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This is written in the book 'Treatise on Thermodynamics' by Max Plank-

the crossing of the critical temperature at any pressure other than the critical pressure differs in no way from the crossing of any other temperature.

But from the phase diagram of the states, it seems that if we change the temperature at a pressure greater than the critical pressure, then on crossing the critical temperature the gas will liquify where as crossing at other temperatures the gaseous state won't change. So is the statement in the book wrong?

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    $\begingroup$ No, the book's statement is correct. If the temperature is below the critical temperature then the critical pressure won't be needed to liquefy the gas. If the temperature is above the critical temperature then no pressure will liquefy the gas. Only at the critical temperature is the critical pressure needed. $\endgroup$ – MaxW Jun 26 at 7:39
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At the critical temperature/pressure, all properties of liquid and gaseous phase mutually converge to the same values. There is just one phase at these temperatures, neither liquid, neither gas. Or, both at the same time.

Changing pressure at temperature above the critical temperature causes continuous changes of the supercritical phase properties without the edge of the gas/liquid phase transition.

Changing temperature at pressure above the critical pressure causes continuous changes of the supercritical phase properties without the edge of the gas/liquid phase transition.

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What they refer to is how properties such as density and viscosity change when you pass through the critical point.

Say you have water initially at 25°C under one atmosphere pressure. You heat it up and, of course, it boils into steam at 100°C. During the boiling process the density and viscosity drop discontinuously from those values for the liquid to the values for the gas, but once you have formed steam any further changes become continuous and at a finite rate. That includes reaching the critical temperature of 374°C, which really isn't critical at all unless you are also at the critical pressure of 218 atmospheres.

So, now try the same experiment but with 218 atmospheres (ctitical) pressure so that the water "boils" at the critical temperature of 374°C. Now you find the density and viscosity decreasing continuously, with no apparent break due to any phase change. But, at precisely the critical point, those properties are momentarily dropping at an infinite rate, much like the cube root of a number dropping at an infinite rate when the number passes through zero. The critical point is the one point where you get this infinite rate of change within a single phase.

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