Diluting a solution adds solvent without adding any more solute — the total number of solute molecules or ions present does not change; only the volume they are dispersed through increases. Since concentration is simply moles (or mass) of solute divided by volume, and the amount of solute stays fixed, this single fact is enough to derive the entire dilution equation without needing to know anything else about the specific chemicals involved.
C₁ and V₁ are the concentration and volume of the concentrated stock solution before dilution; C₂ and V₂ are the concentration and volume after solvent has been added. Because both sides of the equation represent the same fixed quantity of solute (concentration times volume equals amount, on both sides), the units of concentration are free to cancel — molarity, percent by mass, or grams per litre all work equally well, as long as the same unit is used consistently on both sides of the equation. The same logic applies to volume: litres and millilitres both work, provided both volumes use the same unit.
Making up a dilution correctly in practice
The correct technique is to measure the stock solution volume precisely using a pipette or volumetric measuring device, transfer that measured volume into a clean volumetric flask, and then add solvent up to the flask’s calibrated graduation mark — not simply adding a separately-measured volume of solvent to the stock. This distinction matters because volumes of different substances are not always perfectly additive when mixed (dissolving or mixing two liquids can very slightly change total volume due to how molecules pack together), and this small discrepancy compounds into meaningful error exactly in the situations where accuracy matters most — concentrated stock solutions used in analytical or pharmaceutical work, where a small volume error at the concentrated stage translates into a proportionally larger error after dilution.
Serial dilution — reaching concentrations too low to measure directly
Some target concentrations are so far below the stock concentration that attempting the dilution in a single step would require measuring an impractically tiny stock volume, which is both physically difficult and highly inaccurate with standard lab equipment. The standard solution is serial dilution: performing several dilution steps in sequence, each one diluting the result of the previous step. Ten successive 1:10 dilutions compound multiplicatively to give an overall factor of 10¹⁰ — ten billion-fold — a dilution no single step could achieve with any real accuracy, since it would require measuring out roughly one drop of stock into an Olympic swimming pool’s worth of solvent. Each individual stage in a serial dilution carries its own small measurement error, though, so it is important to mix each stage thoroughly before proceeding, and to use a fresh pipette tip at every step to avoid carrying over residual concentrated solution that would throw off the following dilutions.
A safety rule that is not optional: acid into water, never the reverse
Diluting a concentrated acid releases a substantial amount of heat, because the process of the acid molecules interacting with water is itself exothermic. Always add the concentrated acid slowly into the water, never the reverse. Adding water directly onto a concentrated acid — particularly concentrated sulfuric acid — can generate heat fast enough, at the point of contact, to flash a thin layer of water into steam almost instantly. That steam expansion can violently eject droplets of concentrated acid out of the container toward whoever is performing the dilution. This is one of the very few rules in a chemistry lab that has essentially no exceptions, regardless of the specific acid or scale involved.