Beer-Lambert Law Calculator
Solve for absorbance, concentration, or path length with support for non-plasmonic materials including organic dyes, quantum dots, and graphene
Calculation Mode
⚡ Auto-UpdateAlso called molar extinction coefficient. Values depend on wavelength, solvent, pH, and temperature.
A = log₁₀(I₀/I). Ensure baseline correction and proper blank subtraction.
Ensure solute is fully dissolved and no precipitation or aggregation has occurred.
Standard cuvettes are 1 cm. Ensure cuvette is clean and properly positioned.
Wavelength: Report your measurement wavelength for reproducibility. Absorbance and ε are wavelength-dependent.
Typical UV-Vis range: 190–1100 nm. Record for experimental reproducibility.
Results
Understanding the Beer-Lambert Law
The Beer-Lambert law (also called Beer's law or the Beer-Lambert-Bouguer law) describes the linear relationship between absorbance and concentration of an absorbing species in solution. First formulated in the 18th and 19th centuries, it remains the foundation of quantitative UV-Vis spectroscopy.
Key assumption: The Beer-Lambert law assumes monochromatic light, dilute solutions, negligible scattering, and no chemical reactions or aggregation. Deviations occur at high concentrations where molecular interactions become significant.
The Fundamental Equation
The Beer-Lambert law relates absorbance (A) to the molar absorptivity (ε), path length (l), and concentration (c):
A = ε × l × c
Where:
- A = absorbance (dimensionless, though often expressed in AU)
- ε = molar absorptivity or extinction coefficient [L/(mol·cm)]
- l = optical path length [cm]
- c = concentration [mol/L or M]
Absorbance and Transmittance
Absorbance relates to transmittance (T) through a logarithmic relationship:
A = -log₁₀(T) = log₁₀(I₀/I)
Where I₀ is incident light intensity and I is transmitted intensity. This means:
- A = 0 corresponds to T = 100% (no absorption)
- A = 1 corresponds to T = 10% (90% absorbed)
- A = 2 corresponds to T = 1% (99% absorbed)
- A = 3 corresponds to T = 0.1% (99.9% absorbed)
Molar Absorptivity
The molar absorptivity (ε) is an intrinsic property of the absorbing species at a given wavelength. It quantifies how strongly a substance absorbs light:
- High ε (>10⁴): Strong absorbers like organic dyes, quantum dots
- Medium ε (10²–10⁴): Many organic compounds, metal complexes
- Low ε (<10²): Weak absorbers, forbidden transitions
Molar absorptivity depends on wavelength, solvent, pH, temperature, and ionic strength. Always use values determined under conditions matching your experiment.
Wavelength matters: Always record and report the wavelength at which ε was determined. Using an ε value from a different wavelength will give incorrect concentration results.
Limitations and Deviations
The Beer-Lambert law is an idealization. Real deviations include:
- High concentration: Molecular aggregation, dimerization, and refractive index changes cause negative deviations above A ≈ 1–2
- Polychromatic light: Real light sources have finite bandwidth, causing apparent deviations if ε varies significantly across the band
- Scattering: Particles, precipitates, or turbid samples scatter light and inflate apparent absorbance
- Fluorescence: Fluorescent samples emit light that can reach the detector, reducing apparent absorbance
- Chemical equilibria: pH-dependent speciation or complexation changes ε with concentration
- Stray light: Instrument limitations at very high absorbance values
Optimal Absorbance Range
For most reliable quantification:
- Ideal range: A = 0.2 – 0.8 (minimum noise, maximum linearity)
- Acceptable range: A = 0.1 – 1.0
- Questionable: A = 1.0 – 2.0 (verify linearity with standards)
- Unreliable: A > 2.0 (dilute sample and remeasure)
Monitor the Linearity indicator above to check whether your absorbance falls within the reliable range.
Molar Absorptivity Reference
Common absorbers and their approximate molar absorptivity values. Actual values depend on experimental conditions.
| Compound | λmax (nm) | ε [L/(mol·cm)] | Application |
|---|---|---|---|
| Rhodamine 6G | 530 | 1.05×10⁵ | Fluorescence standard, laser dye |
| Fluorescein | 490 | 7.2×10⁴ | Biological labeling, pH indicator |
| Cy5 | 650 | 2.5×10⁵ | Near-IR labeling, DNA sequencing |
| NADH | 340 | 6.22×10³ | Enzyme assays, metabolism studies |
| dsDNA (per bp) | 260 | ~50 ng/µL per AU | DNA quantification |
| BSA protein | 280 | 4.3×10⁴ | Protein quantification |
| CdSe QD (5 nm) | 1st exciton | ~7×10⁵ | Nanoparticle quantification |
| Graphene oxide | 230 | 2.46×10³ L/(g·cm) | 2D material characterization |
| p-Nitrophenol | 400 | 1.82×10⁴ | Enzyme substrates, colorimetry |
| Methylene Blue | 664 | 7.4×10⁴ | Redox indicator, biological stain |
Note: These values are approximate and vary with solvent, pH, temperature, and ionic strength. For accurate quantification, determine ε experimentally using your specific conditions and reference standards.
Special Considerations for Nanomaterials
Quantum Dots
Semiconductor quantum dots have size-dependent optical properties. The molar absorptivity at the first exciton peak scales with particle volume (approximately r³). Empirical relationships exist for CdSe, CdS, CdTe, and other QD systems:
- ε increases dramatically with QD size
- The first exciton wavelength red-shifts with increasing size
- Use published size-dependent ε calibrations for concentration determination
Graphene and 2D Materials
For 2D materials like graphene oxide, absorptivity is often reported per unit mass rather than per mole (L/(g·cm) instead of L/(mol·cm)). Key considerations:
- Scattering contribution may be significant for large flakes
- Oxidation state affects absorption spectrum
- Sheet size distribution affects optical properties
Organic Dyes
Organic dyes can exhibit concentration-dependent spectra due to:
- H-aggregates (blue-shifted absorption)
- J-aggregates (red-shifted absorption)
- Solvatochromism (solvent-dependent spectra)
- pH-dependent protonation states
For nanomaterials: Size distributions, aggregation states, and surface chemistry all affect optical properties. Validate your ε values against a standard method (e.g., ICP-MS for metal concentration, TGA for mass) when possible.
References
AI & Computational Tools for Researchers
Explore our curated guides to the best free AI tools for literature discovery, data analysis, computational modeling, and more.
