Re: CCL:critical point

Critical points for pure fluids and binaries are discussed in most
 chemical engineering thermodynamics texts.
 In a fluid system consisting of a single component, there is a
 unique critical POINT.  That means that the critical T, P, and
 density are unique values for that fluid. For T>Tc, the substance is
 called a "fluid", since it has both gas-like and liquid-like
 (In the culture of the field, a "supercritical fluid" often means that
 T>Tc AND P>Pc.)  For T<Tc, and above the triple point
 temperature, the fluid is called a vapor if it is a gas (engineers call
 it "superheated vapor"); otherwise it is a liquid (called a
 liquid" or "compressed liquid").
 Mathematically, the dimension of the critical point is a matter of
 the "state postulate" for pure fluids, an extended form of which is
 the "phase rule" for multicomponent systems.  The state postulate
 for a pure fluid states that the number of independent intensive
 variables for pure-phase behavior of a single substance is 2.  This
 is a _postulate_, based on experimental observations -  NOT a
 theorem.  There are 2 critical conditions, resulting in a critical
 In a mixture, it depends on the number of components.  For a
 binary mixture, there are critical LINES, which are curves in (P,T,x)
 space.  There are also special critical points called tricritical
 points, as well as other more complex phenomena.  For some of
 the mathematical discussion, see, for example,
 I. Nezbeda, J. Kolafa, and W. R. Smith, J. Chem. Soc. Faraday
 Trans. 93, 3073 (1997)
 J. Kolafa, I. Nezbeda, J. Pavlicek, and W. R. Smith, Phys. Schem.
 Chem. Phys. 1, 4233 (1999)
 	For ternaries and higher, it gets more complicated.  In general,
 the dimension of the critical point space is N-1, where N is the
 number of species in the mixture.  The coordinates of the critical
 manifold are (T,P,x1,x2,...x_{N-1}).
 	If there are chemical reactions occurring, the dimension of the
 critical point manifold is decreased by one for each linearly
 independent reaction that occurs in the system.  See, for example,
 Y. Jiang. G. R. Chapman, and W.R. Smith, "On the geometry of
 chemical reaction and phase equilibria", Fluid Phase Equilibria,
 118, 77-102 (1996).
 On 19 Oct 00, at 11:45, Artem R. Oganov wrote:
 > BlankDear CCLers,
 > I wonder if it is possible to say that in a fluid system there is a
 > unique value of the critical density, above which the system is
 > necessarily liquid, and below which it is gaseous - at temperatures
 > below the critical, and arbitrary pressure. This seems to be the case,
 > but is there any mathematical proof that this critical density will be
 > the same at all pressures and all subcritical temperatures? I'll
 > summarise if requested. Thanks a lot,
 Best Regards,
 W. R. Smith, Professor
 Dept. of Mathematics and Statistics and School of Engineering
 Room 546 MacNaughton Building
 University of Guelph
 Stone Road and Gordon Streets
 Guelph, Ontario, CANADA N1G 2W1
 Tel: 519-824-4120, ext. 3038; FAX: 519-837-0221;