Empirical Determination of the Internal Energy of Polyethylene Based on the Pressure-Volume-Temperature-Entropy Equation of State. II. The Revised Internal Energy Based on the Zero Kelvin Isotherm

Abstract

In our previous reports, two types of the internal energy for polyethylene (PE) were derived, which were based on the thermodynamic relation that $d{U}_{TH}=C_{V }dT+\left({\gamma }_{V}T-P\right)dV$ and based on the partition function, that $U_{PF}=R{T}^2\frac{\partial \ln \hspace{0.17em}f}{\partial V}$ where $f=\sum e^{-\frac{{\theta }_i}{T}},$ C_V is the isochoric heat capacity, ${\gamma }_V$ is the thermal pressure coefficient and $e^{-\frac{{\theta }_i}{T}}$ is the Boltzmann factor. In this work, a revised internal energy based on the U_TH, U_rev.TH, was derived by introducing a zero Kelvin internal energy. A revised internal energy based on the U_PF, U_rev.PF, was also derived by introducing the zero Kelvin internal energy. The zero Kelvin internal energy, U(V, T = 0 K), was determined based on a zero Kelvin isotherm, which was determined from the P-V-T equation of state derived in our previous work. In this report, the three-dimensional (3D) plot, 3D plot, with the three axes, x = T_U, y = P_U, z = V_U expressed by the 3D (T_U, P_U, V_U) and the 3D (T_U, V_U. S_U) at constant U in PE were determined by the U_rev.TH where the processes were characterized by decreasing of the volume and the entropy with increasing of the pressure, while the temperature in the iso-internal process had a maximum with increasing the pressure, which were different from those determined by the U_TH in our previous work. The revised internal energy U_rev.PF was expressed by the summation of the terms of T ² and T ⁴, the Boltzmann factors $e^{-\frac{i {\theta }_1}{T}}$ and the U(V,T = 0 K) where the temperature dependence of the T ⁴ term and the Boltzmann factors $e^{-\frac{i {\theta }_1}{T}}$ with i = 6 and 3 were dominant in the U_rev.PF at V = 1.04 cm³/g, while at V = 0.85 cm³/g in the range of high pressure, the U(V,T = 0 K) became a controlling factor. In the case of the U_rev,PF at constant temperature T = 320 K, the volume dependence of the T ² term and the U(V,T = 0 K) were dominant factors.

Keywords:

• Internal energy
• partition function
• polyethylene
• pressure-volume-temperature-entropy equation of state
• zero Kelvin isotherm