Dilution Calculator
Calculate solution dilutions using the M₁V₁ = M₂V₂ formula. Enter any three values to find the fourth. Works with any concentration unit (M, mM, µM, %, mg/mL, ppm) as long as both sides use the same unit.
How the Dilution Formula Works
The dilution equation is:
M₁V₁ = M₂V₂
The logic is conservation of solute: no solute is added or removed when you dilute a solution, you only add solvent. So the moles of solute are the same before and after:
- M₁ × V₁ = moles of solute in the aliquot taken from the stock
- M₂ × V₂ = moles of solute in the final diluted solution
- Since no solute is added or removed: M₁V₁ = M₂V₂
Assumptions: this equation assumes the solute quantity is conserved — that is, no chemical reaction occurs during dilution (no precipitation, decomposition, or reaction with the solvent). For strong acids/bases, volumes are also assumed to be additive (V_total ≈ V_stock + V_solvent), which holds for dilute solutions but can fail at very high concentrations due to volume-of-mixing effects.
The four rearrangements, one per unknown:
- Find C₁: C₁ = (C₂ × V₂) ÷ V₁
- Find V₁: V₁ = (C₂ × V₂) ÷ C₁
- Find C₂: C₂ = (C₁ × V₁) ÷ V₂
- Find V₂: V₂ = (C₁ × V₁) ÷ C₂
Worked Example
Problem: You have a 10 M stock solution and you need to prepare 100 mL of a 0.5 M working solution. How much stock do you need?
Given:
- M₁ = 10 M (stock concentration)
- M₂ = 0.5 M (final concentration)
- V₂ = 100 mL (final volume)
Find: V₁ (volume of stock to pipette)
Rearrange the formula:
- V₁ = (M₂ × V₂) ÷ M₁
- V₁ = (0.5 × 100) ÷ 10
- V₁ = 50 ÷ 10 = 5 mL
Procedure: Pipette 5 mL of the 10 M stock into a 100 mL volumetric flask, then add solvent (water) up to the 100 mL mark. Mix thoroughly. You now have 100 mL of 0.5 M solution.
Dilution Factor
The dilution factor (DF) tells you how many times the solution was diluted. It can be calculated two equivalent ways:
- DF = V₂ ÷ V₁ (from volumes)
- DF = C₁ ÷ C₂ (from concentrations)
In the worked example above, a 10 M stock diluted to 0.5 M gives DF = 10 ÷ 0.5 = 20× — the solution is 20-fold diluted. You can also express this as a ratio: 1:20 (one part stock to twenty parts total volume).
Common shorthand: "1:10 dilution" means one volume of stock added to nine volumes of solvent, for a total dilution factor of 10.
Serial Dilutions
A serial dilution is a sequence of dilutions where each step dilutes the previous solution by the same factor. Serial dilutions are used when you need a very large overall dilution or a ladder of standards (e.g., calibration curves, dose-response assays).
Example — 1:10 serial dilution:
- Start with stock: 1 M
- Step 1: transfer 1 mL into 9 mL of solvent → 0.1 M (10× dilution)
- Step 2: transfer 1 mL of step 1 into 9 mL of solvent → 0.01 M (100× from stock)
- Step 3: transfer 1 mL of step 2 into 9 mL of solvent → 0.001 M (1000× from stock)
The overall dilution factor is the product of the individual factors: 10 × 10 × 10 = 1000. This calculator handles one step at a time — for multi-step serial dilutions, repeat the calculation using the output of one step as the input of the next.
Frequently Asked Questions
What is the dilution formula?
The dilution formula is M₁V₁ = M₂V₂ (also written C₁V₁ = C₂V₂), where M₁ and M₂ are the initial and final concentrations and V₁ and V₂ are the initial and final volumes. It works because the moles of solute are conserved during dilution — adding solvent does not change the amount of solute.
What does M₁V₁ = M₂V₂ mean?
The moles of solute before dilution (M₁ × V₁) equal the moles after dilution (M₂ × V₂). This works because adding solvent does not change the amount of solute, only how spread out it is. The formula is a direct statement of conservation of mass for the solute.
Can I use any concentration units?
Yes — molarity (M, mM, µM), percent (%), mg/mL, ppm, or any other concentration unit works, as long as C₁ and C₂ use the same unit. Similarly, V₁ and V₂ must use the same volume unit. The formula is dimensionally flexible because the units cancel out on both sides.
What is a serial dilution?
A serial dilution is a sequence of dilutions where each step dilutes the previous solution by a fixed factor. It is common in microbiology (bacterial counts), pharmacology (dose-response curves), and analytical chemistry (calibration standards). A 1:10 serial dilution reduces concentration 10-fold at each step, so after three steps the concentration is 1/1000 of the original.
How do I calculate dilution factor?
Dilution factor (DF) = V₂ ÷ V₁ = C₁ ÷ C₂. A 1:10 dilution means a dilution factor of 10 — the concentration is reduced 10-fold. For example, diluting a 10 M stock to 0.5 M gives DF = 10 ÷ 0.5 = 20, so the solution is diluted 20-fold.
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All calculations on CoolConversion use formulas and constants documented by internationally recognised standards bodies (such as IUPAC and NIST).
