What is Beer's Law (Beer-Lambert Law)?
The Beer-Lambert law — also known as Beer's law or the Beer-Lambert-Bouguer law — describes the relationship between the concentration of a solution and the attenuation of light as it passes through it. It states that when a beam of electromagnetic radiation passes through a sample, its absorbance depends on the concentration of the absorbing species and the path length of the beam through the sample.
Beer's Law Equation
When a light beam of intensity I₀ passes through a solution of path length l, the transmitted intensity I is reduced by absorption. The absorbance A is defined as:
A = log₁₀(I₀ / I) = ε × l × c
Where:
- A — Absorbance (dimensionless)
- ε (epsilon) — Molar absorption coefficient, or molar absorptivity (L·mol⁻¹·cm⁻¹)
- l — Path length of the light beam through the sample (cm in metric; inches in imperial)
- c — Molar concentration of the absorbing species (mol/L)
The transmittance T is the fraction of light that passes through the sample: T = I / I₀. The relationship between absorbance and transmittance is: A = log₁₀(1 / T), so T% = 10^(2 − A).
How to Use the Beer-Lambert Law Calculator
This calculator solves for any one of the four Beer-Lambert variables when the other three are known:
- Find Absorbance: Enter molar absorptivity (ε), path length (l), and concentration (c) → get A
- Find Concentration: Enter A, ε, and l → get c (useful for quantitative analysis)
- Find Path Length: Enter A, ε, and c → get l
- Find Molar Absorptivity: Enter A, l, and c → get ε
- Convert Transmittance ↔ Absorbance: Enter T% to get A, or enter A to get T%
Example: A solution with ε = 8,400 L/(mol·cm), l = 1 cm, and c = 4.33 × 10⁻⁵ mol/L has absorbance A = 8400 × 1 × 0.0000433 = 0.3637, which corresponds to transmittance T = 43.30%.
Unit Systems Supported
The calculator supports both measurement systems:
- Metric system: path length in centimeters (cm) — the standard unit in spectroscopy
- Imperial (American) system: path length in inches (in) — automatically converted to cm for calculation
Concentration is expressed in mol/L (molarity) in both systems, as this is the universal standard in analytical chemistry.
Absorbance to Transmittance Conversion
The transmittance T is the ratio of transmitted light intensity to incident light intensity: T = I / I₀. In spectrophotometry, results are often reported as percent transmittance (T%). The conversion formulas are:
- A = −log₁₀(T%) + log₁₀(100) = log₁₀(100 / T%)
- T% = 100 × 10⁻ᴬ = 10^(2 − A)
An absorbance of 0 means 100% transmittance (no absorption), while A = 2 means 1% transmittance.
Absorbance Interpretation Guide
| Absorbance (A) | Transmittance (%) | Measurement Quality |
|---|---|---|
| < 0.1 | > 79% | Very low — consider increasing concentration or path length |
| 0.1 – 0.3 | 50 – 79% | Low — still measurable but less accurate |
| 0.3 – 1.0 | 10 – 50% | Optimal range — best accuracy (±1% photometric error) |
| 1.0 – 2.0 | 1 – 10% | High — approaching limits of linearity |
| > 2.0 | < 1% | Very high — consider diluting the sample |
Applications of the Beer-Lambert Law
- Spectrophotometry: measuring protein, DNA, and RNA concentrations in biological samples
- Environmental monitoring: analyzing pollutant concentrations in water and air
- Food & beverage industry: measuring color intensity, turbidity, and concentration
- Pharmaceutical analysis: quantifying drug concentrations in formulations
- Clinical diagnostics: blood glucose, hemoglobin, and bilirubin measurements
- Chemical research: reaction kinetics, equilibrium constants, and purity testing
Frequently Asked Questions
What are the units of molar absorptivity (ε)?
Molar absorptivity has units of L·mol⁻¹·cm⁻¹ (also written M⁻¹·cm⁻¹). It is a property of the absorbing molecule at a specific wavelength and temperature — it does not depend on concentration or path length. Values typically range from 10 (weak absorbers) to over 100,000 (strong absorbers, such as many dyes).
When does Beer's law break down?
Beer's law assumes a linear relationship between absorbance and concentration. This linearity holds only at low to moderate concentrations. At high concentrations, intermolecular interactions between solute molecules alter their absorptivity, causing deviation from linearity. Stray light in the instrument and polychromatic radiation can also cause deviations. The optimal range for measurements is typically A between 0.1 and 1.0.
What is the difference between absorbance and transmittance?
Transmittance is the fraction of light that passes through the sample (T = I/I₀), while absorbance is the logarithmic measure of how much light is absorbed (A = log₁₀(1/T)). Transmittance ranges from 0 to 100%, while absorbance ranges from 0 to infinity. Absorbance is preferred in quantitative analysis because it has a linear relationship with concentration (Beer's law).
What wavelength should I use?
For Beer-Lambert law measurements, use the wavelength of maximum absorption (λmax) for the substance being analyzed. At λmax, the molar absorptivity ε is at its highest, giving the greatest sensitivity. The λmax value is substance-specific and must be determined experimentally or found in reference tables.