Diluting a solution adds solvent but never removes solute, so the number of moles of solute stays the same before and after. Moles are concentration times volume, which gives the dilution equation C₁V₁ = C₂V₂: the stock concentration times the volume of stock you take equals the final concentration times the final volume. Know any three of the four quantities and the fourth follows.
The most common question is “how much stock do I need?” – for example, 10 mL of a 1 M stock diluted to 0.5 L gives 20 mM. The same equation answers how far a given amount of stock will stretch, what concentration you end up with after topping up, or what the original concentration was when you know how a sample was diluted.
Any concentration unit works, as long as C₁ and C₂ are expressed in the same unit – the unit menus here convert molar units for you. The calculator also gives the volume of solvent to add and the dilution factor, often written as 1 : X.
How to use the Solution Dilution Calculator
- Solve for
- Stock concentration (C₁)
- Stock volume used (V₁)
- Final concentration (C₂)
- Final volume (V₂)
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose what to solve for and fill in the other three values. The calculator rearranges C₁V₁ = C₂V₂, then reports the solvent volume as V₂ − V₁ and the dilution factor as C₁ ÷ C₂. For accuracy, measure the stock with a pipette and make up to the final volume in a volumetric flask rather than adding a calculated volume of solvent.
Worked example
You need 500 mL of 20 mM solution and have a 1 M stock. V₁ = C₂V₂ ÷ C₁ = 0.020 M × 0.500 L ÷ 1 M = 0.010 L = 10 mL of stock. Add solvent to bring the volume to 500 mL (about 490 mL). The dilution factor is 1 ÷ 0.020 = 50 (1 : 49 stock to solvent), and 0.010 mol of solute is transferred. Solving the other way: 10 mL of 1 M stock diluted to 250 mL gives C₂ = 1 × 10 ÷ 250 = 0.04 M.
Mass concentration and molarity
To use the equation with a stock given in g/L, convert with c = ρ ÷ M (mass concentration divided by molar mass): 5.844 g/L of NaCl (58.44 g/mol) is 0.1 M. Both sides of C₁V₁ = C₂V₂ must use the same kind of concentration.
Tips
- Use the final volume of the solution, not the volume of solvent added, for molarity.
- For dilutions, C₁V₁ = C₂V₂ only holds when both concentrations use the same unit.
- Add acid to water, never water to acid.
- Everything is calculated in your browser.
Common mistakes to avoid
- Mixing w/w, w/v and v/v percentages.
- Forgetting that pH is logarithmic – one unit is a tenfold change in [H⁺].
- Using pKa where Ka is asked for (or the other way round).
Inputs explained
| Field | What to enter |
|---|---|
| Solve for | Choose one: Stock volume needed (V₁), Final volume (V₂), Final concentration (C₂), Stock concentration (C₁) |
| Stock concentration (C₁) | Units: mol/L (M), mmol/L (mM), µmol/L (µM), nmol/L (nM), mol/m³ |
| Stock volume used (V₁) | Units: mL, L, µL, cm³ (cc), fl oz (US) |
| Final concentration (C₂) | Units: mol/L (M), mmol/L (mM), µmol/L (µM), nmol/L (nM), mol/m³ |
| Final volume (V₂) | Units: mL, L, µL, cm³ (cc), fl oz (US) |
What each result means
- Final volume (V₂) = [V2]
- Final concentration (C₂) = [C2]
- Stock concentration (C₁) = [C1]
- Solvent to add = [V2] – [Stock volume needed (V₁)]
- Dilution factor = [C1]÷[C2]
- Parts of solvent per part of stock (1 : x) = [V2]÷[Stock volume needed (V₁)] – 1
- Amount of solute transferred = [C1]×[Stock volume needed (V₁)]
Example values
How the main result changes when Stock concentration changes and everything else stays at the example values above:
| Stock concentration (C₁) (mol/L (M)) | Stock volume needed (V₁) (mL) |
|---|---|
| 0.5 | 20 |
| 0.75 | 13.3333 |
| 1 | 10 |
| 1.25 | 8 |
| 1.5 | 6.6667 |
| 2 | 5 |
Formula
C₁ × V₁ = C₂ × V₂V₁ = C₂ × V₂ ÷ C₁ · V₂ = C₁ × V₁ ÷ C₂ · C₂ = C₁ × V₁ ÷ V₂ · C₁ = C₂ × V₂ ÷ V₁Solvent to add = V₂ − V₁ · Dilution factor = C₁ ÷ C₂ = V₂ ÷ V₁Worked example
Inputs — Solve for: Stock volume needed (V₁); Stock concentration (C₁): 1 mol/L (M); Final concentration (C₂): 0.02 mol/L (M); Final volume (V₂): 500 mL.
Results — Stock volume needed (V₁) = 10 mL; Solvent to add = 490 mL; Dilution factor = 50 ×; Parts of solvent per part of stock (1 : x) = 49; Amount of solute transferred = 0.01 mol.
Key terms
- Stock solution
- Concentrated solution used to prepare dilutions.
- C₁, V₁
- Concentration and volume of the stock solution used.
- C₂, V₂
- Concentration and volume of the diluted solution.
- Dilution factor
- Ratio C₁ ÷ C₂ (or V₂ ÷ V₁).
- Make up to volume
- Add solvent until the total volume reaches the mark of a volumetric flask.
Frequently asked questions
How do I use C₁V₁ = C₂V₂?
Multiply the final concentration by the final volume and divide by the stock concentration to get the stock volume. For 0.5 L of 20 mM from 1 M: V₁ = 0.020 × 0.5 ÷ 1 = 0.010 L = 10 mL.
How much water do I add?
V₂ − V₁. In the example, 500 − 10 = 490 mL – in practice, make up to the 500 mL mark.
Can I use percentages or mg/mL instead of molarity?
Yes, as long as C₁ and C₂ use the same unit; the equation only uses their ratio.
What does a dilution factor of 50 mean?
The final solution is 50 times weaker; 1 part stock in 50 parts total (1 : 49 stock to solvent).
What if C₂ is bigger than C₁?
It is impossible – dilution can only lower a concentration. Check the units of both values.
How do I make 250 mL of 0.5 M from a 2 M stock?
V₁ = 0.5 × 250 ÷ 2 = 62.5 mL of stock, made up to 250 mL with 187.5 mL of solvent (dilution factor 4).
Is it better to do one big dilution or several small ones?
For factors above about 100, two or more serial steps are more accurate because the volumes stay easy to measure.
What formula does the Solution Dilution Calculator use?
C₁ × V₁ = C₂ × V₂.