# 100 Nano-Stories: Beer-Lambert Law!

## Episode #67: Liquid & Gas + Absorption Coefficients!

# Preface! ✨

It’s your favorite material science & nanotechnology enthusiast! It’s time to introduce another concept in creating transparent aerogels!

For all the new readers, I highly recommend checking out the previous article on the single-scattering albedo to understand some of the topics we will be discussing!

Don’t worry, the article is only 3 minutes long! 😃

A brief overview of the single scattering albedo is:

**Single-Scattering Albedo**is used to define the scattering and absorption in a medium/material.- Equation →
**ωλ = σ(s)/ β(e)** **ω = Single-Scattering Albedo symbol.****σ(s) = The Scattering Coefficient.****β = The Extinction Coefficient. (sum of absorption and scattering coefficient).****λ = Wavelength Symbol.**

But for now, let’s move on to the Beer-Lambert Law!

# Beer-Lambert Law Explained! 💡

## Definitions! 🔑

**The Beer-Lambert Law **relates 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.

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.

## Abbsorbant Coefficient Equations! 🔑

However, the absorption coefficient is composed of many other constants that we need to be wary of.

The absorption coefficient is made up of this equation:

**β(a) = A/d**

**β(a) → **Absorption Coefficient (determines how absorbent the material will absorb certain wavelengths of light). Usually measured in* ***cm^-1.**

**A → **Absorbance in the material. The equation for absorbance is **log10(I/I(0)).**

**I(0) → **Original intensity of the light (before it passed through the material).

**I →** Intensity of the light after it passed through the material.

**d → **Distance/Length the light traveled through the material.

Basically, if the Absorbant 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.**

**The smaller the number for absorption, the more transparent our aerogels can be! 😁**

## Transmissivity Equation! 🔑

Now that we know what the meaning of our absorption coefficient is about, we can move on to the bigger picture of the Beer-Lambert Law!

We mentioned in the absorbance coefficient that we need to divide the final intensity of a specific wavelength of light over the initial intensity of a specific wavelength of light.

The (x) located to the final intensity (I) means it is flowing in the x-direction (x can mean either up, down, left, right, etc., you can decide the direction if it makes it easier for you to understand 😄)

In this case, we can have two ways of expressing the transmission equation (shown below):

But in the end, we end up with the transmissivity **(t) **of a certain wavelength of light. The **(x) **means the distance traveled. **(t) **is the transmittance, which is the light that was able to be directly transferred to the other side of the material without scattering or being absorbed.

## Aerogel Applications! 🔑

With these equations, we can calculate the haze in any aerogel, whether it is transparent, silica, polymer, etc.

To calculate haze, we would use this equation:

**T → **Transmissivity.

If you notice, it follows a similar equation as the absorbance **(log10 (I)/I(0)).**

What is happening in this equation is the haze that an aerogel that scatters more than absorbs will result in a higher haze than a partially absorbing medium at the same optical depth (“distance”).

The reason this phenomenon occurs is that absorption does not contribute to the haze/opaqueness in the aerogel (because once absorbed by the particles in the aerogel, it can no longer escape/scatter around).

This way, we can calculate the diffuse intensity of the aerogel, and the overall transmittance, or the light that makes it through the other side without being scattered!

# Closing Thoughts! 💭

Well, that was a little tough to write, but as* **Manasi Patil** **once said:*

*“It's the hard that makes it beautiful!”*

I will be talking a bit more about organic chemistry for the next article! See you soon! ✌🏽

# Vocabulary! 📓

** 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.

** Absorption Coefficient** →

**(β(a))**

** Scattering Coefficient** →

**(β(s))**

*Wavelength Symbol → *λ

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

**τ.**

** Transparency → **All 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.

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

** 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.

# Bonus Resources! 💻

# Previous “100 Nano-Stories!” 🔖

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