# Beer-Lambert Law Calculator - Calibration Curve, LOD & Unknowns

> Free Beer-Lambert calculator: solve A = εbc for any variable, convert absorbance and %T, and fit a calibration curve with R², unknowns, 95% CI and LOD/LOQ.

- URL: https://qilingtools.com/en/math/beer-lambert
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Chemistry](https://qilingtools.com/en/tags/chemistry)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

The Beer–Lambert law, A = ε·b·c, says that the absorbance of a dilute solution is proportional to the molar absorptivity ε (L·mol⁻¹·cm⁻¹), the path length b (cm) and the concentration c (mol/L); absorbance and transmittance are related by A = −log T = 2 − log(%T). For example NADH has ε = 6220 at 340 nm, so A = 0.622 in a 1 cm cuvette means c = 0.622 / 6220 = 1.00 × 10⁻⁴ M = 100 μM. In the lab you usually work from a calibration (standard) curve instead: standards of 0, 2, 4, 6 and 8 mg/L reading 0.002, 0.151, 0.298, 0.452 and 0.598 give the least-squares line A = 0.07465c + 0.0016 with R² = 0.99995. This tool lets you type or paste standards from Excel or a CSV, exclude outliers or force the line through zero, and reports the slope and intercept with standard errors, the residuals, the residual standard deviation s(y/x), and LOD = 3.3·s/slope and LOQ = 10·s/slope (ICH approach). Enter any number of unknowns with replicate readings and a dilution factor to get the back-calculated concentration, its standard deviation s(x₀) from the standard formula and a 95% confidence interval, with a warning when a result is extrapolated beyond the standards. There is also an A = εbc solver for any one variable (M, mM, μM or mg/L, with molar mass from a formula), a pipetting table for preparing the standards from a stock solution, and notes on why real data deviate from Beer's law. Presets cover KMnO₄ at 525 nm, Fe²⁺–phenanthroline at 510 nm, the Bradford protein assay and NADH at 340 nm.

## How to use

1. Pick a preset or concentration unit and enter your standards (concentration and absorbance), or paste two columns from Excel
2. Read the regression line, R², residuals and plot, then enter unknown absorbances and dilution factors to get concentrations with confidence intervals
3. Switch to the A = εbc solver or the standards-preparation table if needed, and export the data as CSV and the plot as PNG or SVG

## Features

- A = εbc solver for any variable, concentration in M, mM, μM or mg/L, molar mass from a formula
- Absorbance ↔ percent transmittance converter
- Least-squares calibration line, optional fit through the origin and per-point exclusion
- Slope and intercept ± standard error, R², r, residuals and s(y/x)
- Unknown concentrations with s(x₀) and 95% CI, replicates and dilution factors
- LOD and LOQ, with warnings for results outside the calibrated range
- Pipetting table for preparing standards from a stock (C₁V₁ = C₂V₂)
- SVG calibration plot with PNG, SVG and CSV export

## FAQ

### What is the Beer–Lambert law and what are the units?

A = ε·b·c, where A is absorbance (dimensionless), ε is the molar absorptivity in L·mol⁻¹·cm⁻¹, b is the path length in cm and c is the concentration in mol/L. If your concentration is in mg/L, divide by 1000 and by the molar mass to get mol/L. Absorbance relates to transmittance by A = −log T = 2 − log(%T), so A = 1 means 10% of the light is transmitted.

### How do I find an unknown concentration from a calibration curve?

Measure 4–6 standards of known concentration, plot absorbance against concentration and fit A = a·c + b. Measure the unknown under the same conditions and solve c = (A − b)/a, then multiply by any dilution factor. The tool also reports the uncertainty s(x₀) = s(y/x)/|a|·√(1/m + 1/n + (Ā − ȳ)²/(a²·Sxx)) and a 95% confidence interval using the t distribution.

### What R² is acceptable, and should I force the line through zero?

Spectrophotometric calibrations commonly aim for R² ≥ 0.999, but follow your lab's or method's acceptance criteria. A high R² does not rule out curvature, so also check that the residuals are randomly scattered. Force the line through the origin only when the blank has been subtracted and the intercept is not significantly different from zero; otherwise keeping the intercept is safer.

### Why does my calibration curve bend at high concentrations?

Beer's law holds strictly only for dilute solutions and monochromatic light. Above an absorbance of about 1.0–1.5 very little light is transmitted and stray light dominates; a wide slit, chemical equilibria such as dimerisation or acid–base shifts, solute interactions above about 0.01 M, and turbidity also cause deviations. Keep readings roughly between 0.1 and 1.0 and dilute samples that read higher.

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