Absorption cross section of light in air

[Big Bird]

When I woke up today in the morning, I had the stupid idea of trying to remember some of my knowledge from university. As it turned out, this was easier thought than done, especially given my still drowsy state of mind.

I want to roughly estimate the penetration depth of visible light in air. First obstacle: what should I assume for the absorption cross section? Should I assume for its diameter the wavelength (600 nm on average), or should I rather assume the bond length (~100 pm)? I know that an accurate analysis would have to take into account the details of the interaction, but as I said, I only want a rough estimate (orders of magnitude).

Anyways if I take the bond length (which would be the best case w.r.t. transparency), I arrive at a cross section of order ~10^-20 m².

Next I considered the number of molecules in a cubic meter of air: N=1000 l/(22.4 l/N_A)~50 N_A~10²⁶.

Now, this seems to lead to a macroscopic cross section of that cubic meter of air of ~10⁶m² as opposed to its geometrical cross section of 1m². That is, the air should be totally opaque already at distances of 1 meter, or the penetration depth would be ~1 micron.

What is wrong about my argument? Or can it even be done on this general level of investigation (i.e. without considering absorption levels of molecules)?

 

[mfb]

Why would the bond length in molecules have any relation to the absorption cross section?

 

[Big Bird]

@mfb: Because it relates to the dimensions of the charge distribution? But as I said, I only meant it as a kind-of best case estimate. If you know any better way of guessing the cross section without the need for detailed quantum mechanical calculations, I'd be glad.

Maybe I should ask the other way around: can we give any general justification for the apparent fact that the cross sections of N₂ and O₂ molecules are much smaller than the charge distribution would indicate (i.e. that the individual molecules are somewhat transparent for visible light)?

Maybe something like: the next resonance is so and so far away from the visible spectrum and the Lorentz oscillator model implies that the absorption is so and so much weaker than if there was a resonance in the visible band itself?

 

[mfb]

You won't get a useful estimate for the cross section without detailed simulations or looking it up.

can we give any general justification for the apparent fact that the cross sections of N2 and O2 molecules are much smaller than the charge distribution would indicate (i.e. that the individual molecules are somewhat transparent for visible light)?

There is no available transition in the range of visible light. You only get these in the vacuum UV range (under 200 nm). That's why it is called vacuum UV - it's strongly absorbed by air.