Beer-Lambert Law: Equation, Derivation & Applications

Beer-Lambert Law or Beer’s Law states that energy absorbed or transmitted by a solution is directly proportional to the molar absorptivity of the solution and the concentration of solute. The law is used in analytical chemistry to measure the absorbance of several samples. Beer-Lambert Law is a combination of two laws: Beer’s Law and Lambert’s law. Beer-lambert law describes the link between the attenuation or weakening of intensity of light through any substance and the properties of the medium through which it travels. 

• The application of beer lambert law ranges from pharmaceutics to organic chemistry analysis and measurements.
• The law explains the direct relationship between the path length and concentration of a sample under study with the absorbance of the incident light ray. 
• The lambert beer law was first stated by August Beer and is given by the formula: I = I_oe^-μ(x). 

Read More: Rectilinear Propagation of Light

Key Terms: Beer-Lambert Law, Spectrophotometer, Spectroscopy, Absorbance, Concentration, Absorption Coefficient, Transmittance, Optical Coefficient

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Beer-Lambert Law

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Beer-Lambert law states that- 

The absorbance of a solution is proportional to its concentration, molar absorption coefficient, and optical coefficient.

t is also known as the Beer-Lambert–Bouguer law, or Beer's law. Some important characteristics of the law are:

• Beer Lambert’s law states that when monochromatic light passes through a homogeneous medium, the intensity of the transmitted radiation decreases at a constant rate as the thickness of the medium increases.
• The concentration of the solution varies directly with the intensity of received radiation.
• Beer-Lambert Law states a linear relationship between the absorbance and concentration of the solution enables the calculation of the concentration of the solution using its absorbance.
• The absorption of a quantity of light by a material dissolved in a fully transmitting solvent is directly proportional to the concentration of the substance and the path of the light through the solution.

The more the number of molecules that absorb light of a particular wavelength, the higher will be the peak intensity in the absorption spectrum. If fewer molecules absorb this radiation, the total absorption of energy is lowered and we get a low-intensity peak

The Beer Lambert law was discovered in 1729 by Pierre Bouguer.

• In 1760, Johann Heinrich Lambert quoted Pierre Bouguer's discovery by stating that the absorbance of a given sample is proportional to the path length of light
• In 1852, August Beer discovered a corresponding law which stated that the absorbance is proportional to the concentration of a sample.

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Beer Lambert Law Equation

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The Beer Lambert law equation is given as:

I = I_oe^-μ(x)

Where,

• I = Intensity
• I_o = Initial Intensity
• μ = Coefficient of absorption 
• x = Depth (meter)

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Beer-Lambert Law Derivation

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When the radiation of light travels through a solution, the quantity of light transmitted or absorbed is an exponential function of the molecular concentration of the minor component or the solute and a function of the length of the path of radiation through the sample. Therefore, Beer Lambert law derivation can be derived by the following steps:

log I₀/I = εcl

• Where I₀ = Intensity of the incident light 
• I = Intensity of light transmitted through the sample solution
• c = concentration of the solute in mol l^-1
• L = path length of the sample in cm
• ε = molar absorptivity constant

The ratio I/I₀ is known as transmittance T and absorbance is the logarithm of the inverse ratio I₀/I.

- Log I /I₀ = - log T = ε c L

Log I₀/I = A = ε c L

Therefore, the Beer-Lambert law equation is A = εLc

Transmittance and Absorbance

The relationship between the amount of light transmitted to the detector after passing through the sample (L) and the initial amount of light, is known as transmittance.

ε = \({A \over Lc}\)

Therefore, A = log₁₀\({I_o \over I_T}\)

The concentration can be estimated if you know the absorption coefficient for a certain wavelength and the thickness of the path length for light transmitted through the solution by Beer-Lambert Law.

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Applications of Beer-Lambert Law

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Beer-Lambert Law finds application in the field of analytical chemistry which deals with the separation, measurement, and identification of matter. This is mostly done by the use of spectrophotometry. The applications of beer lambert law are listed below:

• Spectroscopy: Beer Lambert law is the basic governing principle behind Beer – Lambert Law.
• Blood Analysis: Pre-processing of material is not required instead a spectrophotometer can be used to determine the bilirubin count in a blood sample.
• Atmospheric Phenomena: Beer Lambert’s law is used to describe solar or stellar radiation in the atmosphere.
• Qualitative and quantitative examination: Biological and dosimetric materials that may contain organic or inorganic elements can be analysed.
• Measuring Concentration: By analyzing the absorption spectra of various chemicals in cell structures, Beer-Lambert Law can help determine the concentration.

Beer-Lambert Law Graph

Beer Lambert Law applications are also there in our atmosphere. For instance, solar radiation in the atmosphere can be described using the Beer Lambert law. The equation of the Beer Lambert law in atmospheric applications is as given:

\(\begin{array}{l}\large T=e^{-m(T_{a}+T_{g}+T_{RS}+T_{NO2}+T_{w}+T_{O3}+T_{r}+..)}\end{array}\)
Where,

• a = aerosols
• g = mixed gases
• RS = Raman scattering effect.
• NO2 = Nitrogen dioxide
• w = water vapour absorption
• O3 = Ozone
• r = Rayleigh scattering

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Limitations of Beer-Lambert law

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The limitation of the Beer-Lambert law is based on the few situations where the law maintains linearity:

• As the molecules of the analyte have stronger intermolecular and electrostatic interactions due to the smaller amount of space between them, the law will produce false results at high concentrations.
• This can alter the analyte's molar absorptivity.
• High concentrations alter not just molar absorptivity but also the refractive index of the solution, resulting in deviations from the Beer-Lambert law.

For instance, when the concentration is low, i.e. < 10mM, it becomes simpler to analyze the absorptivity coefficient of the sample using this rule, but as the concentration rises, i.e. >10mM, there is a deviation due to the increase of the electrostatic interactions.

Also Read: Refraction at Spherical Surfaces & Lenses

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Things to Remember

• Beer-Lambert law states that the absorbance of a solution is proportional to its concentration, absorption coefficient, molar, and optical coefficient.
• Beer law states that concentration and absorbance are exactly proportional to one other.
• Lambert's law states that absorbance and path length are exactly related.
• Beer-Lambert law Equation: I = I_oe^-μ(x)
• Beer-Lambert's law can be expressed as A= εLc, where A refers to the absorbance, ε denotes the molar extinction coefficient, L denotes path length, and c denotes concentration.
• The limitations of the Beer-Lambert law claim that, as the concentration of the sample rises, there is a deviation due to the increase of the electrostatic interactions.

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