How Does Exponential Decay Explain Light Absorption in Solutions?

[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%).