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Beer-Lambert Law

The Beer-Lambert law says that the amount of light absorbed by a sample is directly related to the volume of sample the light passes through and the concentration of the sample. It is also referred to as Beer’s Law.

What is the Beer-Lambert Law?

The Beer-Lambert law relates the concentration of a sample to the amount of light the sample absorbs as it passes through the sample. The equation for the Beer-Lambert Law is generally written as:

A= ϵLc

A= Absorbance

ϵ = Molar extinction coefficient

L = Path length

C = Concentration of the sample

The absorbance is related to the ratio of the intensity of light that enters the sample and leaves the sample.

A = log10 (I0/I)

I0 = Incident Light-Intensity of light before sample

I = Transmitted Light – Intensity of light after sample

Incident light passing into a cuvette will be greater than the transmitted light,
As light passes through a sample some of the light will be absorbed by the sample.

The Beer-Lambert Law is commonly used in absorption and transmission measurements on samples and can be used to determine the concentration of a sample. In an absorption measurement, light passes through a cuvette filled with a sample. The intensity of the light after the cuvette is compared to the light before passing through the cuvette. The size of the cuvette determines the path length (L). (A cuvette is a special piece of glassware.) The wider the cuvette, the more sample the light will pass through, and the the transmitted light will be lower. This explains why the equation is dependent on path length (L).

Illustration of path length size on absorption in the beer-lambert law
As the path length (L) gets larger, the amount of transmitted light decreases. Therefore, the absorption increases.

What is the Molar Extinction Coefficient?

The molar extinction coefficient is specific to every chemical and an important variable in the Beer-Lambert law. The molar extinction coefficient measures how much light a substance absorbs and is wavelength specific. It is also sometimes referred to as the molar absorption coefficient or molar absorptivity. In equations, it is most often symbolized as epsilon, ϵ.

The units of the molar extinction coefficient are most commonly M-1cm-1. The units should match the units of the path length and sample concentration. That way the absorbance results in a unitless number. On a graph, the absorbance is often written with units of A.U., which stands for arbitrary units.

Beer-Lambert Law Graph

A typical graph illustrating the Beer-Lambert law will be linear and positively correlated. The x-axis will have units of concentration and the y-axis will be absorbance. This indicates that the other two variables in the equation, molar extinction coefficient and path length, are held constant. As the concentration increases, the absorbance will also increase. This pattern makes sense because if the concentration increases, there are more molecules present to absorb light and cause an increase in absorption.

Below is a graph similar to one you might see demonstrating the Beer-Lambert Law. Several different concentrations are measured. Then fit a line to these points. The slope of the line will be the path length times the molar extinction coefficient. If you know the path length, the molar extinction coefficient can easily be determined. The molar extinction coefficient will be the slope of the line divided by the path length.

Plot of sample concentrations vs absorbance as a demonstration of the beer-lambert law

Applications of the Beer-Lambert Law

The Beer-Lambert law is commonly used for determining the concentration of a sample of unknown concentration. To do this, first absorbance of multiple samples of known concentration are measured. A spectrometer makes this measurement. These points fit to a line. The line will have a slope of the molar extinction coefficient times the path length. Dividing this by the path length gives the molar extinction coefficient. The absorption of the unknown sample can then be measured. The absorption divided by the path length times the molar extinction coefficient will then give the concentration of the sample.

Limitations of the Law

The law tends to become inaccurate at high concentrations. This is due to a combination of different factors. The refractive index of the solution may deviate. There are saturation and aggregation effects possible due to the molecule of interest interacting with each other (not just solvent as is the situation at low concentrations). An excellent way to test the limitations of the Beer-Lambert Law is to make a plot of concentration verse absorption at increasingly high concentrations for a sample. The plot should be linear, but at high concentrations will stop being linear. At this point, high concentrations are causing the law to be inaccurate.  

An excellent way to test the limitations of the Beer-Lambert Law is to make a plot of concentration verse absorption at increasingly high concentrations for a sample. The plot should be linear, but at high concentrations will stop being linear. At this point, high concentrations are causing the law to be inaccurate.  

Example Problems

Example Problem #1: You have a solution of rhodamine dye of unknown concentration. Using a spectrometer you measure the absorption to be 9048. You know the molar extinction coefficient of rhodamine is 116000 cm-1 M-1. The cuvette you used has a path length of 1 cm. What is the concentration of your sample?

Example Solution #2: Here we are trying to determine the value of C in the Beer-Lambert Law. So we start by rearranging the equation to solve for the variable we are looking for

A = ϵLc

c = A / ϵL

Then we can start plugging in values. Make sure to pay attention to units so that our concentration comes out with units of molarity.

c = 9048 / (1 cm * 116000 cm-1 M-1 )

c = 9048 / 116000 M-1

0.078 M = c

The concentration of the unknown solution is 0.078 M.