Beer-Lambert Law Absorbance Calculator

Beer-Lambert Law A = ε × l × c A = absorbance (dimensionless, also written as OD, for optical density) ε = molar absorptivity (L·mol⁻¹·cm⁻¹), a property of the molecule at a given wavelength l = path length (cm), the distance light travels through the sample c = concentration (mol/L = M) The law was developed by Pierre Bouguer (1729, France), Johann Heinrich Lambert (1760, Germany), and August Beer (1852, Germany).

Absorbance and Transmittance Transmittance T = I/I₀ (fraction of light that passes through) Absorbance A = −log₁₀(T) = log₁₀(I₀/I) At A = 0: T = 100% (all light transmitted, a blank solution) At A = 1: T = 10% (90% absorbed) At A = 2: T = 1% (99% absorbed) At A = 3: T = 0.1% (99.9% absorbed)

Absorbance is logarithmic, which is the part that surprises people. Going from A = 1 to A = 2 does not absorb twice as much light. It cuts what still gets through by a further factor of ten, from 10% down to 1%.

Molar Absorptivity (ε) ε is a molecular property: how strongly the molecule absorbs at a specific wavelength. High ε (>10,000): strong absorber (chromophore, aromatic ring, conjugated system) Low ε (<100): weak absorber (most saturated organics in UV) DNA at 260 nm: ε ≈ 6,600 L·mol⁻¹·cm⁻¹ (per nucleotide) Myoglobin at 410 nm (Soret band): ε ≈ 128,000 L·mol⁻¹·cm⁻¹ p-Nitroaniline at 380 nm: ε ≈ 13,200 L·mol⁻¹·cm⁻¹

Common Values for Proteins Protein concentration (A280/ε) from absorbance at 280 nm: IgG (antibody): ε ≈ 210,000 at 280 nm, molecular weight ≈ 150,000 → 1 A₂₈₀ = 0.71 mg/mL BSA (bovine serum albumin): ε ≈ 43,824 at 280 nm, molecular weight 66,463 → 1 A₂₈₀ = 1.52 mg/mL That BSA figure is often quoted as 1.51, from the older E1% = 6.6 convention. The two disagree in the second decimal place and nobody minds. Use online tools (ProtParam) for exact ε from amino acid sequence.

Worked Example: a 0.1 mM Sample in a Standard Cuvette

Take ε = 10,000 L·mol⁻¹·cm⁻¹, a standard 1 cm cuvette, and c = 1 × 10⁻⁴ mol/L.

A = 10,000 × 1 × 0.0001 = 1.0000

At A = 1 the transmittance is 10^(−1) × 100 = 10%, so 90% of the light is absorbed. That sits inside the 0.1 to 1.5 window most instruments are happiest in, though it is near the top of it. If you wanted more headroom you would dilute twofold, landing at A = 0.5 and T = 31.6%.

Now run it backwards. A reading of A = 0.500 on the same instrument with the same molecule gives c = 0.500 ÷ (10,000 × 1) = 5 × 10⁻⁵ mol/L, which is half the original. That is the whole working method of quantitative spectrophotometry: measure A, divide by ε and l, get concentration.

Watch the size of ε when you sanity-check a number. With ε = 10,000 a millimolar sample would read A = 10, which no instrument can measure. Strong absorbers get measured at micromolar concentrations, not millimolar.

Why the optimal window is 0.1 to 1.5

Below A = 0.1 the sample is barely dimming the beam, and the reading is competing with the instrument’s own baseline noise. Above about 2, so little light reaches the detector that the electronics start guessing. Stray light inside the monochromator, a fraction of a percent that never touched the sample, becomes a large share of what the detector sees.

Photometric error is smallest around A = 0.4 in classical treatments. In practice anywhere from 0.1 to 1.5 is fine on a modern instrument. If your reading is above 2, dilute the sample and multiply the answer back, rather than trusting the number.

Limitations Beer-Lambert Law is valid only at low concentrations (usually < 0.01 M). At high concentrations, intermolecular interactions and saturation effects cause deviations. Scattering by particles also adds apparent absorbance, and a cloudy sample can read high with nothing actually absorbing. Fluorescence can reduce apparent absorbance.