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Consider the following figure:

PT Diagrame

Let us suppose we were at some point in gaseous phase. Can we go from gaseous to liquid phase by moving about critical point continuously?

Why isn’t any critical point in solid liquid coexistence curve? Can we move from liquid phase to solid phase continuously? If yes, how? And if not, why not?

What is the difference between two cases?

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You can go from a gas to a liquid by raising maneuvering at high pressure and temperature to go around the critical point. That is what the phase diagram (whose qualitative form can be derived, by applying the Maxwell construction, from the van der Waals equation of state) indicates, since there are paths from the gas phase to the liquid phase that do not cross the phase transition curve. One way to understand why this is possible is that there is no qualitative difference between the gas and liquid phases. Both of them are fluid, and while a liquid tends to by much less compressible than a gas, that is a difference only in degree.

The case with the solid-fluid transition is different, because there is a qualitative difference between the phases. The solid phase has the atoms laid out in a lattice, which the fluid phases completely lack. The presence of the lattice has direct physical implications, both at the macroscopic and microscopic levels. In particular, a solid can always be distinguished from a fluid by the behavior of the sound waves. Sound waves in a fluid are pure density waves, but sound waves in a solid may be either longitudinal (density) waves—in which the displacement of the atoms from their equilibrium positions is parallel to the direction of propagation—or transverse waves—for which the displacement is perpendicular to the direction the sound wave is propagating.* This qualitative difference in the number of sound wave modes means there has to be discontinuous behavior at the solid-fluid boundary.

*For systems in just one spatial dimension, where there can be no transverse waves, the solid-fluid boundary is actually ambiguous, and there is no qualitative distinction between treating a one-dimensional system as a solid (e.g., a Wigner crystal) or a fluid (e.g., a Luttinger liquid).

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    $\begingroup$ To add to this, the fundamental difference between a liquid and a gas is arguably the positive surface tension for the former. We can gradually go from a positive to a negative surface energy (the zero point is marked by the critical point), whereas we can't transition smoothly from a lack of long-range rearrangement (solid) to the presence of long-range rearrangement (liquid); that's a dichotomy. $\endgroup$ – Chemomechanics Aug 3 '20 at 4:44
  • $\begingroup$ is this "Sound waves in a fluid are pure density waves" true for honey and other very viscous fluids, too? $\endgroup$ – hyportnex Aug 3 '20 at 14:42
  • $\begingroup$ What about symmetry? How that play a role here? $\endgroup$ – Young Kindaichi Aug 3 '20 at 18:28
  • $\begingroup$ @hyportnex High viscosity is not enough to have transverse waves. You need a restoring force that acts even when displaced atoms are not moving, which shear viscosity does not provide. $\endgroup$ – Buzz Aug 4 '20 at 16:44

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