Free Volumes and Free Angle Ratios of Molecules in Liquids

Abstract

The free volume is defined as the total integral over that part of the potential energy of the molecule in the liquid which is due to thermal displacements of the center of gravity of the molecule from its equilibrium position. The free angle is the corresponding integral over angular displacements of the molecule. The free volume, V_f is related to the velocity of sound, u, in the liquid by $$\mathrm{u(}\mathrm{liquid}\mathrm{)=u\; (}\mathrm{gas}{\mathrm{)\; (V/V}}_f{)}^{\frac{1}{3}}.$$ This equation is used to derive a formula connecting the sound velocity in the liquid with its thermal conductivity. The quantity, RT/p exp (ΔH/RT), gives the product of the free angle ratio and the free volume, rather than the free volume itself. Here p is the vapor pressure and ΔH is the heat of vaporization. The free volumes from sound velocities agree with those determined by independent methods. Specific heats, entropies of vaporization, the spectroscopic observations of Cartwright, and the differences between sound velocity, free volumes and the function RT/p ×exp (ΔH/RT) are examined from the point of view of restricted rotation of the molecules in liquids. Lack of free rotation of molecules suffices to explain the abnormalities of the liquids examined. The dielectric constants of polar liquids are interpreted assuming restricted rotation and the following formula is derived: $${\mu }_l^2{\mu }_g^{\text{−2}}\text{=1−((1−}\cos {\theta }_1{\text{)/2)}}^2,$$ where μ_l is the apparent dipole moment of the molecule in the liquid, μ_g is the dipole moment as determined from measurements in the gas phase, and θ₁, is the polar angle related to the free angle ratio, δ₂, by $${\delta }_2\text{=(1−}\cos {\theta }_1\text{)/2.}$$ The concept of restricted rotation of the molecules in the liquid accounts satisfactorily for the observed dielectric polarizations of water and methyl alcohol.