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# 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 (in fact, most of the permutations of these three names appear somewhere in literature) is an empirical relationship that relates the absorption of light to the properties of the material through which the light is travelling.

## Equations

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

$A=\alpha \, l\,c \,$

$T = {I_{1}\over I_{0}} = 10^{-A} = 10^{-\alpha \, l \, c}$

where,

$A = \log_{10} \left( \frac{I_0}{I_1} \right)$
$\alpha = \frac{4 \pi k}{\lambda}$

Here,

• $A \,$ is the absorbance of the sample
• $I_0 \,$ is the intensity of the incident light
• $I_1 \,$ 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
• $\alpha \,$ is the absorption coefficient or the molar absorptivity of the absorber
• $\lambda \,$ is the wavelength of the light
• $k \,$ is the extinction coefficient
• $T \,$ is the transmittance

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 $\alpha \,$ are known, the concentration of a substance can be deduced from the amount of light transmitted by it.

The units of absorber concentration ($c \,$) and absorption coefficient ($\alpha \,$) 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 as a mole fraction i.e. a dimensionless fraction. The units of the absorption coefficient are thus reciprocal length (e.g. cm−1). If the concentration is expressed in moles per unit volume, $\alpha \,$ is a molar absorptivity (usually given the symbol ε) in units of mol−1 cm2 or sometimes L·mol−1·cm−1.

In the case of a gas, the concentration may be expressed as a number density (e.g. cm−3), in which case $\alpha \,$ is an absorption cross-section and has units of area (e.g. cm2).

The value of the absorption coefficient $\alpha \,$ varies between different absorbing materials and also with wavelength for a particular material. It is usually determined by experiments.

In spectroscopy and spectrophotometry, the law is almost always defined in terms of common logarithm. In optics, the law is often defined in an alternate exponential form,

${I_{1}\over I_{0}} = e^{-\alpha' l c} ,$
$A' = \alpha' l c = -\ln \frac{I_1}{I_0} .$

The values of $\alpha' \,$ and A' are ln 10 (approximately 2.3) times larger than the corresponding values of $\alpha \,$ and A defined in terms of common logarithm. Note though that when c is given as a number density and $\alpha' \,$ as an area it is often denoted by $\sigma \,$ and represents the "true" cross section of the absorber (as can be seen from the derivation below). Therefore, care must be taken when interpreting data that the correct form of the law is used.

In molecular absorption spectrometry, the absorption coefficient α' is expressed in terms of a linestrength, S, and an (area-normalized) lineshape function, Φ. The frequency scale in molecular spectroscopy is often in cm-1, wherefore the lineshape function is expressed in units of 1/cm-1, which can look funny but is strictly correct. Since c is given as a number density in units of 1/cm3, the linestrength is often given in units of cm2cm-1/molecule. A typical linestrength in one of the vibrational overtone bands of smaller molecules, e.g. around 1.5 μm in CO or CO2, is around 10-23 cm2cm-1, although it can be larger for species with strong transitions, e.g. C2H2. The linestrengths of various transitions can be found in large databases, e.g. HITRAN. The lineshape function often takes a value around a few 1/cm-1, up to around 10/cm-1 under low pressure conditions, when the transition is Doppler broadened, and below this under atmospheric pressure conditions, when the transition is collision broadened. It has also become commonplace to express the linestrength in units of cm-2/atm since then the concentration is given in terms of a pressure in units of atm. A typical linestrength is then often in the order of 10-3 cm-2/atm. Under these conditions, the detectability of a given technique is often quoted in terms of ppm•m.

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, α, 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, and A and dz as 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, αAc dz, divided by the area of the slab, or αc dz. Expressing the number of photons absorbed by the slab as dIz, and the total number of photons incident on the slab as Iz, 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 Iz as a function of z

$\ln(I_z) = - \alpha cz + C . \,\!$

For a slab of real thickness, ℓ, the difference in light intensity I0 at z = 0, and I1 at z = ℓ, is given by

$\ln(I_0) - \ln(I_\ell) = (- \alpha 0 c + C) - ( - \alpha \ell c+ C) = \alpha \ell c , \,\!$

or

$\mbox{Transmittance} = \frac{I_1}{I_0} = e ^ {- \alpha \ell c} .$
$\mbox{Absorbance} = - \log_{10}\left( \frac{I_1}{I_0} \right) = \epsilon\ell c ,\mbox{ where }\epsilon = \alpha / 2.303$

It is instructive to consider the consequences of error in an assumption that is implicit in this derivation, namely that every absorbing particle behaves independently with respect to the light. Error is introduced when particles interact by lying along the same optical path such that some particles are in the shadow of others. The assumption approaches accuracy only in very dilute solutions, and it becomes increasingly inaccurate with increasingly concentrated solutions or long optical paths.

In practice, the accuracy of the assumption is better than the accuracy of most spectroscopic measurements up to an absorbance of 1 (or : I1 / I0 = 0.1) and to a good approximation, measurements of absorbance in this range are linearly related to the concentration of absorbing substances in solution. At higher absorbances, concentrations will be underestimated due to this shadow effect unless one employs a nonlinear relationship between absorbance and concentration.

## Beer-Lambert law in the atmosphere

This law is also applied to describe the attenuation of solar or stellar 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

$I = I_0\,\exp(-m(\tau_a+\tau_g+\tau_{\rm NO_2}+\tau_w+\tau_{\rm O_3}+\tau_r)),$

where each τx is the optical depth 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 (CO2) and molecular oxygen (O2) which only absorb)
• NO2 is nitrogen dioxide, mainly due to urban pollution (absorption only)
• w is water vapour absorption
• O3 is ozone (absorption only)
• r is Rayleigh scattering from molecular oxygen (O2) and nitrogen (N2) (responsible for the blue color of the sky).

m is the optical mass or airmass factor, a term approximately equal (for small and moderate values of θ) to 1 / cos(θ), where θ is the observed object's zenith angle (the angle measured from the direction perpendicular to the Earth's surface at the observation site).

This equation can be used to retrieve τa, 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 discovered by Pierre Bouguer before 1729. It is often mis-attributed to Johann Heinrich Lambert, who cited Bouguer's “Essai d'Optique sur la Gradation de la Lumiere” (Claude Jombert, Paris, 1729) — and even quoted from it — in his “Photometria” in 1760. Much later, August Beer extended the exponential absorption law in 1852 to include the concentration of solutions in the absorption coefficient.