Dilutions

Since moles of solute don't change when you add solvent, the moles before dilution must equal the moles after dilution. Using $n=cV$ for both the concentrated (initial) and diluted (final) solution and setting them equal gives the dilution equation. Notice what the equation says physically: concentration and volume are inversely proportional when the amount of solute is fixed. Double the volume and the concentration halves; make the volume ten times larger and the concentration falls to one-tenth. Because the same unit of volume appears on both sides, the units cancel, which is why you may leave both volumes in millilitres. The equation works for any concentration unit that is 'amount per volume' (mol/L, g/L, %m/v, and ppm for dilute aqueous solutions), but it only applies when NOTHING reacts — the solute simply spreads out.

• $C_1$ = initial (stock) concentration, $V_1$ = volume of stock solution taken.
• $C_2$ = final (diluted) concentration, $V_2$ = final total volume after dilution.
• Any consistent concentration and volume units work, as long as they're the same on both sides.

The dilution factor tells you how many times more dilute the new solution is. Diluting 10.0 mL of stock to a final volume of 100.0 mL is a tenfold (1 in 10) dilution, so the concentration drops to one-tenth. This gives a fast check on any answer: the diluted concentration must be SMALLER than the stock concentration, and the final volume must be LARGER than the volume of stock taken. If your calculation says otherwise, $V_1$ and $V_2$ (or $C_1$ and $C_2$) have been swapped. The same mole-counting idea handles mixing two solutions of the same solute: find the moles in each with $n=cV$, add them, and divide by the total volume.

1. Calculate $V_1$ with the dilution equation.
2. Rinse a volumetric pipette with a little of the stock solution (so leftover water inside cannot dilute it), then use a pipette bulb to draw up exactly $V_1$.
3. Deliver the stock into a volumetric flask whose calibration volume equals $V_2$, already partly filled with distilled water.
4. Add distilled water until the bottom of the meniscus sits on the calibration line at eye level, adding the last few millilitres dropwise.
5. Stopper the flask and invert it several times so the concentration is uniform throughout.

Sometimes the required dilution is so large that a single step is impractical. To make a 1.0 ppm standard from a 1000 ppm stock in a 100.0 mL flask, you would need only 0.100 mL of stock — far too small to measure accurately with ordinary glassware. Instead, chemists perform a serial dilution: a series of smaller dilutions, each using the product of the previous step as its stock. The overall dilution factor is the PRODUCT of the individual factors, so three 1-in-10 steps give 10 × 10 × 10 = 1000. Environmental labs prepare the calibration standards they compare water samples against this way when measuring trace contaminants such as lead or nitrate.