Law of Beer-Lambert

Statement

That is to say an electromagnetic radiation of Wavelength λ (p. e.g. light), crossing a transparent medium. The intensity of this radiation undergoes an exponential reduction according to the distance covered and in the density of the species absorbing in this medium.

$I (\ lambda, X) = I_0 (\ lambda) \ cdot e^ {- \ alpha X R}.$

I is the intensity of the incidental light.
I is the intensity of the outgoing light.
α is the Coefficient absorption (in m²·mol^-1 or in m³·mol^-1·cm^-1).
X is the length of the optical way (in m or cm).
R is the density of the absorbing entity, atoms or molecules, in the medium (in mol·m^-3).

The value of the absorption coefficient α varies between various materials and also with the wavelength for a particular material. It is defined by the equation:

$\ alpha = \ frac {2 \ Omega K} {C} = \ frac {4 \ pi K} {\ lambda}.$

ω is the angular pulsation electromagnetic radiation.
K is the linear Coefficient of extinction, it expresses the attenuation of the energy of the electromagnetic radiation through the medium.
C is the celerity electromagnetic radiation in the vacuum.

Derivation

The absorption of a Monochromatic beam of light in a Homogeneous medium and Isotrope is proportional to the length of the optical way followed by this radiation and the concentration, in solution, or the partial pressure, in gas phase, of the absorbing species.

The law of Beer-Lambert can be expressed as follows:

I = I_0 \ cdot e^ {- \ alpha \ ell C}.

Or:

A_ \ lambda = - \ log_ {10} \ frac {I} {I_0} = \ varepsilon_ \ lambda \ cdot \ ell \ cdot C.

$I/I_0$ is the Transmittance of the solution (without unit).
has is the Absorbance or optical Densité with a wavelength λ (without unit).
ε is the molar coefficient of extinction (in L·mol − 1 ·cm − 1 ). It depends on the wavelength, the chemical nature of the entity and the temperature.

\ varepsilon = \ frac {\ alpha} {2,303}.

ℓ is the length of the optical way in the crossed solution, it corresponds to the thickness of the tank used (in cm).
C is the molar concentration of the solution (in mol. L − 1 ). In the case of a gas, C can be expressed like a density (reciprocal units of length to the cube, cm − 3 ).

This equation is very useful for analytical chemistry. Indeed, if ℓ and ε are known, the concentration of a substance can be deduced from the quantity of light transmitted by it.

Additivity

With a given wavelength λ , the absorptance has of a mixture of N absorbing species is the sum of the individual absorptances:

$A = \ sum_ {i=1} ^n A_i (\ varepsilon_ {\ lambda, I}, \ ell, C_i).$

; Example of application That is to say a solution containing a mixture of Nor (II) and of Co (II), one seeks to determine their respective concentrations C and C by applying the law of Beer-Lambert.

For this, one measures the absorptance of the solution with two longeurs of different waves λ = 393 Nm and λ = 510 Nm which correspond repectivement to the maximum absorptances of the two entities in solution taken each one separately. The following equations then are established:

$A_a = \ varepsilon_ {has, Neither} \ cdot C_ {Nor} + \ varepsilon_ {has, Co} \ cdot C_ {Co} \; \; \; (\ lambda_a; \ ell=1).$

A_b = \ varepsilon_ {B, Neither} \ cdot C_ {Nor} + \ varepsilon_ {B, Co} \ cdot C_ {Co} \; \; \; (\ lambda_b; \ ell=1).

Knowing has and ε in each case, one can determine the relative concentrations of each species by simple resolution of this Système of equations.

N.B. : contrary to the absorptances, transmittances of several entities are not additive sizes.

Law of Beer-Lambert in the atmosphere

The law of Beer-Lambert can be applied to describe the attenuation of the solar Rayonnement through the atmosphere. In this case, part of this light is diffused whereas another is absorbed by the various components. The law is expressed as follows:

I = I_0 \, \ exp (- m (\ tau_a+ \ tau_g+ \ tau_ {NO2} + \ tau_w+ \ tau_ {O3} + \ tau_r)).

In this equation,

\ tau_x

is the coefficient of transparency of a component X of the atmosphere, which can be:

has refers to the Aérosol S which absorb and diffuse the light,
G is a uniform mixture of Gaz (mainly, the Carbon dioxide (CO) and the Dioxygène (O). They are only absorbents, respectively, of the Ultraviolet S and of the Infrarouge S),
NO is the Dioxide nitrogen (coming mainly from the urban pollution, it only absorbs),
W is absorption due to the Steam,
O is the Ozone (it absorbs only part of the ultra-violets),
R is the Diffusion Rayleigh due to oxygen (O) and the Azote (NR) (this last is at the origin of the blue color of the sky),
m is the optical factor of the mass of the air, equal to 1/cos θ where θ is the angle which form incidental solar rays with the zenith.

This equation is used to determine the optical Depth aerosols, in order to study their effects on the climate, or to correct the abbérations in the images obtained by the satellites.

Amusing checking

If you make infuse a sachet of in a pan containing an ebullient water bottom and withdraw it then, you will have the surprise to see that the addition of an unspecified quantity of pure water then does not deteriorate of anything coloring liquid, as against-intuitive as that appears. The reason is that the parameter of concentration decreases exactly by the same value that increases that from thickness to be crossed.
One second reflection makes the thing obvious, owing to the fact that the quantity of pigment crossed by the light remains necessarily the same one because of the Parallélisme of the walls of the pan.

History

This law was discovered by Pierre Bouguer in 1729 and was published in its work “Test of Optics on the Gradation of the Light” (Claude Jombert, Paris, 1729), then taken again by Johann Heinrich Lambert in 1760 and finally August Beer in 1852 introduced the concentration there, giving him the form in which it is generally used.

References

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