Principle of Corresponding States in the Phase Transition of Solid Methanes

Abstract

Quantum corrections for the orientational distribution function of tetrahedral molecules in a potential field are presented. By taking the primary correction into account and applying the resulting distribution function to the theory of phase transitions based on the Hartree approximation, we predict $T_C\mathrm{\hspace{0.17em}+\hspace{0.17em}\Theta \hspace{0.17em}=\hspace{0.17em}}\mathrm{const}$ for the upper transition of various isotopic methanes, where $T_C$ is the transition temperature and $\Theta$ a quantum mechanical parameter dependent on the moment of inertia: $k_B{\mathrm{\Theta \hspace{0.17em}=\hspace{0.17em}\hslash }}^2{\mathrm{\hspace{0.17em}/\hspace{0.17em}I,\; (\hslash }}^2{\text{\hspace{0.17em}/\hspace{0.17em}3) (2\hspace{0.17em}/\hspace{0.17em}I}}_1{\text{\hspace{0.17em}+\hspace{0.17em}1\hspace{0.17em}/\hspace{0.17em}I}}_3{\mathrm{),\; (\hslash }}^2{\text{\hspace{0.17em}/\hspace{0.17em}3) (1\hspace{0.17em}/\hspace{0.17em}I}}_1{\text{\hspace{0.17em}+\hspace{0.17em}1\hspace{0.17em}/\hspace{0.17em}I}}_2{\text{\hspace{0.17em}+\hspace{0.17em}1\hspace{0.17em}/\hspace{0.17em}I}}_3)$ , respectively, for spherical, symmetric, and asymmetric rotors. This prediction agrees very well with measured results. For the lower transition, we predict $T_C\text{\hspace{0.17em}+\hspace{0.17em}2Θ\hspace{0.17em}=\hspace{0.17em}}\mathrm{const}$ , which is rather close to a linear equation, $T_C\text{\hspace{0.17em}+\hspace{0.17em}(5\hspace{0.17em}/\hspace{0.17em}3)Θ\hspace{0.17em}=\hspace{0.17em}}\mathrm{const}$ , to be expected from measured results. The concentration dependence of the upper transition temperature of mixed solids CH₄–CD₄ is studied, in excellent agreement with the measurements.