Beer-Lambert Law

Beer’s Law and Beer-Lambert 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:

$A = \varepsilon \cdot c \cdot l$

• 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

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:

1. Transmission and Absorbance:

   • • $I = I_0 \cdot e^{-\alpha l}$
   • The intensity of light decreases exponentially as it passes through an absorbing medium:
   • where α is the absorption coefficient.

2. Logarithmic Relationship:

   • • $\ln\left(\frac{I_0}{I}\right) = \alpha l$
   • Taking the natural logarithm:

3. Introducing Molar Absorptivity:

   • • $\alpha = \varepsilon c$
   • Where c is concentration.

4. Final Form:

   • • $A = \varepsilon c l$
   • 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.

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