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 =

where

T = I / I

where I is the light intensity after it passes through the sample and I

A = -log T = - log (I / I

*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.

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/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}) = *sigma* * (6.023x10^{20} / 2.303) * c * b

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

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

Typical cross-sections and molar absorptivities are:

sigma(cm^{2})epsilon(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}

- 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

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Copyright © 1996 by Brian M. Tissue

updated 2/22/96