100 Nano-Stories: Is Beer’s Laws Breakable (Part 1)?

Episode #74: Beer-Lambert Law x Transmittance!

Carlos Manuel Jarquín Sánchez
5 min readApr 17, 2021

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!

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

Don’t worry, the article is only a 5-minute read! 😁

TL;DR → Beer-Lambert Law (Part 2)! 🔑

To calculate the total transmittance of an aerogel with an optical depth of less than 1 (τ < 1), we can use the equation below:

In some cases, τ is almost equal to 0 in some transparent aerogels, but the reason why sometimes the optical depth (τ) doesn’t reach zero is that some of the light was absorbed or scattered. However, if the single-scattering albedo (ω = 0) is equal to zero, the light that enters the aerogel can only have one of two outcomes:

  • The light passes through the aerogel directly or by diffusing through the network of particles in the aerogel.
  • The light is absorbed by the aerogel particles.

When ω = 0, we can calculate the total absorption of a 100% transparent aerogel via this equation:

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

t = Total Transmissivity of light in the aerogel.

Perfect! Hope that cleared up everything! In the meantime, let's move more into how the Beer-Lambert Law is not enough to solve for haze/opaqueness in aerogels!

Beer-Lambert Law Broken! 💡

Absorption v. Scattering! 🔑

The reason why I never brought up equations for scattering in the TL;DR of the previous is that scattering contributes to blurriness/haze in the aerogel.

Absorption and direct/diffused transmission of light don’t contribute to haze because either of them can’t return into the aerogel once it’s left the material or it can’t escape because it has been absorbed by the aerogel particles.

But there is one factor that can render the properties of Beer’s Law invalid results: Incident Radiation (the light/radiation that is entering the aerogel).

Breaking Beer’s Law! 🔑

There are 3 ways the incident radiation can destroy the Beer-Lambert Law to calculate transmissivity of the transparent aerogel:

  • The incoming wavelengths of light that enter the aerogel are not one wavelength of light only (monochromatic). (Ex: Silica Aerogel can become more blurry than what it is if wavelengths of violet light enter the aerogel because they could be scattered throughout the network of aerogel particles).
There is some violet light on the upper half of the silica aerogel!
  • 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. If confused, look at the example below:
Option A is the beams of light that will minimize scattering/haze in the aerogel.
  • While the light is trying to get through to the other side, the beams of parallel rays of light should be traveling at the same length from one another. This can be taken care of if the absorbing particle inside the aerogel is distributed evenly throughout the aerogel. If you're confused by this, check the example below:

This aerogel is distributed evenly throughout the air (no carving or fancy shapes), where scattering can happen.

This aerogel contains so much scattering because there are many bumps and shapes, so the aerogel is not distributed evenly throughout the air, where scattering can happen.

Closing Thoughts! 💭

Now we understand the importance of The Beer-Lambert Law and how it can be violated!

A bonus violation of Beer’s Law was that it can be broken if the aerogel particles depended on one another to scatter light. However, that doesn’t occur in aerogels, the aerogel particles will scatter the light at their own will.

See you tomorrow for The Equations demonstrating the violations of Beer’s Law! ✌🏽

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.

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