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Derivation of Beer-Lambert Law

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Beer Lambert Law Statement

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The Beer-Lambert law states that the absorption of energy in a solution is directly proportional to the molar absorptivity and the solute concentration of the given solution. Beer-Lambert law is the combination of 2 laws given by Johann Lambert and August Beer, hence it came to be known as Beer-Lambert law.

Lambert’s law states that the Absorption of light in a homogenous solution is directly proportional to the length of the sample through which it passes.

A = log10(Io/I) ~ l

where,

A = Absorption

l = Length of the sample through which light passes

Beer’s law said that the absorption of light in a solution is directly proportional to the sample concentration in which the light passes.

A = log10(Io/I) ~ C

Where, 

A = Absorption

C= Concentration of the solution

This law is used in chemical analysis and the understanding of attenuation in optics. 


Beer-Lambert Law Formula

Lambert said that the loss in the intensity of light is directly proportional to the path length. Later, in 1852, Beer extended the law and found out that the transmittance remains constant if the product of concentration and the length of the sample through which the light passes remains constant. 

The modern-day definition of Beer-Lambert law combines the two definitions and says that the attenuation of the transmitted light is directly proportional to the length of the sample solution and its concentration.

Mathematical derivation of the Beer-Lambert law is as follows.

Let I and I0 be the transmitted and the incident ray of light, respectively.

The incident light I0 falls upon the sample and the transmitted light I is radiated out.

Here, 

A = I0/I

T = I/I0

Where,

A = Absorbance 

T = Transmission

We get,

A = log10(1/T)

A = log10(Io/I)

So according to Beer-Lambert Law:

A = log10(Io/I) ~ l

A = log10(Io/I) ~ C

Mathematically, Beer-Lambert Law is given as follows.

A = εlC

Where,

A = Absorbance

l = Length of the sample solution

C = Concentration of the solution


Derivation of Lambert's law

Lambert’s law states that the absorbance of the solution is directly proportional to the length of the sample of the solution through which the light passes.

So mathematically,

A = log10(Io/I) ~ l

A = εl

Where,

A = Absorbance

l= Length of the sample solution

ε = Absorptivity or molar attenuation coefficient


Derivation of Ber's law

Beer’s law states that the absorbance of the solution is directly proportional to the concentration of the solution.

So mathematically,

A = log10(Io/I) ~ C

A = εC

Where, 

A = Absorbance

C = Concentration of the solution

ε = Absorptivity or molar attenuation coefficient


Importance of Concentration of the Solution

Let us discuss how the concentration of the solution affects its absorbance. The amount of incident light absorbed will depend on how many molecules it interacts with in the solution. In the presence of a concentrated solution, there will be a number of molecules to interact with the light. However, in the case of a dilute solution, the number of molecules to react with the incident light will be far less, hence, the attenuation of the incident light will be less. This is why absorbance is directly proportional to the concentration of the solution.


Importance of the Length of the Sample

The length of the sample interacting with the incident light amounts to the number of molecules coming in contact with the light. So, if the path length of the incident light is increased or decreased the number of molecules interacting with it will also increase and decrease respectively. This is the reason for the directly proportional relationship between the absorbance and length of the sample.


Conditions Required for Beer-Lambert’s Law

To avoid deviations, certain conditions are to be fulfilled for the Beer-Lambert law to be valid. These conditions are as follows.

  • There should be no electromagnetic coupling present and the attenuators must act independently of each other.

  • The sample to be tested must be homogenous. The solution should be homogenous where the light interacts with the solution for obvious reasons because any variation in the sample will affect the attenuation.

  • The incident radiation should have parallel rays travelling the same length along with the sample solution.

  • The incident radiation should not be polychromatic. It should be monochromatic radiation, preferably.

  • The light should not influence the atoms or molecules as it will change the sample, and the attenuation may vary due to optical saturation.

  • The wave properties of the light used should be negligible, so it does not vary the attenuation.

FAQ (Frequently Asked Questions)

Question 1: Explain absorbance and transmission mathematically in the case of opaque and transparent solutions.

Answer: An incident light I0 falls on the sample and the transmitted light I comes out. 

In the first case where the solution is transparent, the sample is non-absorbent. 

So, I = I(transmitted light is equal to the incident light).

So, the transmission here is 100%.

In the second case where the solution is opaque, the sample is completely absorbing.

So, I = 0 (no transmitted light comes out).

So, the transmission here is 0%.

Question 2: What are the deviations of Beer-Lambert law?

Answer: There are some fundamental deviations that occur in the law. Other than the attenuation due to how the measurements were made and the instruments, the deviation of the beer-lambert law are as follows. 

  • The chemical change arising when the sample associates, disassociate or reacts with the solvent itself bringing about the change in the absorbance of the solution.

  • The presence of stray radiation, radiation other than the one being measured or calculated.

  • Beer-Lambert law is fulfilled only in monochromatic radiation, so, in the presence of polychromatic radiation, a deviation is observed.