Empirical Determination of the Helmholtz Free Energy and the Gibbs Free Energy of Polyethylene Based on the Pressure–Volume–Temperature–Entropy Equation of State and the Partition Function

Abstract

The Helmholtz free energy, A, and the Gibbs free energy, G, of polyethylene (PE) were determined based on its empirical pressure–volume–temperature–entropy equation of state, P-V-T-S Eos, and the empirical partition function, which we reported previously. The P_A-V_A-T_A-S_A Eos for the iso-Helmholtz free energy, iso-A, was determined based on the P-V-T-S Eos where the pressure (P_A), volume (V_A), temperature (T_A), and entropy (S_A) satisfied the condition that A is constant. The P_G-V_G-T_G-S_G Eos for the G being constant was also determined based on the P-V-T-S Eos. The three-dimensional (3D) plots, 3D plot, with the three axes, x = T_A, y = P_A, z = V_A expressed by the 3D (T_A, P_A, V_A) and the 3D (T_A, V_A, S_A) at constant A in PE were determined, where the iso-A processes were characterized assuming that the volume and the entropy in the iso-A process decreased with increasing pressure and temperature, while in the 3D (T_G, P_G, V_G) and 3D (T_G, V_G, S_G) plots at constant G in PE, the volume and the entropy in the iso-G process were increasing with increasing pressure and temperature. The empirical partition function for the Helmholtz free energy was expressed by the summation of the terms of T, T², and T⁴, the Boltzmann factors $e^{-\frac{i {\theta }_1}{T}}$ with i = 1, 2, 3, and 6 and ${\theta }_1=71.5$ and the U(V,T = 0 K) term for the zero Kelvin internal energy where the T⁴ term and the Boltzmann factors $e^{-\frac{i{\theta }_1}{T}}$ with i = 6 and 3 were dominant negative factors in the A-T relation at the constant V = 1.04 cm³/g. In case of the Gibbs free energy where the partition function was expressed by the summation of the terms of T, T², and T⁴, the Boltzmann factors $e^{-\frac{i {\theta }_1}{T}}$ with i = 1, 2, 3, and 6, the U(V,T = 0 K) term, PV and PV(T = 0) terms, the T² term and the Boltzmann factors $e^{-\frac{i {\theta }_1}{T}}$ with i = 6 and 3 were dominant negative factors in the G-T relation at 1 atm (1.01 × 10⁻⁴GPa), while at high pressure, 1.0 GPa, the PV and PV(T = 0) terms were dominant positive factors.

Keywords:

• Gibbs free energy
• Helmholtz free energy
• partition function
• polyethylene
• pressure–volume–temperature–entropy equation of state