# Chemical Sciences: A Manual for CSIR-UGC National Eligibility Test for Lectureship and JRF/Beer–Lambert law

In optics, the **Beer-Lambert law**, also known as **Beer's 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[edit]

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

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle A=\alpha lc \,}**

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle {I_{1}\over I_{0}} = 10^{-\alpha l c}}**

Here:

- A is absorbance
*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- 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. cm^{2}). If concentration *c* is expressed in moles per unit volume, α is a molar absorptivity given 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:

The values of α' and *A'* are approximately 2.3 (≈ln 10) times smaller 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.

## Beer-Lambert law in the atmosphere[edit]

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

- ,

where each is an extinction coefficient whose subscript identifies the source of the absorption or scattering it describes:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): a**refers to aerosols (that absorb and scatter)**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): g**are uniformly mixed gases (mainly carbon dioxide () and molecular oxygen() which only absorb)- is nitrogen dioxide, mainly due to urban pollution (absorption only)
- is water vapour absorption
- is ozone (absorption only)
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): r**is Rayleigh scattering from molecular oxygen (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle O_2}**) and nitrogen () (responsible for the blue color of the sky).

is the *optical mass*, a term basically equal to where is the solar azimuth (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 , 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[edit]

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