The one fact behind every dilution
To dilute a dilution is just to add more solvent. Here's the only fact you need: adding solvent does not add or remove any solute. You still have exactly the same number of dissolved particles — they're just spread through more liquid, so the concentration drops. It's like the sugar cube moving from teaspoon to bathtub: the sugar is unchanged, the sweetness fades.
Because the solute amount is fixed, concentration and volume trade off in lockstep. Double the volume and the concentration halves; cut the volume to a tenth and the concentration multiplies by ten. That inverse seesaw is the whole story of dilution.
The dilution factor
We package that seesaw into a single number: the dilution factor, how many times more dilute the result is than the start. Take 1 part of a stock and add solvent up to 10 parts total, and you've diluted tenfold — a dilution factor of 10. The new concentration is the old one divided by 10. People often write this as "1 in 10" or "1:10".
The C₁V₁ = C₂V₂ rule
Since solute amount = concentration × volume, and that amount is conserved when you dilute, the amount before equals the amount after: C₁ × V₁ = C₂ × V₂. Here C₁, V₁ are the concentration and volume you start with, and C₂, V₂ are what you want to end with. Three of the four are usually known, so you solve for the fourth. Most often you want a final concentration and volume and need to find V₁, the volume of stock to measure out.
Worked example: you have a 2.0 M stock and need 100 mL of 0.10 M. Rearranging, V₁ = C₂V₂ ÷ C₁ = (0.10 × 100) ÷ 2.0 = 5 mL. So measure 5 mL of stock with a pipette into a 100 mL volumetric flask and top up with solvent to the mark. The units of volume cancel as long as both volumes use the same unit, so you needn't convert to litres.
Serial dilution: reaching the truly tiny
Some target concentrations are so faint you couldn't reach them in one step — the volume of stock you'd need to measure would be a fraction of a drop, far too small to do accurately. The cure is a serial dilution: dilute, then dilute the result, then dilute again, each step a manageable factor. The factors *multiply*. Three tenfold steps give 10 × 10 × 10 = 1000-fold overall, a thousand times more dilute than you started.
- Label tubes 1, 2, 3 and put 9 mL of solvent in each.
- Add 1 mL of stock to tube 1 and mix: that's a tenfold dilution.
- Transfer 1 mL from tube 1 into tube 2 and mix: now a hundredfold overall.
- Transfer 1 mL from tube 2 into tube 3 and mix: a thousandfold overall, reached safely.