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Entropy 2020, 22, 437 18 of 26 4.2. Heat Capacity Definitions Thermodynamic state functions, internal energy (U) and enthalpy (H), and hence also the heat capacities Cv and Cp, are defined for the two-phase region, i.e., at T < Tc, according to the lever rule: ∂U Cv = ∂T V ∂H = xCv(liq) + (1 − x)Cv(gas) (5) = xCp(liq) + (1 − x)Cp(gas) (6) Cp = ∂T were x is the temperature-dependent mole fraction of liquid defined by the experimental density and coexisting densities: p x(T)=ρρ −ρ (7) liq gas ρ ρ−ρ liq gas Whereas Cv does not diverge either below or above Tc, the heat capacity at constant pressure Cp diverges both in the two-phase region and in the supercritical region, i.e., as T → Tc both above and below Tc, we obtain: ∂p ∂V →0 T as most of the heat added is converted into work of expansion. An experiment in the two-phase region, in a single cell that measures reversible heat added with increments of T, i.e., Qrev/∆T, therefore, requires the definition of a heat capacity at saturation of gas or liquid, which is usually designated Cσ. This can be calculated from the heat capacity Cp if the variation in thermal pressure γσ = (dp/dT)σ along the coexistence line is known. Cσ(liq) and Cσ(gas) are defined as the heat to reversibly increase the temperature of that phase in coexistence. Cσ for gas or liquid can be expressed in terms of available properties Cp, αp and γσ: Cσ =Cp+TVαpγσ, (8) where αp is the thermal expansivity, defined by: 1 ∂V αp = V ∂T . p Thus, in order to interpret 2-phase isochoric heat capacity experiments [56–58], and the NASA space shuttle microgravity experiments [60], a fourth heat capacity has been defined by the lever rule [60,61] and designated Cλ, |∂x| Cλ = xCσ(liq) + (1 − x)Cσ(gas) + ∆Hv ∂T (9) where ∆Hv is the latent heat of evaporation. Cσ for both liquid and gas, and enthalpies of coexisting phases, and hence Cλ, can be obtained for most pure atomic and molecular fluids from the NIST fluid property data bank [4]. Thermal expansivities in Equation (8) are calculated from Joule–Thompson coefficients: ∂p VTαp −1 μJ−T = ∂T = C (10) Hp σPDF Image | Supercritical Fluid Gaseous and Liquid States
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