Dilution Calculations: M1V1 = M2V2
Dilution is the process of lowering the concentration of a solution by adding solvent, and the amount of solute stays the same, so M1V1 = M2V2. In brief: diluting 25.0 mL of 12.0 M HCl with water to a total volume of 500.0 mL gives a final concentration of (12.0 × 25.0) / 500.0 = 0.600 M.
Laboratories usually keep concentrated stock solutions and dilute them to the concentration that is needed. This saves space, and it improves accuracy, since it is easier to measure a few milliliters of stock than to weigh tiny amounts of a solid. This article explains why the dilution equation works, shows how to apply it, covers serial dilutions and mixing solutions, and lists the rules for safe dilution.
Why does M1V1 = M2V2 work?
The number of moles of solute is the product of molarity and volume: n = M × V. Adding solvent does not change the number of moles of the solute, so n before the dilution equals n after it:
M1V1 = M2V2
M1 and V1 are the molarity and volume of the concentrated stock, and M2 and V2 are the molarity and total volume of the diluted solution. The equation also works with other concentration units, as long as both sides use the same unit, for example percent or mg/L. V1 and V2 must be in the same unit, but it does not matter which one.
How do you carry out a dilution calculation?
- Identify the three known quantities and the unknown.
- Check that the volume units agree.
- Solve for the unknown: V1 = M2V2 / M1, M2 = M1V1 / V2, and so on.
- Check that the diluted solution is less concentrated and that V2 is larger than V1.
- To prepare it, add V1 of stock to a volumetric flask and make up to V2 with solvent. The volume of solvent added is V2 − V1 only approximately.
Worked examples
Example 1: finding the new concentration
A 25.0 mL sample of 12.0 M HCl is diluted to 500.0 mL. M2 = 12.0 × 25.0 / 500.0 = 0.600 M. The dilution factor is 500.0 / 25.0 = 20, and the concentration falls by the same factor: 12.0 / 20 = 0.600 M.
Example 2: finding the volume of stock
How much 2.00 M NaCl stock is needed to prepare 250.0 mL of 0.100 M solution? V1 = M2V2 / M1 = 0.100 × 250.0 / 2.00 = 12.5 mL. Place 12.5 mL of stock in a 250 mL volumetric flask and fill to the mark.
Example 3: finding the final volume
To what volume must 40.0 mL of 0.800 M KNO3 be diluted to give 0.250 M? V2 = M1V1 / M2 = 0.800 × 40.0 / 0.250 = 128 mL.
Example 4: dilution factors
| Dilution | Stock volume | Final volume | Final concentration |
|---|---|---|---|
| 1 : 2 | 50.0 mL | 100.0 mL | 0.500 M |
| 1 : 5 | 20.0 mL | 100.0 mL | 0.200 M |
| 1 : 10 | 10.0 mL | 100.0 mL | 0.100 M |
| 1 : 20 | 5.00 mL | 100.0 mL | 0.0500 M |
| 1 : 100 | 1.00 mL | 100.0 mL | 0.0100 M |
The notation 1 : 10 here means one part of stock in a total of ten parts of solution. Some texts use “1 : 10” to mean one part of stock plus ten parts of solvent, which is an 11-fold dilution, so check the convention in use.
What is a serial dilution?
In a serial dilution, a solution is diluted in several steps, with each step using the previous solution as the stock. Starting with 1.00 M and carrying out three successive 1 : 10 dilutions gives 0.100 M, 0.0100 M and 0.00100 M. The total dilution factor is 10 × 10 × 10 = 1000. Serial dilutions are used when a very large dilution factor is needed, which would require an impractically small volume of stock in a single step, and in microbiology to count bacteria in a sample and in making calibration standards.
How do you calculate the concentration after mixing?
When two solutions of the same solute are mixed, the moles add up. Add 100.0 mL of 0.50 M NaCl to 200.0 mL of 0.20 M NaCl.
- n1 = 0.50 × 0.1000 = 0.050 mol.
- n2 = 0.20 × 0.2000 = 0.040 mol.
- Total n = 0.090 mol in 0.3000 L, so M = 0.090 / 0.3000 = 0.30 M.
The final concentration lies between those of the two original solutions. This calculation assumes that volumes are additive, which is a good approximation for dilute solutions.
How do you dilute safely?
Diluting concentrated sulfuric acid releases a lot of heat. The rule is “always add acid to water”, never the reverse: if water is poured into concentrated acid, it may boil at the surface and splash the acid. Add the acid slowly in small amounts to most of the water, with stirring, let the solution cool, and then make it up to the final volume. Wear goggles and gloves. The same care applies to concentrated bases, such as sodium hydroxide, which also release heat on dissolving.
Where did dilution techniques develop?
Volumetric analysis was developed in the late 18th and early 19th centuries, and Joseph Louis Gay-Lussac made major contributions in the 1820s by improving the burette and the pipette, and by introducing standardized solutions for the titration of silver and chlorine. Accurate dilution in volumetric glassware has been central to chemical analysis ever since, and today it is automated in many laboratories by pipetting robots.
Which mistakes are common?
- Treating V2 as the volume of water added: V2 is the total final volume.
- Mixing units: if V1 is in mL, V2 must be in mL.
- Using the equation for reacting solutions: it applies only to dilution, where the amount of solute does not change.
- Pouring water into concentrated acid: add acid to water.
- Misreading dilution ratios: check whether 1 : 10 refers to the total or to the solvent.