Beer–Lambert law

In optics, the Beer-Lambert law, also known as Beer's law or the Lambert-Beer law or the Beer-Lambert-Bouguer law is an empirical relationship that relates the absorption of light to the properties of the material through which the light is travelling.

Equations

File:Beer lambert.png

Diagram of Beer-Lambert absorption of a beam of light as it travels through a cuvette of size l.

There are several ways in which the law can be expressed:

$A=\alpha lc\,$

${I_{1} \over I_{0}}=10^{-\alpha lc}$

$A=-\log _{10}{\frac {I_{1}}{I_{0}}}$

$\alpha ={\frac {4\pi k}{\lambda }}$

Here:

• A is absorbance
• I₀ is the intensity of the incident light
• I₁ is the intensity after passing through the material
• l is the distance that the light travels through the material (the path length)
• c is the concentration of absorbing species in the material
• α is the absorption coefficient or the molar absorptivity of the absorber
• λ is the wavelength of the light
• k is the extinction coefficient

In essence, the law states that there is a logarithmic dependence between the transmission of light through a substance and the concentration of the substance, and also between the transmission and the length of material that the light travels through. Thus if l and α are known, the concentration of a substance can be deduced from the amount of light transmitted by it.

The units of c and α depend on the way that the concentration of the absorber is being expressed. If the material is a liquid, it is usual to express the absorber concentration c as a mole fraction i.e. a dimensionless fraction. The units of α are thus reciprocal length (e.g. cm^-1). In the case of a gas, c may be expressed as a density (units of reciprocal length cubed, e.g. cm^-3), in which case α is an absorption cross-section and has units of length squared (e.g. cm²). If concentration c is expressed in moles per unit volume, α is a molar absorptivity (usually given the symbol ε) in units of mol^-1 cm^-2 or sometimes L mol^-1 cm^-1.

The value of the absorption coefficient α varies between different absorbing materials and also with wavelength for a particular material. It is usually determined by experiment.

In spectroscopy and spectrophotometry, the law is almost always defined in terms of common logarithms and powers of 10 as above. In general optics, the law is often defined in an alternate exponential form:

${I_{1} \over I_{0}}=e^{-\alpha 'lc}$

$A'=\alpha 'lc=-\ln {\frac {I_{1}}{I_{0}}}$

The values of α' and A' are approximately 2.3 (≈ln 10) times larger than the corresponding values of α and A defined in terms of base-10 functions. Therefore, care must be taken when interpreting data that the correct form of the law is used.

The law tends to break down at very high concentrations, especially if the material is highly scattering. If the light is especially intense, nonlinear optical processes can also cause variances.

Derivation

Assume that particles may be described as having an area, $\alpha$, perpendicular to the path of light through a solution, such that a photon of light is absorbed if it strikes the particle, and is transmitted if it does not.

Define $z$ as an axis parallel to the direction that photons of light are moving, '$A$' and '$dz$' are the area and thickness (along the $z$ axis) of a 3-dimensional slab of space through which light is passing. We assume that $dz$ is sufficiently small that one particle in the slab cannot obscure another particle in the slab when viewed along the $z$ direction. The concentration of particles in the slab is represented by ‘$c$’.

It follows that the fraction of photons absorbed when passing through this slab is equal to the total opaque area of the particles in the slab, $\alpha Ac\,dz$, divided by the area of the slab, or $\alpha cdz$. Expressing the number of photons absorbed by the slab as $dI_{z}$, and the total number of photons incident on the slab as $I_{z}$, the fraction of photons absorbed by the slab is given by

${\frac {dI_{z}}{I_{z}}}=-\alpha c\,dz$

The solution to this simple differential equation is obtained by integrating both sides to obtain $I_{z}$ as a function of $z$

$\ln(I_{z})=-\alpha cz+C\,$

For a slab of real thickness, ‘$l$’, the difference in light intensity $I_{0}$ at z=0, and $I_{1}$ at z=l, is given by

$\ln(I_{0})-\ln(I_{l})=(-\alpha 0c+C)-(-\alpha lc+C)=\alpha lc\,$

or

$\mathrm {Absorbance} =-\ln \left({\frac {I_{1}}{I_{0}}}\right)=\alpha lc$

$\mathrm {Transmittance} ={\frac {I_{1}}{I_{0}}}=e^{-\alpha lc}$

Beer-Lambert law in the atmosphere

This law is also applied to describe the attenuation of solar radiation as it travels through the atmosphere. In this case, there is scattering of radiation as well as absorption. The Beer-Lambert law for the atmosphere is usually written

$I_{n}=(I_{o}/R^{2})\,\exp(-(k_{a}+k_{g}+k_{NO2}+k_{w}+k_{O3}+k_{r})m)$ ,

where each $k_{x}$ is an extinction coefficient whose subscript identifies the source of the absorption or scattering it describes:

• $a$ refers to aerosols (that absorb and scatter)
• $g$ are uniformly mixed gases (mainly carbon dioxide ($CO_{2}$) and molecular oxygen ($O_{2}$) which only absorb)
• $NO_{2}$ is nitrogen dioxide, mainly due to urban pollution (absorption only)
• $w$ is water vapour absorption
• $O_{3}$ is ozone (absorption only)
• $r$ is Rayleigh scattering from molecular oxygen ($O_{2}$) and nitrogen ($N_{2}$) (responsible for the blue color of the sky).

$m$ is the optical mass, a term basically equal to $1/\cos(\theta )$ where $\theta$ is the solar zenith angle (the solar angle with respect to a direction perpendicular to the Earth's surface at the observation site).

This equation can be used to retrieve $k_{a}$, the aerosol optical thickness, which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate.

History

The law was independently discovered (in various forms) by Pierre Bouguer in 1729, Johann Heinrich Lambert in 1760 and August Beer in 1852.

External links

• Beer-Lambert Law Calculator

See also

• Logarithm
• Scientific laws named after people