I think that something is invisible if it's isolated particles are smaller than the wavelength of visible light. Is this correct?
Why is air invisible? What about other gases and fumes which are visible?
I think the pithy answer is that our eyes adapted to see the subset of the electromagnetic spectrum where air has no absorption peaks. If we saw in different frequency ranges, then air would scatter the light we saw, and our eyes would be less useful.
I don't think one can just state that particle size smaller than the wavelength of light implies no interaction with light. You have to look at the quantum mechanical modes of the atoms/molecules. If they have modes with frequencies in the wvaelength range you are interested in, then you will get interaction/absorption. I think that even clean air does have nonzero absorption in the visible. Have a large enough column density along a beam of light, and there will be some absorption.
Also do note, that the speed of light is slower in air than in a vacuum, so the air does have some effect. You do get effects of refraction, such as mirages, and heat waves seen looking across convection over a hor surface. Also if you point a telescope at a star, you see the mess the atmosphere makes of the image.
Air is mostly composed of first two row small molecules like O2 and N2, and atoms like Argon. For all these, the absorption is in the deep ultraviolet. Molecules which absorb in the visible have smaller energy differences in their first absorption bands, although chlorine(2) and Iodine(2) molecules are visible. This answer is meant to be complementary to and supportive of the evolutionary answer given above
Even without any absorption peaks, there is Rayleigh scattering
which works on particles much smaller than wavelength of light, such as gas molecules. Rayleigh scattering is basically why sky is blue (and why it would be blue even if the air contained no oxygen). The scattering is inversely proportional to wavelength to 4th power, so the blue light is scattered a lot more than red light.
Air is very sparse compared with liquids and solids, so there's just much less material to block light. Secondly, air is very uniform and homogenous so there are no edges to cause reflections and scattering, except in the extreme case of a explosion shockwave. For the same reason, still water is mostly invisible except at the surface, while frothy water is very visible. Thirdly, the molecules that comprise air are very simple and don't have a lot of lines in the visible range. This is in contrast to complex organic molecules with a lot of vibrational and rotational modes. Also, our eyes evolved to see in a range where air is transparent.
The air has purely real rifraction indix $\sim 1$, no dispersive, "like vacuum". So there is not absorption, and the air is transparent for visible wavelenght, like X rays for metals over $plasm frequency$.
Simplistic answer: But it isn't invisible; it can bee seen from outside - just look at photos from the ISS or shuttle. It is just that the observer standing inside it doesn't see it unless something unusual occurs such as fire or fumes.
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