The van der Waals equation of state was a breakthrough of a genius. However, while it was a masterpiece from a conceptual point of view, it is not quantitatively accurate. Many equations of state have been proposed to date, the accuracy of which has been improved by introducing temperature-dependence of coefficients, and various ways to treat the volume of the molecules themselves. Long time ago, the author analyzed the mathematical aspects of equations of state and proved that the general form must be P = A(V)T- B(V), where P is the pressure, T is the absolute temperature, V is the volume, and A and B are functions of V only. Thus, the corrections to the van der Waals equation should depend on the volume only and not on the temperature. Practical implications of this conclusion are demonstrated.

Another major problem is the construction of equations of state of mixtures. The prevalent approach is to define mixing rules for the parameters of the equation of state, by which the mixture is treated as if it is a single component with some averaged parameters of the individual components. However, this approach leads to discontinuities in the isotherms. A new approach that avoids this problem is suggested and demonstrated.

Definition of frame-invariant Soret coefficients for ternary mixtures

The definition of the Soret coefficient of a binary mixture includes a concentration prefactor, x(1-x) when mol fraction x is used, or w(1-w) when mass fraction w is used. In this presentation the physical reasons behind this choice are reviewed, emphasizing that the use of these prefactors makes the Soret coefficient invariant upon change in the reference frame, either mass or molar. Then, it will be shown how this invariance property can be extended to ternary mixtures by using an appropriate concentration prefactor in matrix form. The presentation will be completed with some considerations of general non-isothermal diffusion fluxes, binary limits of the concentration triangle, selection of the dependent concentration in a ternary mixture, and, finally, extension to multi-component mixtures.
References
[1] J. M. Ortiz de Zárate. Definition of frame-invariant thermodiffusion and Soret coefficients for ternary mixtures. Eur. Phys. J. E 42(2019), 43

When thermodynamics is understood as the science (or art) of constructing effective models of natural phenomena by choosing a minimal level of description capable of capturing the essential features of the physical reality of interest, the scientific community has identified a set of general rules that the model must incorporate if it aspires to be consistent with the body of known experimental evidence. Some of these rules are believed to be so general that we think of them as laws of Nature, such as the great conservation principles, whose ‘greatness’ derives from their generality, as masterfully explained by Feynman in one of his legendary lectures. The first law and second law are universally contemplated among the great laws of Nature. In the logical development of thermodynamic theory they support the definitions of the energy and the entropy of every state of the modelled system, respectively. The recent paper https://dx.doi.org/10.1098/rsta.2019.0168 shows that in the past four decades, an enormous body of scientific research devoted to modeling the essential features of nonequilibrium natural phenomena has converged from many different directions and frameworks towards the general recognition (albeit still expressed in different but equivalent forms and language) that another rule is also indispensable and reveals another great law of Nature. We call it the ‘fourth law of thermodynamics’ and state it as follows: every non-equilibrium state of a system or local subsystem for which entropy is well defined, must be equipped with a metric in state space with respect to which the irreversible component of its time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints. A powerful feature of the fourth law is that it provides (nonlinear) extensions of Onsager reciprocity and fluctuation-dissipation relations to the far-non-equilibrium realm.