A Necklace of Pearls as a One-Dimensional Fluid Andras Baranyai and lmre Ruff L. Ebtvos University, H-1088 Budapest, Muzeum krt. 6-8, Hungary
Owing to the considerable development in the technique of diffraction experiments on liquids as well as in the theory of fluids based on integral equations and computer simulations, our knowledge of the structure of fluids has been aonearine in graduate courses. This helm to chanee the tkkitional proportions of physical chemistry-treatGg the gaseous state a t ereat leneth. the solid state a t some 1enzt.b. and the liquid ;ate praitidally not at all-to more even proportions, thus serving the needs of chemists for whom liquids and solutions are, perhaps, the most important states of matter. The complicated nature of the mathematical problems associated with the theory of liquids is one of the main reasons why they are still rather neglected. Although the level of mathematical analysis required, say, to solve the Ornstein-Zernike integral equation is rather high, it is still not higher than the level needed to solve the Schrodinger equation. As one would not expect every chemistry student to solve the latter for a molecule more complicated than H2+ at most. one should expect also onlv the minimum, hut essential, knowledge of the statistical dechanics of fluids. In thisuauer we discuss the structure of a hard-core model whose th&&odynamics was treated by Runnels in this Journal.'
of a one-component real fluid. I t is also well known that, after reaching a definite number density, the hard-core model shows the same kind of phase transitions of freezing or melting.=
The Palr Correlation Function
The structure of disordered systems can be characterized by the statistical average of the surroundings of all particles each taken as the origin of a coordinate system. The function givinginformation on this average structure is called the pair correlation function gg(lrj- ril), or its simpler version used for the case of radially symmetric intermolecular potentials gU(rij)where r; and r, are the vectors pointing to the spatial position of an ith and jth type of particle, while rij is the distance between two particles. The physical significance of this function, which can he deduced in ageneral way from the statistical theory of fluids, can be easily visualized as follows. I t represents the probability of finding a jth kind of particle at a distance ri, from an ith kind of particle. This probability is normalized to the number density of particle type j in the entire phase. The pair correlation functions are not applied to crystalline solids, since they give less information than available from diffraction exneriments (viz. aneular correlations). The gv, of gases, on the other hand, is uninformati\.e due to their hiehdisorder. This is made auiteobviouj bv thegirl's shown i n - ~ i ~ u r1e for a hypothe&cal ideal crystal, r e d crystal, liquid, real gas, and perfect gas, respectively. The Hard-core Fluid
It is well known that the short-range order in dense fluids is mainly due to the steep repulsion wing of the pair potential. that is. to the excluded volume effect. The&) u . . function for a fluid consisting of uniform hard spheres freeof interactionsisagood approximation o f f h r pair correlation function 'Runnels, L. K. Chem. Educ. 1970,47,742. Croxton. C. A. Liquid State Physics: Cambridge University: 1974; Hansen, J. P.; McDonald. L. R. Theory of Simple Liquids; Academic: New York. 1976. 400
Journal of Chemical Education
Fog,re
1 Schsmat8c pair correlat,on lunn~ons(a) for an ,deal crystal !the
venm mes represen! Dwac funclmns 01 (01 n le ne ghtl, lo) lor a rea Crysta (C) for a liquid, id)fora real gas, and(@ for the perfect g s .
.
I t is thus quite obvious why so much effort has been made in the last decades to derive the exact pair correlation function and equation of state of the hard-core fluid. The most important achievement was the analytical solution of the Percus-Yevick e q ~ a t i o nThis . ~ (mathematically complicated) solution yields a simple, closed, formula for the equation of state of the hard-core fluid, which is, however, not exact, and the final results differ depending on the choice whether the virial or the compressibility equation is used for closure:
Figure 2. Theeven distribution of the pointlike particles of the OnedimenJionai perfect gas (see text).
given arrangement, since the particles are undistinguishable. The nth particle can he selected in N different ways, while the number n - 1before the nth can he selected in
(z)
Still they proved to be good approximations, because they flank closely the function fitting computer simulation results that can be considered exact4:
where q = so3pl6 is the so-called packing density in which o is the diameter of the rigid spheres and p the number density. The equation of state of a hard-core fluid is the first auantitative and rieorous theoretical step from the perfect gas toward real dense tluids. As such, it Ean be regaided as the theoretical determination of the semi-empirical van der Waals b coefficient. From the solution of the Percus-Yevick equation, the&) function can be calculated analytically in definite ranges. It can also be obtained numerically by various methods. The Pair Correlatlon Function of a Necklace of Pearls The advantages of the one-dimensional hard-core fluid, whose mechanical model is a necklace of pearls threaded together in a more or less dense way, in comparison with a three-dimensional one are as follows. (1) Both the equation of state and the pair correlation function are easily visualized and derived for anyone less familiar with mathematical methods. ('2) Exact resultscan beobtained. (3) foragiven set of parameters theg(r1 functioncan he calculated even witha prorrammnhle pocket calculator, and thus it is exceptionally &
