F. 1. Pilar University
of
N e w Hampshire Durham 03824
The Critical Temperature: A Necessary Consequence of Gas Non-Ideality
A n ideal or perfect gas, i.e., one described bv the well-known eauation of state PV = nRT, is characterized by the lack of interactions among particles which themselves have no finite volume. If such gases mere actually to exist they would exhibit no condensation phenomena and, furthermore, could be compressed to zero volume. It is the interparticle interactions which account for the condensed states of matter, and it is the finite volumes of the particles which account for the essential incompressibilities of such condensed states. Not so well known (at least to beginning students of chemistry) is the closely related fact that any gas composed of mutually interacting particles of finite volume must exhibit a critical temperature. The existence of such a critical temperature can be deduced from the mathematical properties of equations of state which contain parameters analogous to the van der Waals a and b parameters; however, the necessity of a critical temperature can be made plausible by the use of much simpler and more easily visualizable arguments. It is the purpose of this paper to illustrate this point in a simple, essentially non-mathematical fashion suitahle for presentation to beginning students of chemistry. As a model of a real gas we choose N spherical particles assumed to he mutually interacting but bereft of internal structure. Neither the geometry of such particles nor their lack of internal structure (electronic,vibrational, rotational, etc.) is relevant to the qualitative aspects of the illustration. At a low enough pressure and/or a high enough temperature any such gas behaves approximately as the hypothetical ideal gas. The total energy E of the system of N particles may be written as the sum of a kinetic energy term K and a potential energy term V ,
N
V
=
C Vij
.-.,
(3)
>,:
Each term V, represents an interaction between an arbitrary pair of particles labelled i and j. All we need to know about V,, is that it represents an attraction (V,, < 0) when the particles are far apart and a repulsion (Va > 0) when the particles are very close together and that it goes through a minimum at some finite value of the interparticle distance r,,. A plot of V , versus r,, must show that Vij goes to infinity as r,, approaches zero (repulsion due to resistance of two particles occupying the same region of space), that V,, asyrnptotically approaches some constant limiting value (equal to zero by convention) as r,j approaches infinity (attraction between particles decreases as the distance apart increases), and that V,, has a minimum value at some intermediate value of r u (point at which repulsion begins to negate attraction). Extensive experimental and theoretical investigations are in clear accord with the above form of Vq even though many of the details of the interaction are not known. We shall further assume that the V/s are temperature independent-an assumption which is not strictly valid but which simplifies the illustration without affectingthe qualitative results. It is convenient to characterize the physical state of the sample in terms of the energy ratio defined by
r
When >> 1, the system behaves as a typical gas, i.e., the kinetic energy of the random translational motions essentially swamp out any orientation tendencies due to interparticle interactions. If, however, 1, the
T, one can only decrease /VI so that f cannot fall below unity. It is this temperature T,which one calls the cdical temperature. Rigorous quantitative verification of the above model for the critical region of a gas is complicated by several factors, but it is possible to illustrate some qualitative aspects by means of very simple calculations. The average molar potential energy of a liquid is given a p proximately by
-.
