Serial dilution
= Serial dilution
{title2=$c_k=c_0\prod_{j=1}^k(v_j/V_j)$}
<Serial dilution> repeatedly transfers a measured portion of one well-mixed <chemical solution> and dilutes that portion to a new final <volume (thermodynamics)>. With transferred <volume (thermodynamics)> $v_j$ and final <volume (thermodynamics)> $V_j$ at step $j$, conserved <solute> gives
$$
n_j=c_{j-1}v_j,\qquad c_j=\frac{n_j}{V_j}=c_{j-1}\frac{v_j}{V_j},\qquad
c_k=c_0\prod_{j=1}^k\frac{v_j}{V_j}.
$$
This <product> follows by repeated substitution. Each step conserves the <solute> in the portion transferred into its receiving flask, not the entire inventory in the preceding flask. Uniform mixing makes the selected fraction representative.
Continue the original <solution mass and volume bookkeeping> example. A $10.00\,\mathrm{mL}$ portion of the $0.2000\,\mathrm{mol\,L^{-1}}$ stock contains $0.002000\,\mathrm{mol}$ of analytical <sodium chloride>, corresponding to $0.11688\,\mathrm g$ before rounding. After <dilution> to $100.0\,\mathrm{mL}$, its <analytical concentration> is $0.02000\,\mathrm{mol\,L^{-1}}$. A new $5.000\,\mathrm{mL}$ portion of that mixed <chemical solution> contains $0.0001000\,\mathrm{mol}$; <dilution> to $50.00\,\mathrm{mL}$ gives $0.002000\,\mathrm{mol\,L^{-1}}$. The two factors $10.00/100.0$ and $5.000/50.00$ multiply to $1/100$.
\Image[/volumetric-solution-dilution-accounting.png]
{title=Final <chemical solution> <volume (thermodynamics)> and transferred <solute> in a <serial dilution>}
{id=image-volumetric-solution-dilution-accounting}
{description=Original illustrative account, not experimental data. Calibration marks denote final contained <volumes>; arrows denote measured portions transferred. The first arrow carries $0.002000\,\mathrm{mol}$; the second carries $0.0001000\,\mathrm{mol}$. Each receiving flask conserves the inventory in its incoming portion. Equal dilution factors give successively tenfold lower <analytical concentrations>.}
The remaining stock holds $0.05000-0.002000=0.04800\,\mathrm{mol}$; the remaining first <dilution> holds $0.002000-0.0001000=0.001900\,\mathrm{mol}$. Including the final flask restores the starting inventory:
$$
0.04800+0.001900+0.0001000=0.05000\,\mathrm{mol}.
$$
This is a direct check that making a dilute final sample does not destroy the <solute>. Most of it remains in earlier vessels.
<Serial dilution> makes a small final <concentration> accessible using measurable portions instead of a single extremely small transfer. Its limitation is that systematic errors propagate: if each transferred <volume (thermodynamics)> is $1\%$ below its intended value while final <volumes> are correct, two steps give $0.99^2=0.9801$ times the intended <concentration>, a $1.99\%$ deficit. Repeated readings alone do not remove that bias. The gain in manageable <volume (thermodynamics)> therefore depends on calibrated transfer and representative mixing.