Beer-Lambert Law

The Beer-Lambert law relates the amount of light absorbed by a sample to the concentration of the absorbing species:

A = εbc

where:

• A = absorbance (unitless).
• ε (epsilon) = molar absorptivity (M⁻¹ cm⁻¹), how strongly the molecule absorbs at the given wavelength.
• b = path length (cm), the distance light travels through the sample (typically 1 cm for standard cuvettes).
• c = concentration (M).

The law is linear: double the concentration, double the absorbance. This linearity makes UV-Vis an ideal quantitative tool.

Transmittance vs. Absorbance

Transmittance (T) is the fraction of light that passes through the sample: T = I / I₀, where I₀ is the incident light intensity and I is the transmitted intensity.

Absorbance (A) is the negative log of transmittance:

A = -log(T) = -log(I / I₀)

Conversions:

• A = 0 → T = 1 (100% transmittance, no absorption).
• A = 1 → T = 0.1 (10% transmittance, 90% absorbed).
• A = 2 → T = 0.01 (1% transmittance, 99% absorbed).
• A = 3 → T = 0.001 (99.9% absorbed).

Molar Absorptivity (ε)

Molar absorptivity is a characteristic of the molecule and wavelength. It ranges from:

• ε < 10 (weak absorber, like an n → pi* transition in an isolated carbonyl).
• ε ~ 1,000 (moderate, like alkenes).
• ε > 10,000 (strong, like conjugated systems and aromatic compounds).
• ε > 100,000 (very strong, like extended chromophores).

A molecule with ε = 10,000 and concentration 0.0001 M in a 1 cm cuvette gives A = 10,000 × 1 × 0.0001 = 1 (90% of light absorbed).

Linearity Limits

Beer-Lambert is only linear for DILUTE solutions (typically A < 2). At higher concentrations:

• Molecules interact with each other, shifting the effective ε.
• Scattering and other non-absorbing processes become significant.
• The detector saturates (cannot measure A > ~3 reliably).

For accurate measurements, dilute the sample to give A between 0.1 and 1 (well within the linear range).

Calculation Examples

Example 1: A protein sample absorbs 0.5 at 280 nm in a 1 cm cuvette. The protein’s extinction coefficient is 50,000 M⁻¹ cm⁻¹ at 280 nm. What is the concentration?

c = A / (ε b) = 0.5 / (50,000 × 1) = 10⁻⁵ M = 10 μM.

Example 2: NADH has ε = 6220 M⁻¹ cm⁻¹ at 340 nm. A reaction starts with 0.100 mM NADH. The initial absorbance at 340 nm is:

A = 6220 × 1 × 0.000100 = 0.622.

As NADH is consumed by an enzyme, the absorbance decreases. If A drops to 0.311 at time t, the remaining NADH is:

c = 0.311 / 6220 = 5.0 × 10⁻⁵ M = 0.050 mM (half the starting value).

Applications

• Protein quantification: measure A at 280 nm, use typical ε for protein (around 1 mg/mL per 1 A for generic protein, or specific ε for known protein).
• DNA quantification: measure A at 260 nm. 1 A = 50 μg/mL of double-stranded DNA.
• Enzyme kinetics: monitor the disappearance of a substrate or appearance of a product at a specific wavelength; convert ΔA/Δt to reaction rate.
• Standard curve method: prepare known concentrations, measure A, build a linear fit, then interpolate unknown concentrations.

Calculator

Slide the molar absorptivity, path length, and concentration; the cuvette color and the A vs c chart update instantly so you can see how each variable changes the measured absorbance and percent transmittance.

Practical Considerations

• Match the wavelength: measure at the λmax of the chromophore for maximum sensitivity.
• Zero the baseline: use a blank (buffer only, no sample) to account for any background absorbance from the solvent.
• Keep path length constant: most cuvettes are 1 cm, but other path lengths exist (0.5 cm for small volumes, 10 cm for very dilute samples).