How probable is it that a gas giant planet made of "air" (roughly the same composition as Earth's atmosphere) could exist?
What would it be like? How would it differ from Earth?
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There are two parts to this question:
Giant planets relatively close to the central star (but beyond the frost line) will accrete large quantities of hydrogen and helium, which were the major components of the initial gas and dust that surrounded the Sun when it first formed. These planets are massive enough that it is simple for them to retain their atmospheres of these light gases.
However, less massive planets (such as the terrestrial planets) will lose any initial hydrogen and helium envelopes via atmospheric escape. The only planets that can possibly retain these gases are the most massive giant planets, which then amass enormous atmospheres. On these planets, there is no possibility of having an Earth-like atmosphere.
Planets further out can have interesting compositions.1 If you move far enough away from the central star, ice giants can forms. Uranus and Neptune are examples (they are not, technically, gas giants). These giant planets are composed of heavier elements than hydrogen and helium - ices, which may include oxygen, methane, sulfur and nitrogen. They also, however, have hydrogen and helium, which, though composing much of their volume, only contribute about 1/5 of their total mass.
Ice giants are better targets for the kind of atmosphere you want. The issue, though, is that the oxygen and nitrogen you want may be in solid form, and even if some of it is gaseous, there's still a large atmosphere of hydrogen and helium to deal with.
The good thing is that exoplanets are not just either terrestrial planets or gas/ice giants. There are several intermediate classes that could help you:
All of those planets sound similar, if not the same. That's because not much is known about them, and their characteristics are largely hypothetical, though some candidates have been observed. Tweak some things and maybe, just maybe, you can figure out a way for one of them to have "air."
1 Although they may have formed further in and migrated outwards.