Is there a phase transition between a gas and plasma? Does a phase transition occur as a gas is heated to create a plasma? If so, is this a first or second order phase transition?
Also, does the presence of a phase transition depend on the pressure or composition? It seems to me that in the dilute limit (i.e. low pressures), no phase transition should occur because the fraction of atoms that are ionized will follow a Boltzmann distribution, which is a smooth function of temperature. However, the presence of phase transitions in Debye-Hückel theory seems to suggest that a gas-plasma transition could occur at higher pressures.
 A: The short rough answer is no. The transition between gaseous state and plasma is continuous and gradual. Phase transition typically happens at constant temperature for given pressure, which doesn't happen for plasma. Have a look here.
Some references classify the transition from gaseous state to plasma as a special type of phase transition called second order phase transition.The difference between the second order and first order (standard well known phase transition) is that second order is gradual while first order is sudden. Have a look here.
So if you are referring to standard definition of phase transition, the answer is no.
Hopefully that helped
A: The plasma properties already become palpable at a low degree of ionization. On the other hand the degree of ionization never reaches 100% in a macroscopic plasma (where thermal collisions occur): there will always be some electrons and ions recombining somewhere (equilibrium). So it seems like the 'perfect' plasma state is only asymptotic as $T\rightarrow\infty$. For practical purposes, a plasma with a high degree of ionization is considered fully ionized.
A: The obvious order parameter is something like the fraction of atoms or molecules ionized, say "x'. Simple  stat mech says that at any finite temperature that will be in the range (0,1). The question is whether, for some density of particles, there's a discontinuity as a function of T in either x (first order) or dx/dT (second order). There's a reason some such discontinuity could happen. As more ions form, the Debye screening cloud lowers the free-energy of ion formation. That cooperative phenomenon, like many others, e.g. spin-spin interactions in magnets,can in principle lead to runaway feedback to a new phase. It's a quantitative question. A very quick first calculation indicates that unless the density of ions becomes larger than ~(kT/e^2)^3 (in cgs units), there won't be a phase transition. (Here k is Boltzmann's constant, T is temperature, and e is the ion charge.) Since x doesn't get large until T is pretty high, (Boltzmann factor), and since you don't have a gas to start with unless the overall concentration is pretty low, I think you typically don't get a phase transition. Somebody with more detailed knowledge should check that. Of course, the quantitative comparisons are completely different for say quark-gluon matter, etc.
A: This is a very interesting question and since I have thought before about it, I would like to share my answer.
As far as I am concerned, there has been no empirical evidence of coexistence between a fully ionized plasma and a neutral gas as separated phases in contact such as ice and water at $0^{\circ}$C. As @Gotaquestion correctly pointed out, the transition between gaseous and plasma state is continuous and gradual.
However, the heat capacity at constant volume, $C_{V}$, and also the heat capacity at constant pressure, $C_{p}$, exhibit peak values in some temperature intervals where atom/molecule ionization due to energy exchange gets more probable. In the figure below, extracted from an old paper form Phys. Fluids by Drellishak et al, $C_{V}(T)$ and $C_{p}(T)$ curves were calculated using thermodynamics principles and the partition function of diatomic nitrogen and diatomic oxygen.


In these figures, the peaks show the effect of very strong increase in the specific heats mainly due to energy loss by electron ionization. After each peak, the curve decreases to a minimum which is always higher than the previous. This happens because after a peak occurs, the corresponding ionized electrons are introduced in the system, giving their own contribution to the specific heat by increasing the degrees of freedom of the system.
Note that as the peaks get narrower they should resemble more and more a second order phase transition. Second order phase transitions are characterized by continuous $G(T, P), S(T,P), V(T,P)$, but discontinuous constant pressure heat capacity $C_{p}(T,P)$ (a good reference may be find here).
Finally, I must point some caveats in this answer. Here I considered what is called "classic plasma", which uses classical statistical mechanics in the treatment of Debye-Hückel.
I am not a specialist in quantum mechanical plasmas, but maybe in such systems other kind of phase transition may occur.
