Dilution Calculator

Calculate concentrations and volumes for simple dilutions and serial dilutions.

e.g. 1:10 = 10-fold
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Why the Equation Works in Any Units

C1V1 = C2V2 states that the amount of solute is unchanged by adding solvent. Because both sides express the same quantity, the units cancel — molarity, mg/mL, percent or ppm all work, provided both concentrations use the same one.

C1V1 = C2V2     solvent to add = V2 − V1

The most frequent practical error is treating V2 as the volume of solvent to add. It is the final total volume. Making 100 mL from 10 mL of stock means adding solvent up to the 100 mL mark — about 90 mL, not 100.

Serial Dilution and Error Accumulation

1:10 stepsTotal factorFrom 1 mg/mL
110×100 μg/mL
31,000×1 μg/mL
6106×1 ng/mL
9109×1 pg/mL

Reaching a millionfold dilution in one step would mean pipetting 1 μL into a litre, where measurement error dominates. Serial dilution replaces one impossible measurement with several manageable ones — but errors accumulate.

A random 2% pipetting error over six steps compounds to about 5%, growing as √n. A systematic 2% bias — a miscalibrated pipette used throughout — compounds to about 12%, growing as n. This is why calibration matters more than technique in serial work.

Worked Examples

Example 1: Make 100mL of 0.01M from 1M stock
C2=1×10/100
Result: C2=0.1M — but entered V1=10mL
V1=C2×V2/C1=0.01×100/1=1mL of stock
Example 2: 1:10 serial, 6 steps from 1mg/mL
C after 6 steps=1/10^6
Result: Final=1×10⁻⁶ mg/mL=1 ng/mL
Used for antibody or drug concentration-response curves
Example 3: Solvent to add
25 mL of stock diluted to 250 mL final
Result: Add about 225 mL
V2 is the final total. Adding 250 mL of solvent would give 275 mL and a concentration about 10% too low.
Example 4: Error type matters
Six 1:10 steps, 2% error per step
Result: 5% if random, 12% if systematic
Random errors partially cancel and grow as √n. A miscalibrated pipette biases every step the same way and grows as n.

Common Mistakes

⚠️
Treating V2 as solvent added

V2 is the final total volume. Solvent to add is V2 minus V1, which is why volumetric flasks are filled to a mark rather than by measuring solvent.

⚠️
Mixing concentration units across the equation

Both concentrations must use the same unit. Converting between molarity and mg/mL requires molar mass, which the equation itself cannot supply.

⚠️
Using the same pipette throughout a serial dilution

A systematic calibration error compounds linearly across steps. Alternating pipettes or verifying calibration reduces accumulated bias.

⚠️
Not mixing between serial steps

Each dilution must be homogeneous before the next aliquot is drawn. Incomplete mixing is a large and often invisible source of error.

Frequently Asked Questions

Why serial dilution?
Extends concentration range by factors of 10^n. 6 steps of 1:10 = 10⁶ range. Economical: small volume of stock + many diluent volumes. Used in microbiology (plate counts), pharmacology (dose-response), biochemistry (enzyme kinetics), ELISA calibration curves.
Dilution accuracy?
Use volumetric glassware for accuracy. Pipette error propagates through serial dilution: 1% error per step × 6 steps → compounding. Gravimetric dilution (by mass) is more accurate than volumetric for critical work.
Does it matter which concentration units I use?
No, provided both sides use the same unit. Molarity, percent, mg/mL and ppm all work because the units cancel.
Is V₂ the amount of solvent I add?
No — it is the final total volume. The solvent to add is V2 minus V1.
Why use serial dilution instead of one large step?
Because very large dilutions require impractically small volumes in a single step. Several moderate steps give far better accuracy.
How do errors accumulate in serial dilution?
Random errors grow as the square root of the number of steps; systematic errors grow linearly. Six steps at 2% give about 5% random or 12% systematic.

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.

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