Beer's Law states that the absorbance, A(λ), of a species at a particular wavelength of electromagnetic radiation, λ, is proportional to the concentration, c, of the absorbing species and to the length of the path, l, of the electromagnetic radiation through the sample containing the absorbing species.  This can be written in the form:

A(λ) = e(λ) l c

The proportionality constant e(λ) is called the absorptivity of the species at the wavelength, λ.

[ If the concentration is measured in moles/liter, e(λ) is called the molar absorptivity . ]

It is common to use the energy carried by the radiation per unit area per unit time, which is called the intensity, I, to measure of the "amount" of electromagnetic radiation impinging on a surface.  For a partially transparent sample, we can consider the fraction of the intensity that is permitted to pass through the sample as a measure of the transmittance of the sample.  In fact, we define the percent transmittance, %T , of a sample in terms of the intensity of the light incident on the sample, I0 , and the light transmitted through the sample, It as:

%T = 100 It / I0
A completely transparent sample will have It = I0, and its percent transmittance will be, appropriately, 100%.  Similarly, a sample which permits no radiation of a particular wavelength to pass through it will have It  = 0, and a corresponding percent transmittance of  0%.

Since the most interesting materials are those that absorb electromagnetic radiation at some frequencies, we define absorbance of light of wavelength λ by a sample in terms of the percent transmittance.  The amount of radiation absorbed can vary over an extremely wide range, so it is useful to define absorbance logarithmically (much the same way as we define pH).  The absorbance of a sample is defined in terms of percent transmission as follows:

A(λ) = log ( 100 / %T ) = 2.000 - log ( %T  )     ( The 2.000 in this expression is simply log10(100) )

The absorbance of a sample exhibiting a percent transmittance of 100 (a completely transparent sample) will be 0.  The logarithmic dependence of absorbance means that a sample that absorbs 10 % will have A = 1; 1% transmittance corresponds to A = 2; 0.1% transmittance to A = 3, etc.

The way in which e(λ) depends on wavelength defines the spectrum of the substance in question.  Most substances show a maximum in e(λ) over a sufficiently broad wavelength range.  The wavelength at that maximum value is called the analytical wavelength of the substance and denoted λmax.  Normally, Beer's law is applied at the analytical wavelength of a given material.  The sensitivity to concentration differences should be largest at that wavelength.

The experimental procedure for using Beer's Law to measure concentrations generally involves the following:
1.)    Determine the analytical wavelength of the substance whose concentration is desired, λmax`
2.)    Prepare a series of samples of known concentration of the substance & calibrate the spectrophotometer
3.)    Measure the absorbance of each of the solutions of known concentration at the analytical wavelength.
 Concentration (mg/L) Absorbance 0 0 1.2 X 10-3 0.15 2.4 X 10-3 0.31 4.8 X 10-3 0.70 9.6 X 10-3 1.25 1.9 X 10-2 2.62
4.)    Plot the values of the absorbance as a function of the concentration.
5.)    Verify that, within experimental error, the absorbance is a linear function of the concentration. [ What can cause non-linearity? ]

In the above graph, the value R2 is a measure of the "goodness of fit" of the data to a straight line.  A value of 1 indicates a perfect fit.

6.)    If linearity is confirmed, determine the slope of the best straight line through the experimental points in the absorbance vs concentration plot.  Call this the Beer's Law slope.  The Beer's law slope has the value e(λmax) l, where l is the path length through the sample.

In the above plot, the Beer's law slope is the value of the coefficient in the equation y = 136.67 x.  To the appropriate number of significant figures, the slope is 137.  I.e.,  A = 137 c, where c is the concentration.

7.)    If the same experimental arrangement ( same spectrometer, same cell ) will be used for subsequent absorbance measurements, the value of the slope can be used to determine the concentration corresponding to a given absorbance by samples of unknown concentration.
E.g., suppose in the above example, we measure the absorbance of a solution of unknown concentration to be 1.84.
The concentration of that solution will be 1.84 / 137 = 1.34 X 10-2 (in the same units of concentration that were used in the determination of Beer's Law).

[Note that the value of the slope of the Beer's Law plot depends on the units in which concentration is measured and its units are the reciprocal of that concentration unit.]

If a different experimental arrangement will be used (e.g., a different cell), the Beer's Law slope will need to be adjusted for the difference in pathlength from that of the cell used in the Beer's Law determination, if known.  If the same (or matched) cells are used for Beer's Law determination and subsequent analysis, knowing the slope of the Beer's Law plot (the product of the absorptivity and the path length) suffices.

A known path length provides the value of the absorptivity, e(λmax), which is characteristic only of the substance and the wavelength, and independent of the experimental arrangement used to determine it.

When cylindrical cells are used, such as in our exercise, path length is a complex quantity. (See here.)

Non-linearity in Beer's Law Plots:

Several phenomena can cause a deviation from Beer's Law.

1.) Improper cuvette rinsing

Suppose the absorbances of the Beer's Law dilutions are measured in the same cuvette, in the order of decreasing concentration and without proper rinsing of the cuvette between dilutions. I.e., some of the previously measured solution is retained and mixed with the new solution before measuring absorbance.

The Beer's Law plot for such a case is shown at the right. It assumes that each dilution involves an error in concentration of 15% and that the circled point is the result for a pure stock solution (and therefore has no dilution error). The curvature in the plot for the dilutied points is apparent.

The best straight line through the experimental points including the origin shows a slope of 2.16 which is 8% above the expected (true) value of the slope. Note that the goodness of fit criterion, R2, has the very respectable value of 0.98.

2.) The material undergoes a chemical change when exposed to the electromagnetic radiation at the analytical wavelength.

If the material is not susceptible to change under exposure to some different wavelength, we could try the different wavelength to see if Beer's law is followed.

3.) The material is involved in concentration dependent equilibria.
Suppose, for example, that the material can form a dimer with a different analytical wavelength than the monomer
[  B2 ]
2 B    =    B2        with an equilibrium constant , Keq    =    ----------
b - 2x       x                                                                      [ B ]2

Assuming the total concentration of B in either form is b, the concentration of monomer will be
[ B ] =  b - 2x  =  b - 2 Keq [ B ]2
from which we can deduce that, in terms of the total concentration of B, b, the concentration of the monomer will be a relatively complex function of the total amount of B, namely
[ B ] =  ( 1 / 4 Keq )(( 1 + 8 Keq b)1/2 - 1)
The chart below shows a plot of the absorbance of a colored substance that undergoes such a reaction as a function of the total concentration of the substance. The dimer is assumed to show no absorbance at this wavelength. Note the curvature of the Beer's Law Plot.

4. The material decomposes slowly in solution
Since the determination of Beer's Law takes time, the concentrations of successive dilutions prepared to determine absorbances may no longer have their nominal concentrations by the time they are measured.

The only remedy would be to prepare each dilution and measure it as soon as it is prepared.