Beer-Lambert Law

Beer’s Law and Lambert’s Law

• Beer’s Law: Beer’s Law, also known as Beer-Lambert Law, relates the absorption of light to the properties of the material through which the light is traveling.

Mathematical Expression:

• Where:

  • A = Absorbance (no units)

  • ε = Molar absorptivity or extinction coefficient (L·mol⁻¹·cm⁻¹)

  • c = Concentration of the absorbing species (mol·L⁻¹)

  • l = Path length of the sample cell (cm)

Lambert’s Law: 

• Lambert’s Law states that absorbance is directly proportional to the path length of the sample cell.

Combined Beer-Lambert Law: 

• Combines both Beer’s and Lambert’s laws to provide a comprehensive relationship between absorbance, concentration, and path length.

Derivation of Beer-Lambert Law

Starting Point:

• The law is derived from the principles that:

  • Each molecule has a probability of absorbing light proportional to its concentration.

  • The absorbance is cumulative over the path length.

Derivation Steps:

Transmission and Absorbance:

• The intensity of light decreases exponentially as it passes through an absorbing medium:

• where α is the absorption coefficient.

Logarithmic Relationship:

• Taking the natural logarithm:

Introducing Molar Absorptivity:

• Where c is concentration.

Final Form:

• Substitute α\alphaα into the equation:

Key Assumptions:

• The system is homogeneous.

• The absorbers do not interact with each other.

• The incident light is monochromatic and collimated.

Graph Description:

• If you were to plot absorbance (A) on the vertical axis (y-axis) against concentration (c) on the horizontal axis (x-axis) for a system that follows the Beer-Lambert law perfectly, you'd expect a straight line passing through the origin.

• The slope of this line would be equal to c × ε × l.