# 100 Nano-Stories: Is Beer’s Laws Breakable (Part 2)?

## Episode #75: Direct Transmittance Equations x Beer’s Law!

5 min readApr 17, 2021

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# Preface! ✨

It’s your favorite material science & nanotechnology enthusiast! Today, we will cover how The Beer-Lambert Law can be “violated” when calculating the total transmittance of light from a material/aerogel using direct transmittance equations!

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

## TL;DR → Beer’s Law Violations (Part 1)! 🔑

• The incoming wavelengths of light that enter the aerogel are not one wavelength of light only (monochromatic).
• All the light that enters the aerogel should be in a parallel beam/parallel rays, to minimize the amount of scattering that could happen when the light is trying to directly transmit through the aerogel.
• The absorbing particles inside the aerogel are distributed evenly throughout the aerogel without bumps or fancy shapes!

Now that we have the main violations of the Beer-Lambert Law for aerogels, we can move into a brief explanation of the equations that govern the direct transmittance of light in aerogels (if the properties are not violated, of course 😉)!

# Transmittance x Beer’s Law Equations! 💡

## Concepts! 🔑

The equations that I will show below are under the Beer-Lambert Law. The Beer-Lambert Law is a relation of the logarithmic dependence of loss of radiant energy/intensity through a medium/material and the properties of the material the light is passing through.

The second thing that you should know is that the equations that I will show under The Beer-Lambert Law are only looking for Direct/Diffuse Transmissivity. Scattering/Absorption can cause a change in the results calculated for transmissivity/transparency in the aerogel.

## Equation 1! 🔑

When looking for the total transmissivity of light (the light that passes through the aerogel without being scattered or absorbed), we can use the equation listed above for the parallel beam of monochromatic light.

I = Final intensity of the monochromatic light after it transmits through the aerogel.

I(0) = Initial intensity of the monochromatic light before it transmits through the aerogel.

exp = Euler’s Number (“e”). e/exp ≈ 2.72.

β(ext) = Extinction Coefficient (The sum of the absorption coefficient and the scattering coefficient in the aerogel).

x = Distance/Length of the aerogel.

Beer’s Law applies to this equation because there is a dependence on the beam of light (parallel rays of light only), and the properties of the aerogel for the light to directly transmit, absorb, or scatter the light. This is where Beer’s Law could get “violated” due to the potential hindrances listed above in [TL;DR Section].

## Equation 2! 🔑

The second equation is not controlled/applied by Beer’s Law because it is only looking for total transmittance. We are looking for the Transmission Coefficient, which is a measure of how much light can pass through a material (in this case, aerogel!).

To calculate the transmission coefficient, we will be calculating it via the intensity of the wavelength of the light that is passing through the aerogel (also, because we used the wavelength intensity in the first equation to apply it with Beer’s Law).

The equation for the transmission coefficient is shown below:

t(direct) = Direct Transmissivity.

exp = Euler’s Number (“e”). e/exp ≈ 2.72.

β(ext) = Extinction Coefficient (The sum of the absorption coefficient and the scattering coefficient in the aerogel).

x = Distance/Length of the aerogel.

The main reason why Beer’s Law doesn’t apply to the transmission coefficient is that this equation can be used to solve for the direct beam of the parallel ray of light (Equation 1).

In reality, the original equation to calculate the direct/parallel beam of light intensity transmission can be changed from this:

to this:

## I(x) = I(0) * t(direct)

t(direct) = exp (-β(ext (x)))

# Closing Thoughts! 💭

There we have it! Two different equations were brought together to explain how the violation of The Beer-Lambert Law can be destroyed if not for the combination of both the overall intensity of the wavelengths of light and the transmission coefficient!

See you tomorrow to review more of the transparent aerogel research papers! ✌🏽

# 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.

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.

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

Diffuse TransmittanceLight passes through the aerogel and is scattered throughout the network of pores. The light will leave the aerogel at an angle rather than straight through the aerogel.

The Beer-Lambert Law A relation of the logarithmic dependence of loss of radiant energy/intensity through a medium/material and the properties of the material the light is passing through.

Transmission CoefficientA measure of how much light can pass through a material (in this case, aerogel!) by calculating via the intensity of the wavelength of the light.

# cjarquin0005@gmail.com

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