Equation of State ofHe3Close to the Critical Point

Abstract

We present measurements of the equation of state of ${\mathrm{He}}^{3}$ with an impurity of 250 ppm ${\mathrm{He}}^{4}$ near the critical point. In these experiments the density is measured by a dielectric-constant technique. The density cell is a horizontal capacitor to reduce the gravitational effects. The temperature gradient between the cell and 4 \ifmmode^\circ\else\textdegree\fi{}K occurs along a horizontal capillary. The results are analyzed in terms of the critical exponents which are found to have values similar to those of other simple fluids. From the coexistence curve and the compressibility on the critical isochore above ${T}_{c}$, we get $\ensuremath{\beta}=0.361$, $B=1.31$, $\ensuremath{\gamma}=1.18$, $\ensuremath{\Gamma}=2.48\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ ${\mathrm{Torr}}^{\ensuremath{-}1}$, ${T}_{c}=3.3105$ \ifmmode^\circ\else\textdegree\fi{}K, ${\ensuremath{\rho}}_{c}=0.04145$ g/${\mathrm{cm}}^{3}$, and ${P}_{c}=860.5$ Torr. From the compressibility along the coexistence curve, we obtain ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.12$ and ${\ensuremath{\Gamma}}^{\ensuremath{'}}=6.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ ${\mathrm{Torr}}^{\ensuremath{-}1}$ for both the liquid and the vapor sides. An analysis of the critical isotherm using the conjugate variables $\ensuremath{\mu}$ and $\ensuremath{\rho}$ gives $\ensuremath{\delta}=4.21\ifmmode\pm\else\textpm\fi{}0.10$ and $\ensuremath{\nabla}=2.77$. The critical isobar gives the exponents ${\ensuremath{\pi}}_{+}(\ensuremath{\rho}g{\ensuremath{\rho}}_{c})=4.21\ifmmode\pm\else\textpm\fi{}0.10$ and ${\ensuremath{\pi}}_{\ensuremath{-}}(\ensuremath{\rho}l{\ensuremath{\rho}}_{c})=4.35\ifmmode\pm\else\textpm\fi{}0.10$. The study of the critical isochore shows $\frac{{d}^{2}P}{d{T}^{2}}$ to be always positive. Combination of these data with specific-heat results by Moldover indicate no discontinuity in $\frac{{d}^{2}\ensuremath{\mu}}{d{T}^{2}}$ within experimental error. We find at the critical point ${({P}_{c}{V}_{c})}^{\ensuremath{-}1} \frac{{d}^{2}\ensuremath{\mu}}{d{T}^{2}}=\ensuremath{-}0.22$ ${(\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{K})}^{\ensuremath{-}2}$. The isotherms expressed by the conjugate variables $\ensuremath{\mu}$, $\ensuremath{\rho}$ are fitted to the scaling equation of state proposed by Missoni et al. A good fit was obtained with the following parameters: ${E}_{1}=2.53$, ${E}_{2}=0.44$, $\ensuremath{\delta}=4.23$, ${x}_{0}=0.475$, and $\ensuremath{\gamma}=1.17$. Results of a less extensive study of ${\mathrm{He}}^{3}$ with an impurity of 10 ppm are presented. They give critical indices and ${P}_{c}$, ${T}_{c}$, and ${\ensuremath{\rho}}_{c}$ in substantial agreement with the less pure ${\mathrm{He}}^{3}$. A comparison of the present data with previous work is presented.