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The dilution equation C₁V₁ = C₂V₂ is one of the most fundamental and frequently used formulas in chemistry, biology, and medicine. It describes the relationship between concentrations and volumes before and after dilution: the amount of solute remains constant, but the volume increases and the concentration decreases. This calculator solves for the volume of stock solution needed to prepare a desired diluted solution.
Whether you are diluting concentrated acids in the chemistry lab, preparing serial dilutions in microbiology, making buffer solutions from stock concentrates, or adjusting drug concentrations in pharmaceutical compounding, this equation is your essential tool. Simply enter the stock concentration (C₁), desired final concentration (C₂), and desired final volume (V₂), and the calculator determines how much stock solution to use.
The calculator also shows the dilution factor (C₁/C₂) and the volume of solvent to add (V₂ − V₁), giving you everything needed to prepare the solution in one step. The key assumption is that volumes are additive — this is a good approximation for most aqueous dilutions but may not hold for mixing organic solvents or highly concentrated solutions.
The dilution equation is derived from the conservation of solute:
C₁ × V₁ = C₂ × V₂
Where:
Solving for V₁ (the volume of stock to pipette):
V₁ = (C₂ × V₂) / C₁
The volume of solvent (diluent) to add is simply: Vsolvent = V₂ − V₁
The dilution factor is the ratio of initial to final concentration: DF = C₁ / C₂. A dilution factor of 10 means the solution is diluted 10-fold (1:10).
Important notes for accurate dilutions:
The calculator shows three key results: the volume of stock solution to pipette (V₁), the volume of solvent to add, and the dilution factor. For example, to prepare 100 mL of 1 M HCl from 10 M stock, you need 10 mL of stock and 90 mL of water — a 10× dilution.
If V₁ is greater than V₂, the calculation is physically impossible — you cannot dilute to a concentration higher than the stock. If V₁ is very small (less than your pipette accuracy), consider making an intermediate dilution first.
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To prepare 500 mL of 1 M HCl from 12 M concentrated HCl: measure 41.67 mL of concentrated HCl and slowly add it to approximately 400 mL of water in a 500 mL volumetric flask, then bring to volume. Always add acid to water.
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To make 250 mL of 1× PBS buffer from a 10× stock: pipette 25 mL of 10× PBS into a graduated cylinder, then add 225 mL of deionized water. This is a standard 10-fold dilution.
Yes, as long as both concentrations are in the same units. You can use molarity (M), millimolar (mM), percent (%), ppm, g/L, or any other unit. The equation works because it is based on conservation of solute mass (or moles) — the units cancel as long as they match.
Simply rearrange: C₂ = (C₁ × V₁) / V₂, or V₂ = (C₁ × V₁) / C₂. This calculator defaults to solving for V₁ (the most common use case), but you can enter V₁ and leave other parameters to derive the result you need.
The equation assumes volumes are additive (V₁ + Vsolvent = V₂), which is only approximately true. When mixing concentrated solutions, molecular interactions can cause volume contraction or expansion. For example, mixing ethanol and water produces a total volume slightly less than the sum of individual volumes. For precise work with concentrated solutions, use mass-based calculations instead.
A serial dilution is a sequence of stepwise dilutions, each diluting the previous solution by the same factor. It is used when a very large dilution factor is needed (e.g., 10⁶-fold) and a single dilution would require pipetting impractically small volumes. For example, six consecutive 10-fold dilutions achieve a 10⁶-fold total dilution more accurately than pipetting 1 μL into 1 L.
If you are adding a solid solute or a very viscous liquid, the solute itself occupies volume. In this case, dissolve the solute first in less than the final volume of solvent, then add solvent to reach the exact final volume using a volumetric flask. The C₁V₁ = C₂V₂ equation is for liquid-to-liquid dilutions where both C₁ and C₂ refer to solutions.
Always add the acid to water, never water to acid. Pour the calculated volume of acid slowly into a container that already holds most of the required water, while stirring continuously. The reaction is exothermic, so allow the solution to cool before adjusting to the final volume. Wear appropriate PPE: safety goggles, gloves, and lab coat. Work in a fume hood for volatile acids like HCl and HNO₃.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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