Thermodynamic Fit Functions of the Two-Phase Fluid and Critical Exponents

Author(s)
Albrecht Elsner

Abstract

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*=1－2*β* and those of the entropy *s* and internal energy *u* are 2*β*, while that of the reciprocal isothermal compressibility κ^{﹣1}_{T} is *γ*=1. It is thus found that in the case of the two-phase fluid the Rushbrooke equation conjectured *α* + 2*β* + *γ*=2 combines the scaling laws resulting from the two relations *c*=d*u*/d*T* and κ_{T}=dln*ρ*/d*p*. 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, ﹣d^{2}*μ*/d*T*^{2} diverges as *c*, while d^{2}*p*/d*T*^{2} 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.

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

Keywords

Critical Condition*U* = 0,
Critical Opalescence,
Rushbrooke Equation,
Thermodynamic Fit Functions for Saturated Water,
Vapor and Liquid Volumes,
Vapor Pressure,
Chemical Potential,
Entropy,
Internal Energy,
Free Energy,
Heat Capacity

Critical Condition

Cite this paper

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

Elsner, A. (2014) Thermodynamic Fit Functions of the Two-Phase Fluid and Critical Exponents.

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