100 Nano-Stories: Beer-Lambert Law Properties (Part 2)!

Episode #73: Total Transmittance x Total Absorbance!

Carlos Manuel Jarquín Sánchez
5 min readApr 17, 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 absorbance 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 articles only take 5 minutes to read each!

TL;DR

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.

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).

Enough scattering all the information! Time to absorb the knowledge you all came for! 😉

Total Transmittance x Absorbance Explained! 💡

Concepts! 🔑

In the last article, we touched upon the optical depth of certain materials. The Optical Depth is the quantity of light that has been removed due to absorption, scattering, and reflection; τ.

The equation to calculate optical depth is shown below:

The equation means you have to divide the intensity of the light after it passed the material (I) over the intensity of the light before it passed through the material (I(0)).

The other way to write the equation is e^-τ. In this case, τ = 1.

So this means e^-1 = 1/e. In terms of optical depth, 1/e is approximately equal to 37% transmissivity loss. Transmissivity is the overall light that passes/diffuses through a medium/material.

To make transparent aerogels, we need the optical depth or “τ” to be less than the value of 1. This can result in the object having more transparency, but to make the most transparent aerogels for real-world engineering applications like rockets, solar panels, and Mars Homes, “τ” has to be less than 0.3.

Beer-Lambert Law Property! 🔑

This is a transparent aerogel. This aerogel has an optical depth that is less than 1, τ < 1.

So how do we calculate the total transmittance, Carlos?

Our aerogel sample is transparent, and thanks to the transmission of light passing through the material undergoing direct transmission, we can calculate the total transmittance via this equation:

What this equation means is that the total transmittance is equal to 1 minus the optical depth (τ) times the side of the aerogel where light enters the material (s1) and the exiting side of the aerogel where transmitted light leaves the aerogel (s2).

The 1 comes from the original value of τ, but it can also play a role in calculating the total absorbance of light in the aerogel! This is where the loss of intensity can play a role, but absorbance does not play a role in haze or opaqueness in a material because the light absorbed by the aerogel particles can no longer escape and interact with the rest of the aerogel.

Absorbance Property (if τ < 1)! 🔑

There have been instances where we have achieved a near-perfect transparent aerogel, where the optical depth is at 0, and the single-scattering albedo is also zero.

The single scattering albedo can equal zero if the scattering coefficient equals zero. If you are confused, this is the equation I am talking about, reader!

ωλ = σ(s)/ β(e)

ω = Single-Scattering Albedo Symbol.

σ(s) = The Scattering Coefficient.

β = The Extinction Coefficient.

λ = Wavelength Of Light.

If the scattering coefficient equals zero, then ω = 0. This means that all of the light that doesn’t directly transmit through the aerogel has to be absorbed by the particles of the transparent aerogel.

In this case, the absorbance of a 100% transparent aerogel/material equals this equation:

a = Absorbance (of the light in the aerogel).

t = Total Transmissivity of light in the aerogel.

As I mentioned earlier, the 1 comes from the original value of τ.

Closing Thoughts! 💭

This was a fun article to write! It now explains how we can bring Single-Scattering Albedo, Optical Depth, Transmissivity, Absorbance, and Scattering into 5 beautiful equations!

See you tomorrow to return to the research papers and explain more of the new optical equations in the research of creating optically transparent aerogels! ✌🏽

Vocabulary! 📓

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

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

Absorption The light is absorbed by the aerogel particles.

IsotropicA specific property of a medium/material is the same (or uniform) in all directions.

The Extinction Coefficient → A sum of the absorption coefficient and the scattering coefficient; β(e).

Single-Scattering Albedo Defines the scattering and absorption in a medium/material.

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

Direct TransmittanceLight passes through the aerogel without scattering.

TransmissivityThe overall light that passes/diffuses through a medium/material.

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© 2021 by Carlos Manuel Jarquin Sanchez. All Rights Reserved.

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