Electron gas in metals: more like a liquid? I'm trying to form a figure in my head about how electron bindings look like in metals. The electrons in metals can move freely, as opposed to their pair-bindings in molecules, and form what is often referred to as an "electron gas".
I'm puzzled as to this choice of words. I assume this "gas" in metal can't be compressed much - one can't put twice as many electrons in a piece of metal as there are normally in it, as one could with regular gas. Is this assumption correct?
If so, why is it called a gas, and not a liquid? I'm wondering specifically because the analogy seems to be obvious:


*

*electron solids = molecules

*electron liquids = metals

*"electron gas" = free electrons


Is the choice of words here simply historical or is there a deeper reason why the gas analogy is more appropriate for electrons in metal?
 A: Quick answer : electrons are a liquid, but you can replace them with fake electrons which behave like they were in a gas, so both formulations are correct and meaningful.
Now for a longer answer. 
Indeed, the valence electrons in (most) metals form a state which is closer to a liquid than to a gas. This liquid of interacting electrons is called a Fermi liquid (because electrons are fermions). 
On the one hand, electrons have to respect Pauli exclusion principle. The Hamiltonian describing the electrons has eigenstates corresponding to different energies, and the fundamental state of the system is obtained by filling these states with electrons, from the lowest energy to the point where there are no electron left. The object you get is called a Fermi sea, the surface of which (called the Fermi surface) separates filled states from empty states. 
On the other hand, when you want to describe the electrons in a metal, what you really want to do is to describe some properties of them, typically their conductivity, which characterizes their response to an electric field (applied e.g. by a voltage source). It appears that to describe a lot of such properties, only the "elementary excitations" of the system are needed. Landau theory of the Fermi liquid shows that in a wide range of systems these excitations happen only near the Fermi surface, where you can create, for example, an electron-hole pair (see e.g. this answer), which behave like particle (e.g. an electron), even if they are in fact a complicated mix of all electrons. Such objects are called quasiparticles, and I can now get to your question : these quasi-particles behave like almost free particles, and as such, they form a gas. Quasi-electrons, for example, have the same charge than an electron, but not the same mass, because of all the interactions between real electrons, so you can describe a metal like a gas of "electrons" with some effective mass, and forget about all the underlying description. Hence the name.
Now, beware : even if this Landau theory of the Fermi liquid is extremely efficient, it does not always work. When interactions between electrons are strong, this descriptions breaks down, and you get more complex yet more interesting physics (Mott insulators, Luttinger liquids, etc.).
A: I suggest that we need 'electron gas' for metals instead of 'electron liquids' because electrons in metals move much more like particles of gas than particles of liquid.
The big difference between gas and liquid is that gas fills its container and the particles can all travel throughout the volume of the gas and (for an ideal gas) the spacings between particles are much larger than the dimensions of the particles so they roam very freely in the volume and only rarely bump into each other. - For liquids the particles are constantlu in contact with other particles and they can move, but their movement is much more restricted by collisions.
Your suggestion is very good though and people do think about 'electron sea' or 'a sea of electrons' in the metal. A google search on 'electron sea' will pull up lots on this.
ultimately the model of a 'gas', 'liquid' or 'sea' is not perfect and we need to think about quantum mechanics and the Pauli exclusion principle to describe/understand electrons in a metal.  
