# What is Beer-Lambert law?

by egpat        Posted on 05 Jun 2019

What happens when an electromagnetic radiation falls on a matter? Suppose, if the matter is transparent to radiation, nothing absorbed and entire radiation is transmitted through the medium. Conversely, if the matter absorbs some energy it only transmits remaining energy which is of course with reduced intensity. Can we measure that absorption? Yes, that's possible with Beer-Lambert law.

## Interaction of EMR with matter

What happens when a radiation is absorbed by a medium? Where the absorbed energy utilized?

To answer these questions, first we have to see how absorption of EMR takes place. Suppose you have green ball in your hand. Can you see the ball in a dark room? Certainly you can’t. You can see the green ball either in day light or under the electric light.

So the ball is visible when visible radiation falls on the green ball. That’s fine, but why it is appearing as green colour. Here the absorption comes in to play.

When visible radiation falls on the ball, it absorbs some part of energy and then transmits the remaining part of the energy which is perceived by our eye as green color.

Within this process, the absorbed energy is utilized for electronic transitions such as π → π* and  n→ π*  in the molecules of the green ball. These two transitions are mostly responsible for the absorption of radiation within the UV-visible region.

## Estimation of absorption

Now let’s come to the point that how can we measure this absorption.

Suppose we are sending an electromagnetic radiation of intensity Io through the analyte. Analyte is placed in a cell. What happens to the light we send? We can observe any of three processes such as reflection, scattering and absorption.

If surface of the cell is not that much smooth, a part of the radiation we send may be reflected back.

Similarly if any small solid particles are present in the solution, the radiation may be scattered to the different angles.

Finally, if the analyte is able to absorb the radiation, some part of the energy is absorbed and remaining radiation is transmitted with intensity It.

If first two processes are eliminated we really observe the absorption of radiation by the analyte only. This can be achieved by using highly polished surfaces of cells to eliminate reflection and thoroughly filtered solutions as analyte to eliminate scattering.

Now if the reflection and scattering are eliminated then intensity of the radiation that is absorbed is given by

Iab=Io - It

But this is not the absorption we measure practically in UV-visible spectroscopy. This indicates the intensity of the radiation that is absorbed by the analyte. Instead of going with difference in the intensities of radiation before and after the sample, we measure the ratio of the corresponding intensities and label as absorption.

So absorbance is the ratio of logarithms of intensities of incident radiation to the transmitted radiation.

Absorbance=A=Io / It

## Beer-Lambert’s law

Now we can see how Beer-Lambert law can be used for estimation of absorbance.

### Beer’s law

Suppose you have a dark colored solution in a test tube. You have made serial dilutions and filled into the other test tubes each with same volume. Now which absorbs more energy? Undoubtedly, it is the dark colored solution. By with each successive dilution, the intensity of color of the solution is reduced and even become transparent after sufficient dilutions.

It indicates that the absorption depends on the concentration of the analyte. Higher the concentration, higher will be the absorption.

Beer’s law states this phenomenon in terms of intensity and can be states simply as follows.

The intensity of a beam of parallel monochromatic radiation decreases exponentially with the no. of absorbing molecules.

It α Io e-c

Where It=Intensity of transmitted light

Io=Intensity of incident light

C=Concentration of the analyte

To be valid, here we should use monochromatic radiation

### Lambert’s law

Now consider another situation where the dark colored solution is filled in a test tube with 10 cm  diameter and another test tube with 50 cm diameter. Which absorbs more radiation?

The two test tubes have same solution with equal concentration and they only differ in volume. As the EMR travels more in test tube 2 , the energy may be more absorbed in this test tube.

Stated in another way, the absorption depends on the total path travelled by EMR. Therefore, just like Beer’s law, Lambert’s law can be defined as follows

The intensity of a beam of parallel monochromatic radiation decreases exponentially as it passes through a medium of homogenous thickness.

It α Io e-b

Where b is the path length of the analyte through which EMR travels

Here again the radiation should be monochromatic and analyte should be of uniform thickness.

Combining these two laws,

It α Io e-bc

It=Io e-kbc

Where K is a constant

Since the equation is in exponential form, let’s take logarithm and change the base to 10.

Log (It/Io)=-abc

Where a  is another constant obtained by conversion of base of logarithm to 10.

a=K * 2.303

Rearranging the equation to eliminate negative sign,

Log (Io/It)=abc

This is the absorbance.

A=Log (Io/It)=abc

Here, a=absorptivity

b=Path length

c=Concentration