 mL of each mixed together measures roughly 965 mL, not 1000 mL, a contraction of about 3.6% near the middle of the range. See why alcohol plus water is less than the sum for the full curve and what it means for hitting a target ABV; the ethanol-water dilution calculator blends from measured density data instead of assuming additive volumes. Concentrated acids in water contract too, on top of releasing real heat (more below).
For a plain aqueous dilution — salt, sugar, most lab reagents in water — the additive assumption this calculator makes is fine for everyday work. It's specifically liquid-in-liquid mixes at high concentration where the gap gets large enough to matter, and where "dilute to the final volume in a graduated vessel" beats "add this many mL of water" as a technique.
Temperature
Every concentration figure that goes into this calculator is implicitly tied to whatever temperature the solution was at when it was measured or prepared, because both liquid volume and — for w/v and molarity — the density link between mass and volume shift with temperature. A stock measured warm and diluted, then compared later against a spec written for room temperature, won't match exactly. It's a second-order effect for a quick working solution, but not for anything that needs to hold to better than a percent or two.
Concentrated stock solutions can generate real heat
Diluting a concentrated acid or base is often exothermic — concentrated sulfuric acid strongly so. The order you combine them in is not a formatting preference: add the concentrated stock to water, slowly and with stirring, never the reverse, because pouring water onto concentrated acid can flash-boil at the point of contact and throw acid back out of the container. See the acid dilution safety guide before diluting anything above dilute bench-strength acid, and this site's sulfuric acid and hydrochloric acid concentration calculators for the density and molarity that go with a given strength.
Common mistakes that aren't the arithmetic
Once C₁, V₁, C₂, and V₂ are all in the same basis, the arithmetic almost never goes wrong — people don't typically fail at cross-multiplying. What actually produces a bad batch happens upstream of the equation:
- Adding water to acid instead of acid to water. Reversing the order on a concentrated acid or base dilution doesn't change the final concentration, but it changes whether the heat release stays controlled or throws hot, corrosive liquid back at you. See the section above and the acid dilution safety guide.
- Adding V₂ of water instead of V₂ − V₁. The stock being diluted already accounts for part of the final volume — measuring out a full V₂ of solvent on top of it overshoots the target and under-strengths the batch. This calculator's "Add solvent" chip already does the subtraction; working it by hand, don't skip that step.
- Treating % w/w and % w/v (or v/v) as the same number. They agree closely only in dilute, near-water solutions. Feed a concentrated reagent's % w/w spec into a target that expects % w/v, and the miss can approach a factor of two — see the concentration-basis section above and dilution math that survives the shop floor.
- Ignoring density when converting between mass and volume. Any time a calculation crosses from grams to milliliters or back — reading a supplier's % w/w spec into a volumetric recipe, for instance — a density figure has to enter the math somewhere. Skipping it and assuming 1 g ≈ 1 mL is close enough for water and dilute solutions, and increasingly wrong for anything concentrated.
Frequently asked questions
What does C₁V₁ = C₂V₂ actually mean?
It's a statement of mass balance: the amount of solute in solution doesn't change when you add solvent — only its concentration and the volume it's spread through change, and they change in inverse proportion. C₁V₁ (stock concentration × stock volume) is the amount of solute before dilution; C₂V₂ (final concentration × final volume) is that same amount after. Since nothing was added or removed, the two products must be equal, which lets you solve for whichever of the four values you don't already know.
Why do C₁ and C₂ have to use the same concentration unit?
Percent, molarity, and mg/mL aren't interchangeable without extra information — converting between them needs a molar mass (for percent ↔ molarity) or a solution density (for percent ↔ mg/mL), neither of which this calculator collects. So the tool keeps C₁ and C₂ locked to one shared unit at a time: change either dropdown and both update together. If your stock and target are specified in different unit families, convert one of them to match the other first.
Is the amount of water I add the same as the final volume V₂?
No — this is the most common mistake. You don't add V₂ of water; you add the difference between the final and stock volumes (V₂ − V₁), because the stock solution you're diluting already accounts for part of that final volume. The calculator shows this as "Add solvent" in the results so you don't have to subtract it yourself.
My calculated stock volume is tiny (a few microliters) — is that a problem?
Very small volumes are hard to pipette accurately, and a small absolute error becomes a large relative error at that scale. Two common fixes: prepare a larger total batch so every volume scales up proportionally, or make an intermediate dilution first and dilute that instead of pulling straight from the concentrated stock — which is exactly what a serial dilution does.
Can I use this for percent, molar, or fold (X) concentrations?
Yes. The math only depends on concentration and volume being paired consistently, not on which unit family you pick — % w/v stock diluted to a lower %, molar stock diluted to a working molarity, or a 10X buffer diluted down to 1X all use the identical C₁V₁ = C₂V₂ relationship.
How do I avoid wasting concentrated stock solution?
Calculate the exact volume you need here rather than rounding up generously — add only a small margin (5–10%) to cover pipetting dead volume, not double. It also helps to keep your main stock in smaller aliquots instead of one large bottle, so repeated opening, temperature swings, or contamination only affect a portion of your supply rather than the whole batch.
What's the real difference between % w/v and % w/w, and why should I care?
% w/w (weight-in-weight) is grams of solute per 100 g of solution — mass all the way through, and what a reagent bottle or SDS almost always means by a bare "%". % w/v (weight-in-volume) is grams of solute per 100 mL of solution — mass over volume. They're equal only for a solution with a density of exactly 1.00 g/mL, so the gap grows with concentration and density: 98% w/w sulfuric acid (density ≈1.84 g/mL) works out to roughly 180% w/v. Feed one basis into a formula expecting the other and the miss can approach a factor of two, not a rounding error. See the dilution basis section above and the guide to dilution math that survives the shop floor for the conversion formula.
Does a 1:10 ratio mean 1 part in 10 total, or 1 part added to 10?
Both conventions exist, and they give slightly different answers — "1 part in 10 total" is 10% of the mix, while "1 part added to 10 parts diluent" is about 9.1% of an 11-part mix. This page's ratio table uses the more common parts-of-total convention (the one CDC chlorine-disinfectant guidance uses for its 1:10 and 1:100 figures). When a ratio comes from somewhere else, check which convention it means rather than assuming — see the bleach dilution ratios guide for a full worked-through case of exactly this ambiguity.
If I mix 100 mL of stock with 400 mL of water, do I actually get exactly 500 mL?
For a dilute aqueous solution, close enough for most work — this calculator assumes final volume equals stock volume plus added solvent, and that additive approximation is what the live beaker diagram above draws. It stops being a safe assumption for concentrated liquid-in-liquid mixes: 500 mL of ethanol plus 500 mL of water measures about 965 mL, not 1000 mL, because the two liquids' molecules pack together more tightly than either does alone. See why alcohol plus water is less than the sum for the full curve. When it matters, dilute to the target volume in a graduated vessel rather than adding a calculated volume of water.
Is it safe to dilute a concentrated acid or base using this calculator's numbers?
The volume math is the same regardless of what's in the beaker, but concentrated acids and bases — sulfuric acid especially — release real heat as they dilute. Always add the concentrated stock to water, slowly and with stirring, never water onto the concentrated stock: pouring water onto concentrated acid can flash-boil at the point of contact and throw acid back out of the vessel. See the acid dilution safety guide before diluting anything above dilute bench-strength acid, and this site's sulfuric acid and hydrochloric acid concentration calculators if you also need the density or molarity of a given strength.