Dilution Calculator
Apply C₁V₁ = C₂V₂ to plan any dilution: find final volume, final concentration, stock concentration, or a full serial dilution table.
What is Dilution?
Dilution is the process of reducing the concentration of a solute in a solution by adding more solvent. When you dilute a solution, the amount of solute stays constant — you only add solvent. This is captured by the fundamental equation:
C₁V₁ = C₂V₂
where C₁ is the initial concentration of the stock solution, V₁ is the volume of stock taken, C₂ is the desired final concentration, and V₂ is the total final volume. The equation simply states that moles of solute are conserved: C₁ × V₁ = moles in stock taken = C₂ × V₂ = moles in final solution.
Dilution calculations are essential in chemistry, biology, pharmacology, and food science. A laboratory routinely prepares working solutions from concentrated stock reagents. A pharmacist dilutes concentrated drug solutions to clinical dosage concentrations. Microbiologists serially dilute samples by factors of 10 to count colony-forming units.
The dilution factor (V₂/V₁) describes how many times more dilute the final solution is compared to the stock. A 1:10 dilution (dilution factor = 10) means 1 part stock in 10 parts total (9 parts solvent added).
Formula
Dilution equation (C₁V₁ = C₂V₂):
Solve for any unknown:
- V₂ = C₁ × V₁ / C₂ (find total volume needed)
- C₂ = C₁ × V₁ / V₂ (find final concentration)
- C₁ = C₂ × V₂ / V₁ (find stock concentration)
- V₁ = C₂ × V₂ / C₁ (find volume of stock to use)
Dilution factor: DF = V₂ / V₁ = C₁ / C₂
Solvent to add: V_solvent = V₂ − V₁
Serial dilution (n steps, factor f each): Cₙ = C₀ / f^n
Variables:
- C₁, C₂ — concentrations (any consistent unit: M, %, mg/mL, ppm)
- V₁ — volume of stock solution taken
- V₂ — final total volume of diluted solution
- f — dilution factor per serial dilution step
- n — number of serial dilution steps
How to Use
- Find V₂ — enter stock concentration C₁, volume V₁ to take, and target concentration C₂. The calculator returns the required total volume V₂, the volume of solvent to add, and the dilution factor.
- Find C₂ — enter stock concentration C₁, volume V₁, and final volume V₂. Returns the resulting concentration after dilution.
- Find C₁ — enter the diluted concentration C₂, the volume of stock taken V₁, and the final volume V₂. Returns the original stock concentration.
- Serial Dilution — enter stock concentration, dilution factor per step, and number of steps (1–12). A table shows the concentration after every step and the cumulative dilution factor.
- Volume of solvent — in Find V₂ mode, the calculator always shows both the final volume V₂ and the amount of solvent to add (V₂ − V₁) for practical lab use.
Example Calculations
Example 1 — Preparing 500 mL of 0.1 M NaCl from 5 M Stock
C₁ = 5 M, C₂ = 0.1 M, V₂ = 0.5 L → find V₁
1
C₁V₁ = C₂V₂ → V₁ = C₂ × V₂ / C₁ = 0.1 × 0.5 / 5 = 0.01 L = 10 mL
2
Solvent to add = V₂ − V₁ = 500 − 10 = 490 mL
Measure 10 mL of 5 M stock, add to a 500 mL flask, fill with distilled water to the 500 mL mark. Dilution factor = 50×.
Try this example →Example 2 — Finding Concentration After Dilution
50 mL of 2 M HCl diluted to 200 mL → find C₂
1
C₂ = C₁ × V₁ / V₂ = 2 × 0.05 / 0.2 = 0.5 M
2
Dilution factor = V₂ / V₁ = 200 / 50 = 4×
Final concentration = 0.5 M HCl. Moles check: 2 M × 0.05 L = 0.1 mol = 0.5 M × 0.2 L ✓
Try this example →Example 3 — Serial 10× Dilution (6 steps)
Stock 1 M, dilution factor 10×, 6 steps
1
Step 1: 1 M / 10 = 0.1 M (10× dilution)
2
Step 2: 0.1 M / 10 = 0.01 M (100× overall)
3
Steps 3–6 continue: 10⁻³, 10⁻⁴, 10⁻⁵, 10⁻⁶ M
After 6 serial 10× dilutions: final concentration = 10⁻⁶ M (1 µM). Overall factor = 10⁶.
Try this example →❓ Frequently Asked Questions
What is the dilution formula C₁V₁ = C₂V₂?
C₁V₁ = C₂V₂ expresses conservation of solute: the moles of solute in the stock volume (C₁ × V₁) equal the moles in the diluted solution (C₂ × V₂). Since you only add solvent (not more solute), the total moles remain constant. Rearrange to find any one unknown: V₂ = C₁V₁/C₂, C₂ = C₁V₁/V₂, C₁ = C₂V₂/V₁, V₁ = C₂V₂/C₁.
What is a dilution factor?
The dilution factor (DF) = V₂ / V₁ = C₁ / C₂. It tells you how much more dilute the final solution is than the stock. A DF of 10 means the final concentration is 1/10 of the stock. A 1:10 dilution (in ratio notation) adds 1 part sample to 9 parts solvent (total 10 parts), giving DF = 10. A 1:2 dilution (DF = 2) takes 1 part sample and 1 part solvent.
How do you perform a serial dilution?
A serial dilution applies the same dilution factor repeatedly. For a 10× series: take 1 mL of stock into 9 mL of diluent (tube 1, now 10× diluted). Take 1 mL from tube 1 into 9 mL (tube 2, now 100× diluted). Continue for n steps: final concentration = C₀ / 10^n. Serial dilutions are used when you need to span many orders of magnitude in concentration.
What units can C₁ and C₂ use?
Any concentration unit works in C₁V₁ = C₂V₂ as long as C₁ and C₂ use the same unit. Common choices: molarity (M = mol/L), percent (w/v: g/100 mL; v/v: mL/100 mL), mg/mL, ppm, ppb. Similarly, V₁ and V₂ must use the same volume unit (both mL, both L, etc.) — but the volume unit does not need to match the concentration unit.
How do you prepare a 1:5 dilution?
A 1:5 dilution (DF = 5) means the final volume is 5× the stock volume taken. Example: take 10 mL of stock and add diluent to reach a total of 50 mL (add 40 mL of diluent). The final concentration is C₁ / 5. Note: 1:5 in US lab notation usually means 1 part sample + 4 parts diluent = 5 parts total (DF = 5). Some sources define 1:5 as 1 part sample + 5 parts diluent (DF = 6) — always clarify which convention is used.
Can you dilute a solution below zero concentration?
No — concentration cannot go below zero, and you cannot concentrate a solution by diluting it (C₂ cannot exceed C₁). The dilution equation always gives C₂ ≤ C₁. If you need a higher concentration, evaporate solvent (rotary evaporation, lyophilization) or start with a more concentrated stock. The calculator flags it as an error if C₂ > C₁.
What is the difference between a dilution and a solution preparation?
A solution preparation starts from a pure solid or liquid solute: weigh out the required mass, dissolve, and make up to volume. A dilution starts from an existing solution (the stock) and adds solvent. Use the Molarity Calculator to compute mass for a solution preparation; use this Dilution Calculator once you have the stock. Combining both: first prepare a concentrated stock from solid, then dilute to working concentrations.
How do you calculate the number of moles in a diluted solution?
Moles = concentration × volume. For the diluted solution: moles = C₂ × V₂. Since C₁V₁ = C₂V₂, the moles in the diluted solution equal C₁ × V₁ — exactly the moles taken from the stock. Example: 10 mL of 5 M stock gives 5 × 0.01 = 0.05 mol. Whether diluted to 100 mL (0.5 M) or 500 mL (0.1 M), the sample still contains 0.05 mol.
What is the significance of the 10-fold dilution series in microbiology?
In microbiology, 10× serial dilutions allow counting of bacteria or fungi at manageable concentrations. A sample might have 10⁸ CFU/mL (colony-forming units per mL). After 6 serial 10× dilutions, a 100 µL aliquot plated gives about 10 colonies — countable by eye. The original count is then 10 × 10⁶ = 10⁷ CFU/mL. This technique underlies the standard plate count method for food safety, water quality testing, and fermentation monitoring.
How accurate is a dilution in practice?
Accuracy depends on measuring equipment. Volumetric glassware (volumetric flasks, Grade A pipettes): ±0.01–0.05% error. Graduated cylinders: ±0.5–1%. Plastic transfer pipettes: ±2–5%. For analytical-grade work, use volumetric flasks and calibrated pipettes. For serial dilutions, errors compound: a 2% error per step becomes a 12% error after 6 steps. Calibrate pipettes regularly and use certified reference materials for critical applications.