The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres

Equations of state have been derived for one, two and three-dimensional gases of hard elastic spheres of finite size. For the linear gas the phase integral is directly integrable, and by using the virial the equation was found to be $fl=\frac{\mathrm{NkT}}{(1-\theta )}$, $f$ being the force of the gas, $l$ the length of the gas, $N$ the total number of atoms, and $\theta$ the fraction of the gas length occupied by the atoms themselves. By two tests a linear gas consisting of a single atom was found on a time average basis to obey statistical laws, the first showing that the equation of state is unchanged if $l$ be subdivided into $N$ equal compartments with an atom in each; and the second showing that the length of the single atom gas consisting of one atom bounded by its two neighbors obeys the Boltzmann equation. This is significant for the Debye-Hückel theory of electrolytes. For the plane gas, Boltzmann's method of correcting the virial is used for low $\theta$, the fraction of the surface which would be covered in close triangular packing, giving $\tau a=NkT(1+1.814\mathrm{}\theta +2.573\mathrm{}{\theta }^2+\cdots )$, where $\tau$ is the surface tension and $a$ the area of the gas. For $\theta$ near unity, a combination treatment involving both the virial and an extension of the concepts developed for one dimension leads to $\tau a=\frac{\mathrm{NkT}}{(1-{\theta }^{\frac{1}{2}})}$. The adsorption isotherm of the plane gas is derived from the equation of state and is compared with those for films in which the adatoms occupy fixed positions on the surface. In three dimensions the same methods give $pv=NkT (1+2.962\mathrm{}\theta +5.483\mathrm{}{\theta }^2+\cdots )$ and $pv=\frac{\mathrm{NkT}}{(1-{\theta }^{\frac{1}{2}})}$, respectively. Expressions which match these near $\theta =0\mathrm{}$ and $\theta =1\mathrm{}$ but cover the complete range are given.