# 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 / 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.

### 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:

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

dI / Iz = - sigma * N * dz

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

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

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

or - log(I / Io) = 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

`updated 2/22/96`