On the Interpretation of Atomic Distribution Curves for Liquids

It is shown that if we approximate to a liquid by an Einstein model, in which each atom has a restricted region of motion, wherein it moves independently of its neighbors, and is surrounded by coordination shells of other atoms; and if we denote the density distribution of the atoms by $\rho \mathrm{}\left(r\right)\mathrm{}$, where $r$ is the radial distance from any given atom; then the contribution to $\rho \mathrm{}\left(r\right)\mathrm{}$ made by any coordination shell, e.g., the $i$th is a function ${\rho }_i\mathrm{}\left(r\right)\mathrm{}$ for which $r{\rho }_i\mathrm{}\left(r\right)\mathrm{}$ is symmetrical about its corresponding maximum value. The complete distribution curve, $r\rho \mathrm{}\left(r\right)\mathrm{}$ against $r$, is the sum of peaks of equal width and similar shape. A semi-empirical application of this theory to liquid sodium (a reapplication of C. N. Wall's theory) gives a latent heat of melting in fair agreement with experiment. The model suggests a change of structure on melting, since agreement with the experimental distribution curve is impossible if the number of atoms in the first coordination shell is that of solid sodium. It is also shown that if the parameters in a partition function developed in this way are chosen to give agreement with any one physical property, then it is incorrect to add to the partition function terms representing "communal entropy."