### Statistical Mechanics Lecture 8 of 29

April 13, 2012 by Antonello Scardicchio

A. Scardicchio, ICTP

This lesson continues the study of phase transitions focusing in the critical point for the vapor - liquid transition. Remembering that the critical point is the state where the vapor-liquid transition line in the P - T plane ends. This means that as the substance approaches critical temperature (Tc), the properties of its gas and liquid phases converge, resulting in only one phase at the critical point: a homogeneous supercritical fluid. This was proved in this lesson, were it was shown that below the critical temperature there is a discontinuity in density at the phase transition and that this difference in density goes to zero as the temperature converge to the critical temperature. This is done with the help of the reduced form of the van der Waals equation of state expanded near the critical point using a third order in volume polynomial function.

It was also shown that the vaporization heat decreases as the critical point is approaches from below so no work is needed to pas from the gas phase to the supercritical fluid phase. In the second part of the lesson the minimal principle of the Gibbs free energy (G) is used to explain the absence of the coexistence of the two phases for T > Tc. For this porpoise G was expanded in powers of the volume till the 4th power on volume near the critical temperature and it was shown that for T > Tc the G has just one minimum for all variation of pressure (dp) and volume (dv) so just one solution is possible. For T = Tc then the G have a very flat minimum near to dv. Now if T< Tc then G has two minimum meaning that the two phases gas and liquid may coexist in a small range of pressure (dp < 4dt) if dp > 4 dt then one phase is favored and becomes the global minimum and just one phase exist.

As the <span class="clickable"><span class="q">disappearance</span></span> of the liquid and vapor phases is really an amazing phenomena, here we propose the videos of two more experiments.

Professor Martyn Poliakoff demonstrates supercritical fluids in his office at the University of Nottingham.

Professor Martyn Poliakoff demonstrates supercritical fluids in his office at the University of Nottingham.

Close look at supercritical carbon dioxide CO2.Close look at supercritical carbon dioxide CO2.

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