# Beer–Lambert Law Calculator

> Solves the Beer–Lambert law A = ε·l·c for the selected quantity (absorbance, concentration, molar absorptivity or path length) from the other three, and reports the percent transmittance T = 100 × 10^(−A).

- Calculator id: `beer-lambert` · Category: Chemistry (`chemistry`) · Tool name: `calculate_beer_lambert_law`
- Canonical page: https://tttkmbb.com/chemistry/beer-lambert · This document: https://tttkmbb.com/chemistry/beer-lambert.md · JSON definition: https://tttkmbb.com/chemistry/beer-lambert.json

## Purpose

Solves the Beer–Lambert law A = ε·l·c for the selected quantity (absorbance, concentration, molar absorptivity or path length) from the other three, and reports the percent transmittance T = 100 × 10^(−A).

**Use when:** You have a spectrophotometer absorbance and need the concentration of the absorbing species from its molar absorptivity, the expected absorbance of a solution, or the molar absorptivity or cuvette path length from a calibration measurement.

**Do not use when:** Absorbance is above about 2, or the sample is concentrated, turbid or chemically interacting (the law is linear only for dilute, non-scattering solutions in monochromatic light); for concentration from a weighed mass use molarity.

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `mode` | enum: concentration \| absorbance \| molar_absorptivity \| path_length |  | optional, default "concentration" | Quantity to solve for; the other three must be given (path_length_cm defaults to a 1 cm cuvette). |
| `absorbance` | number |  | optional | Blank-corrected absorbance (optical density, base-10, dimensionless). Leave empty when solving for it. (min 0, max 10) |
| `molar_absorptivity_l_mol_cm` | number | L/(mol·cm) | optional | Molar (decadic) absorption coefficient at the measurement wavelength, e.g. 6220 for NADH at 340 nm. Leave empty when solving for it. (> 0, max 10000000) |
| `path_length_cm` | number | cm | optional, default 1 | Optical path length through the sample (inner cuvette width); standard cuvettes are 1 cm. Ignored when mode = path_length. (> 0, max 1000) |
| `concentration_mol_l` | number | mol/L | optional | Molar concentration of the absorbing species. Leave empty when solving for it. (> 0, max 100) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `absorbance` | number |  | Base-10 absorbance, given or solved. |
| `molar_absorptivity_l_mol_cm` | number | L/(mol·cm) | Molar absorption coefficient, given or solved. |
| `path_length_cm` | number | cm | Optical path length, given or solved. |
| `concentration_mol_l` | number | mol/L | Molar concentration, given or solved. |
| `concentration_scientific` | string |  | The concentration in scientific notation, mol/L. |
| `concentration_umol_l` | number | µmol/L | The same concentration in micromoles per litre (µM). |
| `transmittance_percent` | number | % | 100 × 10^(−A): share of the incident light that passes through the sample. |
| `solved_for` | string |  | Name of the output that was computed: absorbance, molar_absorptivity_l_mol_cm, path_length_cm or concentration_mol_l. |

## Formula

`A = molar_absorptivity_l_mol_cm × path_length_cm × concentration_mol_l; c = A / (ε × l); ε = A / (l × c); l = A / (ε × c); transmittance_percent = 100 × 10^(−A)`

A is the decadic absorbance log10(I0/I) and ε the molar decadic absorption coefficient, so ε is in L·mol⁻¹·cm⁻¹ with l in cm and c in mol/L. Linearity assumes monochromatic light and a dilute (below about 0.01 mol/L), homogeneous, non-scattering sample; above A ≈ 1.5–2 stray light and detector limits make readings unreliable.

## Data Sources

- Wikipedia – Beer–Lambert law — https://en.wikipedia.org/wiki/Beer%E2%80%93Lambert_law (reference, retrieved 2026-09-24)
- IUPAC Gold Book – Beer–Lambert law — https://goldbook.iupac.org/terms/view/B00626 (standard, retrieved 2026-09-24)
- LibreTexts – The Beer-Lambert Law — https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Spectroscopy/Electronic_Spectroscopy/Electronic_Spectroscopy_Basics/The_Beer-Lambert_Law (textbook, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/beer-lambert?`
- `POST https://tttkmbb.com/api/v1/calculate/beer-lambert` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/beer-lambert · OpenAPI operationId `calculate_beer_lambert_law` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "beer-lambert", "inputs": {…}}`

## Example

- Absorbance of 1.0×10⁻⁴ M NADH at 340 nm (ε 6220, 1 cm cuvette): inputs `{"mode":"absorbance","molar_absorptivity_l_mol_cm":6220,"path_length_cm":1,"concentration_mol_l":0.0001}` → `{"absorbance":0.622,"transmittance_percent":23.88,"concentration_umol_l":100,"solved_for":"absorbance"}`
- Concentration from A = 0.500 with ε 6220 in a 1 cm cuvette: inputs `{"mode":"concentration","absorbance":0.5,"molar_absorptivity_l_mol_cm":6220}` → `{"concentration_mol_l":0.000080386,"concentration_scientific":"8.039e-5","concentration_umol_l":80.386,"transmittance_percent":31.62,"solved_for":"concentration_mol_l"}`

```
GET https://tttkmbb.com/api/v1/calculate/beer-lambert?mode=absorbance&molar_absorptivity_l_mol_cm=6220&path_length_cm=1&concentration_mol_l=0.0001
```

## Limitations

Absorbance is above about 2, or the sample is concentrated, turbid or chemically interacting (the law is linear only for dilute, non-scattering solutions in monochromatic light); for concentration from a weighed mass use molarity. A is the decadic absorbance log10(I0/I) and ε the molar decadic absorption coefficient, so ε is in L·mol⁻¹·cm⁻¹ with l in cm and c in mol/L. Linearity assumes monochromatic light and a dilute (below about 0.01 mol/L), homogeneous, non-scattering sample; above A ≈ 1.5–2 stray light and detector limits make readings unreliable. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**What if my ε is in mM⁻¹cm⁻¹ or the concentration in mg/L?**

Multiply an ε in mM⁻¹·cm⁻¹ by 1000 to get L·mol⁻¹·cm⁻¹ (NADH: 6.22 mM⁻¹cm⁻¹ = 6220). For mass concentrations divide by the molar mass to get mol/L, or use a specific absorbance in L·g⁻¹·cm⁻¹ and read the result as g/L.

**Why is absorbance above 2 unreliable?**

At A = 2 only 1 % of the light reaches the detector, so stray light and detector noise dominate and the response flattens; dilute the sample or use a shorter path length to bring A into the 0.1–1 range.

**Is absorbance the same as optical density?**

In practice yes for clear solutions: both are log10(I0/I). Optical density also includes light lost by scattering, which is why turbid samples deviate from the Beer–Lambert law.

## Related

- [Molarity Calculator](https://tttkmbb.com/chemistry/molarity.md) — Prepare a standard of known molarity for the calibration.
- [Dilution Calculator](https://tttkmbb.com/chemistry/dilution.md) — Dilute a sample whose absorbance is outside the linear range.