Beer–Lambert law calculator
Leave one value blank, get it from the other three.
What is the Beer–Lambert law?
A = ε b c: absorbance equals the molar absorptivity ε times the path length b times the concentration c. Absorbance is also A = −log₁₀ T, where T is the fraction of light transmitted.
Example: finding ε
A solution of 5.00 × 10⁻⁴ M in a 1.00 cm cell has an absorbance of 0.338: ε = 0.338 ÷ (1.00 × 5.00 × 10⁻⁴) = 676 M⁻¹ cm⁻¹.
Example: finding concentration
With ε = 676 M⁻¹ cm⁻¹ and a 1.00 cm cell, an absorbance of 0.228 means c = 0.228 ÷ 676 = 3.37 × 10⁻⁴ M.
How do absorbance and transmittance relate?
A = −log₁₀ T. Transmitting half the light (T = 0.500) is an absorbance of 0.301; A = 1 means 10% of the light gets through, A = 2 means 1%.
When does the law stop working?
At high concentrations and with stray light or chemical changes the straight-line relationship bends, so labs check it with a calibration curve.
What are the three kinds of deviation from Beer’s law?
Harvey’s Analytical Chemistry 2.1 (section 10.2) separates them. Fundamental: Beer’s law is a limiting law, valid only for low concentrations, because at higher ones the particles interact and the refractive index changes. Chemical: if the absorbing species is in an equilibrium, such as a weak acid and its conjugate base, the absorbance no longer follows the total concentration unless the pH is held constant or both forms absorb equally. Instrumental: light that is not a single wavelength, and stray light reaching the detector, both pull the absorbance below the straight line at higher concentrations.
What if the sample contains more than one absorbing species?
If the components do not interact, their absorbances at a given wavelength add: A = ε₁bc₁ + ε₂bc₂ + … (Harvey, section 10.2). That is why a mixture can be analyzed by measuring at two wavelengths, but the calculator here handles one absorbing species at a time.
The equation and its units
Transmittance T is the ratio of the light power leaving the sample to the power entering it, and absorbance is A = −log T, a base-10 logarithm with no units. Beer’s law states that absorbance is proportional to concentration: A = εbc, where ε is the molar absorptivity in M⁻¹cm⁻¹, b is the path length of the cell in centimeters, and c is the concentration in mol/L (Harvey, Analytical Chemistry 2.1, section 10.2). Because ε depends on the substance and on the wavelength, quote it together with the wavelength it was measured at. The calculator solves for whichever of A, ε or c you leave blank; the path length must always be entered.
Worked example
The two Harvey examples in the first answers have ε = 676 M⁻¹cm⁻¹ in a 1.00 cm cell. Absorbance scales linearly with concentration, but transmittance does not: 5.00 × 10⁻⁴ M gives A = 0.338, which passes 45.9% of the light, while twice the concentration, 1.00 × 10⁻³ M, gives A = 0.676 and only 21.1%. Going from transmittance to absorbance, T = 0.500 gives A = 0.301, and a sample with A = 1.27 passes only 5.4% of the light (Harvey’s Example 10.2.1 and Exercise 10.2.1). To reproduce the first line, leave A blank and enter ε = 676, b = 1 and c = 5e-4.
Common mistakes
The unit of the path length is the commonest source of error: the formula uses centimeters, so a 1 mm cell is 0.1 cm. Entering 1 in that case would make the same absorbance of 0.338 correspond to a concentration ten times too small; with the correct 0.1 cm it gives 0.005 M. Second, mixing up transmittance and absorbance: a 50% transmittance reading is A = 0.301, not 0.5, so convert with A = −log T before using A = εbc. Third, extending the straight line beyond the range where it holds; check against a calibration curve of known standards, and be wary of high absorbances where stray light matters. Fourth, using ε measured at one wavelength for readings taken at another.
Sources: Harvey, Analytical Chemistry 2.1 (LibreTexts), section 10.2 “Spectroscopy Based on Absorption” (transmittance, absorbance, Beer’s law, limitations, Examples 10.2.1 and 10.2.2 and the practice exercises). Related: molarity calculator, dilution calculator, serial dilution calculator and ppm calculator.