C1V1 = C2V2 is the core equation behind nearly every dilution calculation in chemistry and biology. Understanding the c1v1 c2v2 formula means recognizing that it expresses one simple fact: the total amount of solute in a solution does not change when you dilute it, only the volume it is dissolved in changes.
Breaking Down Each Term
In the equation, C1 is the initial, more concentrated concentration, V1 is the initial volume of that concentrated solution you plan to use, C2 is the final, more dilute concentration you want to achieve, and V2 is the final total volume after dilution. Multiplying concentration by volume on each side gives the total moles (or mass, or any consistent unit of amount) of solute, and that amount is identical before and after dilution because you are only adding solvent, not removing or adding solute.
Why the Equation Works
Concentration is defined as amount of solute divided by volume of solution, so amount of solute equals concentration multiplied by volume. When you dilute a solution, you add more solvent, which increases the volume, but you do not add or remove any solute molecules. Since the amount of solute is unchanged, C1V1 (the amount before) must equal C2V2 (the amount after). This conservation of solute is the entire logic behind the formula, and it holds true regardless of what units are used, as long as they are consistent on both sides.
Solving for an Unknown Volume
The most common use of the formula is finding how much stock solution to measure out. Rearranging gives V1 = (C2 × V2) / C1. For example, to prepare 500 mL of a 0.1 mol/L solution from a 2 mol/L stock, V1 = (0.1 × 500) / 2 = 25 mL. You would measure 25 mL of the stock and add diluent until the total volume reaches 500 mL, not add 475 mL of diluent to the 25 mL blindly, since the final total volume is what matters for C2.
Solving for an Unknown Concentration
If you already know how much stock and diluent you combined, you can instead solve for the resulting concentration: C2 = (C1 × V1) / V2. This direction is useful when checking a dilution that has already been prepared, or when a protocol specifies the volumes to use and you need to confirm what concentration will result.
A Dilution Calculator handles either direction of this equation instantly, which is useful for double-checking a calculation before committing reagent.
A Common Mistake to Avoid
A frequent error is adding V1 and the diluent volume together without treating that sum as V2 correctly, or confusing V2 with the amount of diluent added rather than the final total volume. V2 is always the complete final volume, not the amount added to V1. Keeping a clear note of which volumes are “amount taken” versus “total after dilution” avoids most mistakes when applying this formula.
Units Must Match
C1 and C2 must be expressed in the same concentration units (both in mol/L, or both in mg/mL, for instance), and V1 and V2 must be in the same volume units as each other, though the concentration units and volume units do not need to match each other. Mixing molarity on one side with mass per volume on the other will give a meaningless result.
Extending the Formula to Multiple Steps
When a dilution is carried out across more than one stage, the same equation can be applied once per stage, using the output concentration of one stage as the input concentration of the next. Alternatively, the overall effect of several stages can be combined into a single equivalent dilution factor, then applied once to the original concentration to find the final result directly. Both approaches give the same answer, so the choice is largely a matter of convenience depending on whether the intermediate concentrations are also needed for the experiment.
Checking a Result for Reasonableness
After calculating V1 or C2, it is worth checking that the answer makes physical sense: V1 should always be smaller than V2 in a true dilution, and C2 should always be smaller than C1. If a calculation produces V1 larger than V2, or C2 larger than C1, that is a sign a value was entered incorrectly or the equation was rearranged the wrong way.
Frequently Asked Questions
Does C1V1 = C2V2 work for any type of concentration unit?
Yes, it works for molarity, percent concentration, mass per volume, or any other consistent concentration unit, as long as the same unit type is used on both sides of the equation.
What if I am diluting in multiple steps?
Apply the formula once per step, using the result of one step as C1 for the next, or calculate the overall dilution factor across all steps and apply it directly to the original concentration.
Can this formula be used for solids dissolved to make a solution?
The formula applies to diluting an existing solution. For dissolving a solid directly, you instead calculate concentration from the mass weighed out and the volume of solvent used.
