Comparison of Some Exact and Approximate Results for Gases of Parallel Hard Lines, Squares, and Cubes

Publisher Website
Google Scholar
Abstract
The first seven virial coefficients for hard parallel lines, squares, and cubes, as derived from approximations of the ring and watermelon type, are compared with the exact coefficients. These approximations give no useful information as to the sign or magnitude of the virial coefficients. A Cartesian distribution function depending upon only one space coordinate arises naturally for the line, square, and cube molecules. The first four terms of the exact number density expansion of this function are presented and compared with results obtained by iteration from the Percus—Yevick, Kirkwood, and convolution integral equations. The Percus—Yevick equation yields a distribution function which closely resembles the exact result at low densities. Virial coefficients are obtained from the approximate distribution functions by means of the Ornstein—Zernicke relation and the virial theorem, as well as from a relation between the potential of mean force at zero separation and the virial coefficients. This last relation (which is valid for hard spheres as well as lines, squares, and cubes) has an interesting graphical interpretation and leads to correct values for the third virial coefficient from the Kirkwood equation, but not from the Percus—Yevick or convolution equations.