Beer–Lambert law

The Beer–Lambert law, also known as Beer’s law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law, is a fundamental relation in optics and analytical chemistry that describes the exponential attenuation of a beam of monochromatic electromagnetic radiation as it passes through a homogeneous absorbing medium. In its most widely used decadic form, the law states that the absorbance \(A\) of a sample is directly proportional to the concentration \(c\) of the absorbing species and to the path length \(l\) of the radiation through the sample:

\[
A = \varepsilon c l
\]

where \(\varepsilon\) is the molar attenuation coefficient. The law combines Lambert’s law, which states that absorbance is proportional to path length, and Beer’s law, which states that absorbance is proportional to concentration. It is the quantitative basis of absorption spectroscopy and is used extensively in chemistry, biochemistry, environmental science, atmospheric physics, and medical diagnostics.

Definition and statement

When a beam of light of intensity \(I_0\) enters an absorbing medium, and the transmitted intensity after a path length \(l\) is \(I\), the transmittance \(T\) is defined as

\[
T = \frac{I}{I_0}.
\]

The absorbance \(A\), also called optical density in older literature, is defined in terms of base-10 logarithms:

\[
A = -\log_{10} T = \log_{10}\left(\frac{I_0}{I}\right).
\]

The Beer–Lambert law states that for a given wavelength and under appropriate conditions,

\[
A = \varepsilon c l.
\]

Here \(c\) is usually expressed in moles per cubic decimetre (mol dm⁻³, equivalent to molarity), \(l\) in centimetres, and \(\varepsilon\) therefore has units of dm³ mol⁻¹ cm⁻¹, commonly written as M⁻¹ cm⁻¹. Because the absorbance is defined through a logarithm, it is dimensionless. If \(A = 1\), only 10% of the incident light is transmitted; if \(A = 2\), 1% is transmitted; and so on.

For a solution containing several non-interacting absorbing species, the total absorbance at a given wavelength is additive:

\[
A_{\text{total}} = l \sum_i \varepsilon_i c_i.
\]

This additivity is the basis for multicomponent spectrophotometric analysis.

Mathematical formulation

The law may be expressed in equivalent forms. In decadic or base-10 form,

\[
I = I_0 \, 10^{-\varepsilon c l},
\]

and therefore

\[
A = \varepsilon c l.
\]

In natural-logarithm form, often used in physics and atmospheric science,

\[
I = I_0 \exp(-\alpha l),
\]

where \(\alpha\) is the absorption coefficient. The two forms are related by

\[
\alpha = \varepsilon c \ln 10.
\]

If the absorbing species is described by a number density \(N\) and a molecular absorption cross section \(\sigma\), the same law is written as

\[
I = I_0 \exp(-\sigma N l).
\]

The dimensionless quantity \(\tau = \sigma N l = \alpha l\) is called the optical depth. The decadic absorbance and optical depth are related by

\[
A = \frac{\tau}{\ln 10}.
\]

Derivation

The Beer–Lambert law follows from the assumption that the fractional loss of intensity in an infinitesimally thin layer of the medium is proportional to the intensity itself and to the amount of absorbing material in that layer.

Consider a beam of intensity \(I(z)\) travelling through a homogeneous medium. Across a small thickness \(dz\), the decrease in intensity is

\[
dI = -\alpha I \, dz,
\]

where \(\alpha\) is the absorption coefficient. Integration from \(z=0\) to \(z=l\) gives

\[
\int_{I_0}^{I(l)} \frac{dI}{I} = -\alpha \int_0^l dz,
\]

so that

\[
\ln\left(\frac{I(l)}{I_0}\right) = -\alpha l,
\]

or

\[
I(l) = I_0 \exp(-\alpha l).
\]

For a solution in which the absorbing centres are independent of one another, the absorption coefficient is proportional to concentration:

\[
\alpha = k c.
\]

Thus

\[
I = I_0 \exp(-k c l).
\]

Converting to decadic logarithms gives

\[
\log_{10}\left(\frac{I_0}{I}\right) = \frac{k c l}{\ln 10}.
\]

Defining the molar attenuation coefficient as \(\varepsilon = k / \ln 10\) yields the usual statement of the law,

\[
A = \varepsilon c l.
\]

History and nomenclature

The historical development of the law is usually attributed to three scientists. Pierre Bouguer described the exponential attenuation of light with distance in a medium in 1729 in his Essai d’optique sur la gradation de la lumière. Johann Heinrich Lambert in 1760, in Photometria, formulated the dependence of attenuation on path length more explicitly. August Beer, in 1852, extended the relation to solutions by showing that the attenuation depends on the concentration of the absorbing solute.

Because of these contributions, the law is sometimes called the Beer–Lambert–Bouguer law. In many analytical chemistry contexts it is simply called Beer’s law, although strictly Beer’s law refers only to the concentration dependence, while Lambert’s law refers to the path-length dependence. The older term “optical density” for absorbance is now discouraged, and the term “molar extinction coefficient” for \(\varepsilon\) is still encountered but is considered less precise than “molar attenuation coefficient” or “molar absorption coefficient”.

Validity and assumptions

The Beer–Lambert law is an idealisation that holds exactly only under several conditions:

• the incident radiation is monochromatic;
• the beam is collimated and perpendicular to the sample;
• the medium is homogeneous and non-scattering;
• the absorbing species do not interact chemically or physically;
• the concentration is sufficiently low that each absorbing centre behaves independently;
• the incident light does not cause saturation, nonlinear optical effects, or significant photochemical change;
• fluorescence or phosphorescence from the sample does not reach the detector.

In practice, absorbance measurements are usually made against a blank to subtract reflection, solvent absorption, and cell losses. The law is most reliable for absorbance values roughly between 0.1 and 1, although modern spectrophotometers can often measure accurately up to 2 or higher.

Deviations from the law

Deviations from the Beer–Lambert law are commonly classified as real, chemical, or instrumental.

Real deviations

Real deviations arise when the fundamental assumptions of the law are violated. At high concentrations, the molar attenuation coefficient may change because chromophores are close enough to interact electronically or electrostatically, altering their absorption properties. The refractive index of the solution can also change with concentration, which affects the local light intensity and hence the measured absorbance. For this reason the law is strictly valid in the dilute limit.

Chemical deviations

Chemical deviations occur when the absorbing species participates in a chemical equilibrium, such as acid–base dissociation, dimerization, complexation, or solvation changes. In such cases the concentration of the actual absorbing form is not simply proportional to the total analytical concentration. For example, a pH-dependent indicator may exhibit strong absorbance changes even when the total amount of indicator is fixed, because the acid and base forms have different spectra. These effects do not represent a failure of the law itself but rather a failure to use the correct concentration of the absorbing species.

Instrumental deviations

Instrumental deviations arise from the measurement system rather than the sample.

• Polychromatic radiation: Real spectrophotometers use a finite spectral bandwidth. If \(\varepsilon\) varies appreciably across that bandwidth, the measured transmittance is an average of transmittances over different wavelengths. Because transmittance is an exponential function of \(\varepsilon\), the average of exponentials is not the exponential of the average, producing a nonlinear calibration curve.
• Stray light: Light that reaches the detector without passing through the sample, or scattered light within the monochromator, adds to the measured intensity and causes the absorbance to plateau at high values. The measured absorbance may also be lower than the true absorbance.
• Non-parallel light: If the beam is not perpendicular to the sample or passes through different optical paths, the effective path length is uncertain.
• Cell mismatch or contamination: Imperfections in cuvettes, fingerprints, or refractive boundaries can introduce errors.

Applications

The Beer–Lambert law is the basis of quantitative absorption spectroscopy. Some common applications include:

• Analytical chemistry: Determination of analyte concentration by measuring absorbance at a suitable wavelength and using a calibration curve or known molar attenuation coefficient.
• Biochemistry and molecular biology: Quantification of proteins and nucleic acids; for example, protein absorbance at 280 nm and nucleic acid absorbance at 260 nm are used to estimate concentration and purity.
• Clinical and pharmaceutical analysis: Measurement of drug concentrations, metabolites, and enzymatic reaction rates.
• Environmental monitoring: Measurement of pollutants such as nitrogen dioxide, ozone, and organic compounds in air and water.
• Atmospheric science: The law, usually in its natural-logarithm or optical-depth form, is used to calculate atmospheric transmittance, aerosol optical depth, and trace gas column amounts from remote sensing.
• Colourimetry and spectrophotometric pH measurement: Quantitative analysis using coloured complexes or indicator dyes.
• Pulse oximetry: In medical devices, modified forms of the Beer–Lambert law are used to estimate oxygen saturation of blood from absorption at multiple wavelengths.

Related quantities

Several related quantities are commonly used with the Beer–Lambert law:

• Transmittance \(T = I/I_0\).
• Absorbance \(A = -\log_{10} T\).
• Molar attenuation coefficient \(\varepsilon\), characteristic of a substance at a given wavelength.
• Absorption coefficient \(\alpha\), defined with respect to natural logarithms.
• Absorption cross section \(\sigma\), often used for gases and individual molecules.
• Optical depth \(\tau = \alpha l\), commonly used in atmospheric physics and radiative transfer.

For a substance with a decadic molar attenuation coefficient \(\varepsilon\) expressed in M⁻¹ cm⁻¹, the corresponding molecular absorption cross section \(\sigma\) in cm² is given by

\[
\sigma = \frac{1000 \ln 10}{N_A} \varepsilon,
\]

where \(N_A\) is the Avogadro constant. This relation is useful for converting between solution-phase and gas-phase absorption data.

The Beer–Lambert law is a limiting law. Although its simple linear form may break down under extreme conditions, it remains one of the most important quantitative relationships in the experimental sciences and a cornerstone of spectroscopic analysis.