Coexistence curve, compressibility, and the equation of state of xenon near the critical point

Abstract

Optical interferometric measurements which determine the equation of state of xenon in the neighborhood of the critical point are described. Analysis of Fraunhofer interference patterns from a thin slab of fluid yields data pairs: optical phase ${\ensuremath{\psi}}_{+}=\ensuremath{\rho}\ensuremath{-}{\ensuremath{\kappa}}_{T}\ensuremath{\mu}$ and isothermal compressibility ${\ensuremath{\kappa}}_{T}$, along isotherms in the temperature range $\ensuremath{-}{10}^{\ensuremath{-}4}<\ensuremath{\epsilon}<{10}^{\ensuremath{-}4}$, where $\ensuremath{\epsilon}=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. Experimental data are analyzed in terms of a new parametric transformation of thermodynamic variables, based on the static scaling hypothesis of Widom, which requires that $\frac{d\mathrm{ln}{\ensuremath{\psi}}_{+}}{d\mathrm{ln}{\ensuremath{\kappa}}_{T}}=\ensuremath{-}(\frac{\ensuremath{\beta}}{\ensuremath{\gamma}})W(\ensuremath{\theta})$, where $\ensuremath{\theta}=\ensuremath{\epsilon}{\ensuremath{\kappa}}_{T}^{\frac{1}{\ensuremath{\gamma}}}$. On the critical isotherm, $\ensuremath{\epsilon}=0$, we expect that $\mathrm{ln}{\ensuremath{\psi}}_{+}=\mathrm{const}\ensuremath{-}(\frac{\ensuremath{\beta}}{\ensuremath{\gamma}})\mathrm{ln}{\ensuremath{\kappa}}_{T}$. This accords with observation and yields a sharp determination of ${T}_{c}$ which is decoupled from other parameters. The data are well represented by the bilinear form $W(\ensuremath{\theta})=\frac{(1\ensuremath{-}\frac{\ensuremath{\theta}}{{\ensuremath{\theta}}_{x}})}{(1\ensuremath{-}\frac{\ensuremath{\theta}}{{\ensuremath{\theta}}_{0}})}$ where $\ensuremath{\theta}={\ensuremath{\theta}}_{0}$ on the critical isochore and ${\ensuremath{\theta}}_{x}$ on the coexistence boundary. This is integrated to yield the parametric equation of state ${\ensuremath{\psi}}_{+}={Y}_{0}^{\ensuremath{\beta}}{R}^{\ensuremath{\beta}}{(1\ensuremath{-}\frac{\ensuremath{\theta}}{{\ensuremath{\theta}}_{0}})}^{\ensuremath{\beta}\ensuremath{\Delta}}$, where $R={\ensuremath{\kappa}}_{T}^{\ensuremath{-}\frac{1}{\ensuremath{\gamma}}}$. A six-parameter fit to 1200 data points yields ${T}_{c}={T}_{c}(\mathrm{lab})\ifmmode\pm\else\textpm\fi{}0.0001\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$, $\ensuremath{\beta}=0.3583\ifmmode\pm\else\textpm\fi{}0.0002$, $\ensuremath{\gamma}=1.2296\ifmmode\pm\else\textpm\fi{}0.0005$, ${\ensuremath{\theta}}_{0}=0.1101\ifmmode\pm\else\textpm\fi{}0.0003$, ${Y}_{0}^{\ensuremath{\beta}}=0.4203\ifmmode\pm\else\textpm\fi{}0.0004$, and $\ensuremath{\Delta}=3.869\ifmmode\pm\else\textpm\fi{}0.001$. This implies $\ensuremath{\beta}\ensuremath{\Delta}=1.386\ifmmode\pm\else\textpm\fi{}0.001$, which differs significantly from the value $\ensuremath{\beta}\ensuremath{\Delta}=\frac{3}{2}$ implied by a five-parameter transformation suggested by Ho and Litster. The coexistence curve is measured in the range ${10}^{\ensuremath{-}5}<|\ensuremath{\epsilon}|<5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, and fitted by the power law $\frac{({\ensuremath{\rho}}_{L}\ensuremath{-}{\ensuremath{\rho}}_{G}}{{\ensuremath{\rho}}_{c}}=B{(\ensuremath{-}\ensuremath{\epsilon})}^{\ensuremath{\beta}}$, with the result $\ensuremath{\beta}=0.344\ifmmode\pm\else\textpm\fi{}0.003$ and $B=3.51\ifmmode\pm\else\textpm\fi{}0.05$. Systematic deviations indicate that $\ensuremath{\beta}$ increases for large $|\ensuremath{\epsilon}|$. A fit with the form $\frac{({\ensuremath{\rho}}_{L}\ensuremath{-}{\ensuremath{\rho}}_{G})}{{\ensuremath{\rho}}_{c}}=B{(\ensuremath{-}\ensuremath{\epsilon})}^{\ensuremath{\beta}}+A{(\ensuremath{-}\ensuremath{\epsilon})}^{{\ensuremath{\beta}}^{\ensuremath{'}}}$ yields significant improvement, with $\ensuremath{\beta}=0.332\ifmmode\pm\else\textpm\fi{}0.001$, $B=3.042\ifmmode\pm\else\textpm\fi{}0.03$, ${\ensuremath{\beta}}^{\ensuremath{'}}=0.61\ifmmode\pm\else\textpm\fi{}0.02$, and $A=0.93\ifmmode\pm\else\textpm\fi{}0.04$. The disagreement between this $\ensuremath{\beta}$ and the $\ensuremath{\beta}$ obtained from fitting the Fraunhofer data will be discussed in the text. The coefficient of isothermal compressibility on the critical isochore ${P}_{c}{K}_{T}$ is measured in the range $2.7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}<\ensuremath{\epsilon}<4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$, and fitted by the equation ${\ensuremath{\kappa}}_{T}={P}_{c}{K}_{T}=\ensuremath{\Gamma}{\ensuremath{\epsilon}}^{\ensuremath{-}\ensuremath{\gamma}}$. Over the measured range, the data indicate $\ensuremath{\gamma}=1.260\ifmmode\pm\else\textpm\fi{}0.002$ and $\ensuremath{\Gamma}=0.056\ifmmode\pm\else\textpm\fi{}0.001$. There is evidence that $\ensuremath{\gamma}$ depends on the range of fit, and we find $\ensuremath{\gamma}=1.232\ifmmode\pm\else\textpm\fi{}0.006$ for $\ensuremath{\epsilon}<{10}^{\ensuremath{-}3}$, which agrees well with the $\ensuremath{\gamma}$ determined from the near-critical Fraunhofer data.