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

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<sub style="margin-left:-10px;"><i>T</i></sub> 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.