100 Nano-Stories: Beer-Lambert Law Properties (Part 1)!

Episode #72: Optical Depth x Total Transmittance!

Carlos Manuel Jarquín Sánchez
5 min readApr 16, 2021

Preface! ✨

It’s your favorite material science & nanotechnology enthusiast! Today, we will cover a few more properties of the Beer-Lambert Law and how it can determine the optical depth and total transmittance throughout a material!

If you need a reminder of the Beer-Lambert Law, I highly recommend you read this article!

Don’t worry, the article only takes 5 minutes to read!

To briefly define what the Beer-Lambert Law is, The Beer-Lambert Law relates the logarithmic dependence of loss of radiant energy/intensity through a medium/material. In other words, it is a relationship between the intensity of light through a material and the properties of the material.

But for now, enough chit-chat! Time to explain more about the Beer-Lambert Law! 😄

Optical Depth x Total Transmittance Explained! 💡

Concepts! 🔑

The Beer-Lambert Law states that there is a dependence between the transmission of energy (light) through the aerogel and the product of the absorption coefficient (all the light that is absorbed by the aerogel) and the length/distance of the material from one end to the other.

Thanks to the Beer-Lambert Law and Optical Depth, we have the transmittance at a value of almost zero to one.

Basically, if the Absorbance in the material results in 0, the transmissivity of light/transparency in a material is 100%. If the absorbency ends up in 1, the transparency of the material is 10%. This means that absorbance has a logarithmic dependence on transmissivity/transparency.

But because of this logarithmic dependence, we want to use the Beer-Lambert Law Properties to find the answer to Optical Depth and Total Transmissivity!

Equations! 🔑

One section of determining optical depth and transmittance is called the extinction coefficient. The Extinction Coefficient is a sum of the absorption coefficient and the scattering coefficient in a material; β(e).

Absorption Coefficient(β(a))

Scattering Coefficient(β(s))

In certain materials, the extinction coefficient is constant throughout the material, which means that there is no change in how much light is absorbed or scattered in a certain section of the material.

If this happens, we can find the optical depth via this equation:

Here are some definitions of the letters and symbols in this equation:

  • τ → Optical Depth: The quantity of light that has been removed due to absorption, scattering, and reflection.
  • s(2) → The side of the material where the transmitted light exits the material.
  • s(1) → The side of the material where the incident light enters the material.

If you are confused about the extinction coefficient (since it's the sum of the absorption coefficient and the scattering coefficient), this is another way to look at the equation!

τ = [(β(a) + (β(s))] (s(2) - (s(1))

[(β(a) + (β(s))] = (β(e))

Property Revealed! 🔑

Since the optical depth is equal to the equation below:

τ = [(β(a) + (β(s))] (s(2) — (s(1))

Any dimensionless value of optical depth in this equation will reduce the optical depth of the intensity of the wavelength of the light to e^-1 / 37% loss. This will also increase the transmissivity of the material!

But Carlos, what equations govern this change between the material, optical depth, and transmissivity?

Optical Depth and Transmissivity can have multiple sections [s(1), s(n)], where the optical depth/transmissivity may change. So we can use these equations to calculate both the optical depth and transmissivity!

Equation 1 → Optical Depth.

Equation 2 → Transmissivity.

If it helps, the total optical depth is the sum of adding all the individual optical depths that we decided to calculate throughout the material (Equation 1).

The total transmissivity is the product of multiplying all the individual transmittances that we decided to calculate throughout the material (Equation 2).

The reason we add the individual optical depth and multiply the individual transmittances is that the Beer-Lambert Law Property of the transparent aerogel having a homogeneous medium full of aerogel particles distributed quite evenly to not let scattering affect the optical depth and the Beer-Lambert Law Property.

Closing Thoughts! 💭

Now we have explained one property of the Beer-Lambert Law, which is the material must be homogenous or have the particles evenly distributed to allow for the equations and the properties of The Beer-Lambert Law to solve for optical depth and transmissivity!

See you tomorrow on how the homogeneous medium and an optical depth of less than 1 can be applied to the current transparent aerogel research! ✌🏽

Vocabulary! 📓

Optical Depth → The quantity of light that has been removed due to absorption, scattering, and reflection; τ.

The Extinction Coefficient → A sum of the absorption coefficient and the scattering coefficient. The volume of the aerogel explains how easily the material can be penetrated by a beam of light, sound, particles, or other energy or matter; β(e).

eEuler’s Number, an irrational number that can be used as an exponential number/constant; approximately 2.72.

Absorption The light is absorbed by the aerogel particles.

HazeIn aerogel, haze is defined as a lack of transparency, or that the aerogel looks somewhat cloudy rather than clear.

TransparencyAll the light will pass through a material, and it won’t be reflected, absorbed, or scatter in a material.

Absorption Coefficient(β(a))

Scattering Coefficient (β(s))

Wavelength Symbol → λ

The Beer-Lambert Law A relation of the logarithmic dependence of loss of radiant energy/intensity through a medium/material. The direction of the energy can be ignored. In the case of aerogels, the energy comes from the light itself.

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