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Study Notes: Beer’s Law

We have previously seen how light can be absorbed or transmitted by transparent objects such as glass and plastic. This phenomenon also occurs when light is shone on a solution containing a chemical species. Absorbance and transmission are spectroscopic measures that are used to make quantitative determinations of the species.

Let us consider visible or white light. This has a continuous spectrum the main identifiable colours of which are red, orange, yellow, green, blue and violet (ie the colours of the rainbow). Removal of colours from this mixture occurs by absorption. To the eye, the observed colour will be a mixture of the remainder.

For example, a copper sulfate solution appears blue because it transmits blue coloured light and strongly absorbs yellow (ie red and green) light. Yellow is the complementary colour that is absorbed.

Wavelength absorbed
Colour absorbed
Colour observed
400 nm violet yellow-green
435 nm blue yellow
495 nm green violet
560 nm yellow blue
590 nm orange green-blue
650 nm red blue-green

For analytical purposes, the wavelength of the incident light beam is chosen so as to give strong absorption. A blue solution strongly absorbs yellow and transmits blue and so would be analysed with yellow light.

The concentration of the solution would not affect the amount of blue light that is passing through the sample but it would greatly affect the amount of yellow light that is absorbed.

absorbance (A) is directly related to the concentration (c) and optical path length (l) of the solution through which the beam of light must pass. Beer’s Law describes this relationship:

A sample container or cell of width 'l' and containing a solution of sample concentration 'c'.

epsilon is the molar absorptivity constant for the analytical species and is valid at a specific wavelength only (epsilon is also called the extinction constant or extinction coefficient). The concentration is expressed in mol L-1 and the optical path length in cm. The unit of epsilon is cm-1 mol-1 L but is often expressed without units. Note that absorbance is unitless.

The absorbance is directly proportional to the concentration of the solution.

Light passes through a solution resulting in the transmitted light being less intense than the incident light. Some light is absorbed as it passes through a solution so less light is transmitted.

Increasing the concentration of the solution results in even less light being transmitted. Doubling the concentration doubles the absorbance.

Increasing the path length of the solution also results in less light being transmited. Doubling the path length doubles the absorbance.

For a fixed path length, a graph of absorbance (A) against concentration (c) is a straight line through the origin.

A straight line graph through the origin showing the direct relationship between absorbance and concentration. Absorbance is directly proportional to the concentration of the absorbing solution.

By measuring absorbance and knowing the values for epsilon and l, the concentration of the chemical species in solution can be determined.

The value of epsilon depends on the absorbing species in the sample. It will also alter if a different wavelength of light is used. For this reason, Beer’s Law really only holds true for monochromatic (single wavelength) light and care must be taken to ensure that a narrow bandwidth of analytical light is selected on the spectrometric instrument.

The radiant power (ie intensity) of the light beam incident upon a sample (P0) is related to the radiant power of the emergent beam (P) by:

T equals P divided by P subscript zero.

where T, the fraction of the radiant energy transmitted is called transmittance.

Transmittance (T) and absorbance (A) are inversely related as described by the following equations:

A = - log T and T = 10-A

when A = 0.0, T = 100%
  A = 0.2, T = 64%
  A = 0.4, T = 40%
  A = 0.8, T = 16%
  A = 1.0, T = 10%
  A = 2.0, T = 1%

Notice that absorbance and transmittance move in opposite directions – the more a solutions absorbs light, the less it transmits.

Sample problem:

A tea solution in a 0.5 cm sample cell gave an absorbance reading of 0.64 in blue light (420-440 nm).
If the molar absorptivity for the tea is 120 cm-1mol-1L, using the same light filter, calculate the concentration and the transmittance of the solution.

Solution

C equals A divided by (epsilon times l), which equals 0.64 divided by (120 times 0.5), which equals 0.011 Molar. T = 10-A = 10-0.64 = 0.23

ie 23% of the power of the incident beam is transmitted.

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