Beerâ€™s Law and Multicomponent Samples

**Absorbance and Concentration: Beerâ€™s Law**

When monochromatic electromagnetic radiation passes through
an infinitesimally thin layer
of sample, of thickness *dx, *it
experiences a decrease
in power of *dP *(Figure 10.21). The fractional decrease
in power is proportional to the sampleâ€™s
thick- ness and the
analyteâ€™s concentration, *C; *thus

10.3

where *P *is the power incident
on the thin layer of sample, and Î± is a proportionality constant. Integrating the left side of equation
10.3 from *P *=
*P*_{0} to *P *= *P*_{T}, and the
right side from *x *= 0 to *x *=
*b, *where *b *is the
sampleâ€™s overall thickness

Converting from ln to log, and substituting equation 10.2, gives

*A *=
*abC â€¦â€¦â€¦â€¦. 10.4*

where *a *is the analyteâ€™s absorptivity with units of cmâ€“1 concâ€“1. When concentration is expressed using molarity, the absorptivity is replaced by the molar
absorptivity, Îµ (with units
of cmâ€“1
Mâ€“1)

*A *=
Îµ*bC â€¦â€¦â€¦â€¦. 10.5*

The absorptivity and molar absorptivity give, in effect,
the probability that the ana- lyte
will absorb a photon of given energy.
As a result, values for both *a *and
Îµ depend on the
wavelength of electromagnetic radiation.

Equations 10.4 and 10.5, which establish the linear relationship
between absorbance and concentration, are known as the Beerâ€“Lambert law, or more commonly, as **Beerâ€™s law. **Calibration curves
based on Beerâ€™s
law are used routinely in quantitative
analysis.

Beerâ€™s law can be extended to samples containing several
absorbing components provided that there
are no interactions between the components. Individual ab-
sorbances, *A _{i}, *are additive. For a two-component mixture of X and Y, the total
ab- sorbance,

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Modern Analytical Chemistry: Spectroscopic Methods of Analysis : Absorbance and Concentration: Beerâ€™s Law |

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