# How do I use the Beer-Lambert law?

Solves the Beer–Lambert law, A = ε × l × c, for the absorbance, molar absorptivity, path length or concentration from the other three, with the transmittance.

- Page: https://www.acalculator.org/chemistry/beer-lambert-calculator
- JSON spec: https://www.acalculator.org/chemistry/beer-lambert-calculator.json
- Version: 71afeb69e46c

## Default answer

Example with the default inputs (Absorbance (A) 0.37, Molar absorptivity ε (L/(mol·cm)) 138,000, Path length (cm) 1): A = 0.37, ε = 138,000 L/(mol·cm), l = 1 cm and c = 0.00000268116 mol/L.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| a | Absorbance (A) | The absorbance read on the spectrophotometer, log₁₀(I₀ ÷ I); it has no unit. |
| e | Molar absorptivity ε (L/(mol·cm)) | The molar absorption coefficient of the substance at this wavelength, in liters per mole per centimeter. |
| l | Path length (cm) | The distance the light travels through the sample; a standard cuvette is 1 cm. |
| c | Concentration (mol/L) | The molar concentration of the absorbing substance, in moles per liter. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| A | Absorbance (A) | ε × l × c, with no unit. |
| e | Molar absorptivity ε (L/(mol·cm)) | A ÷ (l × c). |
| l | Path length (cm) | A ÷ (ε × c). |
| c | Concentration (mol/L) | A ÷ (ε × l). |
| T | Transmittance (%) | 100 × 10^−A: the percent of the light that passes through the sample. |

## Method

A = ε × l × c; ε = A ÷ (l × c); l = A ÷ (ε × c); c = A ÷ (ε × l); transmittance T = 100 × 10^−A %.

## Assumptions

- The light is monochromatic, the solution is clear and dilute, and the substance does not react or scatter light, so A is proportional to c.
- The path length is in centimeters and the concentration in moles per liter, so ε is in L/(mol·cm).
- The absorbance is the decadic one, A = log₁₀(I₀ ÷ I), so T = I ÷ I₀ = 10^−A.

## Worked examples

1. a = 0.37, e = 138,000, l = 1 gives c = 0.000003, T = 42.657952. Source: OpenStax, Organic Chemistry, §14.7 Ultraviolet Spectroscopy (A = log(I₀ ÷ I); ε = A ÷ (c × l), c in mol/L and l in cm; β-carotene, ε = 138,000, absorbance 0.37 in a 1.0 cm cell: c = 2.7 × 10⁻⁶ mol/L), https://openstax.org/books/organic-chemistry/pages/14-7-ultraviolet-spectroscopy (retrieved 2026-10-18).
2. a = 0.5, l = 1, c = 0.0001 gives e = 5,000. Source: IUPAC Compendium of Chemical Terminology (Gold Book), Beer–Lambert law: A = log₁₀(P₀ ÷ P) = εcl, ε the molar (decadic) absorption coefficient, https://doi.org/10.1351/goldbook.B00626.
3. e = 138,000, l = 1, c = 0.000004 gives a = 0.552, T = 28.054336. Source: OpenStax, Organic Chemistry, §14.7 Ultraviolet Spectroscopy (A = log(I₀ ÷ I); ε = A ÷ (c × l), c in mol/L and l in cm; β-carotene, ε = 138,000, absorbance 0.37 in a 1.0 cm cell: c = 2.7 × 10⁻⁶ mol/L), https://openstax.org/books/organic-chemistry/pages/14-7-ultraviolet-spectroscopy (retrieved 2026-10-18).
4. a = 1, e = 2,000, c = 0.005 gives l = 0.1, T = 10. Source: IUPAC Compendium of Chemical Terminology (Gold Book), Beer–Lambert law: A = log₁₀(P₀ ÷ P) = εcl, ε the molar (decadic) absorption coefficient, https://doi.org/10.1351/goldbook.B00626.

## FAQ

### What is the Beer-Lambert law?

It says that the absorbance of a dilute solution is proportional to the path length of the light through it and to the concentration of the absorbing substance: A = ε × l × c. The constant ε, the molar absorptivity, depends on the substance and the wavelength.

### How do I find the concentration from the absorbance?

Divide the absorbance by the molar absorptivity times the path length: c = A ÷ (ε × l). β-carotene has ε = 138,000 L/(mol·cm); an absorbance of 0.37 in a 1 cm cell gives c = 0.37 ÷ 138,000 = 2.68 × 10⁻⁶ mol/L.

### How do I calculate molar absorptivity?

Measure the absorbance of a solution of known concentration and divide: ε = A ÷ (l × c). A 0.1 mM (0.0001 mol/L) solution that reads 0.5 in a 1 cm cuvette has ε = 0.5 ÷ (1 × 0.0001) = 5,000 L/(mol·cm). The units are often left off.

### How are absorbance and transmittance related?

Transmittance is the fraction of the light that gets through, T = I ÷ I₀, and the absorbance is A = log₁₀(I₀ ÷ I) = −log₁₀ T. In percent, A = 2 − log₁₀(%T): 10% transmittance is an absorbance of 1, and 1% is an absorbance of 2.

### What units does the calculator use?

Path length in centimeters and concentration in moles per liter, so ε is in liters per mole per centimeter, L/(mol·cm), also written M⁻¹ cm⁻¹. Absorbance has no unit.

### When does the Beer-Lambert law not hold?

IUPAC states the law for a collimated beam of monochromatic light in a uniform (homogeneous) sample. It does not hold when the band of wavelengths is wide compared with the absorption band, or when the sample scatters light, so use a clear sample at a single wavelength.

## Sources

- OpenStax, Organic Chemistry (McMurry), §14.7 Ultraviolet Spectroscopy: absorbance A = log(I₀ ÷ I); molar absorptivity ε = A ÷ (c × l), with c in mol/L and l in cm; β-carotene, ε = 138,000 L/(mol·cm), absorbance 0.37 in a 1.0 cm cell, c = 2.7 × 10⁻⁶ mol/L. https://openstax.org/books/organic-chemistry/pages/14-7-ultraviolet-spectroscopy (retrieved 2026-10-18)
- IUPAC, Compendium of Chemical Terminology (the Gold Book), Beer–Lambert law (Beer–Lambert–Bouguer law): A = log₁₀(P₀ ÷ P) = εcl, with ε the molar (decadic) absorption coefficient. https://doi.org/10.1351/goldbook.B00626 (retrieved 2026-10-18)