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# Beer-Lambert Law

Consider the following experiment in which you prepare a chemical solution while working in a chemistry lab. The solute, which we will also refer to as the sample, absorbs ultraviolet-visible (UV-vis) light, but the solvent either does not absorb or absorbs weakly within the UV-vis wavelength range (200nm-800nm). Pure water, ethanol, and acetone are examples of solvents that do not absorb or absorb weakly in the UV-vis wavelength range.

Continuing the experiment, you then transfer some of the chemical solutions you just prepared to a quartz cuvette, which we will refer to as the sample cell. In turn, you prepare a separate cuvette in which only pure solvent is as a reference. We will refer to the reference quartz cuvette as the reference cell. As an experimenter, you then transfer these two cuvette cells into a spectrophotometer so that the cuvettes are placed into appropriate slots: one in the sample cell slot and the other in the reference cell slot. Light passes through both slots, and spectrograms are subsequently recorded and compared.

Ultraviolet-visible (UV-vis) spectroscopy uses a spectrophotometer capable of generating and measuring light signals within the UV-vis wavelength range (200nm-800nm) to obtain UV-vis absorbance spectra of solvents and solutes in a chemical solution.

As this experiment relates to UV-vis absorption spectroscopy, we will continue with a discussion of the Beer-Lambert law. In this article, we will discuss:

• The Beer-Lambert law equation – here, we derive the Beer-Lambert equation in relation to the experimental setup.
• The Beer-Lambert law graph – here, we show how to set up a Beer-Lambert graph.
• The Beer-Lambert law units – we present the Beer-Lambert law units used to calculate the absorbance.
• a Beer-Lambert Law example – this example goes over finding the enzyme activity.
• And finally, we present the limitations of the Beer-Lambert law.

## 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

We use the ratio, ${I}_{0}}{I}$, to calculate the absorbance due to the sample via the following formula, the absorbance formula:

$Abs={\mathrm{log}}_{10}\left(\frac{{I}_{0}}{I}\right)$

The absorbance formula considers the light attenuation due to light absorption from the solute relative to the absorption from the solvent. At wavelengths of interest, in which the solvent does not appreciably absorb UV-vis light but the solute does absorb light strongly, the intensity ratio, ${I}_{0}}{I}$, would be a number larger than one. That is, more light reaches the detector when passing through the reference cell yielding a high value for ${I}_{0}$relative to the light reaching the detector from the sample cell which yields a smaller value for the light intensity, I, because the sample has absorbed light that would have otherwise reached the detector. The plot of absorbance, Abs, versus wavelength, the Beer-Lambert graph, yields positive sample absorbance peaks at wavelengths of interest, that is where the ratio, ${I}_{0}}{I}$, is greater than one, ${I}_{0}}{I}>1$:

$Ab{s}_{wavelength}={\mathrm{log}}_{10}\left(\frac{{I}_{0}}{I}\right)>1\underset{}{\to }thesoluteabsorbslightatthatwavelength$

Figure 3: Beer-Lambert Graph

## The Beer-Lambert Law Equation

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

1. The sample absorbance, Abs, is directly proportional to the light path length, $\mathcal{l}$, through the cuvette.
2. The sample absorbance, Abs, is directly proportional to the solute concentration, c.

In symbols:

$Abs\propto \mathcal{l}\propto c\mathcal{l}$

$Abs\propto \mathcal{l}\propto c\mathcal{l}$

$Abs\propto \mathcal{l}\propto c\mathcal{l}$

$Abs\propto \mathcal{l}\propto c\mathcal{l}$

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

$Abs=\epsilon c\mathcal{l}$

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, $\epsilon$, 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, $\mathcal{l}$, through the cuvette:

$Abs={\mathrm{log}}_{10}\left(\frac{{I}_{0}}{I}\right)=\epsilon c\mathcal{l}$

## Beer-Lambert Law Units

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, $\epsilon$, 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

Consider the following Beer-Lambert law example.

A solution of an enzyme of interest was prepared in a water-based buffer. The chemist would like to know how to calculate enzyme activity using the Beer-Lambert law. We noted earlier that the Beer-Lambert law is used in UV-vis spectroscopy to detect a UV-vis active solute within the sample solution.

Thus, a UV-vis active metabolite must be produced by the activity of the enzyme, that is, the metabolite must have an absorbance within the 200nm-800nm range. The requirement for a UV-vis active solute to be in a sample solution is a requirement of UV-vis spectroscopy. As concerns the Beer-Lambert law, another limitation is intermolecular forces between the solute and the solvent may change as the solute concentration changes, and these changes in the type of solute/solvent intermolecular forces may affect the Beer-Lambert law graph in a way that is not linear.

Returning to the experiment, the chemist then transfers the buffered enzyme solution into a quartz cuvette; this is what we will call the reference cell, which is placed into the reference cell slot of the spectrophotometer. Then another portion of the buffered enzyme solution is transferred to a different quartz cuvette, and a compound that is metabolized by the enzyme is added and the whole is immediately placed in the sample cell slot. The absorbances, Abs, of both cells are taken incrementally until there is no change in the metabolite absorbance. The concentration of the sample solution, c, at each time point is then determined by the beer lambert law equation in the following form:

$c=\frac{Abs}{\epsilon ·\mathcal{l}}$

This is an alternate form of the beer lambert equation used to calculate the solute (metabolite) concentration, c, in a sample; where, $\epsilon$, is the molar extinction coefficient of the metabolite in the sample solution and, $\mathcal{l}$, is the cuvette path length.

The calculated metabolite concentration, c, data points are then plotted versus time. A curve is then fitted to the scattered data points. The slope at a region in which the fitted curve is linear is then determined. This value of the slope of a linear region of the fitted curve is the enzyme activity.

## Limitations of the Beer-Lambert Law

Here we discuss some of the limitations of the Beer-Lambert law. The first and foremost limitation is that a UV-vis active solute is required to be in the sample solution for a detection signal to be generated by a UV-vis spectrophotometer.

### Intermolecular Forces

One must keep in mind that intermolecular forces between solute and solvent may change as the solute concentration changes and these changes in solute/solvent intermolecular interactions may affect the beer lambert law graph in a way that is not linear. Therefore, it is important to pick a wavelength, λnm, in the beer lambert law graph that displays a linear response proportionate to changes in the solute concentration. That is, wavelengths of interest display a linear response proportional to changes in the solute concentration.

## Beer-Lambert Law - Key takeaways

• The formula for the absorbance, when the molar extinction coefficient is not known: $Abs={\mathrm{log}}_{10}\left(\frac{{I}_{0}}{I}\right)$$Abs={\mathrm{log}}_{10}\left(\frac{{I}_{0}}{I}\right)$
• This formula is to be used when the molar extinction coefficient is known is: $Abs=\epsilon c\mathcal{l}$
• An alternate form of the Beer-Lambert equation used to calculate the solute (metabolite) concentration, c, in a sample is: $c=\frac{Abs}{\epsilon ·\mathcal{l}}$
• Limitations of the Beer-Lambert law: 1. A UV-vis active solute is required to be in the sample solution for a detection signal to be generated by a UV-vis spectrophotometer. 2. The effects of intermolecular forces between solute and solvent must be linear for at least some of the wavelengths, λnm, in the beer lambert law graph.
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