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Pharmaceutical Calculators · QC & Analytical

Serial Dilution Calculator: Cn = C0 × D^n

Calculate concentration, stock transfer volume, and diluent volume at every step of a serial dilution series. Built for HPLC/LC-MS calibration curves, ELISA standards, MIC testing, and bioanalytical QC — supports 1:2, 1:5, 1:10, 1:100, and custom ratios.

Quick Answer

Serial dilution creates a geometric concentration series by repeating the same fold-dilution at each step: Cn = C0 × D^n, where D is the per-step dilution factor (0.1 for 1:10). Pharma and QC labs use it for HPLC/ELISA calibration curves, MIC testing, and bioanalytical QC levels per ICH Q2(R2). This calculator outputs concentration, stock volume, and diluent volume for each step — chain to the Dilution Calculator for single C1V1=C2V2 steps.

Formula
Cn = C0 × Dn
C0 = Initial stock concentration    D = Dilution factor per step
n = Step number    Cn = Concentration at step n
Each step: Cn = C(n-1) × D

Enter Values

Enter starting concentration, dilution ratio, number of steps, and volume per step to calculate the full dilution series.

Stock solution
Dilution series
Preparation volume
Starting Concentration
—
Total Dilution Factor
—
×
Final Concentration
—
Step Dilution Concentration Stock Volume Diluent Volume

How to Use

1
Enter the starting concentration of your stock solution (C0) and select the unit (mg/mL, μg/mL, CFU/mL, or mol/L).
2
Select the dilution ratio for each step — 1:2, 1:5, 1:10, 1:100, or enter a custom ratio.
3
Enter the number of dilution steps you want to calculate (1 to 12).
4
Enter the total volume to prepare at each step and choose your volume unit.
5
Click Calculate to see the full concentration table with stock and diluent volumes for every step.

Worked Example

Example: 3-Step 1:10 Serial Dilution

Starting with 1 mg/mL, making 3 serial 1:10 dilutions with 1 mL total volume per step:

Step 1: 0.1 mg/mL — take 0.1 mL of stock + add 0.9 mL diluent

Step 2: 0.01 mg/mL — take 0.1 mL of Step 1 solution + add 0.9 mL diluent

Step 3: 0.001 mg/mL — take 0.1 mL of Step 2 solution + add 0.9 mL diluent

Total dilution factor after 3 steps: 1000×

Pharma & QC laboratory context

Serial dilution protocols underpin validated bioanalytical methods, HPLC/UPLC calibration, ELISA standard curves, and microbiological limit testing under GMP. Each calibration level must trace to the reference standard lot with documented preparation date, analyst, and equipment ID — ICH Q2(R2) and FDA bioanalytical guidance expect at least five non-zero calibrator concentrations across the reportable range.

Prepare stock concentration with the Molarity Calculator using COA molecular weight, then build the dilution series here. Verify single-step math with the Dilution Calculator, confirm quantitation limits with the LOD/LOQ Calculator, and convert units with the Unit Converter.

Evidence & sources

Frequently Asked Questions

Serial dilution creates precise concentration series for HPLC and LC-MS calibration curves, ELISA standards, antimicrobial MIC/MBC testing, potency assays, and bioanalytical QC sample preparation. Each step multiplies concentration by a fixed factor D, spanning multiple orders of magnitude from one stock — essential when dynamic range exceeds a single dilution step.
Serial dilution chains steps — each tube uses the previous dilution as stock, so pipetting error can compound. Parallel dilution prepares every concentration independently from the original stock, improving per-point accuracy but requiring more stock and pipetting. Serial is faster for wide-range curves; parallel is preferred when each point must be independently traceable to the primary standard.
For 1:10, transfer 1 part stock into 9 parts diluent (D = 0.1). With 1 mL total per step: take 0.1 mL stock + 0.9 mL diluent. Concentration at step n is Cn = C0 × (0.1)^n. After 3 steps from 1 mg/mL, final concentration is 0.001 mg/mL (1000× total dilution).
Cn = C0 × D^n, where C0 is initial stock concentration, D is the dilution factor per step (e.g., 0.1 for 1:10, 0.5 for 1:2), and n is the step number. Per step: Cn = C(n−1) × D. Total dilution factor after n steps is (1/D)^n — for 1:10 over 5 steps, total factor is 10^5 = 100,000×.
Bioanalytical and compendial practice typically uses 5–7 non-zero calibration levels spanning 2–3 orders of magnitude. FDA bioanalytical guidance and ICH Q2(R2) require demonstrated linearity across the reportable range — often 6 HPLC levels from LLOQ to ULOQ. Choose step count and ratio so the final concentration bracket your LLOQ and highest QC level.
Each individual step satisfies C1V1 = C2V2: source concentration × transfer volume = target concentration × final volume. Serial dilution applies this repeatedly with the prior step as the new C1. This calculator computes transfer and diluent volumes from total volume and dilution factor D = V_transfer / V_total.
1:10 serial dilutions are standard for plate counts — 6 steps from dense culture covers 10^6-fold range. Spread-plate or pour-plate methods target 30–300 colonies per plate for statistically reliable counts. Document diluent (peptone water, saline), mixing, and incubation per USP ⟨61⟩ or EP 2.6.12.
Relative error compounds with each step — a 2% pipetting error per step becomes roughly 10% cumulative over 5 steps. Use calibrated micropipettes, pre-wet tips for viscous solutions, mix thoroughly between steps, and avoid volumes below the accurate range of your pipette (typically ≥10% of nominal volume). For critical assays, prepare parallel points from stock when compounding error is unacceptable.
Yes — serial dilution reduces exposure by working from a single high-concentration stock in a controlled hood. Document OEL/OEB band, PPE, and waste per occupational toxicology SOPs. For occupational health, prefer dilution schemes that keep working concentrations in the µg/mL range rather than weighing nanogram quantities repeatedly.
Half-log dilutions use D ≈ 0.316 (1:3.16), giving equal spacing on a log concentration axis — useful for pharmacology dose-response curves and 4PL IC50/EC50 fitting where inflection sits mid-curve. Eight half-log steps cover roughly 10^4-fold range with denser points near the inflection than 1:10 spacing.
Calibration standards prepared by serial dilution must bracket the limit of quantitation (LOQ) and upper limit of quantitation. After building the series here, confirm LLOQ/ULOQ with the LOD/LOQ Calculator using calibration slope and residual SD per ICH Q2(R2). Standards should be prepared fresh or from qualified frozen aliquots with documented stability.
Use this serial dilution calculator for multi-step geometric series (calibration curves, MIC panels, CFU). Use the single-step Dilution Calculator when you know stock concentration C1 and need one dilution to target C2 in volume V2 (C1V1 = C2V2). Typical workflow: prepare stock with the Molarity Calculator, serial-dilute here, then verify individual steps with the Dilution Calculator if needed.

Related Calculators

C₁V₁ Dilution Calculator M Molarity Calculator LOQ LOD/LOQ Calculator ↔ Unit Converter

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