Modelling of liquid internal energy and heat capacity over a wide pressure-temperature range from first principles

Recently there have been significant theoretical advances in our understanding of liquids and dense supercritical fluids based on their ability to support high frequency transverse (shear) waves. Here, we have constructed a new computer model using these recent theoretical findings (the phonon theory of liquid thermodynamics), to model liquid internal energy across a wide pressure-temperature range. We have applied it to a number of real liquids in both the subcritical regime and the supercritical regime, in which the liquid state is demarcated by the Frenkel line. Our fitting to experimental data in a wide pressure-temperature range has allowed us to test the new theoretical model with hitherto unprecedented rigour. We have quantified the degree to which the prediction of internal energy and heat capacity is constrained by the different input parameters: The liquid relaxation time (initially obtained from the viscosity), the Debye wavenumber and the infinite-frequency shear modulus. The model is successfully applied to output the internal energy and heat capacity data for several different fluids (Ar, Ne, $N_2$, Kr) over a range of densities and temperatures. We find that the predicted heat capacities are extremely sensitive to the values used for the liquid relaxation time. If these are calculated directly from the viscosity data then, in some cases, changes within the margins of experimental error in the viscosity data can cause the heat capacity to exhibit a completely different trend as a function of temperature. Our code is computationally inexpensive, and it is available for other researchers to use.