Attenuation of light as it traverses an absorbing medium depends on the absorptivity of the substance, path length and the quantity of the absorbing substance. In order to find out their interrelationship, let us first derive the Beer-Lambert Law. Let us consider a thin slab of solution of thickness dx that contains dn light-absorbing molecules. Then

dn= c.Na.S.dx (1)

where c is the concentration (in mole per unit volume) of light absorbing molecule, N _a is the Avogadro constant, and S is the cross-section of the incident beam. Each molecule is associated with a molecular absorption cross-section, σ, which measures the photon capture area of the molecule. (Remember that σ is function of wavelength,λ.)The total absorption cross-section of thethin layer is the sum of all molecular cross-sections, i.e. σ.dn.

The probability of photon capture = σ.dn/S= fraction of light absorbed by the thin layer= -dI/I where small light intensity dI is absorbed from the incident light of intensity I. Therefore,

-dI/ I = σ.dn/S = σ. c. Na. dx (2)

Rearrangement and integration, subject to the boundary condition I = I ₀ at x = 0, yields:

ln(I ₀ /I) = σ. c. Na. x (3)

where x is the thickness of the light absorbing sample (sample path length) and Io and I are the incident and transmitted intensities, respectively. This equation is the mathematical form of the Beer-Lambert Law. Following alternative form of the equation is generally used:

log10 (I ₀ /I) = ε c x = optical density (OD) or absorbance (4)

where ε is the decadic molar extinction coefficient or molar absorptivity (in M–1 cm–1), c is the molar concentration (in mole/1000 cm3, M), and x is path length (in cm). Comparison of equations 3 and 4 gives the relationship between the molar extinction coefficient, ε,(in M^–1 cm^–1) and the molecular cross-section for light absorption, σ (in cm ² ):

ε (in M ^–1 cm ^–1 ) = optical density (OD) or absorbance (5)

or, σ (in cm ² ) = 3.82 × 10-21. ε (in M ^–1 cm ^–1 ). (6)

A knowledge of molar absorptivity helps to calculate the absorption cross-section of light absorbing compounds. For example, the molar absorptivities of anthracene are 160,000 and 6,300 M ^–1 cm ^–1 at wavelengths 253 nm and 375 nm, respectively. Thus one calculates cross-sections of 6.1 Å ² and 0.24 Å ² , for wavelengths 253 nm and 375 nm, respectively. If the molecular cross-section of anthracene is assumed to be 12 Å ² , anthracene absorbs about 50% of the photons it encounters at 253 nm and 2% of the photons at 375 nm.

ε _max , the molar absorptivity at the spectral peak frequency (or, at the wavelength of maximum absorption), is often experimentally determined and quoted. The molar extinction coefficient,ε, is characteristic of the chromophore under given conditions (wavelength, solvent and temperature) and is a measure of the strength of electronic transition. The molar extinction coefficient is related to the integral absorption coefficient (α) which relates the experimental spectrum to a theoretical quantity known as the oscillator strength, f _nm . As absorption bands generally spread over a range of frequencies, the integral absorption coefficient , α = ∫bandε(ν)dν (in which integration is taken over the whole band frequency range), is a measure of the probability that an incident photon will be absorbed in a specific transition. The integral absorption coefficient is the sum of the absorption coefficients over the entire spectral band and corresponds to the area under the plot of the molar absorptivity, ε, against frequency. Therefore, α is easily calculated from the experimental spectral band. For a Lorentzian spectral band profile, the integral absorption coefficient is given by

α = ½ π ε _max Г (7)

where Γ is the full bandwidth at half of maximum (FWHM) absorbance. The relationship between the integral absorption coefficient (α) and the oscillator strength, fnm, is given as follows:

fnm = 4.319 × 10 ^-9 mol.L ^-3 .cm ² × α = 4.319 × 10 ^-9 mol.L ^-3 .cm ² × (½ π ε ^max Г) (8)

f _nm is a measure of the strength of an electric dipole transition (proportional to the square of the transition dipole moment μnm) between electronic states n and m with respect to that of a free electron oscillating in three dimensions. In other words, it represents the strength of absorption relative to a completely allowed transition. ε can be experimentally determined as follows. According to Beer–Lambert law,

log ( I ₀ / I) = A = ε c l (9)

where A = absorbance or optical density. Since A vs. c is a straight line equation with slope = (ε.l) and zero intercept, molar absorptivity is best determined from the slope of a linear plot of absorbance versus concentration, called a calibration curve or calibration plot. Calibration graph is constructed by plotting absorbance at a given wavelength versus concentration for a series of standard solutions whose concentrations are accurately known. Since the absorptivity is a function of wavelength, often its value at the highest absorbance, ε max (at the peak position i.e., at the wavelength of maximum absorbance) is determined. One should note that the Beer–Lambert law is obeyed by many substances mainly at low to moderate concentrations; therefore, dilute concentrations of the absorbing species should be measured. Care must be taken to avoid any kind of chemical associations of the absorbing species.