Theoretical Foundation

Basic Principles

The Beer-Lambert law combines two fundamental principles:

1. Lambert's Law: For parallel, monochromatic radiation passing through an absorber of constant concentration, the radiant intensity decreases logarithmically as the path length (l) increases arithmetically.

2. Beer's Law: The transmittance of a stable solution is an exponential function of the concentration (c) of the absorbing solute.

Mathematical Formulation

When both path length and concentration are variable, the combined Beer-Lambert law is expressed as:

I_t = I_o exp(-kcl)

or in logarithmic form:

log_e(I_o/I_t) = kcl

where:

• I_o = incident intensity
• I_t = transmitted intensity
• k = constant (function of wavelength)
• c = concentration
• l = path length

Absorbance and Transmittance

Converting to base 10 logarithm, the equation becomes:

log(I_o/I_t) = A = εcl

where:

• A = absorbance
• ε = molar absorptivity (formerly called extinction coefficient)
• c = concentration (mol/L)
• l = path length (cm)

The transmittance (T) is defined as: T = I_t/I_o

Key Characteristics

1. Linearity: A plot of absorbance versus concentration should yield a straight line passing through the origin with slope = εl
2. Path Length: When l = 1 cm, the slope equals the numerical value of ε
3. Range:
   • Absorbance (A) can range from 0 to infinity
   • Transmittance (T) must be between 0 and 1

Limitations and Deviations

The Beer-Lambert law may show deviations from linearity due to:

1. High Concentrations: At high analyte concentrations, molecular interactions can affect absorptivity
2. Chemical Effects: Association of molecules can change the nature of absorbing species
3. Instrumental Factors:
   • Polychromatic radiation
   • Stray light
   • Detector response
4. Physical Effects: Changes in refractive index at high absorbance
5. Chemical Associations: Molecular interactions affecting the absorbing species