Dilution Calculator

Calculate concentration or volume needed for serial dilutions using C₁V₁ = C₂V₂.

Solve for

Initial concentration C₁ (mol/L, mg/mL, or any unit)

Initial volume V₁ (mL) (mL)

Final concentration C₂ (same unit as C₁)

Final volume V₂ (mL) (mL)

Please check your inputs and try again.

Why C₁V₁ = C₂V₂ Works

The dilution equation follows from a single fact: adding solvent does not change how much solute is present. The amount before and after must be equal, and since amount equals concentration multiplied by volume, the two products must match.

C₁V₁ = C₂V₂

Because both sides are the same quantity, the units cancel. Molarity, mg/mL, percent, or ppm all work equally well — provided both concentrations use the same unit and both volumes use the same unit. This is why the equation is so widely applicable.

Term	Meaning	Common use
C₁	Stock concentration	Usually known from the label
V₁	Volume of stock to take	Most often the unknown
C₂	Target concentration	What you need
V₂	Final total volume	Total, not solvent added

The most frequent practical error is treating V₂ as the volume of solvent to add. It is the final total volume. To make 100 mL of diluted solution from 10 mL of stock, you add solvent up to the 100 mL mark — adding 100 mL of solvent would give 110 mL and the wrong concentration.

Serial Dilution

Reaching very low concentrations in one step is impractical — a 10,000-fold dilution would require measuring 0.1 mL into a litre, where pipetting error dominates. Serial dilution performs several moderate steps instead, and the dilution factors multiply.

Steps of 1:10	Total dilution	From 1 mg/mL
1	10×	0.1 mg/mL
2	100×	0.01 mg/mL
3	1,000×	0.001 mg/mL
6	10⁶×	1 ng/mL

Errors also multiply, which is the trade-off. A random 2% pipetting error compounds to about 5% after six steps, while a systematic 2% error — a miscalibrated pipette — compounds to about 12%. Serial dilution trades one large measurement error for several accumulating smaller ones.

Worked Examples

Example 1: 1 mol/L → 0.1 mol/L in 100 mL

V₁ = 0.1×100/1

Result: 10 mL stock + 90 mL solvent

10-fold dilution

Example 2: Serial: 1 mg/mL stock, take 1 mL to 10 mL

C₂ = 1×1/10 = 0.1 mg/mL then repeat

Result: After 3 serial dilutions: 0.001 mg/mL

10⁻³ dilution factor

Example 3: Preparing a working solution

Need 250 mL of 0.2 M from 2.0 M stock

Result: V₁ = 0.2 × 250 / 2.0 = 25 mL

Take 25 mL of stock and dilute to a final volume of 250 mL — adding about 225 mL of solvent, not 250.

Example 4: Percent solutions

Dilute 70% ethanol to 500 mL of 20%

Result: V₁ = 20 × 500 / 70 = 143 mL

Percent works directly in the equation without conversion, since the units cancel on both sides.

Example 5: Error accumulation

Six 1:10 steps, 2% error each

Result: Total dilution 10⁶, uncertainty 5–12%

Random errors compound as √n times the per-step error, giving about 5%. A systematic bias such as a miscalibrated pipette compounds as n times, giving about 12%.

Common Mistakes

⚠️

Treating V₂ as solvent added

V₂ is the final total volume. The solvent to add is V₂ − V₁. Adding V₂ of solvent overshoots and gives a lower concentration than intended.

⚠️

Mixing concentration units between the two sides

C₁ and C₂ must use the same unit. Mixing molarity with mg/mL requires converting through molar mass first — the equation itself cannot do that.

⚠️

Assuming volumes are exactly additive

For concentrated solutions, mixing can change total volume slightly. This is why volumetric flasks are filled to a mark rather than by adding measured solvent.

⚠️

Ignoring compounding error in serial dilutions

Each step adds uncertainty. Six steps with 2% random error give about 5% total; a systematic 2% bias in the same pipette compounds to about 12%.

Frequently Asked Questions

What is C₁V₁ = C₂V₂?

Conservation of moles during dilution. Moles in = Moles out. Adding solvent changes volume and concentration but not the total amount of solute.

What is a serial dilution?

Successive dilutions where the diluted solution becomes the new stock. Three 1:10 dilutions gives 1:1000 overall. Used in microbiology and analytical chemistry for precise low concentrations.

What is C₁V₁ = C₂V₂?

The dilution equation. It states that the amount of solute is unchanged by adding solvent, so concentration times volume must be equal before and after.

Does it matter which concentration units I use?

No, provided both sides use the same unit. Molarity, mg/mL, percent and ppm all work because the units cancel.

Is V₂ the solvent I add?

No — it is the final total volume. The solvent to add is V₂ minus V₁, which is why volumetric flasks are filled to a mark.

Why use serial dilution?

Because very large dilutions in one step require measuring impractically small volumes. Several moderate steps give better accuracy, though errors accumulate across them.

How do serial dilution factors combine?

They multiply. Three consecutive 1:10 dilutions give a total factor of 1,000, not 30.

Formula Explorer connections

Interpretation: This formula connects solute amount and solution volume, mass or particle count to concentration and colligative behavior. Assumption: Distinguish solution volume from solvent volume, use the stated temperature, and account for dissociation or nonideal activity when required.

Dilution Factor Calculator →Freezing Point Depression Calculator →Molality Calculator →Chemistry Formula Explorer →