The Yang–Yang relation and the specific heats of propane and carbon dioxide

Abstract

The Yang–Yang relation expresses the heat capacity at constant volume, $C_V\mathrm{(T,\rho ),}$ of a fluid linearly in terms of the second temperature derivatives of the pressure and the chemical potential, $p^{″}\mathrm{(T,\rho )}$ and ${\mu }^{″}\mathrm{(T,\rho ).}$ At a gas–liquid critical point $C_V$ diverges so, on approaching $T_c$ from below, either $p_{\sigma }^{″}\mathrm{(T),}$ or ${\mu }_{\sigma }^{″}\mathrm{(T),}$ or both must diverge, where the subscript σ denotes the evaluation of p and μ on the phase boundary or vapor-pressure curve. However, previous theoretical and experimental studies have suggested that ${\mu }_{\sigma }^{″}\mathrm{(T)}$ always remains finite. To test these inferences, we present an analysis of extensive two-phase heat capacity data for propane recently published by Abdulagatov and co-workers. By careful interpolation in temperature and subsequently making linear, isothermal fits vs specific volume and vs density, we establish that the divergence is shared almost equally between the derivatives $p_{\sigma }^{″}\mathrm{(T)}$ and ${\mu }_{\sigma }^{″}\mathrm{(T).}$ A re-examination of the analysis of Gaddy and White for carbon dioxide leads to similar conclusions although the singular contribution from ${\mu }_{\sigma }^{″}\mathrm{(T)}$ is found to be of opposite sign and probably somewhat smaller than in propane.