If we can heat a gas so that the electrons "fly" off, can we heat one so that the protons do?
When you say fly off, what do you mean? In any case a plasma consists of ionized particles and free electrons. If you mean by fly off, can you get a proton out of a nucleus this way - the answer is no, since it takes a lot more energy than simply heating.
Electrons surround the nucleus and can be removed by visible light (several eV) or higher energy photons (10 eV to kev), such as UV, X-ray (keV) or gamma ray(high keV to MeV). If one heats a gas hot enough, then the atomic collisions can cause electrons to be knock off atoms or molecules. The degree of ionization will depend on the density and temperature of the gas/plasma.
A particle kinetic energy of 1 eV corresponds to a temperature of ~11605 K, so 1 keV ~ 11.6 million K, and 1 MeV ~ 11.6 billion K.
Removing nucleons, i.e., protons and electrons, from the nucleus of an atom normally requires nuclear reactions involving nucleons, electrons, and even photons in the MeV range.
Atomic nuclei are made of protons and neutrons, they aren't something that protons and neutrons are attached to. That said, if you get things hot enough, nuclei will start to break apart. This is a process, though, not a state of matter.
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