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 / I_{o}

where I is the light intensity after it passes through the sample
and I_{o} is the initial light intensity. The relation
between A and T is:

A = -log T = - log (I / I_{o}).

*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:

I_{o} is the intensity entering the sample at z=0,
I_{z} 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 / I_{z} = - * N * dz

Integrating this equation from z = 0 to z = b gives:

ln(I) - ln(I_{o}) = - * N * b

or - ln(I / I_{o}) = * N * b.

Since N (molecules/cm^{3}) * (1 mole / 6.023x10^{23}
molecules) * 1000 cm^{3} / liter = c (moles/liter)

and 2.303 * log(x) = ln(x)

then - log(I / I_{o}) = * (6.023x10^{20}
/ 2.303) * c * b

or - log(I / I_{o}) = A = * b * c

where = * (6.023x10^{20} / 2.303)
= * 2.61x10^{20}

Typical cross-sections and molar absorptivities are:

(cm^{2}) (M^{-1}cm^{-1}) absorption - atoms 10^{-12}3x10^{8}molecules 10^{-16}3x10^{4}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:

- deviations in absorptivity coefficients at high concentrations (>0.01M) due to electrostatic interactions between molecules in close proximity
- scattering of light due to particulates in the sample
- fluoresecence or phosphorescence of the sample
- changes in refractive index at high analyte concentration
- shifts in chemical equilibria as a function of concentration
- non-monochromatic radiation, deviations can be minimized by using a relatively flat part of the absorption spectrum such as the maximum of an absorption band
- stray light