EQ-BEER-LAMBERT · Spectroscopy
Beer–Lambert Law
A - c*ell*epsilon = 0
Variables
variable
A
absorbance, A = log10(I_0 / I)
- Object
- solute
- Property
- AbsorbanceDecadic
- Context
- dilute
- Constraint
- at_measurement_wavelength
variable
c
concentration of the absorbing species
- Object
- solute
- Property
- MolarConcentration
- Context
- dilute
variable
ell
optical path length through the sample
- Object
- solvent
- Property
- Length
- Context
- dilute
- Constraint
- optical_path
variable
epsilon
molar absorptivity (extinction coefficient) at the measurement wavelength
- Object
- solute
- Property
- MolarAbsorptivity
- Context
- dilute
Axioms
algebraic classical commutative_factors constant_coefficients deterministic dilute_limit linear
Assumptions
- Dilute solution (c ≲ 10⁻² M); above this, solute–solute interactions distort ε
- Monochromatic light at a wavelength where ε is well-defined
- No fluorescence or scattering that would add photons to the detector
- No concentration-dependent chemical equilibria (e.g., dimerisation) that change the effective absorbing species
Derivation
- Differential form: dI/I = -α c dx (fractional intensity loss proportional to concentration times path increment)
- Integrating over path length: I = I₀ exp(-α c ℓ), then defining A ≡ log₁₀(I₀/I) = (α/ln 10) c ℓ = ε c ℓ
- Bouguer (1729), Lambert (1760), Beer (1852) — all three contributed pieces of the law
References
- Bouguer, Essai d'optique sur la gradation de la lumière, 1729
- Lambert, Photometria, 1760
- Beer, Ann. Phys. Chem. 86 (1852), 78
- Ingle & Crouch, Spectrochemical Analysis, Prentice-Hall, 1988, §13-3