Dilution: lowering concentration with M1V1 = M2V2

What dilution is

Dilution is the simplest operation in solution chemistry: take a concentrated stock solution, add solvent, and get a larger volume of a weaker solution. Labs work this way because one carefully prepared stock bottle can supply months of working solutions at many different concentrations.

The governing principle is conservation of solute. Adding pure solvent changes neither the identity nor the amount of the dissolved substance — it only spreads the same moles through more liquid. Everything about dilution math flows from that single fact.

Deriving M1V1 = M2V2

Write the moles before and after dilution. Before: $n_1 = M_1 V_1$. After: $n_2 = M_2 V_2$. Since no solute is added or removed, $n_1 = n_2$, and therefore:

$$M_1V_1 = M_2V_2$$

By convention, subscript 1 marks the concentrated stock and subscript 2 marks the dilute target. The volumes must be in the same unit on both sides — both in mL, or both in L — but they do not need to be liters, because the units cancel in the ratio.

Worked example: 500 mL of 0.100 M from 1.00 M stock

Problem: prepare 500 mL of 0.100 M NaCl from a 1.00 M stock solution.

1. Identify the knowns: $M_1 = 1.00$ M, $M_2 = 0.100$ M, $V_2 = 500$ mL.
2. Solve for the stock volume: $V_1 = M_2V_2 / M_1$.
3. Substitute: $V_1 = (0.100 \times 500) / 1.00 = 50.0$ mL.
4. Measure 50.0 mL of the 1.00 M stock with a graduated cylinder.
5. Transfer to a 500 mL volumetric flask and add water to the mark.

The check: $1.00 \times 50.0 = 50.0$ millimoles before, $0.100 \times 500 = 50.0$ millimoles after. The water added is $500 - 50.0 = 450$ mL, not 500 mL — a distinction the equation does not spell out but the procedure requires.

Serial dilutions

When the dilution factor is enormous — say a million-fold — no single step is practical, so chemists chain dilutions in series. Each step multiplies the dilution factors: three successive 1:100 dilutions give $(1/100)^3 = 1/1{,}000{,}000$.

Serial dilution is standard in microbiology and analytical chemistry because each transfer uses comfortable, accurately measurable volumes. The trade-off is error accumulation: a 1% pipetting error repeated over five steps compounds, so critical serial dilutions use calibrated micropipettes and the final concentration is verified instrumentally.

Safety: add acid to water

Diluting concentrated acids deserves its own safety rule: always add acid to water, never water to acid. Mixing concentrated sulfuric or hydrochloric acid with water releases substantial heat of dilution. Water poured onto dense concentrated acid can sit on top, boil violently from the heat below, and spatter corrosive liquid.

The safe procedure: put most of the water in the receiving vessel first, then pour the acid in slowly down the wall of the vessel while stirring, allowing heat to dissipate into the large water volume. Wear splash goggles and gloves, work with modest volumes, and let the mixture cool before bringing it to its final volume.

Avoiding the classic errors

Two habits prevent nearly every dilution error. First, label which side is the stock: M1 and V1 always belong to the more concentrated starting solution. Swapping the subscripts inverts the answer. Second, remember that V2 is the final total volume, so the solvent added is V2 − V1.

Unit vigilance matters as well. If M is in mol/L and V2 is in mL, V1 comes out in mL — consistent units on both sides are enough, and converting to liters is unnecessary as long as both volumes match. A final reasonableness check always helps: the stock volume must be smaller than the final volume, since you are diluting.

Variables and units

Common mistakes

Real-world applications

• Preparing working solutions from concentrated laboratory stock bottles

• Serial dilutions in biology and medicine, e.g., antibiotic susceptibility testing

• Titration: diluting a primary standard to the exact working concentration

• Home and pool chemistry: diluting concentrated chlorine or acid safely

• Pharmaceutical compounding: bringing a concentrated drug solution to dose strength

• Environmental labs: diluting samples into an instrument's calibration range