Thermodynamic Fit Functions of the Two-Phase Fluid and Critical Exponents
Albrecht Elsner
Am Mühlbach 14, D-85748 Garching, Germany.
DOI: 10.4236/eng.2014.612076   PDF    HTML   XML   3,265 Downloads   4,328 Views   Citations


Two-phase fluid properties such as entropy, internal energy, and heat capacity are given by thermodynamically defined fit functions. Each fit function is expressed as a temperature function in terms of a power series expansion about the critical point. The leading term with the critical exponent dominates the temperature variation between the critical and triple points. With β being introduced as the critical exponent for the difference between liquid and vapor densities, it is shown that the critical exponent of each fit function depends (if at all) on β. In particular, the critical exponent of the reciprocal heat capacity c﹣1 is α=1-2β and those of the entropy s and internal energy u are 2β, while that of the reciprocal isothermal compressibility κ﹣1T is γ=1. It is thus found that in the case of the two-phase fluid the Rushbrooke equation conjectured α + 2β + γ=2 combines the scaling laws resulting from the two relations c=du/dT and κT=dlnρ/dp. In the context with c, the second temperature derivatives of the chemical potential μ and vapor pressure p are investigated. As the critical point is approached, ﹣d2μ/dT2 diverges as c, while d2p/dT2 converges to a finite limit. This is explicitly pointed out for the two-phase fluid, water (with β=0.3155). The positive and almost vanishing internal energy of the one-phase fluid at temperatures above and close to the critical point causes conditions for large long-wavelength density fluctuations, which are observed as critical opalescence. For negative values of the internal energy, i.e. the two-phase fluid below the critical point, there are only microscopic density fluctuations. Similar critical phenomena occur when cooling a dilute gas to its Bose-Einstein condensate.

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Elsner, A. (2014) Thermodynamic Fit Functions of the Two-Phase Fluid and Critical Exponents. Engineering, 6, 789-826. doi: 10.4236/eng.2014.612076.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Stierstadt, K. (1989) Physik der Materie. VCH Verlag, Weilheim, Chapters 3.4, 10.2, 21.1
[2] Domb, C. (1996) The Critical Point. Taylor & Francis Ltd, London, Chapters 1, 2, 6.2.4
[3] Callen, H.B. (1960) Thermodynamics. John Wiley & Sons, Chapters 8.1-8.3, 10.4, 15, Equation (7.35).
[4] Grigull, U. and Schmidt, E. (1989) Properties of Water and Steam in SI Units. Springer-Verlag, Berlin, T,s-Diagram, p. 205, Equation 1.
[5] Wagner, W. and Pruss, A. (1993) International Equations for the Saturation Properties of Ordinary Water Substance. Revised According to the International Temperature Scale of 1990. Journal of Physical and Chemical Reference Data, 22, 783-787. The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. Journal of Physical and Chemical Reference Data, 31 (2002), 387-535.
[6] Kohlrausch, F. (1996) Praktische Physik. Band 3, Teubner Verlag Stuttgart, Tabelle 3.12.
[7] Stanley, H.E. (1971) Introduction to Phase Transitions and Critical Phenomena. Clarendon Press, Oxford, Chapters 1.1.1, 3.1, 3.4.2, 7.2.
[8] Liu, Y. and Suzuki, M. (1987) Some New Developments of the Scaling Theory of Transitient Phenomena. Phase Transitions, 10, 303-314.
[9] Kadanoff, L.P. (2009) More Is the Same; Phase Transitions and Mean Field Theories. Journal of Statistical Physics, 137, 777-797.
[10] Plascak, J.A. and Martins, P.H.L. (2013) Probability Distribution Function of the Order Parameter: Mixing Fields and Universality. Computer Physics Communications, 184, 259-269
[11] Elsner, A. (2012) Applied Thermodynamics of the Real Gas with Respect to the Thermodynamic Zeros of the Entropy and Internal Energy. Physica B: Condensed Matter, 407, 1055-1067.
[12] Pethick, C.J. and Smith, H. (2004) Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge, Chapter 5, Cover Illustration.
[13] White, J.A. and Maccabee, B.S. (1971) Temperature Dependence of Critical Opalescence in Carbon Dioxide. Physical Review Letters, 26, 1468-1471.
[14] Widom, B. (1965) Equation of State in the Neighborhood of the Critical Point. Journal of Chemical Physics, 43, 3898-3905
[15] Fisher, M.E. (1967) The Theory of Equilibrium Critical Phenomena. Reports on Progress in Physics, 30, 615-730.
[16] Rehr, J.J. and Mermin, N.D. (1973) Revised Scaling Equation of State at the Liquid-Vapor Critical Point. Physical Review A, 8, 472, Equations 2.8 and 5.5.
[17] Heller, P. (1967) Experimental Investigations of Critical Phenomena. Reports on Progress in Physics, 30, 731-826.
[18] Wagner, W. (1973) New Vapour Pressure Measurements for Argon and Nitrogen and a New Method for Establishing Rational Vapour Pressure Equations. Cryogenics, 13, 470-482.
[19] Stewart, R.B. and Jacobsen, R.T. (1989) Thermodynamic Properties of Argon. Journal of Physical and Chemical Reference Data, 18.
[20] Gilgen, R., Kleinrahm, R. and Wagner, W. (1994) Measurement and Correlation of the (Pressure, Density, Temperature) Relation of Argon, II. Saturated-Liquid and Saturated-Vapour Densities and Vapour Pressures Along the Entire Coexistence Curve. Journal of Chemical Thermodynamics, 26, 399-413.
[21] Tegeler, Ch., Span, R. and Wagner, W. (1999) A New Equation of State for Argon Covering the Fluid Region for Temperatures from the Melting Line to 700 K at Pressures up to 1000 MPa. Journal of Physical and Chemical Reference Data, 28, 779-850.
[22] Span, R. and Wagner, W. (1996) A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple Point Temperature to 1100 K at Pressures up to 800 MPa. Journal of Physical and Chemical Reference Data, 25, 1509-1596.
[23] Parola, A. and Reatto, L. (1995) Liquid State Theories and Critical Phenomena. Advances in Physics, 44, 211-298, Table 1, Chapter 5.5, Figure 15.
[24] Zhong, F. and Meyer, H. (1995) Density Equilibration near the Liquid-Vapor Critical Point of a Pure Fluid: Single Phase T > Tc. Physical Review E, 51, 3223-3241.
[25] Rowlinson, J.S. (1959) Liquids and Liquid Mixtures. Butterworths Scientific Publications, London, Chapter 3.4.
[26] Mouritsen, O.G. (1984) Computer Studies on Phase Transitions and Critical Phenomena. Springer-Verlag, Amsterdam.
[27] Potton, J.A. and Lanchester, P.C. (1985) Analysis of Critical Specific Heat Data. Phase Transitions, 6, 43-57.
[28] Landau, L.D. and Lifshitz, E.M. (1980) Statistical Physics. Part 1, Pergamon Press, Oxford, 449.
[29] Levelt Sengers, J.M.H. and Greer, S.C. (1972) Thermodynamic Anomalies near the Critical Point of Steam. International Journal of Heat and Mass Transfer, 15, 1865-1886, Figure 4.
[30] Schomacker, H. (1973) Messungen der inneren Energie und der spezifischen isochoren Warmekapazitat in der Umgebung des kritischen Zustands von Wasser. Thesis Ruhr-Universitat Bochum, Tabelle 1 und 2.
[31] Milonni, P.W. and Eberly, J.H. (2010) Laser Physics. Wiley & Sons, Hoboken, Chapter 16.

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