Beer-Lambert Law

sid_galt
Let's say a compound having a molar absorptivity of 20000 and a concentration of 3e-5 M is kept in a container 5 cm long.
This gives that each centimetre of solution will absorb 50% of the light passing through it since

$$ln\frac{I}{I_0}=-ebc$$
where
I is the final intensity of light, $$I_0$$ is the initial intensity of light, e is the maximum molar absorptivity, b is the path length and c is the concentration.

The molecule is such that 20 J/cm2 of intensity of light is enough to excite all the molecules in a 1 cm range.

The frequency of the light is such that the energy of one photon is just enough to excite the first fundamental frequency of vibration in the compound molecule.

100J/cm2 is the intensity of light actually passed through the solution.

Since according to the beer lambert law, 50% i.e. 50J/cm2 of light has been absorbed by the 1rst cm of solution, where did the rest 30J/cm2 go when we know that its photons do not have enough energy to excite further fundamental frequencies of vibration in the molecule and that only 20J/cm2 is required to excite the first fundamental frequency?

Staff Emeritus
Gold Member
Is this the entire question reproduced verbatim, or is some part of it the question and some of it your doubts ? Also, about the 50%/cm number - is that your calculation or is it given ? You realize that this can be calculated from the given data ? I get 54.9%/cm for this number, so I'm guessing that 50%/cm just a rough approximation. And to get this number, I'm assuming the molar absorbtivity is 20,000 M-1 cm-1 - because the units are not provided.

Please confirm/correct all of the above.

sid_galt

Gokul43201 said:
Is this the entire question reproduced verbatim, or is some part of it the question and some of it your doubts ?

I was reading about the beer-lambert law and this doubt popped up in my mind. I thought the best way to phrase the doubt would be by asking such a question.

Since QM allows only certain vibration modes, I can't understand where the rest of the energy aside from the 20J/cm2 necessary to put the molecules in the first fundamental frequency of vibration is going?

The rest of the photons cannot excite the other vibrational modes as their frequency is resonant for only the first fundamental frequency.

Calculations
The concentration is in Moles, the length in cm and the molar absorptivity in M-1 cm-1

Also, about the 50%/cm number - is that your calculation or is it given ?

My calculation.
I assumed a concentration of 3e-5 and a molar absorptivity of 20000.

I get 54.9%/cm for this number, so I'm guessing that 50%/cm just a rough approximation.

I checked my calculations again. You are correct, it should be 54.9%/cm.

sid_galt

sid_galt
I think this might be a better way to express myself.

If the intensity of a particular frequency of infrared light is so high that although according to the beer lambert law, x% of light should be absorbed, but the actually only y% of the light is necessary to put all the molecules in excited state where y < x, where did the energy of the excess (x-y)% light go?

Staff Emeritus
Gold Member
Your question is interesting, and I believe it probes an unstated assumption of the B-L Law. I'll have to do a quick calculation (just to make sure I'm not talking nonsense) before I go ahead and answer...perhaps later tonight.

Staff Emeritus