{
  "id": "beer-lambert",
  "version": "71afeb69e46c",
  "status": "published",
  "name": "Beer-Lambert Law Calculator",
  "question": "How do I use the Beer-Lambert law?",
  "summary": "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.",
  "category": "chemistry",
  "subcategory": "solutions",
  "url": "https://www.acalculator.org/chemistry/beer-lambert-calculator",
  "markdown": "https://www.acalculator.org/chemistry/beer-lambert-calculator.md",
  "kind": "formulas",
  "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."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "a": {
        "title": "Absorbance (A)",
        "description": "The absorbance read on the spectrophotometer, log₁₀(I₀ ÷ I); it has no unit.",
        "type": "number",
        "minimum": 0,
        "maximum": 1000000000000
      },
      "e": {
        "title": "Molar absorptivity ε (L/(mol·cm))",
        "description": "The molar absorption coefficient of the substance at this wavelength, in liters per mole per centimeter.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 10000000
      },
      "l": {
        "title": "Path length (cm)",
        "description": "The distance the light travels through the sample; a standard cuvette is 1 cm.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 1000
      },
      "c": {
        "title": "Concentration (mol/L)",
        "description": "The molar concentration of the absorbing substance, in moles per liter.",
        "type": "number",
        "minimum": 0,
        "maximum": 1000
      }
    }
  },
  "solve": {
    "fillAny": 3,
    "variables": [
      "a",
      "e",
      "l",
      "c"
    ],
    "inputsOnly": []
  },
  "outputs": {
    "A": {
      "label": "Absorbance (A)",
      "description": "ε × l × c, with no unit.",
      "format": "number"
    },
    "e": {
      "label": "Molar absorptivity ε (L/(mol·cm))",
      "description": "A ÷ (l × c).",
      "format": "number"
    },
    "l": {
      "label": "Path length (cm)",
      "description": "A ÷ (ε × c).",
      "format": "number"
    },
    "c": {
      "label": "Concentration (mol/L)",
      "description": "A ÷ (ε × l).",
      "format": "number"
    },
    "T": {
      "label": "Transmittance (%)",
      "description": "100 × 10^−A: the percent of the light that passes through the sample.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "a": 0.37,
      "e": 138000,
      "l": 1
    },
    "outputs": {
      "c": 0.000002681159420289855,
      "T": 42.65795188015927
    },
    "text": "A = 0.37, ε = 138,000 L/(mol·cm), l = 1 cm and c = 0.00000268116 mol/L."
  },
  "examples": [
    {
      "given": {
        "a": 0.37,
        "e": 138000,
        "l": 1
      },
      "expect": {
        "c": 0.000002681159420289855,
        "T": 42.65795188015927
      },
      "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); hand calculation in content.mdx: 0.37 ÷ (138,000 × 1) = 2.6812 × 10⁻⁶ mol/L; T = 100 × 10^−0.37"
    },
    {
      "given": {
        "a": 0.5,
        "l": 1,
        "c": 0.0001
      },
      "expect": {
        "e": 5000
      },
      "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; hand calculation in content.mdx: 0.5 ÷ (1 × 0.0001) = 5,000 L/(mol·cm)"
    },
    {
      "given": {
        "e": 138000,
        "l": 1,
        "c": 0.000004
      },
      "expect": {
        "a": 0.552,
        "T": 28.05433637951713
      },
      "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); hand calculation in content.mdx: 138,000 × 1 × 0.000004 = 0.552"
    },
    {
      "given": {
        "a": 1,
        "e": 2000,
        "c": 0.005
      },
      "expect": {
        "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; hand calculation in content.mdx: 1 ÷ (2,000 × 0.005) = 0.1 cm"
    }
  ],
  "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)"
  ],
  "related": [
    "molarity",
    "concentration",
    "dilution",
    "serial-dilution"
  ],
  "changelog": []
}