Spectrophotometry Simplified: The Beer-Lambert Law in Spectrophotometry |ChemFam #65|

Greetings to everyone! Ever wondered how scientists accurately determine the concentration of substances in solutions? Enter the Beer-Lambert Law, a brilliant tool that lets us do just that. Named after its founders, August Beer and Johann Heinrich Lambert, this law is like the magic wand of chemistry, allowing us to measure the amount of light absorbed by a substance and convert it into concentration data. In this article, we'll take a friendly tour through the Beer-Lambert Law, demystifying its science and exploring its real-world applications in the fascinating world of spectrophotometry.

Imagine you have a special flashlight that shoots a beam of light through liquid. Now consider this liquid contains all kinds of molecules. Some of these molecules act like light magnets i.e., they absorb light, while others allow it to pass through without a fuss. Spectrophotometry is the science of measuring how much light gets absorbed and how much light comes out the other side. Thanks to the Beer-Lambert law, this measurements of light and concentration is possible.

Cracking the Code: The Beer-Lambert Law

The Beer-Lambert Law gives us a mathematical way to link the concentration of a substance in a solution to the amount of light it absorbs. It's like a recipe for calculating concentration. The Beer-Lambert Law (also called Beer’s Law) is a relationship between the attenuation of light through a substance and the properties of that substance. In this article, the definitions of transmittance and absorbance of light by a substance are first introduced followed by an explanation of the Beer-Lambert Law.

The Beer-Lambert law is frequently used for absorption and transmission measurements of samples and can be used to determine the concentration of samples. In absorbance measurement, light passes through a cuvette containing the sample. The intensity of the light after passing the cuvette is compared to the light before passing through the cuvette. The size of the cuvette determines the path length (l). The wider the cuvette, the more light sample passes through and the lower the transmission.

What are Transmittance and Absorbance

Consider monochromatic light transmitted through a solution; with an incident intensity of I₀ and a transmitted intensity of I.

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The transmittance, T, of the solution is defined as the ratio of the transmitted intensity, I, over the incident intensity, I₀ and takes values between 0 and 1.

T = I/I₀

However, it is more commonly expressed as a percentage transmittance;

T(%) = 100 I/I₀

The absorbance, A, of the solution is related to the transmittance and incident and transmitted intensities through the following relations:

A = -log₁₀ T

A = log₁₀ I₀/I

The absorbance has a logarithmic relationship to the transmittance; with an absorbance of 0 corresponding to a transmittance of 100% and an absorbance of 1 corresponding to 10% transmittance.

Attenuation of a 510 nm laser through three solutions of Rhodamine 6G with different absorbance values at 510 nm
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Mathematical Formulation of Beer-Lambert Law

When a beam of monochromatic radiation of a suitable frequency passes through a solution, it is absorbed by the solution. As a result the intensity of the light when it finally emerges from the solution, is considerably reduced. If I₀ is the intensity of the incident beam and I_t is the intensity of the transmitted beam then the intensity of the light absorbed, I_a is given by,

I_a = I₀ - I_t

The probability that the photons of a beam of intensity I will be absorbed by the sample is directly proportional to the concentration and thickness of the absorbing solution. Mathematically,

dI/I = -αcdx

where dI is the change in intensity produced by the absorption of radiation on passing through a thickness dx of the solution of concentration c and α is a proportionality constant. The negative sign shows that there is a reduction in intensity. On integrating the equation between limits I = I₀ to I at x = 0 to l , we get,

ln(I/I₀) = 2.303 log (I/I₀) = -αlc

On putting α/2.303 = ε and defining log(I₀/I) as absorbance, A of the solution, we get,

A = log(I₀/I) = εlc

This is the Beer-Lambert equation.

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ε is called the absorption coefficient or extinction coefficient of the absorbing medium. It is characteristic of the solute and depends upon the nature of solvent, temperature and the wave length of the radiation employed. If the concentration is exprerssed in terms of mol dm^-3 and the path length l in cm, then ε is referred as molar absorption coefficient or the molar extinction coefficient.

Beer-Lambert-Law Graph

A graph showing the Beer-Lambert law will be linear and positively correlated. The x-axis will have units of concentration and the y-axis will have units of absorbance. This means that the other two variables in the equations (molar extinction coefficient and path length) remain constant. As the concentration increases, the absorbance also increases. 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 the plots you will see in the Beer-Lambert Law. Various concentrations were measured. Then place a line at these points. The slope of the line is the path length multiplied by the molar extinction coefficient. If the path length is known, the molar extinction coefficient can be easily determined. The molar extinction coefficient is the slope of the line divided by the path length.

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Applications of Beer-Lambert Law

The Beer-Lambert law is commonly used for determining the concentration of a sample of unknown concentration, important for experiments such as the Iodine Clock Reaction.

• Lab Marvel: In the lab, scientists use the Beer-Lambert Law to find out how much of a substance is in a solution. It's like being a detective, solving the mystery of "What's in this liquid?"

• Chemistry Adventures: Chemists and biochemists rely on this law to study all sorts of things—like measuring the amount of a drug in your bloodstream, investigating how enzymes work, or checking the concentration of DNA.

• Eco Guardians: Environmental scientists use it to measure pollution levels in the air, water, and soil, helping us keep our planet healthy.

• Quality Check: In industries like pharmaceuticals and food production, the Beer-Lambert Law ensures that medicines are potent, and our food is pure. It's like the quality control superhero of manufacturing.

Limitations of Beer-Lambert 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 versus 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.

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The law is not obeyed if the radiation used is not monochromatic. The temperature of the system should not be allowed to vary to a large extent. This is because the increase in temperature has a bathochromic effect on ions in solution, i.e., the absorption bands shift towards longer wavelength.

Let us now understand the uses of the law through an example.

• Example: A monochromatic radiation is incident on a solution of 0.05 molar concentration of an absorbing substance. The intensity of the radiation is reduced to one-fourth of the initial value after passing through 10 cm length of the solution. Calculate the molar extinction coefficient of the solution.

• Solution:

According to Beer-Lambert Law,

log (I₀/I) = εlc

In this case, I/I₀ = 0.25 = 25%

i.e., I₀/I = 100/25

log(100/25) = ε.10cm.0.05mol dm^-3

ε = 1.204 dm³ mol^-1 cm^-1

Conclusive thoughts

The Beer-Lambert Law, though it may sound complex, is like a trusty calculator for scientists. It allows us to uncover hidden secrets about the concentration of substances in solutions, paving the way for discoveries in medicine, environmental science, and much more. So, the next time you see a spectrophotometer in action, remember the magic of the Beer-Lambert Law, helping us make sense of the invisible world of light and molecules.

Until we meet again :)

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