The Beer-Lambert law (or Beer's law) is the linear relationship
between absorbance and concentration of an absorbing species.
The general Beer-Lambert law is usually written as:
A = a() * b * c
where A is the measured absorbance, a() is a wavelength-dependent absorptivity coefficient, b is the path length, and c is the analyte concentration. When working in concentration units of molarity, the Beer-Lambert law is written as:
A = * b * c
where is the wavelength-dependent molar absorptivity coefficient with units of M-1 cm-1.
Experimental measurements are usually made in terms of transmittance
(T), which is defined as:
T = I / Io
where I is the light intensity after it passes through the sample and Io is the initial light intensity. The relation between A and T is:
A = -log T = - log (I / Io).
Absorption of light by a sample
Modern absorption instruments can usually display the data as either transmittance, %-transmittance, or absorbance. An unknown concentration of an analyte can be determined by measuring the amount of light that a sample absorbs and applying Beer's law. If the absorptivity coefficient is not known, the unknown concentration can be determined using a working curve of absorbance versus concentration derived from standards.
The Beer-Lambert law can be derived from an approximation for the absorption coefficient for a molecule by approximating the molecule by an opaque disk whose cross-sectional area, , represents the effective area seen by a photon of frequency w. If the frequency of the light is far from resonance, the area is approximately 0, and if w is close to resonance the area is a maximum. Taking an infinitesimal slab, dz, of sample:
Io is the intensity entering the sample at z=0, Iz is the intensity entering the infinitesimal slab at z, dI is the intensity absorbed in the slab, and I is the intensity of light leaving the sample. Then, the total opaque area on the slab due to the absorbers is * N * A * dz. Then, the fraction of photons absorbed will be * N * A * dz / A so,
dI / Iz = - * N * dz
Integrating this equation from z = 0 to z = b gives:
ln(I) - ln(Io) = - * N * b
or - ln(I / Io) = * N * b.
Since N (molecules/cm3) * (1 mole / 6.023x1023 molecules) * 1000 cm3 / liter = c (moles/liter)
and 2.303 * log(x) = ln(x)
then - log(I / Io) = * (6.023x1020 / 2.303) * c * b
or - log(I / Io) = A = * b * c
where = * (6.023x1020 / 2.303) = * 2.61x1020
Typical cross-sections and molar absorptivities are:
(cm2) (M-1 cm-1) absorption - atoms 10-12 3x108 molecules 10-16 3x104 infrared 10-19 3x10 Raman scattering 10-29 3x10-9
The linearity of the Beer-Lambert law is limited by chemical and instrumental factors. Causes of nonlinearity include: