🔬 Beer–Lambert Law Calculator – Absorbance, Concentration & Transmittance
The Beer–Lambert Law (also known as Beer's Law or the Beer–Lambert–Bouguer Law) is the fundamental equation of UV-Vis spectrophotometry. It describes the linear relationship between the absorbance of a solution and the concentration of the absorbing species. Whether you're running protein assays in a biochemistry lab, measuring pollutant concentrations in environmental samples, or verifying dye concentrations in quality control, Beer's Law is the equation you reach for first.
The Core Equation
The Beer–Lambert Law is expressed as:
A = ε × l × cWhere:
- A — Absorbance (dimensionless; log₁₀ scale)
- ε — Molar absorptivity or molar extinction coefficient (L·mol⁻¹·cm⁻¹)
- l — Optical path length through the sample (cm)
- c — Molar concentration of the absorbing species (mol/L)
Because this equation has four variables, you can rearrange it to solve for any one of them when the other three are known. This calculator supports all five solving modes — plus a sixth mode for direct absorbance–transmittance conversion.
Absorbance vs. Transmittance
Transmittance (T) is the ratio of transmitted light intensity (I) to incident light intensity (I₀): T = I / I₀. A sample that lets half the light through has T = 0.50 (50 %T).
Absorbance (A) is the negative base-10 logarithm of T: A = −log₁₀(T). Absorbance is preferred for quantitative analysis because it is linearly proportional to concentration, making it straightforward to build calibration curves and apply Beer's Law directly. Transmittance, by contrast, is an exponential function of concentration.
| %T | T (fraction) | A |
|---|---|---|
| 100 % | 1.000 | 0.000 |
| 50 % | 0.500 | 0.301 |
| 25 % | 0.250 | 0.602 |
| 10 % | 0.100 | 1.000 |
| 1 % | 0.010 | 2.000 |
| 0.1 % | 0.001 | 3.000 |
The Five Solving Modes
1. Solve for Concentration (c)
The most common use of Beer's Law in the lab: measure the absorbance of an unknown sample and determine its molar concentration. You need the molar absorptivity (ε) of the substance at the wavelength you're measuring, and the path length of your cuvette (usually 1 cm).
c = A / (ε × l)2. Solve for Absorbance (A)
Predict the expected absorbance for a prepared standard of known concentration. Useful for planning calibration curves and verifying instrument settings before measurements begin.
A = ε × l × c3. Solve for Transmittance (%T)
Convert absorbance to percent transmittance. Useful when reporting results in formats that traditionally express light transmission, or when comparing to older instrument readouts.
T = 10^(−A) → %T = T × 1004. Solve for Molar Absorptivity (ε)
Determine the extinction coefficient of a substance from experimental data. Measure the absorbance of a sample of known concentration in a cuvette of known path length, then back-calculate ε.
ε = A / (c × l)5. Solve for Path Length (l)
Useful when working with non-standard cuvettes, microwell plates, or flow cells where the path length is not exactly 1 cm. Given a known sample (ε and c), back-calculate the effective path length from the measured absorbance.
l = A / (ε × c)Linearity and the A < 2.0 Rule
Beer's Law is reliable only within approximately 0 < A ≤ 2.0. Above this range, stray light in the spectrophotometer, molecular interactions at high concentrations, and instrumental non-linearity cause deviations. For best accuracy, dilute samples to the 0.1–1.0 absorbance range.
Molar Absorptivity (ε) and the Substance Library
Molar absorptivity is a substance-specific and wavelength-specific constant. It reflects how strongly a compound absorbs photons at a given wavelength — the larger ε is, the more sensitive the measurement. Published ε values span many orders of magnitude: NADH at 340 nm has ε ≈ 6,220 L·mol⁻¹·cm⁻¹, while hemoglobin's Soret band (415 nm) reaches ε ≈ 125,000 L·mol⁻¹·cm⁻¹. This calculator includes a built-in library of pre-loaded ε values for common chromophores so you don't need to look them up every time.
Calibration Curves
After calculating a result, the tool automatically generates a calibration curve — a plot of Absorbance (y-axis) vs. Concentration (x-axis) for the entered ε and l values. The dashed orange line at A = 2.0 marks the linearity limit. Real-world calibration involves measuring several known standards and fitting a least-squares regression line; the curve here is the theoretical ideal based on Beer's Law with your specific ε and l.
Unit Support
Concentrations can be entered in mol/L (M), mmol/L (mM),µmol/L (µM), or nmol/L (nM). Path lengths can be incm, mm, or m. The calculator converts everything to SI base units internally (mol/L, cm) and converts results back to your selected units for display.
Common Applications
- Biochemistry — Protein quantitation (Bradford, BCA), enzyme kinetics (NADH assays), nucleic acid quantification (A₂₆₀)
- Environmental Analysis — Measuring nitrate, phosphate, or heavy metal concentrations in water samples
- Pharmaceutical QC — Verifying drug concentrations in formulations against label claims
- Food Science — Colour intensity measurements, anthocyanin or carotenoid quantitation
- Teaching Labs — Introductory analytical chemistry exercises demonstrating spectrophotometry principles