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F.: Broier, A. F.; Skuratov, S. M., Dokl. Akad. Nauk SSSR 173,385 (1966). von Wartenberg, H.; Hanisch, K., Z. Physik. Chem. A161, 413 (1932). Wu, C.; Birkey, M. M.; Hepler, L. G., J. Phys. Chem. 67, 1202 (1963). (Paper 75A2-651) JOURNAL OF RESEARCH of the National Bureau of Standards Vol. 75A, No. 2, March April 1971 A. Physics and Chemistry Expanded Formulation of Thermodynamic Scaling Critical Region Martin J. Cooper Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 (December 18, 1970) A description of the thermodynamic properties in the critical region of a physical system is obtained from a scaled expression for the free-energy F (p. T). In general, a nonsymmetric coexistence curve is predicted, with the symmetric case (e.g., magnets) included as a special example. For fluids, deviations from symmetry give rise to an expression for the average density below the critical point nonlinear in the temperature near 7. (in contrast to the usual "law of rectilinear diameter"); these asymmetries also contribute to the discontinuity in the specific heat along the critical isochore. To lowest order, the formulation reduces to Widom's homogeneous scaling: the classical equations of state of the van der Waals type are incorporated as a special case. Key words: Critical region; equation of state; liquid-gas transition; phase boundary; phase transition; thermodynamic scaling. in the 1. Introduction The Widom-Kadanoff-Domb scaling hypothesis [1-3] provides a qualitative characterization for the thermodynamic features of a second-order phase transition [4]. Sealing incorporates the usual power law forms for the various physical anomalies and more generally serves as a lowest order asymptotic description close to the critical point [5]. Such limiting behavior has been found in various mathematical models and also appears to be valid in real systems. Despite the relative success of the scaling approach, there remains a definite need for a more global picture of the critical region. This is evident during the analysis of bulk thermodynamic data where there exists first the problem of determining those physical quantities which satisfy the basic conditions required by simple scaling. For some systems, this selection is easily based upon some known/intrinsic physical property (symmetry); whereas for others, the actual choice is less certain [5]. The determination is further complicated by the secondary problem, that of estimating the actual range over which scaling remains valid [6]. These difficulties are particularly evident in the application of the scaling description to real fluids near their critical point [4]. Here there is some experimental indication of a partial anti-symmetry about the critical isochore in the chemical potential - number density p representation not present with either the pressure p or the specific volume r coordinates. It is also known, Figures in brackets indicate the literature references at the end of this paper however, that the phase boundary lacks this same symmetry in any set of variables. The coexistence curve is rather asymmetric about the critical isochore with such deviations being described by the empirical "law of rectilinear diameter." Despite the fact that this nonsymmetric behavior cannot be accommodated within the usual lowest order scaling description, the fluid data appear to scale in the preferred -p coordi nates over a rather large region about the critical point (in a density range of ± 30 percent and a temperature width of several percent from their reduced critical values [4]). An understanding and resolution of such behavior is hampered by an almost complete ignorance of what is beyond the lowest order scaling description. Since any results will be biased by the neglect of unknown terms, one cannot hope to extract any more than a simple qualitative picture from the experimental data without some knowledge as to the structure of corrections to the asymptotic scaling form. The present article represents an attempt to gain some insight into the structure of such terms. The idea was to construct a "model" calculation incorporating modification of the usual scaling formulation. Based upon an extension of classical thermodynamic ideas, the scheme describes systems lacking any known or simple symmetry and extends a scaling interpretation over an enlarged region about the transition. The approach allows for the first time discussion of the various contributions arising from asymmetry and higher ordered corrections to the asymptotic scaling behavior and their influence on observable physical properties. (After completing the |