Above the critical point, in the super critical region, the thermodynamic response functions (eg. constant pressure heat capacity (C_P), isothermal compressibility (K_T), volume expansivity (α_P), etc) that are derivatives of state functions, exhibit a maximum. These response functions diverge as the critical point is approached. If one plots the lines of maxima of the different response functions in the super critical region, it can be seen that they approach the critical point asymptotically. This asymptotic line is sometimes called as the Widom line[1]. For a non-associating substance, like propane, the response functions goes through a maximum only once in the super critical region and these lines of maxima approach asymptotically through the Widom line as expected. However, for hydrogen fluoride (HF), when an association model(AEOS-VK)[2] is used to predict the K_T results, it exhibits a maximum in the super critical region more than once. When the system is decompressed, this additional peak extends well into the super heated vapor region. This theoretical prediction is not only obtained from the AEOS-VK model but is also supported by two other models [3,4]. Note that, the experimental values of K_T for HF have not been reported in the literature to date. Considering the caustic and toxic nature of HF, this is not surprising. This scarcity of experimental data on HF has led to the increase in the number of theoretical studies, and to an extent the theoretical predictions have preceded the appearance of experimental data for this substance. This trend is more likely to continue in the future as well.

HF has been a substance of considerable interest because of its unique properties and several useful applications. It has been one of the most of extensively studied substances due to its smaller size and the strength of its hydrogen bond in all phases. This associating behavior makes its unusual physical and chemical properties similar to water instead of other hydrogen halides. However, unlike water, which associates in the liquid phase and shows limited association in the vapor phase, HF associates in both liquid and vapor phases. This unusual vapor phase association has been considered to be the origin for its several anomalies such as high thermal conductivity of up to 0.6 W m-1 K-1, low enthalpy of vaporization (7.5 kJ mol-1 at normal boiling point), high vapor density etc[5]. Much work have been dedicated to explain this vapor phase non-ideality and several models describing the vapor-liquid equilibrium properties have been developed by incorporating association in both phases. Using one such model, the AEOS-VK model, we report this maximum in the K_T that is present in both the super heated vapor region and the super critical region.

Preliminary investigations on this HF K_T maximum suggested no reentrant spinodal and temperature of maximum density unlike liquid water [6,7], and, no Lambda (or higher order phase) transition unlike liquid He [8]. However, this K_T peak is similar to the experimentally supported C_P peak of HF which extends in the super critical and the super heated vapor region. This is understood based on vapor phase clustering in HF [9]. To provide a similar understanding between this K_T peak and the vapor phase clustering in HF, we work with a very simple molecular model called ideal association model (IAM) that allows only association by placing square-well sites on otherwise ideal particles [10, 11]. Using this IAM model we report the change in the K_T behavior with changes in the association patterns as well as their distribution. The mathematical and thermodynamic conditions that are imposed by this K_T maximum were derived and generalized for any association based equation of state. This helps in studying the various individual contributions to this peak and hence aids in the molecular level understanding on the origin of this peak as well as the system itself. Even though there is no K_T experimental data reported for HF, we analyze some of the PVT data that are available in the literature to provide an overview of the K_T behavior in the region of interest and compare them with the model results.

References:

(1) L. Xu, P. Kumar, S. V. Buldyrev, S. H. Chen, P. H. Poole, F. Sciortino and H. E. Stanley, "Relation between the Widom line and the dynamic crossover in systems with a liquid-liquid phase transition", Proc Natl Acad Sci U S A., 102, (2005).

(2) D. P. Visco and D. A. Kofke, "Improved Thermodynamic Equation of State for Hydrogen Fluoride", Ind. Eng. Chem. Res, 38, 4125, (1999).

(3) J. Lee and H. Kim, "An equation of state for hydrogen fluoride", Fluid Phase Equilibria, 190, 47-59, (2001).

(4) M. Lencka and A. Anderko, "Modeling phase equilibria in mixtures containing hydrogen fluoride and halocarbons",AiChE. J, 39, 533-538, (1993).

(5) Quack,M., Schmitt, S., and M. A.Suhm, "Evidence for the (HF)5 Complex in the HF Streching FTIR Absorption Spectra of Pulsed and Continuous Supersoninc Jet Expansions of Hydrogen Ffluoride", Chemical Physics Letters, 208, 446-452, (1993).

(6) S. Sastry, P. G. Debenedetti, S. Francesco and H. E. Stanley, "Singularity-free interpretation of the thermodynamics of supercooled water", Physical Review E, 53, 6144-6154, (1996).

(7) S. Francesco, H. P. Peter, U. Essman and H. E. Stanley, "Line of compressibility maxima in the phase diagram opf supercooled water", Physical Review E, 55, 727-737, (1997).

(8) E. R. Grilly, "Compressibility of Liquid He as a function of pressure", Physical Review, 149, 97-101, (1966).

(9) E. U. Frank and F. Meyer, "Spezifische Warme und Assoziation im Gas bei niedrigen Druck", Z.Elektrochem, 63, 571, (1959).

(10) D. P. Visco and D. A. Kofke, "Modeling the Monte Carlo Simulation of Associating Fluids", J. Chem. Phys, 110, 5493-5502, (1999).

(11) v. Roij, Rene, "Theory of Chain Association versus Liquid Condensation", Physical Review Letters, 76, 3348-3351, (1996).