Serial Dilution Calculator
Enter the initial concentration C₀, dilution factor d and number of steps n; using Cₙ = C₀ × dⁿ the tool instantly computes the final concentration after serial dilution and the per-step volume scheme.
Input Data
Results
At a glance:A serial dilution is a method of repeatedly diluting a solution by the same factor to reach a very low concentration or to build a gradient. The core relation is Cₙ = C₀ × dⁿ: C₀ is the initial (stock) concentration, d the fraction of the previous concentration kept in one step (called the dilution factor, e.g. a 1:10 dilution keeps d = 0.1, a 1:2 dilution keeps d = 0.5), and n the number of repeated steps; the final concentration Cₙ is C₀ times d to the n-th power. Because each step multiplies by d, the concentration falls geometrically (exponentially) with steps — this is why a few steps reach a huge dilution: e.g. three 1:10 steps give 10⁻³ (1:1000), eight give 10⁻⁸. Using this tool's default: C₀ = 100 mg/L, d = 0.1 (1:10 each step), n = 3, C₃ = 100 × 0.1³ = 100 × 0.001 = 0.1 mg/L — i.e. 1:1000. The classic ELISA/microplate two-fold series uses d = 0.5 (each step halves). Why do serial dilution? Preparing 10⁻⁸ directly by adding a tiny drop to a huge volume is error-prone and hard; instead do ten 1:10 steps, each 'take a fixed volume from the previous tube, add diluent to the next' — easy and reproducible. Practical scheme: to make a 1:10 series, each step take 1 part stock + 9 parts diluent (total 10 parts, keeps 1/10); to make a 1:2 series, take 1 part + 1 part diluent. A 'dilution multiple' usually refers to 1/d (e.g. '1000-fold dilution' = 10⁻³). Serial dilution is everywhere: (1) microbiology — to count colony-forming units (CFU) you plate a series and count colonies in the countable range (30–300), then back-calculate the original concentration; (2) immunoassays (ELISA) and PCR — building standard or sample gradients; (3) preparing standard curves for quantitative analysis; (4) toxicology, pharmacy — stepwise dose-gradient screening. Notes: first, d must be 0 < d ≤ 1 (d = 1 means no dilution); n is a non-negative integer. Second, concentration and volume units are consistent (mg/L or mol/L). Third, the geometric result means small d or large n quickly gives extremely low concentration; watch purity of diluent and pipetting error. Fourth, each step's residual or cross-contamination accumulates, so use fresh tips and mix well.
Formula
Final concentration: Cₙ = C₀ × dⁿ.
Dilution factor d: 1:10 → d = 0.1, 1:2 → d = 0.5 (0 < d ≤ 1).
Dilution multiple = 1/d; n steps give (1/d)ⁿ-fold dilution.
Concentration falls geometrically with the number of steps.
$$C_n = C_0 \times d^{\,n}$$How to Use
- Enter the initial concentration C₀ (any unit).
- Enter the dilution factor per step d (1:10 = 0.1, 1:2 = 0.5).
- Enter the number of steps n; the right panel instantly shows the final concentration Cₙ.
Final concentration of a 1:10 serial dilution (C₀ = 100 mg/L)
| Step n | Dilution | Cₙ (mg/L) |
|---|---|---|
| 0 | 1 (undiluted) | 100 |
| 1 | 1:10 | 10 |
| 2 | 1:100 | 1 |
| 3 | 1:1000 | 0.1 |
| 4 | 1:10000 | 0.01 |
Each 1:10 step keeps d = 0.1; concentration falls geometrically, so a few steps reach a very low value quickly.
Case Studies
Ten-fold serial dilution
C₀ = 100 mg/L, each step 1:10 (d = 0.1), n = 3.
C₃ = 100 × 0.1³ = 0.1 mg/L (1:1000).
Practical scheme: each step take 1 mL stock + 9 mL diluent, repeat 3 times.
Two-fold microplate series
C₀ = 64 µg/mL, each step 1:2 (d = 0.5), n = 4.
C₄ = 64 × 0.5⁴ = 64 × 0.0625 = 4 µg/mL.
Typical ELISA gradient: each well take half from the previous, dilute by half each step.
FAQ
What is the dilution factor d, and what value should I use?
d is the fraction of concentration retained in one step: a 1:10 dilution keeps 1/10, so d = 0.1; a 1:2 dilution keeps 1/2, so d = 0.5. d must be between 0 and 1 (d = 1 means no dilution). The 'dilution multiple' people often say is 1/d (e.g. 1000-fold = d = 0.001, = 1:1000).
Why not dilute directly to the target?
A single huge dilution (e.g. 10⁻⁸) needs adding a tiny drop to a huge volume — easy to mis-measure and hard to mix. Serial dilution splits it into many equal, easy steps ('take a fixed volume, add diluent'), more accurate and repeatable, and gives a concentration gradient for titration or standard curves.
Can I use multiplication instead of power?
Yes. Cₙ = C₀ × dⁿ is just multiplying by d, n times. If you prefer the 'dilution multiple' M = 1/d, then Cₙ = C₀ / Mⁿ (e.g. C₀ / 10ⁿ for ten-fold steps). Both give the same result.
How to prepare a 1:10 series in practice?
Each step take 1 part of the previous tube + 9 parts diluent (total 10 parts, keeps 1/10). E.g. 1 mL stock + 9 mL diluent → second tube; take 1 mL from it + 9 mL diluent → third, and so on. Use fresh tips and mix well at each step to avoid carry-over.
What errors accumulate in serial dilution?
Each step's pipetting error and incomplete mixing accumulate and amplify down the series; residual or cross-contamination also propagates. Keep tips clean, mix thoroughly, and avoid too many steps (spread the range with a few larger dilutions) to keep accuracy.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.