"Boiling does not occur when liquid is heated in a closed vessel. On heating continuously vapour pressure increases. At first a boundary is visible between the liquid and vapour phases because liquid is more denser than vapour. As temperature increases more and more molecules go to vapour phase and density of vapours rises. At the same time liquid becomes less dense. It expands because molecules move apart. When the density of liquid and vapours becomes the same; the clear boundary between the two phases disappears. This temperature is called critical temperature"
That is EXACTLY what my textbook says. And that is where I am having doubts. It's because - a point might still be found below the critical temperature where the density of both the phases become equal if the pressure is considerably below its critical pressure ($P_c$) and it has a volume sufficiently larger than its critical volume ($V_c$).
Just consider any point in the Pressure-Volume isotherms of $CO_2$ given above, which is in the region of gas-liquid equilibrium (within the shaded dome shaped curve) but has a volume greater than the critical one. For e.g a point that lies
1. Within the shaded region.
2. On a line passing through B and parallel to the pressure axis.
Let that point represent the state of liquid inside the vessel. From there if we increase the temperature, we would arrive at a point on the boundary of the dome shaped curve, (viz. B) but that point would be on a lower isotherm than the critical isotherm. And it looks like B is also a point where the two densities becomes equal.(isn't it?) Based on that premise, we can't say that the temperature arrived at in the above (heating liquid in a clossed vessel) situation would necessarily be the Critical temperature.
What I want to ask is if my reasoning is correct or am I missing anything?
NOTE: My question can be rephrased in the following manner also:
Is it true that critical temperature is the only temperature at which the density of the gaseous phase of a gas becomes equal to its liquid density?