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A classic example of bioluminescence; the protein luciferin is oxidized by the enzyme luciferase, releasing energy as light. What is it that gives luciferin this ability? Why can't just about any protein display bioluminescence?
Are there more chemical reactions where electromagnetic waves are produced? Also, are there any chemical reactions which produce high energy electromagnetic waves, such as X-Rays?
[...] the protein luciferin is oxidized by the enzyme luciferase
Luciferin is not a protein, but a benzothiazole with a thiazole attached to the carbon atom between nitrogen an sulfur.
Upon oxidation at the thiazole ring, an oxetanone is formed. This four-membered ring breaks, releases carbon dioxide and turns the thiazole into a thiazolone. The resulting molecule is called oxy-luciferine.
Amazingly, the oxy-luciferine generated through this reaction is in an excited state! While typically excited states are the result of the absorption of a photon, here excited state is generated chemically!
Anyway, the fate of this excited state is as to be expected: it relaxes to the ground state by emitting a photon.
In general, these reactions are known as chemoluminescence.
Another known example is the peroxyoxalate reaction of glow sticks.
Phosphorescence at ~450 nm has been observed from the comproportionation of cyclohexanoxy radicals in polymer matrices, when hydrogen atom atom transfer between two radicals produce cyclohexanol and cyclohexanone (in an excited triplet state).
Klaus gave a nice summary on luciferin. The second part of your question was:
Also, are there any chemical reactions which produce high energy electromagnetic waves, such as X-Rays?
Any energy released with an emitted photon must previously have been added into the molecule (or atom) in question. For example, if singlet oxygen is created chemically, it will relax (slowly; the mechanism is, naturally, spin-forbidden) back to triplet oxygen; the energy difference corresponds to red light which is why a faint red glow can be observed with darkness-adapted eyes. For most compounds, the emitted photon corresponds well to the energy difference between a one-electron excited state and the ground state. These can often be calculated quantum-chemically.
For the vast majority of molecules, the energy difference between HOMO and LUMO — the one most likely to get excited by a random proton, and thus also the most likely one to get emitted — corresponds to middle-ultraviolet, near-ultraviolet or visible wavelengths. For example, most phenyl rings have an absorption at a wavelength around $250~\mathrm{nm}$. In some very simple molecules, this difference may extend into the slightly far ultraviolet region.
Typical X-rays have a much smaller wavelength. It helps considering how they are generated in a bench-top manner: a surface (typically a metal) is bombarded with accelerated electrons which knock out core electrons from their atomic orbitals. These holes are filled by the relaxation of valence electrons. For example, if an electron is displaced from the 1s orbital, an electron from a 2p orbital may replace it. If that occurs, energy is released as a photon and the corresponding radiation is termed $\mathrm{K_\alpha}$: $\mathrm{K}$ signifies the shell into which the electron relaxes and $\alpha$ denotes the first relaxation that can fill this void.
The $\mathrm{K_\alpha}$ transition is typically the most important and one of the most intense while all K-series transitions are those with the highest energy. This energy is often close to a wavelength of $0.1~\mathrm{nm}$ — note how far away that is from your average near-UV HOMO-LUMO transition.
Thus, the energy difference that corresponds to X-ray emission is typically that of core orbitals which is much larger than anything present out in valence orbitals even if bonding/antibonding interaction differences are concerned. No chemoluminescent-type emission will be able to emit X-rays unless you’re doing something really bizarre (e.g. bombarding with electrons).
Camara et al. published what can classify as ‘something bizarre’ from the former paragraph: rolling of sticky tape, the researchers found triboluminescence with photon wavelengths right up to ultraviolet ranges — in vacuum.[1] They published a nice graph that reveals how the required peeling force rises and then suddenly drops with concomitant emission of light. Under ambient pressure, the force differences are much smaller and the photon energy released equally. They attribute this behaviour to static electricity discharge, the static electricity building up during the peeling and then suddenly discharging. They were even able to use the X-rays to record an X-ray of one of their fingers.[1]
Remember though that this is only possible in a vacuum ($10^{-3}~\mathrm{torr}$); under atmosphere, the charge separation is probably neutralised much faster and at much lower energies probably due to the interaction of air molecules.
Reference:
C. G. Camara, J. V. Escobar, J. R. Hird, S. J. Putterman, Nature 2008, 455, 1089. DOI: 10.1038/nature07378.
