How To Calculate Dilution

Dilution is the process of lowering a concentration by adding more solvent. No solute is added and none is removed, so the amount of dissolved material stays exactly the same while the volume it is spread through grows. Everything about the calculation follows from that one conservation fact.

The equation is C1 times V1 equals C2 times V2. Each side is a concentration multiplied by a volume, which works out to moles of solute. The left side counts the moles you take out of the stock bottle and the right side counts the moles in the finished solution, and the two must match because the solute never went anywhere.

Rearranging gives the volume of stock you need: C2 times V2 divided by C1. To prepare 100 millilitres of a 0.5 molar solution from 5 molar stock you need 10 millilitres of stock. The remaining 90 millilitres of the 100 is solvent, which is why the diluent volume is the final volume minus the stock volume.

The dilution factor is the ratio of the two concentrations, stock over final. Going from 5 molar to 0.5 molar is a tenfold dilution, written 1 in 10 or sometimes 10x. The same factor also equals the ratio of final volume to stock volume, which is a useful cross-check because both routes should agree.

It is worth being precise about what a tenfold dilution means, because the wording is genuinely ambiguous in practice. A tenfold dilution made in one step takes one part of stock and adds nine parts of solvent, for a total of ten. Confusing that with a ratio of one to ten, which would give eleven parts in total, is a common and avoidable source of error.

The formula only holds when the units match on both sides. Molarity on the left must be molarity on the right, and millilitres must match millilitres. In practice the volume units cancel, so you can use microlitres on one side and millilitres on the other as long as you are consistent within each side of the equation.

Serial dilution extends the idea over several steps. Each step multiplies the dilution factor, so three tenfold steps give a thousandfold dilution overall. Doing it in stages is far more accurate than trying to pipette a tiny volume of stock directly, because measuring 10 microlitres accurately is harder than measuring 100 microlitres three times.

Temperature matters more than most people expect, because volume changes with temperature while the number of moles does not. A solution made up at one temperature and used at another will have a slightly different molarity. Volumetric glassware is calibrated at 20 degrees Celsius, and for work needing four significant figures the difference is real.

Dilution calculations appear constantly in laboratory work: preparing working standards for a calibration curve, reducing a concentrated acid to a safe working strength, or bringing an over-range sample into the measurable window of an instrument. In each case the arithmetic is the same, only the framing changes.

One consequence that catches people out is that the pH does not scale linearly with the dilution factor. Because pH is a logarithmic scale, a tenfold dilution of a strong acid raises the pH by exactly one unit. A hundredfold dilution raises it by two. Doubling the volume, on the other hand, barely moves it.