Beer-Lambert Law: 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

Best Practices

1. Use dilute solutions for better linearity
2. Measure at absorbance maxima
3. Use monochromatic light
4. Maintain constant temperature
5. Use clean, matched cuvettes
6. Properly zero the instrument with blank solution