In this post we are going to understand the ** exponential factor** in Beer – Lambert’s law, by assigning imaginary values to different parameters in the law. This little exercise will help us to understand the law better.

In post 160, we studied an exponential expression for Beer – Lambert’s law –

**I**_{t} = I_{o} e^{ – αc}

_{t}= I

_{o}e

^{ – αc}

Beer Lambert’s law has an exponential /logarithmic term in it which shows us that the intensity of the transmitted light (I_{t}) decreases exponentially with thickness and concentration of the medium. Now, let us try to understand this concept with an example.

**STEP 1**–

Suppose we have a medium with 1 unit thickness and we shine 100 units of light on it.

Thus, the intensity of incident radiation (I_{o}) =100.

Suppose this medium absorbs 50% of the incident light. This means that the remaining 50% of light is transmitted.

Thus, the intensity of transmitted light (I_{t}) = 50.

**STEP 2 –**

If we further pass this 50% transmitted light through 1 unit thickness of the same medium again, 50% of that light will be absorbed by the medium. In this case, I_{o} = 50.

50% of this light will be absorbed.

50% of 50 = 25.

Thus, the intensity of transmitted light ( I_{t} ) = 50-25 = 25.

**STEP 3 – **

Continuing to pass this light with 25% intensity, through 1 unit thickness medium again, we get 50% of 25 = 12.5 % intensity of light absorbed.

Thus, the intensity of transmitted light ( I_{t} ) = 25-12.5 = 12.5 %

**STEP 4 –**

The intensity of light decreases exponentially again and the intensity of transmitted light becomes 50% of 12.5 = 6.25%.

As seen from the above example, the intensity of transmitted light is NOT decreasing linearly. It is an exponential decrease i.e the value decreases by a different amount each time –

Step No | Decrease in I_{t} value |

STEP 1 | 100-50= 50 |

STEP 2 | 50 – 25 = 25 |

STEP 3 | 25-12.5 = 12.5 |

STEP 4 | 12.5-6.25 = 6.25 |

If we plot these values on a graph, we will get an exponential decrease curve as shown below –

In a linear decrease, the value of the intensity of transmission would decrease by a specific amount say 20 units each time. For example, I_{t} values would be 80, 60, 40, 20 for each step respectively. In this case we would get a linear curve like the one shown below-

Can you see the difference between the two graphs shown above? Thus, we now know that the Beer – Lambert’s law is an exponential expression and thus, it will give us an exponential curve.

In the next post we will solve some numericals based on the Beer – lambert’s law. Till then, be a perpetual student of life and keep learning…

Good day !