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Beer-Lambert Law

History

The beer –lambert law is also known as Beer’s law, the Lambert-Beer law or the Beer-Lambert-Bouguer Law. The reason for so many names is because more than one law is involved in it.  In 1729 Pierre Bouger discovered the law and published it in Essai d’optique sur la gradation de la lumiere. In 1760 Lambert quoted the Bouger’s discovery in his Photometria which states that the absorbance of a sample is directly proportional to the path length of light. Lambert did not claim any discovery, but he was often credited with it. In 1852, August Beer discovered that absorbance is proportional to the sample concentration. Generally, beers law relates only to concentration while Beer-Lambert law relates absorbance to both concentration and thickness of a sample.

Definition

Beer Lamberts Law states a relationship between the attenuation of light through a substance and the properties of that substance. It is defined as:

“The path length and concentration of a chemical are directly proportional to its absorption of light.”

The premise is that as a beam of light is passed through a chemical solution that beam of light becomes weaker. The attenuation of light occurs either as a result of distance through a solution or by increasing concentration.

The equation for Beer’s Law

Beers law may be written as:

A= εbc

Where

A= absorbance (no units)

ε = A Greek Letter Epsilon, the molar absorption coefficient with units of

       Mol-1 cm-1

b = the path length of a sample usually expressed in cm

c= the concentration of the compound in solution, expressed in M

while calculating the absorbance of a sample by using this equation, it depends on two assumptions:

  1. the absorbance is directly proportional to the path length of the sample.
  2. The absorbance is directly proportional to the concentration of the sample.

How to use Beer’s Law?

Many modern instruments perform beers law calculations by comparing a blank cuvette with a sample. A graph can easily be prepared by using standard solutions to determine the concentration of a specimen. The graphing method assumes a straight-line relationship between concentration and absorbance, which is valid for dilute solutions.

Beer-Lambert Law 1

Figure 1.1

In above Figure 1.1 graph shows that the concentration is directly proportional to absorbance. Let the concentration =C and the absorbance =A. As C increases, A increases proportionally. Hence

A α C

Beer-Lambert Law 2

Figure 1.2

This second Figure 1.2 graph tells us that as the path length is allowed to travel through the width of the cuvette, increases, the absorbance increases proportionally. The path line is represented by a symbol d.

A α d

Beer-Lambert Law 3

Figure 1.3

Some molecules that absorb a lot of light even at low concentrations are very brightly colored while others are not so good at absorbing light. The molar absorbance coefficient €, which is known as Epsilon is a characteristic for each type of molecule.

A= €Cd

This is known as Beer-Lambert Law.

A steep slope is achieved if € has a large value and it reflects a strong absorbance.

In figure 1.3, A is plotted on the y-axis and C on the x-axis, and the slope achieved is €d =A/C.

Importance of Beers law:

Beers law plays an important role in the fields of chemistry, physics, and meteorology. In chemistry Beers law is used to measure the concentration of chemical solutions, oxidation analysis and to measure the degradation of the polymer. Beer’s law also describes the attenuation of radiation through the Earth’s atmosphere. On the other hand, if applied to light, this law helps scientists to understand the attenuation of particle beams, such as neutrons. In theoretical Physics, the beer lambert law is a solution to the Bhatnagar-Gross-Krook (BKG) operator, which is used in the Boltzmann equation for computational fluid dynamics.

Beer’s Law Example Calculation:

A sample having a maximum absorbance value of 275nm. It has a molar absorptivity of 8400M-1cm-1. 1cm is the width of a cuvette. A spectrophotometer value detected A=0.70. calculate the concentration of the sample.

By using Beer’s law, we will calculate the concentration of the sample.

                        A= εbc

                        0.70 = (8400 M-1cm-1) (1cm) (c)

By dividing both sides of the equation by [(8400 M-1cm-1) (1cm)]

                        c= 8.33 x 10-5 mol/L

Transmittance and absorbance of light using Beers law:

Beer-Lambert Law 4

                                                            Figure 1.4

The transmittance and absorbance of light by a substance was first introduced followed by an explanation of the Beer-Lambert Law.

Consider a solution and a monochromatic light is allowed to pass through it; having an incident intensity of I0 and a transmitted intensity of I (Figure 1.4)

The transmittance, T, of the solution is defined as the ratio of the transmitted intensity, I, divided by the incident intensity, Io.

T=I/Io

It takes the values between 0 and 1. Generally, it is commonly expressed as a percentage transmittance:

T (%)= 100 I/Io

The absorbance A, of the solution, is related to the transmittance and incident and transmitted intensities by the following relations:

A = log10 Io/I

A = -log10 T

There is a logarithmic relationship between the absorbance to the transmittance. The 0 absorbance corresponds to a transmittance of 100% while an absorbance of 1 corresponds to a transmittance of 10%. The table given below tells us the values of transmittance and absorbance.

Absorbance Transmittance
0 100%
1 10%
2 1%
3 0.1%
4 0.01%
5 0.001%
Beer-Lambert Law 5

Attenuation of 510nm laser through three solutions of Rhodamine 6G having different absorbance values at 510nm. A bright yellow glow is the fluorescence emission at ~560nm.

Limitation and Deviation of Beer-Lambert Law:

Beer-Lambert law is unable to maintain a linear relationship between attenuation and the concentration of an analyte. These deviations have been classified into three categories:

  • Real: this deviation is due to the limitation of the law itself.
  • Chemical: this deviation is observed due to specific chemical species of the sample being analyzed.
  • Instrument: this deviation occurs due to how the attenuation measurements are made.

Beer law and lambert law are only able to describe the absorption behavior of the solutions that contain relatively low amounts of solutes dissolved in it i.e. <10mm. the analyte starts behaving differently when the concentration of the analyte in the solution is high i.e. >10mm. this is due to the interaction of the analyte with the solvent and other solute molecules and sometimes even due to hydrogen bonding interactions.

  1. At higher concentrations, solute molecules can cause different charge distributions on their neighboring species in the solution. If the concentration of the solution is more than 10-2M (0.01M) in that case the achieved relation is not linear while plotting a graph between Absorbance V/s concentration. This means that Beer-Lambert’s law deviates.
Beer-Lambert Law 6
  • The molar extinction coefficient ε depends on the refractive index of the absorbing medium. If the initial concentration of the absorbing medium is greater than 10-2M then refractive index changes and hence the ε changes. The refractive index does not change if the concentration is less than 0.01M and hence the ε does not change and the law holds true.
  • The Beer-Lambert law also fails, if the absorbing species react with the solvent i.e. either association or dissociation takes place. For example, the dichromate ions react with water producing chromate ions which are yellow in color.
Beer-Lambert Law 6
  • The law also deviates if non-monochromatic light is used.
  • The change in temperature also leads to the deviation of Beer-lamberts’ law.
  • The deviation may also occur if the width of the instrument is not proper.
  • In instruments where filters are used also leads towards deviation because we do not get monochromatic light through filters.

The Beer-Lambert law is also not compatible with Maxwell’s equation. It only describes the propagation within the medium but doesn’t tell us the transmittance through a medium. It can be able to be compatible with Maxwell’s equation if the transmittance of a sample with a solute is radioed against the transmittance of the pure solvent that describes why it works well in spectrophotometry.

Presently, the Beer lambert law is declared as a limiting law because the absorbance is only nearly linear depending on the concentration. This is the reason that the attenuation coefficient also depends on concentration and density even if there are no interactions. However, these changes are negligible except for high concentrations and large oscillator strength.

Beer-Lambert Law in the atmosphere:

The attenuation of solar or stellar radiation is also described with the help of this law as it travels through the atmosphere. In this case, there is a scattering of radiation as well as absorption. The beer-lambert law for the atmosphere is written as:

Beer-Lambert Law 8

the optical depth for a slant path is τ′ = mτ, where T refers to a vertical path, m represents the relative air mass and a plane parallel atmosphere is given as m=sec θ where θ represents the zenith angle corresponds to the given path.

 Each Tx describes the optical depth where the subscripts tell us the source of scattering or absorption

  • a represents the aerosols that absorb or scatter
  • g represents the uniform mixed gases i.e. carbon dioxide CO2 and molecular oxygen O2 that only absorb
  • NO2 is the nitrogen dioxide due to urban pollution (absorption only)
  • RS tells us the effects due to Raman scattering in the atmosphere
  • W refers to the absorption of water vapor
  • O3 refers to ozone (absorption only)
  • r is known as Rayleigh scattering from molecular oxygen and nitrogen which are responsible for the blue color of the sky.
  • The attenuators selection which needed to be considered depends on the wavelength range and can include many other compounds i.e. HONO, formaldehyde, tetra oxygen, glyoxal, and many other halogen radicals

m refers to optical mass or air mass factor whereas θ represents the observed object’s zenith angle. This equation is used for the correction of satellite images by Ta, the aerosol optical thickness and is also used in predicting the role of aerosols in climate.

References:

https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Spectroscopy/Electronic_Spectroscopy/Electronic_Spectroscopy_Basics/The_Beer-Lambert_Law

https://www.edinst.com/blog/the-beer-lambert-law/

http://www.mathcentre.ac.uk/resources/uploaded/3b2printableversion.pdf

https://www.chemguide.co.uk/analysis/uvvisible/beerlambert.html

https://www.edinst.com/blog/the-beer-lambert-law/

https://www.thoughtco.com/beers-law-definition-and-equation-608172

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