Dilution Calculator

The dilution equation, and what each letter is doing

Every routine dilution in a lab, a brewery, a darkroom or a cleaning cupboard is the same arithmetic: C1V1 = C2V2. C1 is the concentration of the stock you already have, V1 is how much of that stock you will take, C2 is the concentration you want, and V2 is the total volume you want to end up with. Because adding solvent never adds solute, the amount of solute on the left of the equation has to match the amount on the right. Rearranging for the only unknown most people care about gives V1 = (C2 x V2) / C1.

The number the equation does not give you directly is the one you actually pour: the diluent. That is simply V2 minus V1. A calculator that stops at V1 leaves you doing the subtraction at the bench with gloves on, which is exactly where mistakes happen, so this tool prints both numbers and the sentence that combines them.

Worked example: 10 M stock down to 1 M in 100 mL

You have a 10 M NaOH stock and need 100 mL of 1 M. Put C1 = 10, C2 = 1, V2 = 100 mL. Then V1 = (1 x 100) / 10 = 10 mL of stock, and the diluent is 100 - 10 = 90 mL. The recipe reads: add 10 mL of stock to 90 mL of water for 100 mL total. The dilution factor is 10 / 1 = 1:10 — a single comfortable step with no accuracy concerns.

Note the order in that sentence. For a hydroxide it barely matters, but if the stock were concentrated sulfuric acid the wording would be doing safety work: water first, acid into the water, never the reverse. The heat released when strong acid meets water is enough to boil the droplet at the interface and throw it back out of the cylinder.

Worked example: 1:1000 by serial dilution

Say you need a 1:1000 dilution of a primary antibody in 1 mL of buffer. Done in one step, that is 1 µL of stock into 999 µL — a volume at the very bottom of a P2 pipette's range, where relative error is worst and a single air bubble ruins the result. The serial route is three 1:10 steps: 100 µL of stock into 900 µL of buffer gives 1:10; 100 µL of that into 900 µL gives 1:100; 100 µL of that into 900 µL gives 1:1000. Every transfer is 100 µL, a volume any pipette handles well.

The trade-off is that errors multiply rather than add across steps, so sloppy serial work is not automatically better. But a 2% error on three comfortable 100 µL transfers compounds to roughly 6%, while a 10% error on one 1 µL transfer is 10% on its own — and that is before mixing problems with a viscous glycerol stock. As a rule of thumb, split anything above 1:100, and always mix thoroughly between steps because an unmixed intermediate poisons every step after it.

Where pipetting error actually comes from

Manufacturers quote accuracy as a percentage of the set volume, and that percentage grows sharply near the bottom of a pipette's range. A P200 set to 20 µL is typically within about 1%; set to 2 µL it can be off by 5% or more. Beyond the instrument, the usual culprits are reverse-pipetting viscous solutions with forward technique, wet tips carrying extra volume, temperature differences between stock and diluent, and failing to pre-wet the tip. None of these show up in the calculation — the equation is exact, the hands are not.

Two habits fix most of it. Choose the pipette whose range puts your volume in the upper half, and scale the whole preparation up if the calculated V1 lands below about 5 µL. Making 10 mL instead of 1 mL turns a 1 µL transfer into a 10 µL transfer at no extra cost beyond a little more buffer.

Units: molar, percent, mg/mL, or fold

The equation does not care what the concentration unit is, only that both sides use the same one. Molar and millimolar, % w/v, mg/mL, and fold notation such as 10X TBE all work. What breaks the calculation is mixing them: entering C1 as 0.5 (molar) and C2 as 50 (millimolar) gives a nonsense answer because the tool has no way to know you switched scales. Convert first — 0.5 M is 500 mM — then enter both in the same unit. Fold notation is the easiest case of all: 10X stock to 1X working solution is a 1:10 dilution regardless of what is in the buffer.

Writing it into a lab notebook

A reviewable entry records four things: the stock identity and its concentration, the volume of stock taken, the diluent and the volume added, and the final volume. "10 mL of 10 M NaOH + 90 mL deionised water = 100 mL of 1 M NaOH (1:10), prepared 4 March" is complete; "diluted the NaOH 1:10" is not, because it does not say what the starting concentration was or how much you made. The recipe sentence this calculator produces is written in that first form deliberately, so it can be copied straight across.

Limitations worth knowing

C1V1 = C2V2 assumes volumes add up. For dilute aqueous solutions that is close enough to true, but concentrated ethanol and water famously contract on mixing — 50 mL plus 50 mL gives about 96 mL, not 100. For those cases, and for anything prepared to a certified concentration, add the stock to a volumetric flask and top up to the mark rather than measuring the diluent separately. The equation also says nothing about solubility, pH shifts after dilution, or whether a buffer will still buffer at the working concentration. Those are chemistry questions, and the numbers here are only the starting point.