*From*: "Bill Smith" <wsmith %-% at %-% msnet.mathstat.uoguelph.ca>*Organization*: Math & Stats, University of Guelph*Subject*: Re: CCL:critical point*Date*: Thu, 19 Oct 2000 12:15:15 -0400

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 properties. (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 "sub-cooled 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 POINT. 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; http://www.mathstat.uoguelph.ca/faculty/smith/