When the sun is rising/setting, it goes through a phase where the light is bending from the atmosphere. I believe this image will explain much better than I ever could.
Now, if light goes through a prism, it is bent and the different wavelengths bend different amounts, resulting in the rainbow output.
If the light from the sunrise/sunset is bending, then shouldn't we see a rainbow instead of the red tint? At the very least, shouldn't the top of the sun appear more red and the bottom more purple?
Best Answer
First, it must be said that the picture you provided in your question is extreme. The concept of light bending is true, but the amount that the light bends is nowhere near as large as the picture shows it. The quantification of how much light bends when transferring from one medium to another is called the "index of refraction," and air's index of refraction is very very close to that of a vacuum, so the bending of the light is very small, and the spreading apart of the colors in the light from the bending smaller still. Thus, the prism effect of the atmosphere is too small to notice with our eyes.
If you check the sky after the sun has set (but still providing light), you will see different colors the further up you look from the horizon (at least, I can from my home) in a rainbow-like fashion (although much larger than a rainbow). However, I believe this is due to the Rayleigh scattering of light in the atmosphere rather than any prism effects. (Rayleigh scattering is what makes the sky blue and sunsets red).
Here is a Wikipedia article on atmospheric refraction for more information.