Beer-Lambert Law

Introduction

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(lambda) * b * c
where A is the measured absorbance, a(lambda) 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 = epsilon * b * c
where epsilon is the wavelength-dependent molar absorptivity coefficient with units of M^-1 cm^-1.

Instrumentation

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.

Derivation of the Beer-Lambert law

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, sigma, 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 sigma * N * A * dz. Then, the fraction of photons absorbed will be sigma * N * A * dz / A so,

dI / I_z = - sigma * N * dz

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

ln(I) - ln(I_o) = - sigma * N * b

or - ln(I / I_o) = sigma * 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 / I_o) = sigma * (6.023x10²⁰ / 2.303) * c * b

or - log(I / I_o) = A = epsilon * b * c

where epsilon = sigma * (6.023x10²⁰ / 2.303) = sigma * 2.61x10²⁰

Typical cross-sections and molar absorptivities are:

                         sigma (cm²) 
epsilon (M^-1 cm^-1)     
absorption - atoms       10^-12           3x10⁸ 
             molecules   10^-16           3x10⁴ 
             infrared    10^-19           3x10 
Raman scattering         10^-29           3x10^-9

Limitations of the Beer-Lambert law

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

Further Information

http://www.scimedia.com/chem-ed/spec/beerslaw.htm, updated 9/24/96