Derivatives Of Thermodynamic Potential Research Articles

Application of Generalized (Hyper-) Dual Numbers in Equation of State Modeling

The calculation of derivatives is ubiquitous in science and engineering. In thermodynamics, in particular, state properties can be expressed as derivatives of thermodynamic potentials. The manual differentiation of complex models can be tedious and error-prone. In this work, we revisit dual and hyper-dual numbers for the calculation of exact derivatives and show generalizations to higher order derivatives and derivatives with respect to vector quantities. The methods described in this paper are accompanied by an open source Rust implementation with Python bindings. Applications of the generalized (hyper-) dual numbers are given in the context of equation of state modeling and the calculation of critical points.

TFmix: A high-precision implementation of the finite-temperature Thomas–Fermi model for a mixture of atoms

In this work we present a TFmix code intended for numerical calculation of the thermal part of electronic thermodynamic properties of a mixture of elements by the finite-temperature Thomas–Fermi model. The code is based on analytical models for both first and second derivatives of Helmholtz thermodynamic potential. All numerical calculations are made within a controlled high accuracy: tests for thermodynamic consistency give at least 11 coinciding decimal digits. The code calculates thermodynamic functions on a regular grid of isotherms and isochores; at each grid point some extensive parameters and the number of free electrons are output both for the whole mixture and for each component. Other extensive or intensive thermodynamic properties, including pressure, entropy, isochoric and isobaric heat capacities, isothermal and adiabatic sound velocities can be easily calculated from the information available at each grid point. Several unit systems are provided for convenience. A cross-platform graphical user interface is developed to simplify the use of the code. Program summaryProgram Title: TFmix, version 1.0Program Files doi:http://dx.doi.org/10.17632/mc3vj77jfn.1Licensing provisions: GPLv3Programming language: C, PythonNature of problem: Any substance consists of elements so its equation of state contains a contribution of electronic gas. Thermodynamics of the electronic gas in a mixture of ions and electrons has been studied in many approaches. Thermodynamic properties of a uniform ideal electron gas can be calculated using the well-known analytical model of Fermi-gas. On the other hand, models of electron gas which take into account interaction effects are quite complicated and require sophisticated computational techniques. Even a simplified semiclassical Thomas–Fermi model is based upon the numerical solution of a non-linear boundary problem. Two main issues of the Thomas–Fermi model restrict its usage: uncontrolled accuracy of calculated thermodynamic functions (especially second derivatives of a thermodynamic potential), and unphysical behavior of the model at relatively low temperatures.Solution method: Each atom in the mixture is surrounded by a spherical cell. The radii of the cells are fitted to equalize the chemical potentials of all atoms. A guaranteed accuracy of first derivatives of the thermodynamic potential is provided by a transformation of integrals over the Thomas–Fermi potential to a system of differential equations. One of equations in the system is the Thomas–Fermi equation. Second derivatives of the thermodynamic potential are calculated similarly with the only difference that a corresponding derivative of the Thomas–Fermi equation is used in the system of differential equation. To avoid the unphysical behavior of the Thomas–Fermi model at low temperatures we extract a thermal contribution to thermodynamic properties which vanishes at zero temperature. To eliminate the error which appears from the subtraction of the cold part at low temperatures we use asymptotic expressions for thermodynamic functions and the Thomas–Fermi equation. The code calculates regular tables of thermodynamic functions on a grid of isotherms and isochores including second derivatives of a thermodynamic potential. This information is necessary for astrophysical applications, for continuum mechanics simulation of processes in plasma and for the creation of wide-range equations of state. A graphical user interface is provided with the code and allows to specify input parameters, to perform calculations and to plot the results.Additional comments including restrictions and unusual features: GSL library version 1.16 or 2.x is required for compilation; matplotlib Python library is required to run the graphical user interface.

Exploring molecular interactions of binary mixture (dimethyl carbonate + benzyl benzoate): Measurements and correlation

To explore the type of interaction and its behavior with temperature in the binary mixture (dimethyl carbonate+benzyl benzoate), the density (ρ) and speed of sound (u) are measured for the whole mole fraction range at various temperatures T=(308.15, 313.15 and 318.15) K and at atmospheric pressure. Sign with magnitude of computed excess molar volume (VmE) was examined and detailed analysis elucidated not only prevalence of strong molecular interactions between molecules in study but also its variation with temperature. Further, strong interactions are well supported by partial and their excess partial molar volume (V¯m,V¯mE,V¯mE,∞). Furthermore, relative association (RA), Lennard-Jones repulsive power (n), excess isentropic compressibility (κsE), partial and their excess partial molar isentropic compressibility(K¯m, K¯mE), excess values of isobaric thermal expansion coefficient (αPE), isentropic compressibility (κsE), isothermal compressibility (kTE), intermolecular free length (LfE), acoustic impedance (ZE), surface tension (σSE), ultrasonic speed (uE) were evaluated and confirmed the conclusions made for VmE. By Redlich-Kister equation, coefficients with standard deviations are computed to excess parameters. VmE is correlated by Prigogine–Flory–Patterson (PFP) and Topology theories at 308.15K. Extended Langmuir model is examined for influence of the bulk mole fraction on surface tension. At T=308.15K, different derivatives of thermodynamic potentials, densities through Soave-Redlich-Kwong (SRK)/Peng-Robinson (PR) cubic equation of states, excess chemical potential and theoretical speeds using semi-empirical equations are evaluated. Using Hartmann–Balizar and Ballou non-linear equations, cohesive energy (ΔA), van der Wall's constants (a, b), distance of closest approach (d) are estimated through Sehgal's equations at all temperatures.

Investigation of molecular interactions & prediction of calorimetric potentials of a binary liquid system at T = 308.15 K: An insight from physicochemical parameters

Excess values of surface tension/isothermal compressibility/internal pressure/free volume/enthalpy/entropy and Gibbs energy are computed using the experimentally measured density (ρ) and ultrasonic speed (u) for the binary liquid mixtures of dimethyl sulphoxide with acetophenone at T = 308.15 K. Activity coefficients and excess chemical potential are estimated using Morgules, Porter, Van laar and Wilson equations. The studied system shows departure to the ideality indicating the presence of weak interactions. Further, PFP theory is tested to correlative the experimental excess molar enthalpy. Moreover, PFP theory is extended to compute different derivatives of thermodynamic potentials rather calorimetric measurements. The present investigation also encompass of assessment of the acoustic non-linearity parameter (B/A) along with computation of cohesive energy, ΔA, Van der Wall's constants (a,b), distance of closest approach (d).

The part played by magnetoelastic interaction in the forming of thermodynamic functions of ferromagnets: Second thermodynamic derivatives of thermodynamic potential

This is a continuation of the previously [1] started consistent construction of thermodynamics of ferromagnets, the magnetic ordering of which may be considered in the spirit of Landau’s phenomenological approach but in view of magnetoelastic interaction (MEI) between magnetic subsystem and crystal lattice. Thermodynamically exact expressions are obtained within the model for second thermodynamic derivatives of the thermodynamic potential renormalized in view of MEI. Model calculations are performed on the real scale of temperatures in the absence of magnetic field, the results of which demonstrate that the developed approach in view of MEI produces much better qualitative and quantitative agreement with the actually observed behavior of the thermodynamic functions of ferromagnets than the traditional Landau theory.

A new method for critical points calculation from cubic EOS

AbstractDevelopment of accurate and efficient methods for direct calculation of critical points is valuable for phase equilibrium calculations in chemical and petroleum engineering. A new method for critical points calculation is developed. For critical point calculation methods, the spinodal equation is given by setting to zero a determinant of nc (number of components) or nc‐1 order, containing second‐order derivatives of thermodynamic potentials. In this approach, this determinant has a lower order, which does not depend on the number of components in the mixture, but on the number of reduction parameters, with no restrictions on the binary interaction parameters (BIPs). The new method has been tested for several mixtures having various phase diagram shapes, and proved its reliability. The computation effort for each step for correcting temperature and molar volume can be significantly reduced. The proposed method is particularly efficient for mixtures with many components and few nonzero BIPs. Cubic two‐parameter EOS are used; however, the proposed method can be applied for any EOS that observes the restrictions of the Reduction Theorem. © 2005 American Institute of Chemical Engineers AIChE J, 2006