Use the dilution equation C1V1 = C2V2 to calculate the concentration or volume needed when diluting a solution. Solve for any of the four variables.
C₁V₁ = C₂V₂ is the single most-used equation in wet-lab chemistry. It says that the moles of solute don't change when you dilute a solution — only the solvent volume does. Concentration drops in exact proportion to volume increase: 50 mL of 2 M solution diluted to 200 mL ends up at 0.5 M because the moles (0.1) divided by the new volume (0.2 L) give 0.5 M.
This calculator lets you solve for any of the four variables given the other three. Need to dilute a stock to a working concentration? Solve for V₁. Need to know the final concentration after a known dilution? Solve for C₂. Need to figure out what concentration your stock was, given how you used it? Solve for C₁. The math takes seconds; the tricky parts are unit consistency and getting the order of operations right at the bench.
A few practical points: the C₁V₁ = C₂V₂ rule assumes the solute conserves through dilution (true for most ions and small molecules in water), that the volume of solute is negligible compared to the dilution volume (usually fine), and that you're using mL/mL or L/L consistently. For non-ideal solutions (concentrated acid + water shows volume contraction), the calculated final volume is slightly off; use a volumetric flask and "top up to mark" instead of "add a calculated volume of water" for accurate results.
**Scenario:** Household bleach is 5% NaOCl. CDC recommends 0.1% NaOCl for surface disinfection. You want 1 L of working solution. **Calculation:** C₁V₁ = C₂V₂ → V₁ = (0.1 × 1000) / 5 = 20 mL bleach. Add to 980 mL water for 1 L of 0.1% bleach. **Result:** 20 mL bleach + 980 mL water makes 1 L of disinfectant-strength bleach. Mix in a labeled spray bottle; lose effectiveness after 24 hours due to chlorine evaporation. Make fresh daily.
**Scenario:** Your assay needs a 5-point standard curve: 100, 50, 25, 10, 5 µM. You have a 1 mM (1000 µM) stock. **Calculation:** Pick total volume V₂ = 1 mL each. V₁ for 100 µM: (100 × 1000) / 1000 = 100 µL. For 50 µM: 50 µL stock. For 25 µM: 25 µL. For 10 µM: 10 µL. For 5 µM: 5 µL. Each gets topped up with buffer to 1 mL. **Result:** 5 tubes at 1 mL each, with 100/50/25/10/5 µL of stock + 900/950/975/990/995 µL buffer. Mix and read on spectrophotometer. The standard curve A vs concentration should be linear (R² > 0.99) over this range.
**Scenario:** You took 25 mL of a stock acid and diluted it to a final volume of 250 mL. The final solution tests at 0.40 M by titration. What was the stock concentration? **Calculation:** C₁V₁ = C₂V₂ → C₁ = (0.40 × 250) / 25 = 4.0 M. Stock concentration was 4.0 M. **Result:** Useful for verifying labeled commercial reagents or checking that a dilution went as planned. If C₁ from back-calculation doesn't match the stock label, either the label is wrong or the dilution wasn't performed accurately.
**Use C₁V₁ = C₂V₂ in essentially every lab situation involving a known solution:**
- **Preparing reagents from concentrated stocks**: virtually all aqueous lab work. - **Standard curves for spectroscopy and chromatography**: dilute a stock to multiple known concentrations for calibration. - **Buffer prep**: dilute 10× PBS, Tris, citrate to working strength. - **Drug dilution in clinical settings**: stock vials diluted in saline for IV bag delivery. - **Aquarium and pool chemistry**: dosing additives from concentrated solutions. - **Brewing water adjustments**: minerals added as concentrated stock then diluted into batch volume. - **Pesticide and fertilizer dilution**: ag chemistry, label rates apply after dilution to spray volume. - **Photography chemistry**: developer stock diluted to working strength.
**Practical dilution workflow:**
1. **Pick V₂**: usually based on how much working solution you need. 2. **Pick C₂**: based on the experimental requirement. 3. **Solve for V₁**: this is the stock volume. 4. **Solvent volume = V₂ − V₁**: this is what you add to dilute. 5. **In the flask**: add diluent first (or partially), then stock, then top up to V₂. Mix. 6. **Label**: name, concentration, prep date, your initials.
**When C₁V₁ = C₂V₂ doesn't apply:**
- **Reactions during dilution**: if the solute reacts (e.g., dissolving CO₂ from air into a basic solution), concentration changes for reasons other than dilution. - **Two-phase systems**: oil-water mixtures don't obey simple dilution math. - **Non-aqueous solvents with significant solute volume**: dissolving concentrated H₂SO₄ in equal volume of water gives less than 2× the volume due to volume contraction. - **Very concentrated solutions**: at >5 M, ion-ion interactions change effective concentration; molarity-volume relationships drift from ideal.
**Quick reference for common dilution factors:**
| DF | Stock vol | Solvent vol | Total | Use case | |---|---|---|---|---| | 2× | 500 mL | 500 mL | 1000 mL | Halving concentration | | 5× | 100 mL | 400 mL | 500 mL | Half-strength reagent | | 10× | 100 mL | 900 mL | 1000 mL | Most common stock-to-working | | 100× | 10 mL | 990 mL | 1000 mL | Very dilute test | | 1000× | 1 mL | 999 mL | 1000 mL | Trace analysis |
Calculate molarity (moles per liter) and perform dilution calculations (C1V1 = C2V2).
Calculate titration endpoint including molarity, volume, and equivalence point.
Calculate molar mass from element atomic weights and their quantities in a compound.
Calculate boiling point elevation using ΔTb = Kb x m x i.
Calculate freezing point depression using ΔTf = Kf x m x i.
Calculate Gibbs free energy using ΔG = ΔH - TΔS.
Final Volume
200.0 mL
Final Conc.
0.5000 M
Dilution Factor
4.0x
| Parameter | Value |
|---|---|
| Initial Concentration (C1) | 2.0000 mol/L |
| Initial Volume (V1) | 50.00 mL |
| Final Concentration (C2) | 0.5000 mol/L |
| Final Volume (V2) | 200.00 mL |
| Dilution Factor | 4.00x |
| Solvent to Add | 150.00 mL |
| Verification (C1×V1) | 100.0000 |
| Verification (C2×V2) | 100.0000 |