Adding water to a solution spreads the same amount of solute through a larger volume. Nothing is created and nothing is destroyed, which is the entire content of C₁V₁ = C₂V₂ — the moles before equal the moles after.
What each symbol means
| Symbol | Meaning |
|---|---|
| C₁ | Concentration of the stock solution you start with |
| V₁ | Volume of that stock you take |
| C₂ | Concentration you want to end up with |
| V₂ | Total volume after diluting |
Concentration × volume is the amount of solute, so each side of the equation is just “how much solute”, counted before and after. Since diluting only adds solvent, those two numbers must be equal.
The units cancel across the equation rather than within it, so as long as both concentrations use the same unit and both volumes use the same unit, no conversion is needed. Millilitres on both sides is fine; millilitres on one and litres on the other is not.
The distinction that costs the most marks
V₂ is the final total volume, not the volume of solvent you add.
If you take 25 cm³ of stock and the calculation says V₂ = 100 cm³, you add 75 cm³ of water, not 100. In the laboratory you would put the 25 cm³ into a 100 cm³ volumetric flask and top up to the mark — which is why the instruction is always phrased “make up to” rather than “add”.
The phrasing in a question tells you which is meant. “Diluted to 250 cm³” means V₂ = 250. “Diluted with 250 cm³ of water” means you add 250 to whatever you started with, so V₂ is the sum.
A worked example
You have a 2.00 mol/dm³ stock of sodium hydroxide and need 250 cm³ of 0.100 mol/dm³ solution. What volume of stock do you take?
Here C₁ = 2.00, C₂ = 0.100, V₂ = 250 cm³, and V₁ is what we want:
V₁ = C₂V₂ ⁄ C₁ = (0.100 × 250) ⁄ 2.00 = 25 ⁄ 2.00 = 12.5 cm³
So: measure 12.5 cm³ of the stock into a 250 cm³ volumetric flask and make up to the mark with distilled water — adding about 237.5 cm³ in the process.
Check it for sense. The concentration is dropping by a factor of twenty, from 2.00 to 0.100, so the volume must rise by the same factor of twenty. And 12.5 × 20 = 250 ✓. That ratio check is quicker than the formula and catches an upside-down rearrangement instantly.
Where marks get lost
Turning the fraction upside down. Ask yourself whether the answer should be bigger or smaller before you divide. Diluting always means a larger volume and a smaller concentration, so if your numbers move the other way, the rearrangement is inverted.
Mixing volume units across the equation. cm³ on one side and dm³ on the other will be out by a factor of a thousand. Pick one and use it on both sides — and note that 1 dm³ = 1000 cm³ = 1 litre.
Adding V₂ instead of making up to V₂. Covered above, and worth repeating because it is the one that survives into real laboratory work.
Using it on a reaction. This formula only describes dilution — adding solvent. If the solute is reacting, being consumed or being produced, the amount is not conserved and this equation does not apply.
Assuming volumes add exactly. Mixing 50 cm³ of ethanol with 50 cm³ of water does not give 100 cm³, because the molecules pack together differently. For dilute aqueous solutions the error is small enough to ignore, which is why the formula works at this level.
Questions students actually ask
Does C1V1 = C2V2 work with any concentration unit?
Yes, as long as both sides use the same one. Moles per litre, grams per litre, percentage — the equation only cares that the two concentrations match each other and the two volumes match each other.
What is the difference between diluting ‘to’ and diluting ‘with’?
Diluting to 100 cm³ means the final total is 100 cm³. Diluting with 100 cm³ means you add 100 cm³ to what you already had, so the final volume is larger. V₂ in the formula is always the final total.
Can I use this formula for a titration?
Only for the dilution steps. The reaction at the endpoint needs the mole ratio from the balanced equation, because solute is being used up rather than merely spread out.
What is a dilution factor?
The ratio V₂⁄V₁, which equals C₁⁄C₂. Going from 2.00 to 0.100 mol/dm³ is a dilution factor of 20, often written as a 1-in-20 dilution.
Why does the concentration go down when I add water?
Because the same number of solute particles is now spread through more solution. Concentration is amount divided by volume, so increasing the volume while holding the amount fixed must reduce it.
Related formulas
Moles, mass and molar mass · The ideal gas law · Arithmetic progression
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