Gr\"uneisen Parameter for Gases

The Gr\"uneisen ratio ($\Gamma$), i.e.\,the ratio of the linear thermal expansivity to the specific heat at constant pressure, quantifies the degree of anharmonicity of the potential governing the physical properties of a system. While $\Gamma$ has been intensively explored in solid state physics, very little is known about its behavior for gases. This is most likely due to the difficulties posed to carry out both thermal expansion and specific heat measurements in gases with high accuracy as a function of pressure and temperature. Furthermore, to the best of our knowledge a comprehensive discussion about the peculiarities of the Gr\"uneisen ratio is still lacking in the literature. Here we report on a detailed and comprehensive overview of the Gr\"uneisen ratio. Particular emphasis is placed on the analysis of $\Gamma$ for gases. The main findings of this work are: \emph{i)} for the Van der Waals gas $\Gamma$ depends only on the co-volume $b$ due to interaction effects, it is smaller than that for the ideal gas ($\Gamma$ = 2/3) and diverges upon approaching the critical volume; \emph{ii)} for the Bose-Einstein condensation of an ideal boson gas, assuming the transition as first-order $\Gamma$ diverges upon approaching a critical volume, similarly to the Van der Waals gas; \emph{iii)} for $^4$He at the superfluid transition $\Gamma$ shows a singular behavior. Our results reveal that $\Gamma$ can be used as an appropriate experimental tool to explore pressure-induced critical points.