Saturation Line of Ethane in the Renormalization Group Theory Using the Clapeyron–Clausius Equation

Abstract

A system of mutually consistent equations for ethane is developed that describes pressure ${{p}_{s}}$, vapor density ${{\rho }^{ - }}$, liquid density ${{\rho }^{ + }}$, derivative $p_{s}^{'}(T)$, and heat of vaporization $r$ on the phase equilibrium line in the range of the triple point to the critical point. The system also includes apparent heat of vaporization $r\text{*}$, which is associated with heat of vaporization $r$: $r = r\text{*}{\kern 1pt} (1 - {{\rho }^{ - }}{\text{/}}{{\rho }^{ + }})$. It is established on the basis of the thermodynamic analysis that (1) the condition of average diameter ${{d}_{f}} > 0$ is fulfilled at each point of the saturation line except for the critical point, at which ${{d}_{f}} = 0$, and (2) the average diameter is reduced sharply in the interval of ${{T}_{{{\text{tr}}}}} < T < {{T}_{c}}$. The system of mutually consistent equations reproduces the phase equilibrium line of ethane within the experimental uncertainty data of Funke et al. (2002) in the range of the triple point (${{p}_{{{\text{tr}}}}}$, ${{\rho }_{{{\text{tr}}}}}$, ${{T}_{{{\text{tr}}}}}$) to the critical point (${{p}_{c}}$, ${{\rho }_{c}}$, ${{T}_{c}}$). It also reproduces features of the critical point in accordance with the renormalization group (RG) theory developed by Zhou et al. (2022) for a system of asymmetric systems. Based on the Clausius–Clapeyron equation and renormalization group theory, an expression is obtained for the apparent heat of vaporization. Analysis of average diameter ${{d}_{f}} = {{D}_{{2\beta }}}{{\tau }^{{2\beta }}} + {{D}_{{1 - \alpha }}}{{\tau }^{{1 - \alpha }}} + {{D}_{\tau }}\tau$ for two groups of complexes shows that (a) ${{D}_{{2\beta }}} = 0.1$, $\eta = {{D}_{{2\beta }}}{\text{/}}{{D}_{{1 - \alpha }}} = - 0.14$, and $\phi = {{D}_{{2\beta }}}{\text{/}}{{D}_{\tau }} = 0.13$, and (b) ${{D}_{{2\beta }}} = 0.048$, $\eta = - 0.18$, and $\phi = 0.12$, which correspond to values ${{D}_{{2\beta }}}$, $\eta$, and $\phi$ obtained by Wang et al. (2013) in the RG theory and the modeling of experimental data for ethane on the saturation line. Based on the proposed system of mutually consistent equations, average diameter ${{d}_{f}}$of ethane is found for complexes (a) and (b), and it is established that the average diameter determined on the basis of data by Funke et al. (2002) is given most accurately by the system of mutually consistent equations in the range of ${{T}_{{{\text{tr}}}}}$ to ${{T}_{c}}$ with parameters ${{D}_{{2\beta }}} = 0.0039$, $\eta = - 0.14$, and $\phi = 0.13$.