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=1－2β 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.
Highlights
An essential property of matter is its structure, i.e. the distribution of its constituents in space and time as governed by inter-particle forces [1]
The course of a century saw the development of the familiar phenomenological theories of a van der Waals gas, of the stable and unstable thermodynamic equilibrium formulated by Gibbs, of the correlation of fluctuations, of the scaling laws, including the hierarchical reference theory, and of the Monte Carlo computer methods
The thermodynamical physics of critical phenomena above and below the critical point is extensively treated in the literature (e.g. [2] [7]-[10])
Summary
An essential property of matter is its structure, i.e. the distribution of its constituents in space and time as governed by inter-particle forces [1]. An insight into the nature of a fluid in the critical region is afforded by Figure 1, which for water of mass M = 1 [g] and critical density ρc in the volum= e V M= ρc 3.1056 [cm3] shows the different fluid states as a function of the temperature T. The absolute values both of the fluid, X , and of the fluid phases, vapor, Xv , and condensate (liquid, solid), Xl , are proportional to the mass in the volume considered. As extensive quantities they have additive properties, i.e. they satisfy the equations. ∂v) T is a function of the vapor pressure p (T ) , and the quotient ( xvvl − xlvv ) (vv − vl ) =(∂x, v ∂v)T a function of the chemical potential μ (T ) :
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