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I am to show that the number of absorbed photons only is proportional to the optical density at low optical densities.

I do not know how to do this, but this reminds me of the linear range of absorbance vs concentration; at high concentrations, the relationships is no longer linear. I also feel that the "number of absorbed photons" is another way of saying "absorbance", and that "optical density" is another way of saying "concentration times optical path length", but I am not sure. I cannot find references that clarifies this for me, so I would appreciate some guidance.

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Yes, your statement that "the number of absorbed photons only is proportional to the optical density at low optical densities" is correct. (I'll point out that optical density is a depreciated term for what is now referred to as absorbance.)

First let's define some terms.

T: Transmittance, i.e. the fraction of photons not absorbed. So a transmittance of 0.90 would mean that 90% of the photon pass through the sample.

A: Absorbance of sample which is given by the Beer–Lambert law:
$\ce{A = -log_{10}(T)}$ or: $\ce{T = 10^{-A}}$

Now a bit of math swizzling...

For exponentials of $e$ there is a nice series expansion. $$ e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dfrac{x^4}{3!} + ... $$ and if $x < 0 $ then: $$ e^{-x} = 1 - x + \dfrac{x^2}{2!} - \dfrac{x^3}{3!} + \dfrac{x^4}{3!} + ... $$ now if also $|x| << 1$ then the higher order terms can be ignored and the function becomes linear where: $$ e^{-x} \approx 1 - x $$

Now for a bit more math voodoo... Let's let $$10^x = e^y$$ so: $$y = x\mathrm{ln}(10) \approx 2.303x$$

So now substituting $\ce{e^{-2.303A}}$ for $\ce{10^{-A}}$ we get: $$\ce{T = e^{-2.303A}}$$ and thus only when $2.303A << 1$ is the Beer-Lambert law linear.

Now for the rest of the story let's define some more terms:

$\epsilon$ is the molar attenuation coefficient of the attenuating species in the sample;

$c$ is the molar concentration of the attenuating species in the sample;

$l$ is the path length of the beam of light through the material sample.

Now also accoding to the Beer-Lambert law:
$\ce{A = \epsilon c l}$ so $\ce{A}$ is proportional to $\ce{c}$.

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