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When taking a picture with old fashioned film what sets the resolution of the picture? Is it the wavelength, or the chemical makeup of the film?

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Awesome question. There was actually a lot of work done on measuring grain sizes in film under different conditions. I'll write an answer soon, when I'm not on my smartphone. –  Colin K Nov 20 '11 at 1:09
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Both the diffraction limit and the grain size could affect the resolution limit of analog pictures. Let's see how they compare.

Diffraction limit The Abbe diffraction limit states that the size of the spot is $d=\lambda f\#$. The $f$-number will depend widely for different cameras and illumination settings. If, for example, you use $f/10$, this gives about 5μm for green light. See Edgar Bonet's comment below for more details on different lenses.

Grain size From this document, it seems that the mean grain size can go from 500 nm up to 30 microns. It depends on the film and the amount of development. More details can be found in this article

In short, both the diffraction circle and the grain size vary from a fraction of a micron to a few microns. The resolution will be determined by the both of them.

You can also check out http://en.wikipedia.org/wiki/Science_of_photography. There's some further info on this topic.

EDITS: I made several edits thanks to the commenters.

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Assuming the light has a D65 spectrum and the film has the same spectral sensitivity as the human eye, the FWHM of the diffraction spot is $N \times 563.6\;\textrm{nm}$, where $N$ is the F-number. Your assumed diffraction-limited f/1 lens is overoptimistic. Typical ultrafast (D)SLR lenses have F-numbers around 1.2 – 1.4. Pro zooms are mostly 2.8. Consumer zooms are in the range 3.5 – 4.5 or even 5.6. And yet, most lenses are not diffraction-limited until you stop them down to circa f/8. At this point the diffraction spot is 4.5 µm in diameter. –  Edgar Bonet Nov 22 '11 at 11:28
    
@EdgarBonet Ouch! Yes, you are right. I was trying to get an upper bound, but used a low f number instead of a high one. I'm more used to thinking about numerical apertures. I updated my comment. If you want, I can make it a community wiki,and you can add all the details about typical cameras. –  Arnoques Nov 22 '11 at 20:14
    
Just adding more data points... From the tests of dxomark.com and dpreview.com, it seems that the very best SLR lenses, resolution-wise, have an MTF50 in the range 40–50 cycles/mm, often achieved for apertures in the range f/4–5.6. These values of MTF50 suggest that the PSF's FWHM should be about 8–10 µm. The tests were however conducted on cameras with 6.4 µm pixel pitch, thus the real PSF should be smaller than observed, and could well be in the 4–5 µm range. –  Edgar Bonet Dec 2 '11 at 14:54
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