# Beer-Lambert Law

The Beer-Lambert Law is an important concept in UV-vis absorption spectroscopy. In this article, we'll discuss the equation, graph, units, example, and limitations of the Beer-Lambert Law.

The Beer-Lambert Law equation is derived from an experiment in which a chemical solution is prepared and transferred to a quartz cuvette. The solution, or sample, absorbs UV-vis light, while the solvent either does not absorb or absorbs weakly within the UV-vis wavelength range (200nm-800nm). The sample and reference cuvettes are then placed into a spectrophotometer, and light passes through both slots. Spectrograms are then recorded and compared.

The Beer-Lambert Law equation is expressed as A = εbc, where ε is the molar absorptivity of the analyte, b is the path length (the distance the light travels through the solution), and c is the concentration of the analyte. The Beer-Lambert Law graph is used to set up a Beer-Lambert graph. The Beer-Lambert Law units are used to calculate the absorbance. An example of finding the enzyme activity is also presented. Finally, the limitations of the Beer-Lambert Law are discussed.

## The Beer-Lambert Law

When a spectrophotometer light beam passes through a UV-vis active solution, the intensity of the light beam decreases. A decrease in the light intensity of the light beam when passing through a solution is also called attenuation.When the sample absorbs light from a spectrophotometer passing through a sample, we call this sample absorbance. The Beer-Lambert law relates the decrease in the intensity of the spectrophotometer light beam when passing through a sample to the sample absorbance. In addition, the sample absorbance is accounted for by removing the absorbance due to the solvent, as will be explained further below.

A spectrophotometer, a spectrometer able to generate and measure light signals within the 200nm-800nm range, is a machine used to measure the reflectance of a solution. In other words, it is used to measure how much light is reflected/passes through a solution. Sample absorbance is a measure of how much light is absorbed by a sample as it passes through a sample.

The intensity of light passing through the reference cell (consisting of just the solvent), symbolized by I0, is measured at each wavelength, λnm, within the UV-vis spectrometer wavelength range ( Figure 1).

Figure 1: UV-vis experiment set-up for the reference cell.

In turn, the intensity of the light passing through the sample cell (solvent plus solute), symbolized by I, is also measured at each wavelength, λnm, within the spectrophotometer wavelength range (Figure 2).

Figure 2: UV-vis experiment set-up for the sample cell.

The light propagates through each cell onto the detector of the spectrophotometer. The resulting spectrograms of light intensity per wavelength are then data processed and combined to give a Beer-Lambert law graph, as will be discussed in greater detail below. If at a particular wavelength the light intensity through the sample cell, I, is less than the light intensity passing through the reference cell, I0, then the sample has absorbed light at that wavelength.

### The Beer-Lambert Law Graph

When light passes through a solution, the absorbance formula takes into account the attenuation of light due to light absorption from the solute compared to the absorption from the solvent. At wavelengths where the solvent does not significantly absorb UV-vis light but the solute does absorb light strongly, the intensity ratio, symbolized by , is greater than one. This means that more light reaches the detector when passing through the reference cell, resulting in a high value for , compared to the light intensity passing through the sample cell, which yields a smaller value for the light intensity, I, because the sample has absorbed some of the light that would have otherwise reached the detector.

The Beer-Lambert graph plots the absorbance, Abs, against wavelength, showing positive sample absorbance peaks at wavelengths where the ratio, , is greater than one.

Figure 3: Beer-Lambert Graph

### The Beer-Lambert Law Equation

The absorbance due to the sample is assumed to have the following properties:

The sample absorbance, Abs, is directly proportional to the light path length, , through the cuvette. The sample absorbance, Abs, is directly proportional to the solute concentration, c.

In symbols:

This proportionality relation can be converted to an equation by including a constant of proportionality, which we will call the molar extinction coefficient, , giving us an equation for the absorbance:

The molar extinction coefficient is also referred to as the molar absorptivity. This equation for the absorbance can only be used when the molar extinction coefficient, , is known. Putting all of this together yields the Beer-Lambert law equation, which relates the attenuation, or light intensity decrease caused by the sample, to the product of the solute concentration, c, and the path length, ,  through the cuvette: We note further that Beer-Lambert law units for the concentration, c, are typically given in moles per liter (mol/L) and that the cuvette path length is in the centimeters (cm) range. Therefore, the dimensions of the extinction coefficient, , are typically given in L. mol-1 . cm-1. In addition, it should be noted that the absorbance, Abs, is a dimensionless quantity, such that an absorbance is a number greater than zero for peaks of interest.

### Beer-Lambert Law Example

In this Beer-Lambert law example, the chemist wants to calculate enzyme activity using UV-vis spectroscopy. To do so, a UV-vis active metabolite must be produced by the enzyme, with an absorbance within the 200nm-800nm range.

The experiment involves transferring the buffered enzyme solution into a quartz cuvette, which is placed in the reference cell slot of the spectrophotometer. Another portion of the buffered enzyme solution is transferred to a different quartz cuvette, and a metabolite that is metabolized by the enzyme is added, and the whole is immediately placed in the sample cell slot. The absorbances of both cells are taken incrementally until there is no change in metabolite absorbance.

Using an alternate form of the Beer-Lambert equation, the solute (metabolite) concentration, c, in the sample is calculated. The molar extinction coefficient of the metabolite in the sample solution and the cuvette path length are taken into account. The calculated metabolite concentration, c, data points are then plotted against time, and a curve is fitted to the scattered data points. The slope of a linear region of the fitted curve is then determined, which is the enzyme activity.

### Limitations of the Beer-Lambert Law

In summary, there are some limitations to the Beer-Lambert law. The first and most important limitation is the requirement for a UV-vis active solute to be present in the sample solution for a detection signal to be generated by a UV-vis spectrophotometer. Another limitation is that changes in the intermolecular forces between the solute and solvent may affect the Beer-Lambert law graph in a non-linear way. Therefore, it is important to choose a wavelength, λnm, that displays a linear response proportional to changes in the solute concentration. By keeping these limitations in mind, the Beer-Lambert law can be used effectively to determine