Exploring Exponential Decay in Light Absorption

[Moogie]

Hi

I'm self-teaching calculus and I'm looking at exponential growth and decay. The differential equation for relationships like these if this was related to time is
dP/dt = -kP
i.e. the rate of change in P with time decreases at a rate which is proportional to the amount of P present. I can understand this equation if it relates to, say, the change in concentration of a reactant with time during a reaction.

I was then trying to apply the same understanding to the exponential decrease in transmission of light through a sample in a UV spectrometer as concentration increases

If i relate this to the picture I have attached, it looks as if to transmission of light drops 50% for a particular concentration of light absorbing molecules.

For a particular concentration of solution, 50% of the incident light is absorbed. If you then add this amount again, another 50% of the incident light is absorbed (which is 50% of the remaining 50% = 25%, leaving 25% light and 75% absorbed)

I can't really understand the chemical basis for this. Why is the amount of light absorbed proportional to the light that is present? If there are more light absorbing molecules present in solution, why don't they simply 'suck up' all of the light that is available to them. Why isn't there a simple linear decrease in transmission i.e. double the number of light absorbing molecules, double the light absorbed.

many thanks in advance for your help

Note: i think I've added the wrong graph but the one i wanted looks the same but just says % transmission down the y axis

 

[nasu]

You can think in terms of collisions (or interactions) between two kind of particles: photons of light and molecules of absorbent.
The number of interactions in a given time depends on how many of both particles are present. If there are very few photons, there are relatively few interactions even if there are many molecules of absorbent.

 

[Moogie]

Hi

Thanks for your reply. I don't really know how light is quantified. I know the basic EMR e = hf formula but that's it. I believe this gives the energy in one photon of frequency f. The energy in 2 photons is presumably 2hf? So following on from this, is it ok to conceptualise that if there is 100% of an amount of light present there are, for example, 100 photons and if there is 50% of light left there is 50% of the original energy e.g. 50 photons.

In other words the amount of energy/light is directly proportional to the number of photons present?

 

[nasu]

Yes, the energy in a monochromatic beam (all photons have the same energy) is proportional to the number of photons.
It's not necessary to use the quantum picture to understand scattering and absorption.
The wave picture can be used as well. I just thought it may give a better understanding.

You can also think about reflection from a glass surface, for example. The reflection coefficient is about 4% for normal incidence. This means that 4% of the light is reflected. The rest is transmitted into the glass.
No matter how strong or how weak is the beam, 4% is reflected. Even if the beam is very weak, the glass will not reflect it completely (but only 4%).

 

[rwisz]

Hello,

Thank you for sharing your thoughts on exploring exponential decay in light absorption. I can offer some insight into the questions you have raised.

Firstly, it is important to understand that the concept of exponential decay can be applied to various natural phenomena, not just chemical reactions. In this case, we are looking at the decay of light intensity as it passes through a solution containing light-absorbing molecules. The differential equation you have mentioned (dP/dt = -kP) is a general representation of exponential decay and can be applied to different scenarios.

In the case of light absorption, the rate of decrease in light intensity is proportional to the amount of light-absorbing molecules present in the solution. This is because these molecules have the ability to absorb and scatter light, which reduces the overall intensity of the transmitted light. As the concentration of light-absorbing molecules increases, the rate of decrease in light intensity also increases, leading to a non-linear relationship between concentration and light absorption.

To understand why the decrease in light transmission is not linear, we need to consider the behavior of light and how it interacts with matter. When light passes through a solution, it can be absorbed, scattered, or transmitted depending on the properties of the molecules present. In the case of light-absorbing molecules, they have specific energy levels that allow them to absorb certain wavelengths of light. As the concentration of these molecules increases, more and more light is absorbed, leading to a decrease in transmission.

However, there is a limit to how much light can be absorbed by these molecules. As the concentration increases, the available energy levels become saturated, and the rate of light absorption decreases. This is why we see an exponential decay in light absorption rather than a linear decrease.

I hope this helps to clarify some of your questions. Keep exploring and learning about exponential decay and its applications in different fields. Best of luck with your studies!