The Beer–Lambert Law

Introduction

If you double the concentration of a coloured solution, it looks darker. If you look through a longer tube of the same solution, it also looks darker. The Beer–Lambert law turns these everyday observations into a precise equation linking absorbance to the amount of absorbing substance in the light's path. It is one of the most widely used relationships in analytical chemistry, underpinning measurements in hospitals, water laboratories and research labs worldwide.

Core explanation

The law. For monochromatic light passing through a dilute solution:

A = εcl

where - A is the absorbance (no units), defined as log₁₀(I₀/I); - ε (epsilon) is the molar absorption coefficient , in dm³ mol⁻¹ cm⁻¹; - c is the concentration of the absorbing species, in mol dm⁻³; - l is the path length, in cm.

The units cancel: (dm³ mol⁻¹ cm⁻¹) × (mol dm⁻³) × (cm) leaves no units, as required for absorbance.

Two proportionalities in one. Lambert's law states that absorbance is proportional to path length; Beer's law states that it is proportional to concentration. Both follow from the same idea: absorbance depends on the number of absorbing molecules the light meets . Doubling the concentration puts twice as many molecules in the beam; doubling the path length also puts twice as many in the beam. Either way the absorbance doubles.

Why the relationship is linear in A. Each thin slice of solution absorbs the same fraction of the light reaching it. Adding up these fractional losses over the whole sample gives an exponential fall in intensity, I = I₀ × 10^(−εcl). Taking logarithms converts this into the straight-line form A = εcl. This is exactly why absorbance, rather than transmittance, was defined as a logarithm.

The meaning of ε. ε is a property of the substance at a particular wavelength (and in a particular solvent). It measures how effectively one mole of the substance absorbs: a large ε means strong absorption. For a fixed substance and wavelength, ε is constant, so A is directly proportional to c when l is fixed.

The standard cuvette. Most measurements use cuvettes with a path length of exactly 1.00 cm. Glass or plastic cuvettes are suitable for visible light, but quartz (fused silica) is needed below about 340 nm because glass absorbs ultraviolet.

Conditions for the law. The law holds best when: - the light is monochromatic (a single wavelength, usually λmax); - the solution is dilute (typically below about 0.01 mol dm⁻³); - the absorbing species does not react, associate or dissociate as concentration changes; - the solution is clear, not cloudy.

Deviations from these conditions are discussed separately, but they explain why real calibration lines sometimes curve.

Mixtures. Absorbances are additive. If two substances both absorb at the chosen wavelength, the total absorbance is A = (ε₁c₁ + ε₂c₂)l.

Formulae

A = εcl; rearranged, c = A/(εl), l = A/(εc), ε = A/(cl). Units: A none, ε dm³ mol⁻¹ cm⁻¹, c mol dm⁻³, l cm.

Step-by-step reasoning

To predict a new absorbance when conditions change:

1. Write A = εcl and note which quantities stay fixed. 2. Identify the factor by which c or l changes. 3. Multiply the original absorbance by that factor (or by the product of both factors). 4. Check that the answer lies in a sensible range, ideally below about 1.

Visual explanation

Picture a beam of light passing through a row of identical tinted panes. Each pane dims the beam by the same fraction. Adding panes is like increasing path length; making each pane more heavily tinted is like increasing concentration. A graph of absorbance against the number of panes is a straight line through the origin.

Real-world analogy

Walking through a crowd is similar. The more densely packed the crowd (concentration) and the further you must walk (path length), the more people you bump into. The number of collisions is proportional to both, just as absorbance is.

Real-world example

Hospital analysers measure blood glucose by reacting glucose with enzymes to form a coloured product. The absorbance of the product at a fixed wavelength is proportional to its concentration, so the instrument can calculate the glucose level in a patient's sample within seconds.

Why?

Why must the light be monochromatic? ε changes with wavelength. If several wavelengths pass through at once, each is absorbed to a different extent and the combined reading no longer follows a single straight line with concentration.

Common misconception

"The Beer–Lambert law applies to transmittance." Transmittance decreases exponentially with concentration; only absorbance is directly proportional to concentration and path length.

Worked example

Question: A solution in a 1.00 cm cuvette has A = 0.36. What is the absorbance if the concentration is halved and a 2.00 cm cuvette is used?

Reasoning: Halving c multiplies A by 0.5; doubling l multiplies A by 2. The overall factor is 0.5 × 2 = 1.

Answer: A remains 0.36.

Quick check

1. If the concentration of a solution is tripled while the path length stays the same, what happens to the absorbance? Answer: It triples, because absorbance is directly proportional to concentration at constant path length.

Exam focus

Learn A = εcl with the units of every term, and state the conditions (dilute solution, monochromatic light). Examiners often ask you to show that the units of ε cancel with those of c and l, or to predict how absorbance changes when c and l are altered together.

Advanced insight

The law can be derived from probability. The chance of a photon being absorbed in a thin layer dx is proportional to the number of molecules per unit area, so dI/I = −k c dx. Integrating from 0 to l gives ln(I₀/I) = kcl, and converting to base-10 logarithms gives A = εcl with ε = k/2.303.

Summary

The Beer–Lambert law, A = εcl, states that absorbance is directly proportional to concentration and path length. ε is the molar absorption coefficient, a constant for a given substance, wavelength and solvent, with units dm³ mol⁻¹ cm⁻¹. The law arises because each layer absorbs the same fraction of the light. It holds for dilute solutions and monochromatic light, and absorbances of mixtures are additive.

Practice questions

1. State the Beer–Lambert law and give the units of each term. Answer: A = εcl; A has no units, ε is in dm³ mol⁻¹ cm⁻¹, c in mol dm⁻³ and l in cm. 2. Explain why doubling the path length doubles the absorbance. Answer: The light meets twice as many absorbing molecules, so twice the absorbance is produced. 3. Why must quartz cuvettes be used for measurements at 260 nm? Answer: Ordinary glass and most plastics absorb ultraviolet light below about 340 nm, whereas quartz is transparent there. 4. Two dyes both absorb at 500 nm. Write an expression for the total absorbance of their mixture. Answer: A = (ε₁c₁ + ε₂c₂)l, because absorbances of independent species add together.