The formula
Cₙ = C₀ / factorⁿ
What it means
Diluting in series brings a concentration down by many orders of magnitude without pipetting absurd volumes: ten 1:10 dilutions in a row make a ten-billion-fold dilution, and every step handles millilitres. The price is that error accumulates, because each tube inherits the error of all the ones before it. The concentration of tube n is the starting one divided by the factor raised to n, and that is what makes the scale logarithmic.
How to work it out by hand
- Choose the factor for each step: 1:10 divides by ten, 1:2 by two
- Put the diluent in every tube before you start
- Carry the share the factor calls for into the first tube and mix well
- Repeat from the first tube to the second, and so on to the last
What is worth knowing
Mixing between steps is not a detail: a poorly homogenised tube passes its error to every tube after it, and by the tenth that is orders of magnitude. It also pays to change tip at each transfer, because what clings to the previous one biases the next tube upwards. And the final volume in every tube has to be the same for the factor to hold: if one tube ends up with more volume, its dilution is no longer what the table says.