Review of Some Experimental and Analytical Equations of State

Abstract

Four different polynomial equations and seven nonlinear equations, all applicable to both solids and liquids, are compared theoretically and statistically. Detailed curve-fitting results are presented for recent water and Hg isothermal data. Uncommonly used methods of statistical analysis and comparison, including generalized least squares, are described, justified compared to usual methods, and applied. In general, certain polynomial equations are found to yield significantly better fits of many different water and Hg data sets than any nonlinear equation considered. The Tait and Murnaghan equations, in particular, lead to strong systematic behavior of all residuals calculated herein with them, showing that they are inadequate models for all the data considered. Even a nonlinear equation derived from a second-order expansion of the bulk modulus $K$ in powers of the pressure, which is shown to include several frequently used equations as special cases, is inferior to selected polynomial equations but is still the best equation examined when appreciable extrapolation is necessary. The method of volume normalization almost always used heretofore in statistical fitting of equations of state to $P\ensuremath{-}V$ data is shown to be inadequate and two alternative approaches are proposed and employed herein. Critical comparison of previous analyses of water and Hg data is made with the results of the present, more refined approach. The likelihood of important systematic errors in $P\ensuremath{-}V$ data, particularly data derived from ultrasonic measurements on liquids under pressure, is pointed out and high probability of their occurrence in some of the data analyzed is demonstrated. Even the combination of the best data apparently available and the use of better statistical-analysis methods than have been employed before does not yet allow one to obtain highly accurate values of the ${{K}_{0}}^{\ensuremath{'}}$ parameter of water or Hg, and only an order-of-magnitude estimate of the ${{K}_{0}}^{\ensuremath{'}\ensuremath{'}}$ parameter seems currently possible. Nevertheless, it appears that near room temperature ${{K}_{0}}^{\ensuremath{'}\ensuremath{'}}$ is positive for water and probably negative for Hg and that its appreciable magnitude for both materials renders a second-order expansion of $K$ inadequate.