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.
