# Derivation of Beer Lambert Law

Beer-Lambert Law derivation helps us to define the relationship of the intensity of visible UV radiation with the exact quantity of substance present. The Derivation of Beer-Lambert Law has many applications in modern day science. Used in modern-day labs for testing of medicines, organic chemistry and to test with quantification. These are some of the fields that this law finds its uses in.

## Beer-Lambert Law Statement

The Beer-Lambert law states that:

for a given material sample path length and concentration of the sample are directly proportional to the absorbance of the light.

The Beer-Lambert law is expressed as:

 A = εLc

where,

• A is the amount of light absorbed for a particular wavelength by the sample
• ε is the molar extinction coefficient
• L is the distance covered by the light through the solution
• c is the concentration of the absorbing species

Following is an equation to solve for molar extinction coefficient:

 $\epsilon =\frac{A}{Lc}$

But Beer-Lambert law is a combination of two different laws: Beer’s law and Lambert law.

### What is Beer’s Law?

Beer’s law was stated by August Beer which states that concentration and absorbance are directly proportional to each other.

### What is Lambert Law?

Lambert law was stated by Johann Heinrich Lambert which states that absorbance and path length are directly proportional.

## Beer-Lambert Law Formula

 $I=I_{0}e^{-\mu (x)}$

Where,

• I is the intensity
• I0 is the initial intensity
• x is the depth in meters
• 𝜇 is the coefficient of absorption

Following is the table explaining concepts related to Physics laws

## Schematic Diagram of Beer-Lambert Law

Absorption of energy causes the absorption of light as well usually by electrons. Different forms of light such as visible light and ultraviolet light get absorbed in this process. Therefore, change in the intensity of light due to absorption, interference, and scattering leads to:

ΔI = I0IT

The following equations are necessary for us to obtain our ultimate derivative equation. Transmittance is measured as the ratio of light passing through a substance. It can be calculated as IT/I0. To calculate the of transmittance percentage we can do so by:

Percent TransmittanceAnother key metric is absorbance that is defined as the amount of light absorbed. This is usually calculated as the negative of transmittance and is given by:

Absorbance (A) The rate of decrease in the intensity of light with the thickness of the material the light is directly proportional to the intensity of the incident light. Mathematically, it can be expressed as:

As k’= Proportionality constant
Taking in the reciprocal of the equation we get,

Integrating the above equation we also get,

In the above equation, b and C is the constant of integration and IT is the intensity being transmitted at the thickness

In order to solve the above equation with the constant of integration, we then get,

While solving for C in the equation will give us,

Converting to log10 we get,

Rearranging the equation we get,

Lambert Derivation

Thus, Lambert’s law was formed and it states that the monochromatic radiation changes exponentially and decreases when it passes through a medium of uniform thickness.

Beer Derivation

Thus, this concludes the derivation of Beer-Lambert law. This goes to show you that in order to derive a particular law, there are a lot of different equations that need to be found out first, to achieve the ultimate result.