What is the dilution equation?

C₁V₁ = C₂V₂, where C₁ and V₁ are the initial (stock) concentration and volume, and C₂ and V₂ are the final (diluted) concentration and volume. The product C × V represents moles of solute, which is conserved during dilution.

How do I dilute a stock solution?

Calculate V₁ (the volume of stock solution needed) using V₁ = (C₂ × V₂) / C₁. Add that volume of stock to enough solvent to reach the final volume V₂. Use a volumetric flask for accurate "top up to volume" technique.

Do C₁ and C₂ need the same units?

Yes — they cancel in the equation only if matched. Same for V₁ and V₂. Use both M, both mM, or both mg/mL. Same for volumes: both mL, both L, or both µL. Mixed units give correct-looking but wrong answers.

What is a serial dilution?

A sequence of dilutions where each step's output is the next step's input. Common for very large dilution factors (1000× or more) and standard curves. Three serial 1:10 dilutions give an overall 1:1000 dilution with smaller cumulative pipetting error than a single 1:1000 dilution.

How accurate does my dilution need to be?

For lab quantitation: use a volumetric flask (±0.1% accuracy). For routine work: graduated cylinder (±5%). For rough mixing or non-quantitative dilution: a measuring cup or beaker is fine. Match accuracy to experimental need.

Why does my actual concentration differ from calculated?

Common causes: imprecise pipetting (use volumetric pipettes), unstable solute (degrades during dilution), evaporation (cap or work fast), wrong stock concentration (stocks drift over time, especially HCl which loses HCl gas). Always verify by titration if accuracy is critical.

Can I dilute below the calculator's minimum?

Yes, but use serial dilutions for accuracy. Direct pipetting of <1 µL is unreliable. Make a 1:100 intermediate dilution, then dilute that further, rather than pipetting tiny volumes from a concentrated stock.

Dilution Calculator

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.

Inputs

Solve For

Initial Concentration (C1) mol/L

Initial Volume (V1) mL

Final Concentration (C2) mol/L

Final Volume (V2) mL

Results

Final Volume

200.0 mL

Final Conc.

0.5000 M

Dilution Factor

4.0x

Dilution Results

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

Last updated: May 28, 2026

Formula

**The dilution equation:** C₁V₁ = C₂V₂ Where: - **C₁**: initial (stock) concentration - **V₁**: volume of stock taken - **C₂**: final (diluted) concentration - **V₂**: total final volume Both sides equal **moles of solute** — that's the conservation rule. **Solving for each variable:** - V₁ = (C₂ × V₂) / C₁ "How much stock do I need?" - V₂ = (C₁ × V₁) / C₂ "What total volume to dilute to?" - C₁ = (C₂ × V₂) / V₁ "What was my stock concentration?" - C₂ = (C₁ × V₁) / V₂ "What concentration do I end up with?" **Common-sense check:** dilution always decreases concentration (C₂ < C₁) and increases volume (V₂ > V₁). If your answer doesn't follow this pattern, you've made an arithmetic error. **Dilution factor (DF):** DF = V₂ / V₁ = C₁ / C₂ A 10-fold dilution: DF = 10. Concentration drops 10×; volume rises 10×. **Volume of solvent to add:** solvent_added = V₂ − V₁ So a 1:10 dilution at V₂ = 100 mL means V₁ = 10 mL stock and 90 mL solvent. **Worked example: dilute 6 M HCl to 1 M working solution, 250 mL final** - V₁ = (C₂ × V₂) / C₁ = (1 × 250) / 6 = **41.67 mL** of 6 M HCl - Add to ~150 mL water in a 250 mL volumetric flask, then top up to 250 mL. **Serial dilution math (sequential dilutions):** Each step uses the same equation. For three sequential 1:10 dilutions: - Step 1: 10 mL of 1 M into 90 mL water → 100 mL of 0.1 M - Step 2: 10 mL of that into 90 mL water → 100 mL of 0.01 M - Step 3: 10 mL of that into 90 mL water → 100 mL of 0.001 M (1 mM) - Overall: 10⁻³ dilution from initial. 1000× more dilute than start. Serial dilution is more accurate than a single huge dilution for very dilute targets because pipetting error compounds less. **Unit consistency:** - C₁ and C₂ must use the same units (both M, both mM, both mg/mL). - V₁ and V₂ must use the same units (both mL, both L, both µL). - Mixed units give answers that look right but are wrong by a factor of 1000.

How to use this calculator

Identify what you know and what you're solving for. Three of the four variables are needed.
Use the dropdown to pick the unknown.
Enter values. Make sure C₁ and C₂ are in the same units; V₁ and V₂ in the same units.
For dilutions: stock volume V₁ goes first, then diluent (water or buffer) to make total V₂. Always add stock to diluent for acids and bases.
For serial dilutions, run the calculator step by step for each tube.
Use a volumetric flask for any quantitative work — graduated cylinders are ±5% accurate; volumetric flasks ±0.1%.

Worked examples

Diluting bleach for surface disinfection

**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.

Standard curve for spectrophotometer

**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.

Back-calculating what you made

**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.

When to use this calculator

**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 |

Common mistakes to avoid

Mixing units between C₁ and C₂. Using mg/mL for C₁ and µg/mL for C₂ gives an answer that's off by 1000×.
Mixing units between V₁ and V₂. mL/L mixing gives 1000× errors.
Adding water to the stock volume to "make V₂." V₂ is the TOTAL final volume, not the volume of added solvent. Solvent to add = V₂ − V₁.
Using a graduated cylinder when accuracy matters. ±5% error on each measurement; volumetric flasks are ±0.1%.
Diluting too aggressively in one step. A 1:10000 dilution in a single tube introduces large pipette error; do four 1:10 serial dilutions instead.
Forgetting to mix the diluted solution. Solutes don't homogeneously distribute by themselves; always invert/stir after dilution.
Diluting unstable solutions and using them later. Diluted enzymes, vitamins, and many drugs lose activity over hours — dilute fresh.

Frequently Asked Questions

Sources & further reading

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