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From: "Bill Smith" <wsmith-!at!-msnet.mathstat.uoguelph.ca>
Organization: Math & Stats, University of Guelph
To: chemistry <chemistry -8 at 8- ccl.net>
Date: Thu, 19 Oct 2000 12:15:15 -0400
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Subject: Re: CCL:critical point
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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/