Dilution math that survives contact with the shop floor
The formula everyone remembers is right and almost never the thing that goes wrong. What goes wrong is which percent you were quoted, and whether anyone checked the density.
Ask anyone who has done a chemistry class and they'll give you the dilution equation:
C₁V₁ = C₂V₂
Concentration times volume before equals concentration times volume after, because the amount of solute doesn't change when you add water. Want 20 litres of 5% from a 30% stock? V₁ = (5 × 20) ÷ 30 = 3.33 L of stock, topped up to 20 L. Done.
That part is easy and it's rarely where money gets lost. The failures happen one step earlier, in what those percentages actually mean.
Three different percents wear the same symbol
Concentration written as "%" can mean any of three things:
- % w/w — grams of solute per 100 g of solution. The one chemical suppliers usually mean.
- % w/v — grams of solute per 100 mL of solution. Common in labs and biology.
- % v/v — millilitres of solute per 100 mL of solution. Used for liquid-in-liquid mixes like alcohol.
They are not interchangeable, and the gap between them scales with density. Concentrated sulfuric acid at 98% w/w has a density around 1.84 g/mL, so a litre of it holds about 1,800 g of H₂SO₄ — as a w/v figure that's 180%, a number that looks absurd precisely because the bases are different. Use a w/w figure where the recipe wanted w/v and you can be off by nearly a factor of two.
C₁V₁ = C₂V₂ is only valid when C₁ and C₂ are the same kind of percent. Mixing bases inside the equation is the single most common dilution error, and it produces a confident, clean-looking wrong answer.
When you need density (and when you don't)
If everything is already in the same units — molarity to molarity, w/v to w/v, ppm to ppm — the volumes work directly and density never enters.
You need density the moment you cross between mass and volume. Supplier specs are usually mass-based (% w/w); your measuring cylinder is volume-based. To go from one to the other:
grams of solute per litre = 10 × (% w/w) × density in g/mL
For 32% w/w hydrochloric acid at 1.16 g/mL, that's 10 × 32 × 1.16 = 371 g of HCl per litre. Now you can work in litres for the rest of the calculation.
Density also moves with temperature and with concentration itself, which is why concentration tables for the common industrial acids and caustics are published as density-versus-strength curves at a stated temperature. Reading a density off a hydrometer is, in fact, the standard way to check what strength a tank of caustic or acid actually is.
Solution dilution calculator → Handles w/w, w/v, v/v and molarity, and asks for density only when the conversion needs it.Serial dilution, and why it beats one big step
Going from a 1,000 ppm standard to 1 ppm in one step means measuring 1 part in 1,000 — 0.1 mL into a 100 mL flask. Your pipette error on 0.1 mL is a large fraction of 0.1 mL, and it lands directly on the final concentration.
Three tenfold steps instead — 1,000 → 100 → 10 → 1 — mean measuring 10 mL into 100 mL each time, where a small absolute error is a small relative one. The errors do compound across steps, but they compound from a much smaller starting point, and the result is comfortably better than the single-step version.
Rules of thumb worth keeping: no single step much beyond 1:100; mix each step properly before drawing the next (an unmixed layer is a silent factor-of-anything error); and use the same pipette for the same volume through the series so systematic error stays systematic rather than random.
Serial dilution calculator → Plan a dilution series: step factors, volumes at each stage, and the final concentration.Volumes don't always add up
Add 500 mL of ethanol to 500 mL of water and you get roughly 965 mL, not a litre. Molecules of different sizes pack into each other's gaps, and the mixture occupies less space than its parts did. Alcohol-water is the dramatic case, at about 3–4% contraction near the middle of the range; concentrated acids in water do it too, along with a good deal of heat.
This is why a dilution recipe says "dilute to 20 litres", not "add 16.7 litres of water". Make it up to the mark in a graduated vessel and the contraction takes care of itself. Add a calculated volume of water instead and you end up slightly over-strength — and if you were relying on hitting a spec, slightly out.
The safety rule that isn't a formatting preference
Acid into water, never water into acid. Diluting concentrated acid — sulfuric especially — releases a lot of heat. Pour acid into a large volume of water and the water's heat capacity absorbs it and the temperature rise is manageable. Pour water onto concentrated acid and the heat dumps into a small volume at the interface, which can flash the water to steam and throw acid out of the vessel.
Add slowly, stir continuously, and where the batch is large enough for it to matter, cool. Full-face protection, not just glasses. This is the one item on this page where the failure mode isn't a bad number.
Checking your work
Three quick tests that catch most errors before they reach a batch:
- Direction. Diluting means the answer must be less concentrated and more voluminous than what you started with. If your stock volume came out larger than your final volume, you inverted the ratio.
- Order of magnitude. A 10:1 dilution needs about a tenth of the final volume as stock. If the number isn't near that, something's wrong before the decimals matter.
- Mass balance. Work out the grams of solute in what you took, and the grams in what you made. They must match. This is the check that catches a mixed-up percent basis, because it forces both sides onto the same footing.
None of this is difficult mathematics. It's just that the equation everyone remembers assumes you've already sorted out the part that actually bites.