Why does a critical point exist? I still cannot fully comprehend the essence of a critical point on phase diagrams. 
It is usually said in textbooks that the difference between liquid and gaseous state of a substance is quantitative rather than qualitative. While it is easy to understand for a liquid-solid transition (symmetry breaking is a qualitative change), it is unclear to me what meaning does it have for a liquid and its gas: there is always a quantitative difference between a gas at 300 $K$ and at 400 $K$.


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*Is it correct to say just "this substance is in gaseous state"? Shouldn't we also specify the path on the phase diagram by which the substance got in its current state? Did it cross the boiling curve or went over the critical point and never boiled?

*Why does a critical point even exist? Blindly, I would assume that either there is no boiling curve at all - since the difference is quantitative, the density of a substance smoothly decreases with temperatures and increases with pressure; or that the boiling curve goes on to "infinity" (to as high pressures and temperatures as would remain molecules intact). Why does it stop?
 A: For a pure substance that can exist in the solid, liquid, and vapor states (i.e., wood is not in this category), let's assume that a closed container is half full of liquid and half full of vapor.  As the temperature rises, the liquid expands and the liquid density falls.  Also, as the temperature rises, the pressure in the container rises due to the vapor pressure of this material, so the vapor density rises.  At some point, the vapor density becomes equal to the liquid density, and only one phase can exist.  This occurs at the critical temperature and the critical pressure.
The most common example of a material above its critical temperature is air.  No matter how much you compress air, it will not condense at room temperature.  This may just be a personal preference, but I would call air at room temperature a gas (as opposed to a vapor).
A: Good question. I don't have my Widom around, but I'll try to answer from memory.


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*I think the consensus is to say a substance is at its gas state if it could be a liquid at the same temperature. This, as opposed to same pressure, same volume, etc. If the temperature is supercritical, there is no transition between liquid and gas, and the generic term "fluid" should be used.

*You can imagine that at really high temperatures the fluid should be in a highly disordered state. If you start with a dilute gas and pressurize it, you should be able to make it as dense as desired. Hence, there should be no liquid-gas transition at high temperatures. Therefore, if one exists at low temperatures, it must end. Notice a liquid-gas transition could be preempted by a solid-gas one and not exist at all. Also, there is always a solid-gas transition, because that one is due to the molecule's hard cores (ultimately quantum in origin), and largely unaffected by temperature.
A: Attempting to answer the "why" question intuitively:
In a liquid, the molecules experience significant intermolecular force - so much so, that the average energy of the molecules is insufficient to escape the attractive force of the surrounding materials. The result is that it energetically favorable for them to remain close together, even if that means they do not fill all available space (space above the liquid).
As temperature rises, the vapor pressure of the liquid increases as a greater number of molecules reach "escape velocity". In the process they remove energy from the liquid (greater than average velocity needed to escape). However, if you continue to increase the temperature, you will eventually reach a point where the increase in entropy offsets the loss of energy due to vaporization - in other words, there is no longer a "penalty" for a molecule to go from one state to the other, and the distinction between the two phases disappears.
A: 
Why does a critical point even exist?

I think this question is equal to this one:
"Why the width of the two phase region is bigger at lower temperatures and pressures?"

Specific volume of liquids mostly depends on the temperature of them in comparing with their pressure. This means, for a well-defined increment of the pressure, we can neglect its effect on the specific volume of the liquid with respect to increment of the temperature of the liquid. So, by increasing the temperature, specific volume of the saturated liquid will increase.
On the other hand, specific volume of gases mostly depends on the pressure of them in comparing with their temperature. For a well-defined increment of the temperature, the influence of the pressure increment will be more much than temperature. So, by increasing the temperature, specific volume of the saturated vapor will decrease.
Thus, by increasing the temperature, the width of the two phase region will decrease and because of continuity of the properties of the substances, finally this width will be eliminated. I.e. a critical point certainly will exist.
A: I will try to answer these questions from different views.
Macroscopic view
The "quantitative" rather than qualitative difference in a liquid-gas phase transition is due to the fact that the molecules arrangement does not change so much (there is no qualitative difference) but the value of the compressibility changes a lot (quantitative difference). This can be easily seen in the Van der Waals isotherms below the critical temperature,

The phase transition occurs at the dashed line $AD$. For volumes smaller than $V_D$, the high slope of the curve means that one needs a huge amount of pressure in order to decrease a small amount of volume. This characterizes a liquid phase which has a very low compressibility. For the slope is much lesser and the compressibility is high, which characterizes a gas. In between $V_D$ and $V_A$ there is a mixed phase characterizes by a divergent compressibility, i.e. the volume changes at constant pressure. 
Above the critical temperature there is no longer such a radical change in the compressibility. The Van der Waals isotherm is the following

As you mentioned the density continually increase with the pressure. You can also see from the Van der Waals equation, when written as
$$p=\frac{NkT}{V-Nb}-a\frac{N^2}{V^2},$$
that at very high temperatures it behaves like
$$p\rightarrow \frac{NkT}{V-Nb},$$
which is not qualitatively different from an ideal gas isotherm. There is no liquid phase.
Microscopic view
Let us consider a substance below its critical temperature. After a phase transition from gas to liquid, a meniscus (interface) appears between the liquid portion and a vapor (gas) portion which is present due to the kinetic distribution of velocities. The vapor has much smaller density so a molecule in the bulk of the liquid have more bonds than a molecule in the surface (interface). Each bond has a negative binding energy (bonded states) so the molecules in the surface have an excess of energy. 

This gives rise to a (positive) surface energy density which is nothing but the surface tension of the interface. When we increase the temperature, the vapor density increases and at some point it equals the liquid density. At this point the number of bonds for the molecules in the bulk and in the surface equals so that there is no surface tension. This means there is no meniscus and no phase transition. There must be a critical point therefore.
