Quantum nature of the critical points of substances
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Quantum nature of the critical points of substances
S.A. Trigger
Joint Institute for High Temperatures,Russian Academy of Sciences,13/19, Izhorskaia Str. 13 bild. 2,Moscow 125412, Russia;email: [email protected]
Thermodynamics of chemical elements, based on the two-component electron-nuclear plasma model shows that the critical parameters for the liquid-vaportransition are the quantum values for which the classical limit is absent.
For all substances in the vicinity of the critical points of the liquid-vapor transition, theinter-particle interaction is strong. Therefore, theoretical description of critical points seemsto be a very complicated problem. Numerical modeling allows for approximate calculationof the critical parameters for a known inter-particle interaction potential. However, the useof pair short-range potentials acceptable for the matter of low and moderate density (ingaseous state) cannot be justified for the region of the critical point parameters. This meansthat usual numerical calculations of these parameters with the model pair potentials (see,e.g., [1,2]) have empirical nature.To describe the critical parameters (for concreteness, only the chemical elements arediscussed below, although the main statements are universal), we propose to use the pureCoulomb interaction between the electrons and nuclei [3]. This basic model, i.e., the two-component homogeneous and isotropic electron-nuclei Coulomb system (CS) has been re-cently successfully applied to study the properties of dielectric permittivity [4,5]. It was alsoshown that the critical point of the two-component CS is related to the limiting behavior ofthe generalized screening length in the electron-nuclear plasma [6].In this Letter, we focus attention that in two-component electron-nuclear Coulombplasma, when the thermodynamic parameters tend to critical, the sole parameter with en-ergy dimension (excepting the parameters containing the critical ones) is the atomic energyunit me / ¯ h = 2 Ry = 27 ,
21 eV. It immediately follows from this statement that the criticaltemperature T c (in energy units) is given by T c = me ¯ h τ (cid:16) z, mM (cid:17) . (1)Here m , e are the electron mass and charge, respectively, M and ze are the mass andcharge of nuclei of the element under consideration. The function τ ( z, mM ) is the unknowndimensionless function of two dimensionless parameters z (nuclear charge number) and themass ratio m/M . From physical reasons, we can assume that the dependence of the criticaltemperature on the small parameter m/M can be neglected with good accuracy. Then theproblem reduces to the determination of only the function τ ( z ) T c ≃ me ¯ h τ ( z ) . (2)For the critical pressure, on the same basis as above one can write P c = me a ¯ h π (cid:16) z, mM (cid:17) ≃ me a ¯ h π ( z ) , (3)where a = ¯ h /me is the Bohr radius. At last, the critical pressure can be written in theform N c = 1 a n (cid:16) z, mM (cid:17) ≃ a n ( z ) . (4)The introduced dimensionless functions π and n under condition m/M ≪ z .Consideration of the experimental data for the critical points for the liquid-vapor transi-tion shows that the function τ ( z ) is a rapidly varying function of the variable z . One canassume that these rapid ”irregular” changes in the function τ ( z ) have the same physicalnature as spasmodic changes in the first ionization potential I ( z ) and valence v ( z ) which areconditioned by sequential filling of electron shells in the atomic model of matter. However, itshould be emphasized that the ionization potential is an approximate model characteristicrelated to the rarified (gaseous) state of matter. Therefore, the above assumption has alimited and very model nature. At the same time, the representations (1)-(3) are exact,although, in such a general form they cannot provide prediction of the critical points ofsome elements on the basis of the known critical points of other elements. For such predic-tions, methods for calculating or physical models of the functions τ ( z ), π ( z ) n ( z ) should bedeveloped.Useful information can be obtained already within the existing model approximations.As an example, let us consider the Van-der-Waals theory. As is known the relation P c = 3 N c T c , (5)follows from this theory.Although relation (5) is found from the classical and empirical approach, it can be usedwithin the range of its practical applicability to obtain the approximate relation betweenthe functions τ ( z ), π ( z ) n ( z ). The Van-der-Waals model leads to the relation π ( z ) = 38 n ( z ) τ ( z ) . (6)Similarly, the modern model theories of the equation of state and critical points of matter(see, e.g., [7] and references therein) can be used to approximate the unknown functions τ ( z ), π ( z ) n ( z ) according to the known experimental data and the periodic Mendeleev Table ofchemical elements. However, it is not the problem of this Letter.As follows from the basic relations (1)-(3) the parameters of the critical points are thequantum expressions which have no a classical limit or classical analogue . In this connection,the general question, outside the frameworks of this Letter, arises about the classification ofthe physical characteristics of the Coulomb matter into those having a classical limit or not. Acknowledgments
I am thankful to V.B. Bobrov for the useful discussions.This study was supported by the Netherlands Organization for Scientific Research(NWO), grant no. 047.017.2006.007 and the Russian Foundation for Basic Research, projectno. 07-02-01464-a. [1] Lotfi F, Vrabec J, and Fisher J,
Mol. Phys. , 1319 [2] Charpentier I, and Jakse N J, Chem. Phys . 2005, , 204910[3] Kraeft W-D, Kremp D, Ebeling W, and Ropke R, 1986
Quantum Statistics of Charged ParticleSystems (Berlin: Akademie-Verlag)[4] Bobrov V B, and Trigger S A,
Bulletin of the Lebedev Physics Institute , 2010 , No. 2, 35[5] Bobrov V B, Trigger S A, van Heijst G J F, and Schram P P J M, EPL , 2010, J. Phys. A , 2010 (in print)[7] Apfelbaum E M, and Vorob’ev V S,
J. Phys. Chem. B113