Asymptotic Behavior of the Widom Line in Supercritical Fluids: Theory of Complete Scaling and Experimental Verification

1. Isochoric Heat Capacity as a Fundamental Thermodynamic Property for Investigating the Asymptotic Behavior of the Widom Line in Supercritical Fluids

The isochoric heat capacity ($C_{\mathrm{V}}$) of supercritical fluids represents a key thermodynamic property of significant importance for both scientific research and technological applications [1,2,3,4]. In particular, $C_{\mathrm{V}}$plays an important role in studies of critical anomalies in thermodynamic properties, which are induced by large-scale fluctuations of the order parameter (density) and “supercritical phase transitions” between “liquid-like” and “gas-like” states [5]. These transitions are closely associated with the behavior of Widom lines. Among all thermodynamic properties, the isochoric heat capacity exhibits the highest sensitivity to microscopic changes within the system, thereby establishing its theoretical significance as a key thermodynamic characteristic of supercritical fluids.

The thermodynamic properties of supercritical fluids undergo anomalously sharp variations in the vicinity of the Widom line [6], where the transition from a “liquid-like” to a “gas-like” state occurs. Near this line, a crossover between “liquid-like” and “gas-like” regimes take place, resulting in pronounced changes in both thermodynamic and transport properties over narrow temperature intervals [6]. These transitions are driven by strong critical density (order parameter) fluctuations, which lead to significantly elevated heat transfer coefficients (critical enhancement).

This phenomenon has important practical implications. In the vicinity of the Widom line, supercritical fluids exhibit exceptionally high heat capacities, resulting in unusually large energy densities. Consequently, the enthalpy change along an isobar between two isotherms is significantly greater than in other regions of the supercritical state, as clearly illustrated in enthalpy–pressure (h–P) diagrams. This enhanced heat-transfer capability enables substantially more thermal energy to be exchanged per unit mass of fluid, allowing significant reductions in reactor wall thickness and the overall size of high-pressure process equipment, thereby promoting more compact, energy-efficient, and cost-effective process designs.

The present study has two main objectives. First, we experimentally characterize the asymptotic behavior of the isochoric heat capacity maxima loci for chemically and molecularly diverse liquids over a wide range of supercritical conditions. The resulting dataset represents one of the most extensive experimental databases currently available for investigating thermodynamic crossover behavior and the evolution of the Widom line beyond the critical point.

Second, the experimental results are used to critically assess the predictive capabilities of different classes of equations of state (EoS), including conventional multiparameter reference EoS, non-classical crossover EoS, and models based on complete and incomplete scaling theories of critical phenomena. Particular attention is devoted to the asymptotic behavior of the Widom line in the immediate vicinity of the critical point, where long-range density fluctuations dominate the thermodynamic response and conventional EoS often fail to reproduce the correct asymptotical critical behavior.

The obtained experimental database provides a reliable benchmark for validating advanced thermodynamic models and improving predictions of supercritical fluid behavior near the Widom line. Accurate determination of the Widom line is essential for optimizing operating conditions in technologies based on supercritical fluids, including supercritical Brayton power cycles, heat exchangers, compressors, nuclear reactor cooling systems, lithium-ion battery thermal management systems, and other advanced energy-conversion processes [5,6].

The reliability of an EoS should ultimately be evaluated by its ability to reproduce both volumetric ($PVT$) and caloric ($C_{\mathrm{V}}VT,C_{\mathrm{P}}PT$) properties, as well as to accurately describe thermodynamic crossover phenomena associated with the Widom line, where maxima of thermodynamic and transport properties occur. In this work, the behavior of the isochoric heat capacity ($C_{\mathrm{V}}$) maxima loci is predicted using non-classical theories of critical phenomena based on crossover scaling EoS, including incomplete and complete scaling approaches. The theoretical predictions are then systematically compared with directly measured experimental ($C_{\mathrm{V}}$)-maximum loci for various supercritical fluids.

2. Experimental

2.1. High-Temperature and High-Pressure $C_{\mathrm{V}}VT-$Apparatus

The experimental methodology, including the theoretical background, calorimeter design, calibration procedures, and uncertainty analysis, has been described in detail elsewhere [1,7,8,9,10,11,12,13,14,15]. The same high-temperature and high-pressure (HTHP) adiabatic calorimeter has been successfully employed for accurate measurements of the isochoric heat capacity of numerous pure fluids and mixtures over wide temperature and pressure ranges, including the critical and supercritical regions [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Therefore, only a brief description is presented here.

Isochoric heat capacities were measured using a nearly constant-volume adiabatic calorimeter consisting of a thin-walled spherical inner vessel enclosed within a thick-walled outer pressure vessel. Adiabatic conditions were maintained by a highly sensitive Cu₂O semiconductor layer located between the two spheres [22]. Owing to its exceptionally high thermoelectric power (approximately 1150 μV∙ K⁻¹), the Cu₂O layer acts as an integrated thermoelectric sensor capable of detecting temperature differences as small as 10⁻⁶–10⁻⁵ K between the inner and outer vessels. The resulting electrical signal is amplified and fed to a precision temperature controller, which continuously compensates for heat exchange with the surroundings, thereby maintaining true adiabatic conditions.

The Cu₂O layer also transmits pressure from the inner vessel to the outer pressure shell while introducing negligible conductive heat losses because of its low thermal conductivity (≈2.09 W∙ m⁻¹∙K⁻¹). Unlike conventional vacuum adiabatic calorimeters, which monitor temperature differences only at discrete locations, the present design provides integrated control over the entire calorimeter surface. Consequently, the apparatus is suitable for accurate heat-capacity measurements at temperatures up to 1000 K and pressures exceeding 100 MPa [22]. The sample temperature was measured with a calibrated 10 Ω platinum resistance thermometer (PRT-10) installed inside the calorimetric vessel.

2.2. Measurement Principle and Calibration

The isochoric heat capacity was determined from the electrical energy supplied to the sample, the resulting temperature increment, the sample mass, and the heat capacity of the empty calorimeter using [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]

$$C_{\mathrm{V}}={\displaystyle \frac{1}{m}}\left({\displaystyle \frac{\mathsf{\Delta }Q}{\mathsf{\Delta }T}}-C_0\right),$$

(1)

where m is the sample mass, $\mathsf{\Delta }Q=IU\tau$ is the electrical energy supplied by the heater, U and I are the voltage and current, respectively, τ is the heating time, ΔT is the temperature increment, and C₀ is the heat capacity of the empty calorimeter.

The heat capacity of the empty calorimeter was determined by calibration with reference fluids with well-known isochoric heat capacities (water, propane, n-heptane, benzene, etc.) [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]

$$C_0={\displaystyle \frac{\mathsf{\Delta }{Q}_0}{\mathsf{\Delta }{T}_0}}-C_{ref}{m}_0,$$

(2)

The average value of C₀ was approximately 190 J∙K⁻¹, which is much smaller than the total heat capacity of the filled calorimeter. Consequently, the contribution of the empty calorimeter to the overall measurement uncertainty is negligible.

The sample density at each temperature and pressure was calculated from as

$$\rho \left(P,T\right)=m/V_{\mathrm{PT}}\left(P,T\right),$$

(3)

where the pressure- and temperature-dependent calorimeter volume is

$$V_{pT}=V_{p_0{T}_0}\left[1+3\alpha \left(T-T_0\right)+\beta \left(P-P_0\right)\right],$$

(4)

Here, $V_{{\mathrm{P}}_0{\mathrm{T}}_0}=102.51\pm 0.01$ cm³∙g^-1 is the calorimeter volume at 298.15 K and atmospheric pressure, α is the thermal expansion coefficient of stainless steel (10Kh18N9T) [22,26], and β = (21.2 ± 2.4) × 10⁻⁴ cm³∙MPa⁻¹ is the experimentally determined pressure-expansion coefficient [22]. The reference volume was calibrated using the IAPWS reference EoS for pure water [27], whereas the pressure dependence of the calorimeter volume was independently verified using the Love equation for a thick-walled sphere [22,28,29]. The uncertainty of the calibrated calorimeter volume was approximately 0.05%, ensuring accurate density determination throughout the investigated temperature and pressure ranges.

2.3. Experimental Procedure

Isochoric heat capacities were measured along quasi-isochores by continuously heating a sealed calorimeter containing a known mass (m) of the sample. Each experiment was started in the two-phase (liquid–vapor) region and continued into the single-phase liquid region. As the system crossed the liquid–vapor phase boundary at constant density, simultaneous measurements of the isochoric heat capacity and saturation properties ($T_{\mathrm{S}}$,$P_{\mathrm{S}}$,${\rho }_{\mathrm{S}}$) were obtained.

The phase-transition temperature, $T_{\mathrm{S}}$, was determined from the discontinuity in the isochoric heat capacity, $C_{\mathrm{V}}$, at saturation temperature crossing, corresponding to the abrupt change between the single-phase and two-phase heat capacities ($\mathsf{\Delta }{C}_{\mathrm{V}}=C_{\mathrm{V}2}-C_{\mathrm{V}1}$, where $C_{\mathrm{V}1}$and$C_{\mathrm{V}2}$are the single- and two-phase heat capacities at saturation respectively, see Figure 1) [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22], or from the corresponding discontinuity in pressure (P-T isochoric break point [16,26,30,40,41,42,43]). Consequently, a single calorimetric experiment provided simultaneous measurements of the single-phase and two-phase heat capacities together with the corresponding saturation properties ($T_{\mathrm{S}}$,$P_{\mathrm{S}}$,${\rho }_{\mathrm{S}}$). When the calorimeter was equipped with a calibrated pressure transducer (piezo-calorimeter [2,13,16,22]), vapor pressure, single-phase PVT properties, and the thermal-pressure coefficient, ${\gamma }_{\mathrm{V}}={\left({\displaystyle \frac{\partial P}{\partial T}}\right)}_{\mathrm{V}}$, were measured simultaneously [2,13,16,22,31,32,33].

Figure 2. Experimental single- and two-phase isochoric heat capacities, ($C_{\mathrm{V}}$), of CO₂ as a function of temperature in the vicinity of the liquid–vapor phase transition along selected liquid and near-critical isochores [23,24,34]. Symbols denote the experimental data: ○, ($\rho$ = 483.09) kg·m⁻³; ●, ($\rho$ = 763.36) kg·m⁻³; △, ($\rho$= 902.93) kg·m⁻³. Solid lines represent values calculated from the reference fundamental EoS calculated from the NIST/REFPROP [25].

Figure 2. Experimental single- and two-phase isochoric heat capacities, ($C_{\mathrm{V}}$), of CO₂ as a function of temperature in the vicinity of the liquid–vapor phase transition along selected liquid and near-critical isochores [23,24,34]. Symbols denote the experimental data: ○, ($\rho$ = 483.09) kg·m⁻³; ●, ($\rho$ = 763.36) kg·m⁻³; △, ($\rho$= 902.93) kg·m⁻³. Solid lines represent values calculated from the reference fundamental EoS calculated from the NIST/REFPROP [25].

All measurements were performed under quasi-equilibrium conditions using slow continuous heating and cooling runs. The heating rate did not exceed 5×10⁻⁴ K ∙s⁻¹, ensuring that the measured temperature and pressure closely approximated thermodynamic equilibrium.

2.4. Quasi-Static Thermogram and Barogram Techniques

The liquid–vapor phase-transition ($T_{\mathrm{S}}$,$P_{\mathrm{S}}$,${\rho }_{\mathrm{S}}$) and critical ($T_{\mathrm{C}}$,$P_{\mathrm{C}}$,${\rho }_{\mathrm{C}}$) properties were determined using the quasi-static thermogram and barogram techniques developed in our previous studies [2,13,16,22,31,32,33]. These methods are based on continuous recording of temperature (thermograms, T–τ) and pressure (barograms, P–τ) during slow isochoric heating or cooling.

2.4.1. Quasi-Static Thermograms

The thermogram technique utilizes the discontinuous change in the isochoric heat capacity at the liquid–vapor phase transition. Crossing the phase boundary produces a distinct change in the slope of the temperature–time curve,

$$\mathsf{\Delta }{C}_V=k\left[{\left({\displaystyle \frac{\mathrm{d}\tau }{\mathrm{d}T}}\right)}_{V{T}_{\mathrm{S}}-0}-{\left({\displaystyle \frac{\mathrm{d}\tau }{\mathrm{d}T}}\right)}_{V{T}_{\mathrm{S}}+0}\right],$$

(5)

where ${\left({\displaystyle \frac{\mathrm{d}\tau }{\mathrm{d}T}}\right)}_{V{T}_{\mathrm{S}}-0}$and ${\left({\displaystyle \frac{\mathrm{d}\tau }{\mathrm{d}T}}\right)}_{V{T}_{\mathrm{S}}+0}$ are the thermogram slopes before ($T_{\mathrm{s}-0}$) and after ($T_{\mathrm{s}+0}$) the phase transition, respectively, and k depends on the heating power and sample mass. As the critical point is approached, the heat-capacity jump diverges according to asymptotical scaling power law,$\mathsf{\Delta }{C}_{\mathrm{V}}\propto {\left(T-T_C\right)}^{-\alpha }$, where α=0.106 is the universal critical exponent [35,36]. Consequently, the thermogram technique provides highly accurate determination of coexistence curves and critical parameters [2,13,16,22,31,32,33]. It has also been successfully applied to detecting weak first-order phase transitions in complex thermodynamic systems [16,33,37,38,39,44,45].

2.4.2. Quasi-Static Barograms

Pressure was measured before each calorimetric experiment at constant temperature, after which temperature and pressure were recorded simultaneously during continuous heating or cooling. The barogram technique identifies phase transitions through the discontinuity in the slope of the P–T isochore. This method is particularly effective for liquid isochores far from the critical point, where the density difference between the coexisting phases is large. Near the critical point, however, the pressure signal becomes less sensitive because the density difference between the liquid and vapor phases approaches zero (${\rho }_{\mathrm{S}}^{\mathrm{L}}={\rho }_{\mathrm{S}}^{\mathrm{G}}={\rho }_C$). Therefore, the quasi-static thermogram technique was primarily employed for determining near-critical phase-transition properties, whereas the barogram method served as an independent verification of the saturation data.

Simultaneous measurements of temperature and pressure also enabled direct determination of the thermal-pressure coefficient, ${\left(\partial P/\partial T\right)}_{\mathrm{V}}$=$\underset{\mathsf{\Delta }T\to 0}{\lim }{\left(\mathsf{\Delta }P/\mathsf{\Delta }T\right)}_{\mathrm{V}}$, from the slopes of the thermograms ($T=c+d\tau$) and barograms ($P=a+b\tau$) as ${\left(\partial P/\partial T\right)}_{\mathrm{V}}=a/b$.

2.5. Uncertainty Analysis

A comprehensive uncertainty analysis of the experimental method, calibration procedure, correction methods, and propagation of measurement uncertainties has been reported previously [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,31,32,33]. Therefore, only a brief summary is presented here.

The combined uncertainty of the isochoric heat-capacity measurements was evaluated according to the Guide to the Expression of Uncertainty in Measurement (GUM) [46,47,48] using the working equation (Equation (1)). Assuming statistically independent input quantities, the combined standard uncertainty was calculated from

where $u_{C_V}$, $u_{\rho }$, $u_T$, and $u_{rep}$ are the standard uncertainties associated with the heat capacity, density, temperature, and repeatability, respectively; $u\left(X_i\right)$ is the standard uncertainties of the independently measured variables, while the corresponding sensitivity coefficients were obtained from Equation (1). The required partial derivatives ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial T}}\right)}_{\rho }$ and ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_T$ were estimated from the experimental $C_{\mathrm{V}}VT$ data.

The principal sources of uncertainty include the electrical energy supplied to the calorimeter, sample mass, temperature increment, empty-calorimeter heat capacity, and calorimeter volume. The uncertainty of the sample mass was 0.006%, whereas that of the calibrated calorimeter volume was approximately 0.05%. Temperature was measured with an uncertainty better than 15 mK. Thus, the total combined expanded uncertainty (coverage factor k=2 with 95% confidence level) was calculated as $U_r\left(C_{\mathrm{V}}\right)=ku\left(C_{\mathrm{V}}\right)/C_{\mathrm{V}}$. Based on a detailed analysis of all uncertainty sources affecting the determination of $C_{\mathrm{V}}$, the combined expanded uncertainty of $C_{\mathrm{V}}$was estimated to be (3 to 4) % in the critical and supercritical regions (within Widom line range); 1.5% in the liquid phase, and 2-3% in the vapor phase. The corresponding expanded uncertainties of temperature and density were 15 mK and 0.15%, respectively.

3.1. Experimental $C_{\mathrm{V}}$Maxima Loci (Widom Lines) of Supercritical Fluids

As is well known, the isochoric heat capacity, ($C_{\mathrm{V}}$), together with other thermodynamic properties, such as the isobaric heat capacity, ($C_{\mathrm{P}}$), isothermal compressibility ($K_{\mathrm{T}}$), speed of sound, ($W$), and thermal expansion coefficient, (${\alpha }_{\mathrm{P}}$), as well as transport properties, including viscosity, thermal conductivity, and thermal diffusivity, exhibit pronounced anomalies in the near-critical and supercritical regions due to long-range critical density fluctuations [1]. These anomalies generally manifest themselves as pronounced maxima or minima along supercritical isotherms. The loci of maxima form characteristic lines in the supercritical region, commonly referred to as Widom lines. These lines can be regarded as extensions of the critical anomaly into the single-phase supercritical region. Along a Widom line, thermodynamic and transport properties undergo rapid, although continuous, changes associated with the crossover between predominantly “liquid-like” and “gas-like” behavior (see Figure 8, Figure 9, Figure 10 and Figure 11).

Most previous studies have described the Widom line either empirically or using classical multiparameter EoS based on mean-field critical behavior. Such approaches are unable to reproduce the experimentally observed critical anomalies accurately because they do not account for the dominant contribution of long-range density fluctuations in the vicinity of the critical point. Consequently, classical EoS generally fail to predict both the magnitude of the thermodynamic anomalies and the experimentally observed locations of their maxima (see Figure 1, Figure 4 and Figure 6, dotted curves). In contrast, crossover EoS based on the nonclassical scaling theory of critical phenomena explicitly incorporate the effects of critical fluctuations and provide a substantially more accurate description of the observed anomalies throughout the critical crossover region (see Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, solid curves).

Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 present representative experimental supercritical isotherms of the isochoric heat capacity for several molecular fluids. As shown, ($C_{\mathrm{V}}$) exhibits well-defined maxima as a function of density along each supercritical isotherm. This behavior is observed for liquids with widely different molecular structures and intermolecular interactions [2,3,7,8,9,12,19,20,21,34,49,50]. The observed similarity reflects the universal character of critical phenomena. In the immediate vicinity of the critical point, the correlation length of density fluctuations becomes much larger than the characteristic molecular interaction distance, ($\xi >>r_0$). Consequently, the thermodynamic behavior is governed predominantly by universal scaling laws rather than by the microscopic details of the intermolecular potential. For example, the singular contribution to the heat capacity follows a power-law dependence characterized by universal critical exponents, $C_{\mathrm{V}}=A_0{t}^{-\alpha }$, where $t={\displaystyle \frac{T-T_C}{T_C}}$, α = 0.106 [35].

The density corresponding to the ($C_{\mathrm{V}}$) maximum at each fixed supercritical temperature shifts systematically with increasing temperature. The temperature dependence of these density maxima defines the Widom line in the $T-\rho$and $P-T$ projections. In this work, we specifically define the experimental Widom line as the locus of the maxima of the isochoric heat capacity, ($C_{\mathrm{V}}$), along supercritical isotherms. This line characterizes the region of the most pronounced thermodynamic response and provides an experimentally accessible criterion for tracking the continuous crossover from predominantly “liquid-like” to “gas-like” behavior with increasing distance from the critical point.

The experimentally determined temperatures, densities, and pressures corresponding to the ($C_{\mathrm{V}}$) maxima for the selected molecular fluids are summarized in Table 1. The resulting experimental Widom lines are presented in Figure 12, Figure 13, Figure 14 and Figure 15. Their characteristic behavior and asymptotic approach to the critical region are discussed in the following sections (see sec. 4 and 5).

Figure 12, Figure 13, Figure 14 and Figure 15 present the loci of the isochoric heat capacity maxima (Widom lines) for several molecular fluids in the $T-\rho$and$P-T$projections, as determined from the experimental $C_{\mathrm{V}}\rho T$data. The figures also show the corresponding maximum loci predicted by the crossover scaling EoS, as discussed in Section 5, and by the classical multiparameter reference EoS, as discussed in Section 4, allowing direct comparison with the experimental results.

As can be seen from Figure 12, Figure 13, Figure 14 and Figure 15, the experimental $C_{\mathrm{V}}$ maxima originate at the critical density on the critical isotherm and shift monotonically toward lower densities with increasing temperature, then increases sharply at higher supercritical temperatures, and finally approaches the critical isochore again. The minimum density along the Widom line is reached at reduced densities, (${\rho }_{rm}=\rho /{\rho }_C$), ranging from approximately 0.92 for simple molecules, such as methane and propane, to about 0.98 for more complex molecules, such as water and heavy hydrocarbons. In addition, the reduced temperature, ($T_{rm}=T/T_C$), at which the Widom line crosses the critical density for the second time lies in the range 1.01–1.12. These two parameters, (${\rho }_{rm}$ and $T_{rm}$), characterize the main features of the asymptotic behavior of the Widom line associated with the maxima of the isochoric heat capacity in supercritical fluids. The same qualitative behavior is observed for all the fluids studied. However, the physical origin of this characteristic behavior of the Widom line remains unclear. In the $P-T$ projection, the asymptotic behavior of the Widom line is nearly linear (see Figure 15). This can be understood from the fact that, along supercritical isotherms, large changes in density correspond to relatively small changes in pressure. Therefore, most properties of the supercritical fluids less sensitive to the structural changes and cannot be detected in the $P-T$ projection, that is why details of the Widom line features near the critical point most pronounced in $T-\rho$ projection than in$P-T$, i.e., $C_{\mathrm{V}}$ maxima curve more sensitive to variation of density than pressure.

The prediction of the asymptotic behavior of the Widom line using classical multiparameter reference fundamental EoS and nonclassical scaling-type EoS based on the complete scaling theory is discussed below (flat behavior of the supercritical isotherm in $P-\rho$projection).

Figure 16 shows the $T-\rho$and $P-T$phase diagrams of carbon dioxide, together with the loci of the isochoric heat-capacity maxima and minima lines where ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_T=0$, and the region in which the effect of critical fluctuations on the thermodynamic properties is significant. In this region, the contribution of critical fluctuations is assumed to be within ($\mathsf{\Delta }{C}_{\mathrm{V}}^{\mathrm{fluc}}/C_{\mathrm{V}}>1-5\%$).

The Widom line, defined by the loci of the ($C_{\mathrm{V}}$) maxima, together with the loci of the ($C_{\mathrm{V}}$) minima in the $P-T$diagrams of CO₂ and H₂O, is presented in Figure 17. The $C_{\mathrm{V}}$extrema loci (maxima and minima), where ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_{\mathrm{T}}=0$, are directly related to the inflection points of the supercritical$P-T$isochores through the thermodynamic relation ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_{\mathrm{T}}=-{\displaystyle \frac{T}{{\rho }^2}}{\left({\displaystyle \frac{{\partial }^{2}P}{\partial T^2}}\right)}_{\mathsf{\rho }}$=0. Thus, Figure 17 simultaneously illustrates the maximum locus (CA), corresponding to the Widom line, and the minimum locus (AB). At point A, the two loci—the ($C_{\mathrm{V}}$)-maximum (Widom line) line and the ($C_{\mathrm{V}}$)-minimum line—coincide. At temperatures and pressures above point A, ($C_{\mathrm{V}}$) no longer exhibits either a maximum or a minimum and shows regular, monotonic behavior.

It is difficult to identify the inflection points where ${\left({\displaystyle \frac{{\partial }^{2}P}{\partial T^2}}\right)}_{\mathsf{\rho }}=0$ of the $P-T$isochores in the supercritical region from conventional (PVT) measurements, even when high-precision data are available, because the magnitude of the curvature ${\left({\displaystyle \frac{{\partial }^{2}P}{\partial T^2}}\right)}_{\mathsf{\rho }}$of the $P-T$isochores is very small ( $P\propto t^{2-\alpha }$), for example, for carbon dioxide ${\left({\displaystyle \frac{{\partial }^{2}P}{\partial T^2}}\right)}_{\mathsf{\rho }}$is varying within (from -5.3×10^-9 to 1.45×10^-4) MPa⋅K^-2 at supercritical condition. This further demonstrates that the isochoric heat capacity is a highly sensitive thermodynamic property for investigating the qualitative behavior of the thermodynamic (PVT) surface, particularly for identifying Widom lines and characterizing supercritical phase-transition phenomena.

The second temperature derivative of the supercritical $P-T$isochores, ${\left({\displaystyle \frac{{\partial }^{2}P}{\partial T^2}}\right)}_{\mathsf{\rho }}$, exhibits very complex behavior (see Figure 18). The curvature of $P-T$isochores is positive within the isochore inflection curve, and negative outside [155,156,157,158,159,160].

As shown in this figure, the density in the supercritical region of water at which the second temperature derivative of pressure vanishes, ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_{\mathrm{T}}=-{\displaystyle \frac{T}{{\rho }^2}}{\left({\displaystyle \frac{{\partial }^{2}P}{\partial T^2}}\right)}_{\mathsf{\rho }}$=0, and hence the ($C_{\mathrm{V}}$) maxima occur for supercritical isotherms, does not coincide exactly with the critical density. With increasing temperature, the loci of these maxima progressively shift toward lower densities. The same behavior is observed for other supercritical fluids, as illustrated in Figure 19 for carbon dioxide. Figure 19 shows the corresponding ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_{\mathrm{T}}$or ${\left({\displaystyle \frac{{\partial }^{2}P}{\partial T^2}}\right)}_{\mathsf{\rho }}$behavior of CO₂ along critical isotherm.

Comparison of experimentally determined extrema of $C_{\mathrm{V}}$ and their loci, ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_T=0$, with predictions from an equation of state (EoS) provides a stringent test of the accuracy and reliability of the EoS. Numerous $PVT$ EoS have been developed that reproduce thermodynamic properties within their experimental uncertainties. Nevertheless, different $PVT$EoS formulations can predict substantially different, and in some cases even qualitatively contradictory, behavior of the extrema of $C_{\mathrm{V}}$ and their maxima loci in the vicinity of the critical point.

The extrema of the isochoric heat capacity have been investigated in a number of previous studies [56,155,156,157,158,159,160,161,162,163,164,165,166] using different type classical$PVT$ EoS. Most early investigations of the loci of $C_{\mathrm{V}}$ maxima were performed for relatively simple fluids, including noble gases and other low-molecular-weight substances such as Ar, Kr, Xe, Ne, O₂, F₂, ⁴He, and p-H₂. Stephenson [159,160] reported calculations of the maxima and minima of $C_{\mathrm{V}}$ along isotherms for argon using an analytical EoS. Gaddy and White [163] analyzed the experimental $C_{\mathrm{V}}$ data for CO₂ reported by Amirkhanov et al. [23,24] to determine the behavior of the isothermal extrema loci in the vicinity of the critical point and to compare the experimental results with predictions based on a linear model of a scaling EoS.

The loci of $C_{\mathrm{V}}$ extrema for the simple fluids Kr and Ar were investigated theoretically by Bearman et al. [158] using the Percus–Yevick equation. Amirkhanov et al. [162] performed a detailed analysis of experimental maxima and minima of the $C_{\mathrm{V}}$for several hydrocarbons along isotherms and isochores in both $P-T$ and $T-\rho$representations. Theeuwes and Bearman [157] investigated the loci of $C_{\mathrm{V}}$ maxima and minima by analyzing the $PVT$behavior of Kr and Xe in the dense-gas and liquid regions. Magee and Kobayashi [155] and Magee [156] developed an empirical equation for $C_{\mathrm{V}}$ isochores and fitted it to high-precision experimental $PVT$data to investigate the behavior of the isochoric inflection loci, ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_{\mathrm{T}}=-{\displaystyle \frac{T}{{\rho }^2}}{\left({\displaystyle \frac{{\partial }^{2}P}{\partial T^2}}\right)}_{\mathsf{\rho }}$=0, for several fluids and fluid mixtures.

Diller [161] examined characteristic features of the temperature and density dependence of $C_{\mathrm{V}}$ for several simple fluids at high densities (around $2{\rho }_{\mathrm{C}}$), including 4He, Ne, Ar, Kr, p-H₂, O₂, and F2, with particular emphasis on the behavior of the isothermal minima loci of $C_{\mathrm{V}}$ near the $2{\rho }_{\mathrm{C}}$. Gladun [164,165,166] reported systematic studies of the dependence of $C_{\mathrm{V}}$ on molar volume for Ar, Kr, and Ne. In particular, Gladun [164] investigated the isothermal minima of $C_{\mathrm{V}}$ for neon in the temperature range 37–43 K and found that the corresponding locus terminates at the coexistence curve. This observation is in very good agreement with the results obtained in the present study for more complex fluids (see Figure 16). Bagatskii et al. [181] and Voronel’ et al. [182] investigated critical $C_{\mathrm{V}}$enhancement in supercritical Ar and O2, respectively. These studies further demonstrate the importance of accurate characterization of the thermodynamic-property extrema in the critical and supercritical regions.

4. Classical Multiparameter Analytical and Non-Analytical Reference Equations of State

In recent years, accurate multiparameter reference fundamental EoS, such as those implemented in NIST/REFPROP [25], have become a standard database for the calculation of thermodynamic properties of technologically important fluids in both scientific and industrial applications (see also [27,68,72,76,102,103,104,105,106,107,108]). However, most multiparameter EoS, which are essentially analytical at the critical point, do not correctly represent the critical anomalies of supercritical fluids in the immediate vicinity of the critical point, i.e., within the asymptotic region, $t<{10}^{-3}$. Consequently, they also fail to reproduce the correct asymptotic behavior of the Widom line in the immediate vicinity of the critical point (asymptotic range). For example, as shown in Figure 20, the maximum of $C_{\mathrm{V}}$, and therefore the $C_{\mathrm{P}}$, loci, along the critical and supercritical isotherms shift considerably toward lower densities and do not turn toward the high-density region or cross the critical density at elevated temperatures, as observed experimentally (see Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 in $T-\rho$projections).

However, in contrast most non-analytical multiparameter predict the originates of the Widom line at the critical density (along the critical isotherm). With increasing temperature, the line shifts toward lower densities, passes through a minimum in density, as shown in Figure 21, and then the density increases at higher supercritical temperatures, eventually crossing the critical density again.

As is well known, an EoS that is analytical at the critical point cannot reproduce the critical anomalous behavior of the isochoric heat capacity and other thermodynamic properties ($C_{\mathrm{P}}$,$K_{\mathrm{T}}$,$W$, ${\alpha }_{\mathrm{P}}$) in the critical and supercritical regions. Classical (analytical) EoS fail to describe the critical anomalies of thermodynamic properties, which obey simple power-law dependencies along the critical isochore (for example, $C_{\mathrm{V}}=A_0^{+}{t}^{-\alpha }$) and critical isotherm ($C_{\mathrm{V}}=A_1^{+}\mathsf{\Delta }{\rho }^{-\alpha /\beta }$, where α and β are universal critical exponents), as rigorously predicted by the modern non-classical scaling theory of critical phenomena [35,109,110,111,112,113,114,115]. The fundamental limitation of classical analytical EoS is their inability to reproduce the non-analytic power-law singularities that characterize thermodynamic properties in the immediate vicinity of the critical point.

Common formulations for the residual Helmholtz energy ${\alpha }^r(\delta ,\tau )$ in multiparameter EoS are given by Span [108]

$${\alpha }^r(\delta ,\tau )={\displaystyle {\displaystyle \sum _{i-1}^{I_{pol}}}{N}_i{\delta }^{d_i}}{\tau }^{t_i}+{\displaystyle {\displaystyle \sum _{i-I_{pol}+1}^{I_{pol}+I_{exp}}}{N}_i{\delta }^{d_i}}{\tau }^{t_i}exp(-{\gamma }_i{\delta }^{P_i}),$$

(9)

where $\alpha ={\displaystyle \frac{a}{RT}}$, $\tau =T_C/T$, $\delta =\rho /{\rho }_C$, andR=8.31451J⋅mol^-1⋅ K^-1.

When ${\rho }_C$is used to reduce the density, the ${\gamma }_i$ are usually equal to 1. The density exponents $d_i$and $P_i$ have to be positive integer values to ensure that ${\alpha }^r(\delta ,\tau )$and all its derivatives vanish in the limit of zero density ($\rho \to 0$). For $P_i$values up to 8 have been used, and even higher values are common for $d_i$. The $t_i$are mostly positive and may be real numbers, resulting in infinite contributions to the Helmholtz energy at temperatures approaching zero. The main advantage of Equation (9) is that the derivatives required for calculating thermodynamic properties can be expressed analytically for any combination of $d_i$ and $P_i$. To improve the representation of thermodynamic properties in the extended critical region, special terms have been introduced. The equation for pressure can be derived from Equation (9) using the general thermodynamic relation

$${\displaystyle \frac{P}{\rho RT}}=1+\delta {\left({\displaystyle \frac{\partial {\alpha }^r}{\partial \delta }}\right)}_{\tau }.$$

(10)

Different functional forms of terms have been proposed for classical EoS to improve the representation of the critical behavior of the isochoric heat capacity and other thermodynamic properties of supercritical fluids (see, for example, [102,103,104,105,106,107,108]). Haar et al. [117] proposed two-dimensional Gaussian bell-shaped terms to improve the representation of in the supercritical region of pure water. However, these terms distorted the representation of other properties near the critical point. Setzmann and Wagner [72] modified the Gaussian bell-shaped term by introducing an additional parameter. This modification significantly improved the quality of the representation of near-critical behavior for all thermodynamic properties. The residual Helmholtz energy with the modified Gaussian bell-shaped term, introduced to improve the description of $C_{\mathrm{V}}$and $C_{\mathrm{p}}$near the critical point, can be written as [72]:

$${\alpha }^r\left(\tau ,\delta \right)={\displaystyle \sum n_i{\tau }^{t_i}{\delta }^{d_i}+}{\displaystyle \sum n_i{\tau }^{t_i}{\delta }^{d_i}exp\left(-{\gamma }_i{\delta }^{Pi}\right)+}{\displaystyle \sum n_i{\tau }^{t_i}{\delta }^{d_i}exp\left[-{\eta }_i{\left(\delta -{\epsilon }_i\right)}^2-{\beta }_i{\left(\tau -{\gamma }_i\right)}^2\right]},$$

(11)

The isochoric heat capacity can be calculated from the reduced Helmholtz energy function as:

$${\displaystyle \frac{C_{\mathrm{V}}}{R}}=-{\tau }^2\left({\alpha }_{\tau \tau }^0-{\alpha }_{\tau \tau }^r\right),$$

(12)

where ${\alpha }_{\tau \tau }^0=\left({\displaystyle \frac{{\partial }^2{\alpha }^0}{\partial {\tau }^2}}\right)$ and ${\alpha }_{\tau \tau }^r=\left({\displaystyle \frac{{\partial }^2{\alpha }^r}{\partial {\tau }^2}}\right)$are the second temperature derivatives of the reduced Helmholtz energy of the ideal and real gases, respectively. For classical non-analytic EoS, the second temperature derivative of the reduced Helmholtz energy ${\alpha }_{\tau \tau }^r$is infinite at the critical point, ${\alpha }_{\tau \tau }^r\to \infty$. The reduced Helmholtz energy with Gaussian bell-shaped terms, Equation (11), is able to represent experimental data in the critical and supercritical regions within their experimental uncertainties, except for the isochoric heat capacity and speed of sound in the very close vicinity of the critical point ($t$<0.002 and $\mathsf{\Delta }\rho$<0.25, where $t=\tau -1$ and $\mathsf{\Delta }\rho =\delta -1$), where these properties cannot be adequately described by analytical EoS [108].

Span [108] proposed introducing a non-analytical term to correctly represent the sharp anomalous increase (see Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11) of the $C_{\mathrm{V}}$in the critical and supercritical regions. A modified form of the non-analytical terms in the EoS of Equation (11) was used by Span and Wagner [72] for CO₂ and by Pruß and Wagner [27] for H₂O. The modified reduced Helmholtz energy including the non-analytical term is:

$${\alpha }^r\left(\tau ,\delta \right)={\displaystyle \sum n_i{\tau }^{t_i}{\delta }^{d_i}+}{\displaystyle \sum n_i{\tau }^{t_i}{\delta }^{d_i}exp\left(-{\gamma }_i{\delta }^{Pi}\right)+}{\displaystyle \sum n_i{\tau }^{t_i}{\delta }^{d_i}exp\left[-{\eta }_i{\left(\delta -{\epsilon }_i\right)}^2-{\beta }_i{\left(\tau -{\gamma }_i\right)}^2\right]+}{\alpha }_{NA}^r,$$

(13)

where the following nonanalytic terms for the reduced Helmzotz energy were proposed by Span [108]:

$${\alpha }_{NA}^r={\displaystyle \sum n_i\delta {\mathsf{\Delta }}^{b_i}exp(-e_i{(\delta -1)}^2-f_i{(\tau -1)}^2)},$$

(14)

where $\mathsf{\Delta }={\left\{\left(1-\tau \right)+c_i{\left[{\left(\delta -1\right)}^2\right]}^{1/2{\beta }_i}\right\}}^2+d_i{\left[{\left(\delta -1\right)}^2\right]}^{a_i}$. The second temperature derivative of the nonanalytic terms ${\alpha }_{NA}^r$ is [108]:

$$\left({\displaystyle \frac{{\partial }^2{\alpha }_{NA}^r}{\partial {\tau }^2}}\right)=n_i\delta \left[{\displaystyle \frac{{\partial }^2{\mathsf{\Delta }}^{b_i}}{\partial {\tau }^2}}\Psi +2{\displaystyle \frac{\partial {\mathsf{\Delta }}^{b_i}}{\partial \tau }}{\displaystyle \frac{\partial \Psi }{\partial \tau }}{\mathsf{\Delta }}^{b_i}{\displaystyle \frac{{\partial }^2\Psi }{\partial {\tau }^2}}\right],$$

(15)

where $\Psi =exp\left[-e_i{(\delta -1)}^2-f_i{(\tau -1)}^2\right]$, $\theta =(1-\tau )+c_i{\left[{(\delta -1)}^2\right]}^{{\displaystyle \frac{1}{2{\beta }_i}}},$The number of non-analytical terms in Equation (9) varies between 2 and 3. Unfortunately, the non-analytical terms introduced by Span [108] and Span and Wagner [72,98] still do not correctly reproduce the asymptotic critical behavior of $C_{\mathrm{V}}$near the critical point in the form of the theoretically predicted power law, $C_{\mathrm{V}}=A_0{t}^{-\alpha }$with the universal critical exponent $\alpha$=0.106. Span 108] analyzed the functional form of the non-analytical term of the reduced Helmholtz energy and found that the non-analytical term in Equation (13) should satisfy the following conditions to correctly reproduce the scaling asymptotic power laws for $C_{\mathrm{V}}$: $b_i$>0.5 and $a_i<{\left[2{\beta }_i\left(b_i-1\right)\right]}^{-1}$. The second temperature derivative of the non-analytical term in Equation (13) diverges at the critical point in the same manner as the isochoric heat capacity, as $C_{\mathrm{V}}\propto t^{-\alpha }$), if the critical exponent of $C_{\mathrm{V}}$ is $\alpha$=$2-2b_i$ or $b_i$=0.945 ($\alpha$=0.106 universal critical exponent [35,36]). The coefficients $n_i$of the leading non-analytical terms must be negative ($n_i$<0) to result in $C_{\mathrm{V}}=+\infty$at the critical point. The isochoric heat capacities calculated from multiparameter EoS for various pure fluids are shown in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. The uncertainties of the multiparametric reference EoS are approximately 0.2% (up to 0.5% at high pressures) in density, 1% (in the vapor phase) to 2% in heat capacity, 1% (in the vapor phase) to 2% in the speed of sound, and 0.2% in vapor pressure, except in the critical region [25].

To calculate $C_{\mathrm{V}}$in the critical and supercritical regions for pure water, the IAPWS-95 formulation [27] incorporates a special non-analytical term into the Helmholtz energy function that provides infinite values of the second temperature derivative $\left({\displaystyle \frac{{\partial }^2{\alpha }_{NA}^r}{\partial {\tau }^2}}\right)$at the critical point. The experimental basis of the fundamental EoS includes $C_{\mathrm{V}}$ data measured by Baehr and Schomacker [118,119]. As the critical point is approached, this IAPWS formulation [27] exhibits singular behavior with regard to the $C_{\mathrm{V}}$→∞ (as $T\to T_{\mathrm{C}}$ and $\rho \to {\rho }_C$). The non-analytical terms are explicit functions of density $\rho$and temperature T, expressed as differences of parametric scaling functions, and can be evaluated analytically. This is an advantage of the IAPWS EoS for practical applications compared with parametric crossover EoS (see Section 5 below).

Therefore, the IAPWS-95 formulation [27] correctly represents the anomalous behavior of $C_{\mathrm{V}}$in the immediate vicinity of the critical point (t<0.001), although some problems may arise in the form of unphysical predictions [61,183]. Wyczalkowska and Sengers [183] reported an evaluation of the 1995 version of the equation developed by Wagner and Pruss [27] in the critical region. The leading asymptotic behavior of along the critical isochore for this equation is $C_{\mathrm{V}}\propto t^{-0.2}$, which differs from the theoretically predicted power-law behavior of $C_{\mathrm{V}}\propto t^{-0.106}$ [35,36]. This prevents the equation from correctly representing the behavior of the isothermal compressibility and isobaric heat capacity in the asymptotic region, particularly at densities approximately ±0.5% around the ${\rho }_{\mathrm{C}}$at temperatures from $T_{\mathrm{C}}$ up to 5 K above $T_{\mathrm{C}}$ [183].

Thus, the Widom line ($C_{\mathrm{V}}$maximum loci) predicted by analytical multiparameter reference EoS in $T-\rho$ projection is inconsistent with the present experimental results (see Section 3), which demonstrate that the loci of the $C_{\mathrm{V}}$maxima exhibit fundamentally different behavior in the supercritical region. This discrepancy indicates that classical multiparameter reference EoS, despite their high accuracy over a wide range of supercritical conditions, do not qualitatively reproduce the behavior of the $C_{\mathrm{V}}$maxima line in $T-\rho$ projection associated with long-range density fluctuations. Consequently, they fail to predict the correct location and asymptotic behavior of the Widom line, although the corresponding difference is not noticeable in $P-T$projection (see Figure 15 and Figure 16, right).

5. Nonclassical (Scaling -Type) Crossover Equations of State for Supercritical Fluids

As discussed above, classical EoS for supercritical fluids do not ensure a universal description of the thermodynamic behavior of real fluids, particularly $C_{\mathrm{V}}$, in the immediate vicinity of the critical point. In this section, we review scaling-type crossover EoS for pure supercritical fluids that are used to accurately calculate $C_{\mathrm{V}}$and other thermodynamic properties under near-critical and supercritical conditions, as predicted by the modern theory of critical phenomena based on complete- and incomplete-scaling models [120,121,122,123,124,125,126,127,128,129,130,131,132,133].

Figure 22 illustrates the $C_{\mathrm{V}}$maxima loci of n-butane and ammonia predicted using the analytical multiparameter reference EoS [25] (see also [70,135,136]) and crossover EoS [137,138].

As can be seen from Figure 22, the scaling-type crossover EoS qualitatively reproduces the experimentally observed asymptotic behavior of the Widom line ($C_{\mathrm{V}}$maxima curves; see Section 3). In $P-T$projection, the difference between the multiparameter analytical reference EoS and the scaling-type crossover EoS is not noticeable because relatively small changes in pressure can produce large changes in density in the supercritical region due to the large compressibility, $K_T\propto t^{-\gamma }$, where γ=1.24.

5.1. Extended Parametric Scaling Model of the Equation of State

To represent the thermodynamic properties of near-critical and supercritical fluids, an extended parametric scaling-type EoS is used [139,140,141,142,143,144,145,146], which is defined as

$$\mathsf{\Delta }\mu =a{r}^{\beta \delta }\theta {(1-\theta )}^2+c{r}^{\beta \delta +\mathsf{\Delta }}\theta ,$$

(16)

where $\mathsf{\Delta }\mu =\mu -{\mu }_0\left(T\right)$, $\mu$is the chemical potential, ${\mu }_0\left(T\right)$ is the analytical background of the chemical potential, a and c are system-dependent parameters, β, δ, and$\mathsf{\Delta }$ are universal critical exponents, and r and θ are parametric variables. The transformation of the thermodynamic variables (T and ρ) to the parametric variables (r and θ) is defined as

$$t=r(1-b^2{\theta }^2)$$

(17)

where $b^2=(\gamma -2\beta )/\gamma (1-2\beta )$is the universal parameter ($b^2$=1.359) defined by the universal critical exponents (β and γ), $t=(T-T_C)/T_C$, $\mathsf{\Delta }\rho =(\rho -{\rho }_C)/{\rho }_C$, and $k=(b^2-1)/x_0$, $x_0$is the values of scaling variable $x=t{\left|\mathsf{\Delta }\rho \right|}^{-1/\beta }$ on the coexistence curve. The EoS in terms of pressure is derived from Equation (16) as

$$\mathsf{\Delta }P=(1+\mathsf{\Delta }\rho )\mathsf{\Delta }\mu -\Phi (r,\theta )-{\Phi }_0(T),$$

(18)

where

$$\begin{array}{l}\Phi (r,\theta )={\displaystyle \frac{ak}{2b^4}}{r}^{2-\alpha }\left[{\displaystyle \frac{2\beta (b^2-1)}{2-\alpha }}+{\displaystyle \frac{2\beta (2\gamma -1)}{\gamma (1-\alpha )}}(1-b^2{\theta }^2)+{\displaystyle \frac{(2\beta -1)}{\alpha }}{(1-b^2{\theta }^2)}^2\right]-{\displaystyle \frac{kc}{2b^2}}{\displaystyle \frac{r^{2-\alpha +\mathsf{\Delta }}}{1-\alpha +\mathsf{\Delta }}}\left[(1-2\beta )b^2{\theta }^2-{\displaystyle \frac{\gamma +\mathsf{\Delta }}{2-\alpha +\mathsf{\Delta }}}\right]\end{array}$$

(19)

and the analytical function of temperature ${\Phi }_0(T)$ is a regular function of temperature

$${\Phi }_0(T)=a_{1}t+a_2{t}^2+a_3{t}^3.$$

(20)

The scaling behavior of the isochoric heat capacity calculated from this EoS is defined as

$${\displaystyle \frac{\rho C_{\mathrm{V}}}{T}}={\displaystyle \frac{P_C}{T_C^2}}\left[{\displaystyle \frac{ak}{2\alpha b^2}}\gamma (\gamma -1)r^{-\alpha }-{\displaystyle \frac{kc}{2b^2}}(\gamma +\mathsf{\Delta }){\displaystyle \frac{r^{-\alpha +\mathsf{\Delta }}}{1-(1-2\beta )b^2{\theta }^2}}-{\mu }^{"}({\rho }_C,T)\mathsf{\Delta }\rho +f_0(T)\right],$$

(21)

where $f_0(T)=f_1+f_{2}t$ and ${\mu }^{"}({\rho }_C,T)=m_1+m_{2}t$ are analytical functions of the reduced temperature difference. This EoS has been successfully used to represent experimental $C_{\mathrm{V}}$and other thermodynamic properties ($C_{\mathrm{P}}$,$K_{\mathrm{T}}$, $W$,${\alpha }_{\mathrm{P}}$, etc.) of several pure fluids, including CO₂, H₂O, n-alkanes, and alcohols, in the critical and supercritical ranges [141,142,143,144,145,146].

As can be seen from Equation (21), the model correctly reproduces the asymptotic and non-asymptotic scaling laws for the $C_{\mathrm{V}}$ (also other thermodynamic functions) along selected thermodynamic paths, including the critical isochore, critical isotherm, and coexistence curve, in the vicinity of the critical point, as predicted by the scaling theory of critical phenomena:

$${\displaystyle \frac{{\rho }_C{C}_{\mathrm{V}}}{T}}={\displaystyle \frac{P_C}{T_C^2}}\left[{\displaystyle \frac{ak}{2\alpha b^2}}\gamma \left(\gamma -1\right)t^{-\alpha }-{\displaystyle \frac{kc}{2b^2}}\left(\gamma +\mathsf{\Delta }\right)t^{-\alpha +\mathsf{\Delta }}+f_0\left(T\right)\right],$$

(22)

along the critical isochore,$\rho ={\rho }_C$ ($r=t$,$\theta =0$),

$${\displaystyle \frac{\rho C_{\mathrm{V}}}{T_C}}={\displaystyle \frac{P_C}{T_C^2}}\left[{\displaystyle \frac{ak}{2\alpha b^2}}\gamma \left(\gamma -1\right){\left({\displaystyle \frac{b}{k}}\right)}^{1/\beta }\mathsf{\Delta }{\rho }^{-\alpha /\beta }-{\displaystyle \frac{kc}{4\beta b^2}}\left(\gamma +\mathsf{\Delta }\right)\mathsf{\Delta }{\rho }^{\left(-\alpha +\mathsf{\Delta }\right)/\beta }-{\mu }^{"}\left({\rho }_C,T_C\right)\mathsf{\Delta }\rho +f_0\left(T_C\right)\right],$$

(23)

along the critical isotherm, $T=T_C$ ($r={\left({\displaystyle \frac{b}{k}}\right)}^{1/\beta }\mathsf{\Delta }{\rho }^{1/\beta }$,$\theta =\pm b^{-1}$),

$${\displaystyle \frac{\rho C_V}{T}}={\displaystyle \frac{P_C}{T_C^2}}\left[{\displaystyle \frac{ak}{2\alpha b^2}}{\displaystyle \frac{\gamma \left(\gamma -1\right)}{{\left(1-b^2\right)}^{-\alpha }}}{t}^{-\alpha }-{\displaystyle \frac{kc}{2b^2}}{\displaystyle \frac{\left(\gamma +\mathsf{\Delta }\right){\left(1-b^2\right)}^{\alpha -\mathsf{\Delta }}}{1-\left(1-2\beta \right)b^2}}{t}^{-\alpha +\mathsf{\Delta }}-{\mu }^{"}\left({\rho }_C,T\right)\mathsf{\Delta }\rho +f_0\left(T\right)\right],$$

(24)

along the coexistence curve, ($r=t{\left(1-b^2\right)}^{-1}$,$\theta =\pm 1$), $T=T_S$,

where $A_0^{+}={\displaystyle \frac{ak}{2\alpha b^2}}\gamma \left(\gamma -1\right)$ and $A_0^{-}={\displaystyle \frac{ak\gamma {\left(b^2-1\right)}^{\alpha }}{2\alpha b^2}}{\displaystyle \frac{{\left(1-2\beta \right)}^2}{\left(\gamma -1\right)}}$ are the asymptotical critical amplitudes of $C_{\mathrm{V}}$along the critical isochore at $T>T_{\mathrm{C}}$(single –phase) and $T<T_{\mathrm{C}}$ (two-phase), respectively. The isobaric heat capacity $C_{\mathrm{P}}$is defined as

$${\displaystyle \frac{\rho C_P}{T}}={\displaystyle \frac{P_C}{T_C^2}}\left[{\displaystyle \frac{ak}{2\alpha b^2}}\gamma (\gamma -1)r^{-\alpha }-{\displaystyle \frac{kc}{2b^2}}(\gamma +\mathsf{\Delta }){\displaystyle \frac{r^{-\alpha +\mathsf{\Delta }}}{1-(1-2\beta )b^2{\theta }^2}}+{(1+k{r}^{\beta }\theta )}^{-2}{\left\{ak{r}^{1-\alpha }(s_0(\theta )+\gamma {\theta }^2)+a{r}^{\beta \delta -1}\gamma \theta +c{r}^{\beta \delta -1+\mathsf{\Delta }}{\displaystyle \frac{(\gamma +\mathsf{\Delta })\theta }{1-(1-2\beta )b^2{\theta }^2}}+F\right\}}^2\Psi \right],$$

(25)

where$F=ck{r}^{1-\alpha +\mathsf{\Delta }}\left[{\displaystyle \frac{(\gamma +\mathsf{\Delta }){\theta }^2}{1-(1-2\beta )b^2{\theta }^2}}-{\displaystyle \frac{(\gamma +\mathsf{\Delta })}{2b^2(1-\alpha +\mathsf{\Delta })}}\right]-f^{\text{'}}(\tau )$, and

$\Psi =k{r}^{-\gamma }{\left\{a\left(1+{\displaystyle \frac{2\beta \delta -3}{1-2\beta }}{\theta }^2\right)+c{r}^{\mathsf{\Delta }}\left[1+{\displaystyle \frac{2(\gamma +\mathsf{\Delta })b^2{\theta }^2}{1-(1-2\beta )b^2{\theta }^2}}\right]\right\}}^{-1}-(1+k{r}^{\beta }\theta ){\mu }^{"}({\rho }_C,\tau )-f^{"}(\tau )$, where $s_0(\theta )=s_{00}+s_{02}{\theta }^2$, $s_{00}$ and $s_{02}$ are the universal constants [142].

Thus, these relations for $C_{\mathrm{V}}$and $C_{\mathrm{P}}$, derived from the parametric scaling EoS, correctly represent the characteristic critical anomalies of supercritical fluids predicted by the non-classical scaling theory of critical phenomena and, consequently, the behavior of the Widom lines.

5.2. Parametric Crossover Equation of State

The extended parametric scaling model of the EoS described above correctly represents the characteristic features of the critical anomalies of thermodynamic properties of supercritical fluids in the asymptotic critical region around the critical point, namely: $\left|\mathsf{\Delta }\rho \right|\le 0.3$ (for CO₂ this is 327.3 kg⋅m^-3 ≤ ρ ≤ 607.9 kg⋅m^-3) and $-2\cdot {10}^{-3}\le t\le 0.1$ (for CO₂ this is 303.5 K≤ T ≤334.5 K). This range corresponds to pressures from (7.2 to 15.5) MPa. The scaling EoS provides an accurate description of the characteristic power-law behavior of pure fluids governed by universal critical exponents in the asymptotic vicinity of the critical point, as predicted by scaling theory. However, the experimentally observed region in which critical anomalies of the thermodynamic properties of supercritical fluids are significant extends over substantially wider temperature and pressure ranges (see Figure 23) than the range in which the extended parametric scaling model of the EoS is applicable. Far from the critical region, where thermodynamic properties exhibit predominantly regular behavior where the fluctuation contribution to the experimentally observed properties becomes negligible, this EoS does not naturally cross over to well-established classical theoretical EoS, such as the Landau or virial expansions. As discussed above, this limitation of the extended parametric scaling EoS can be overcome using the theory of crossover phenomena [64,78,109,110,111,112,115].

According to the theory of crossover phenomena, a crossover EoS smoothly transforms the thermodynamic behavior of supercritical fluids from the classical regime far from the critical region, where the effects of critical fluctuations can be neglected, to the non-classical (scaling-type) regime, where fluctuations make a significant contribution to the thermodynamic properties. In other words, the crossover describes the smooth transition from the background (mean-field or regular) contribution to the fluctuation (scaling or critical-enhancement) contribution to the thermodynamic properties. This type of crossover model therefore provides a description over the entire supercritical region.

In this section, we consider the parametric crossover model of EoS for pure fluids in the critical and supercritical regions. Within the crossover model based on a parametric EoS, the dimensionless Helmholtz free energy can be expressed as [61] (see also [21,52,64,78,147,148]):

$$\rho \overline{A}(T,\rho )=\mathsf{\Delta }\overline{A}(t,\mathsf{\Delta }\rho )+\overline{\rho }{\overline{\mu }}_0(T)+\overline{A_0}(T),$$

(26)

where the scaling term $\mathsf{\Delta }\overline{A}$ includes the critical contribution (singular part) to the Helmholtz free energy arising from long-range critical fluctuations, while ${\overline{\mu }}_0(T)$and $\overline{A_0}(T)$are analytical functions of temperature represented by truncated Taylor expansions:

$${\overline{\mu }}_0(T)=m_0+{{\displaystyle {\sum }_{j=1}{m}_j\tau }}^i,$$

(27)

In Equations (26) and (27), $A$ is the Helmholtz free energy per mole, $\rho$ is the density, and $T$is the temperature. The thermodynamic quantities are rendered dimensionless using the critical temperature$T_C$, critical density ${\rho }_C$, and critical pressure$P_C$ as

$$\overline{T}={\displaystyle \frac{T}{T_C}}, \overline{\rho }={\displaystyle \frac{\rho }{{\rho }_C}}, \overline{A}={\displaystyle \frac{{\rho }_{C}A}{P_C}} \mathrm{and} t=\overline{T}-1={\displaystyle \frac{T-T_C}{T_C}}, \mathsf{\Delta }\rho =\overline{\rho }-1={\displaystyle \frac{\rho -{\rho }_C}{{\rho }_C}}.$$

(28)

The scaling (fluctuation) part of the Helmholtz free energy, $\mathsf{\Delta }\overline{A}$, which incorporates the contribution of long-range critical fluctuations, is represented in the form

$$\mathsf{\Delta }\overline{A}(r,\theta )=k{r}^{2-\alpha }{R}^{\alpha }(q)\left[a{\Psi }_0(\theta )+{\displaystyle \sum _{i=1}^4{c}_i{r}^{{\mathsf{\Delta }}_i}}{R}^{-{\overline{\mathsf{\Delta }}}_i}(q){\Psi }_i(\theta )\right]$$

(29)

$${\Psi }_i(\theta )={\displaystyle \sum _{j=0}^5{\alpha }_{ij}{\theta }^j}(i=0,1,\dots \dots ,5)$$

(30)

where the parameters $r$ and $\theta$ are defined by

$$t=r(1-b^2\theta ),$$

(31)

$$\mathsf{\Delta }\rho =k{r}^{\beta }{R}^{-\beta +1/2}(q)\theta +d_{1}t,$$

(32)

where α,β, and Δi are the universal critical exponents, $b^2$ is the universal linear-model parameter, k, $d_i$,and $c_i$are system-dependent coefficients, while $R(q)$and ${\Psi }_i(\theta )$ are universal scaling functions [4,21,52,64,78,147].

The crossover function $R(q)$ is defined by

$$R(q)={\left(1+{\displaystyle \frac{q^2}{1+q}}\right)}^2,$$

(33)

where the crossover parameter $q$ is related to the parametric variable r by

$$q={\left(rg\right)}^{1/2}.$$

(34)

The system-dependent coefficient g is proportional to the inverse Ginzburg number $G_i$ [109]. In the critical region at rg<<1 (or $\left|t\right|<<G_i$ at the critical isochore and along the coexistence curve), $R(q)$=1 and Equation (29) reproduces a Wegner expansion for the three-dimensional Ising-like systems [149,150]:

$$\mathsf{\Delta }\overline{A}(\tau ,\mathsf{\Delta }\rho )=k{r}^{2-\alpha }\left[\overline{a}{f}_0(\mathsf{\Delta }\rho /{\left|t\right|}^{\beta })+{\displaystyle \sum _{i=1}^4{\overline{c}}_i{t}^{{\mathsf{\Delta }}_i}}{f}_i(\mathsf{\Delta }\rho /{\left|t\right|}^{\beta })\right].$$

(35)

For large values of the variable q at rg>>1 ($\left|t\right|>>G_i$), the crossover function $R(q)$modifies each term in Equation (29); consequently, all terms become analytical, and Equation (29) is transformed into the extended Landau expansion:

$$\mathsf{\Delta }\overline{A}(t,\mathsf{\Delta }\rho )={\displaystyle \sum _{i=0}^2{\displaystyle \sum _{j=1}^5{\overline{\alpha }}_{ij}}}{t}^i\mathsf{\Delta }{\rho }^j,$$

(36)

which corresponds to classical mean-field theory [151]. In the range where r≈g ($\left|t\right|\approx G_i$), a crossover from singular critical behavior to analytic classical behavior takes place. A good description of experimental data using the parametric crossover model has been achieved for most technologically important supercritical fluids over the wide range of temperatures and densities bounded by

$$t+0.625{(\mathsf{\Delta }\rho )}^2\le 0.25$$

(37)

As can be seen from Figure 24, the range of applicability of the crossover EoS for СО₂ extends very close to the fluctuation range (Widom range), where the effects of long-range fluctuations on the thermodynamic properties are significant.

For H₂O and CO₂, for example, this range extends up to T=808 K and T=380 K, respectively, along the critical isochores and, along the critical isotherms, covers densities within $120{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}\le \rho \le 525{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$ (for H₂O) and $171{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}\le \rho \le 764{\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$ (for CO₂). Within this range, the crossover model accurately reproduces the scaling-type critical anomalies (critical enhancement) of the thermodynamic properties of supercritical fluids, including the behavior of the Widom line (see below), with uncertainties close to the experimental uncertainties.

This crossover model has been successfully applied to describe anomalous thermodynamic properties in near-critical and supercritical fluids, including СО₂, H₂О, methane, ethane, propane, toluene, benzene, methanol, ethanol, and other fluids [21,52,60,63,64,69,78,147,148,152,153,154]. Figure 1 and Figure 3–12 show comparisons between experimental data for pure СО₂, H₂О, and n-hexane and values calculated using the crossover model, Equation (29). The $C_{\mathrm{V}}$extrema curves calculated from the crossover EoS are also presented in Figure 15 and Figure 17–19.

6. Asymptotic Behavior of Isochoric Heat Capacity Maxima in Supercritical Fluids: A Scaling Theory Approach

6.1. Scaling-Type Equation of State for Near-Critical Fluids

Thermodynamic calculations in the near-critical region are most appropriately performed within the framework of a generalized EoS formulated specifically for near-critical fluids. A comprehensive description of the construction and application of such an EoS is provided in Refs. [167,168]. For a one-component fluid, the EoS can be represented in terms of asymptotic expansions of the thermodynamic variables with respect to the scaling fields of an underlying reference model. In general, the EoS is expressed through parametric relationships that relate the thermodynamic quantities to the characteristics of the scaling model:

$$\begin{gathered} \mathsf{\Delta }\overline{\mu }=c_{\mu h_1}{h}_1+c_{\mu h_2}{h}_2+c_{\mu h_3}{h}_3+c_{\mu h_2{h}_2}{h}_2^2+\cdots , \\ t=c_{t{h}_1}{h}_1+c_{t{h}_2}{h}_2+c_{t{h}_3}{h}_3+c_{t{h}_2{h}_2}{h}_2^2+\cdots , \\ \mathsf{\Delta }\overline{P}=c_{P{h}_1}{h}_1+c_{P{h}_2}{h}_2+c_{P{h}_3}{h}_3+c_{P{h}_2{h}_2}{h}_2^2+\cdots , \\ \mathsf{\Delta }\overline{\rho }=c_{\rho {\varphi }_1}{\varphi }_1+c_{\rho {\varphi }_2}{\varphi }_2+c_{\rho h_1}{h}_1+c_{\rho h_2}{h}_2+c_{\rho h_3}{h}_3+c_{\rho {\varphi }_1{\varphi }_1}{\varphi }_1^2+\cdots , \\ \mathsf{\Delta }\overline{s}=c_{S{\varphi }_1}{\varphi }_1+c_{S{\varphi }_2}{\varphi }_2+c_{S{h}_1}{h}_1+c_{S{h}_2}{h}_2+c_{S{h}_3}{h}_3+c_{S{\varphi }_1{\varphi }_1}{\varphi }_1^2+\cdots , \end{gathered}$$

(38)

where $\mathsf{\Delta }\overline{\mu }=\left(\mu -{\mu }_C\right)/R{T}_C$, $t=\left(T-T_C\right)/T_C$,$\mathsf{\Delta }\rho =\left(\rho -{\rho }_C\right)/{\rho }_C$,$\mathsf{\Delta }\overline{s}=\left(s-s_C\right)/{\rho }_{C}R$, and $\mathsf{\Delta }\overline{\mu }=\left(\mu -{\mu }_C\right)/R{T}_C$ denote the deviations of the corresponding thermodynamic variables from their critical-point values, and $R$ is the universal gas constant. Equations (38) are derived within the framework of the complete scaling theory [120,121,122]. The expansion coefficients $c_{ijk}$appearing in these equations are non-universal parameters whose values depend on the particular fluid under consideration.

The underlying thermodynamic model is formulated in terms of three scaling fields (intensive variables), $h_1$, $h_2$, and $h_3$, together with their corresponding conjugate scaling densities,${\varphi }_1$ and ${\varphi }_2$. Hereafter, these quantities are referred to as the scaling variables. All scaling variables vanish at the critical point. Two of the fields, $h_1$ and $h_2$, are treated as independent variables, whereas the field, $h_3\left(h_1,h_2\right)$, represents the thermodynamic potential of the model and is determined by the relation $d{h}_3={\varphi }_{2}d{h}_2+{\varphi }_{1}d{h}_1$.

Fisher and Orkoulas [120] and Orkoulas et al. [122] extended the conventional incomplete scaling theory by incorporating pressure mixing into the ordering scaling field $h_1$. In their formulation, the ordering scaling field $h_1$is expressed as a linear combination of the physical thermodynamic variables, $\mathsf{\Delta }T$,$\mathsf{\Delta }P$, and $\mathsf{\Delta }\mu$- temperature, pressure, and chemical potential [120,122]:

$$h_1=a_1\mathsf{\Delta }\mu +a_2\mathsf{\Delta }T+a_3\mathsf{\Delta }P.$$

(39)

Similarly, the scaling field $h_2$is formulated as a linear combination of the physical field variables $\mathsf{\Delta }T$,$\mathsf{\Delta }P$, and $\mathsf{\Delta }\mu$ [120,122]:

$$h_2=b_1\mathsf{\Delta }\mu +b_2\mathsf{\Delta }T+b_3\mathsf{\Delta }P,$$

(40)

where $a_3$ denotes the pressure- mixing coefficient, representing the additional term introduced in the complete scaling theory, whereas $b_2$b is the temperature -mixing coefficient, which is already present in the incomplete scaling formulation.

Equations (38) provide a general parameterization of the thermodynamic properties in terms of the scaling fields $h_1$ and $h_2$. Because the present analysis is restricted to the immediate vicinity of the critical point (asymptotic range), it is convenient to adopt the Linear Parametric Model (LPM) as the fundamental representation [169,170]. This choice is motivated by the well-established theoretical basis and mathematical simplicity of the LPM. In addition, the model constitutes a phenomenological extension of the three-dimensional Ising model (Liu and Fisher [123,172] and Migdal [173,174]) and is fully consistent with the asymptotic predictions of renormalization-group theory [171].

The LPM is described in terms of two independent variables, $r$ and θ [169,170]. The parameter $r\ge 0$ characterizes the generalized distance from the critical point, whereas $-1\le \theta \le 1$ specifies the thermodynamic state relative to the vapor–liquid coexistence curve. Along the coexistence curve, the condition ${\theta }^2=1$ ($\theta =\pm 1$) is satisfied. Because all non-universal behavior is incorporated into the expansion coefficients appearing in Equation (38), the LPM itself can be formulated in a universal form as

$$\begin{gathered} h_1=a_1{r}^{\gamma +\beta }\theta \left(1-{\theta }^2\right), \\ {h}_2=r\left(1-b^2{\theta }^2\right), \\ {h}_3=r^{2-\alpha }\left(1-a_2{\theta }^2+a_4{\theta }^4\right), \\ {\varphi }_1=r^{\beta }\theta , \\ {\varphi }_2=a_3{r}^{1-\alpha }\left({\vartheta }_S^2-{\theta }^2\right) \end{gathered}$$

(41)

where $b^2={\displaystyle \frac{\left(\gamma -2\beta \right)}{\gamma \left(1-2\beta \right)}}$, ${\vartheta }_S^2={\displaystyle \frac{\gamma \left(\gamma -1\right)}{\left(1-\alpha \right)\left(\gamma -2\beta \right)}}$ and $a_i$ are universal parameters defined in terms of the universal critical exponents $\alpha$, $\beta$, and $\gamma$:

$$a_1={\displaystyle \frac{2\alpha a_3{b}^2}{\left(\gamma -2\beta \right)}},a_2={\displaystyle \frac{a_3\left(\gamma -\alpha \right)}{\gamma \left(1-\alpha \right)}}-a_1,a_3={\displaystyle \frac{\left(2-\alpha \right)}{{\vartheta }_S^2}},a_4={\displaystyle \frac{a_3}{\gamma }}-a_1$$

(42)

The universal critical exponents, $\alpha$=0.109, $\beta$=0.326, and $\gamma =2-\alpha -2\beta$=1.239, are determined solely by the spatial dimensionality and universality class of the three-dimensional Ising model [169,170,172]. The transformation to the LPM changes the parameterization such that all thermodynamic quantities in Equation (38) become functions of the parameters $r$ and $\theta$.

Furthermore, following Ref. [167], we employ an asymptotic EoS in the immediate vicinity of the critical point. According to Equation (38), this equation can be formulated for the characteristic thermodynamic variables by neglecting higher-order correction terms under the condition $-1\le \theta \le 1$. Because the parameter $\theta$ is assumed to be small, it is convenient to explicitly separate its dependence.

$$\begin{gathered} t=c_{t{h}_2}r+a_1{c}_{t{h}_1}{r}^{\gamma +\beta }\theta -b^2{c}_{t{h}_2}r{\theta }^2-a_1{c}_{t{h}_1}{r}^{\gamma +\beta }{\theta }^3, \\ \mathsf{\Delta }\overline{P}=c_{P{h}_2}r+a_1{c}_{P{h}_1}{r}^{\gamma +\beta }\theta -b^2{c}_{P{h}_2}r{\theta }^2-a_1{c}_{P{h}_1}{r}^{\gamma +\beta }{\theta }^3, \\ \mathsf{\Delta }\overline{\rho }=a_3{\vartheta }_S^2{c}_{\rho {\varphi }_2}{r}^{1-\alpha }+c_{\rho {\varphi }_1}{r}^{\beta }\theta +c_{\rho {\varphi }_1{\varphi }_1}{r}^{2\beta }{\theta }^2, \\ \mathsf{\Delta }\overline{s}=a_3{\vartheta }_S^2{c}_{S{\varphi }_2}{r}^{1-\alpha }+c_{S{\varphi }_1}{r}^{\beta }\theta +c_{S{\varphi }_1{\varphi }_1}{r}^{2\beta }{\theta }^2, \end{gathered}$$

(43)

The asymptotic form represented by Equation (43) encompasses both the complete- scaling and the conventional-scaling approaches, provided that $c_{\rho {\varphi }_1{\varphi }_1}=c_{S{\varphi }_1{\varphi }_1}=0$. This formulation was also employed in Ref. [167] and therefore facilitates a direct comparison with the results obtained there. To derive an expression for the isochoric heat capacity, we use the asymptotic of

$$C_{\mathrm{V}}\propto {\left({\displaystyle \frac{\partial \mathsf{\Delta }\overline{s}}{\partial t}}\right)}_{\mathsf{\Delta }\overline{\rho }}={\displaystyle \frac{\left(\frac{\partial \mathsf{\Delta }\overline{s}}{\partial r}\right)\left(\frac{\partial \mathsf{\Delta }\overline{\rho }}{\partial \theta }\right)-\left(\frac{\partial \mathsf{\Delta }\overline{s}}{\partial \theta }\right)\left(\frac{\partial \mathsf{\Delta }\overline{\rho }}{\partial r}\right)}{\left(\frac{\partial t}{\partial r}\right)\left(\frac{\partial \mathsf{\Delta }\overline{\rho }}{\partial \theta }\right)-\left(\frac{\partial t}{\partial \theta }\right)\left(\frac{\partial \mathsf{\Delta }\overline{\rho }}{\partial r}\right)}},$$

(44)

Since Equation (43) define the dependence of the thermodynamic quantities on the parameters $r$ and $\theta$, the corresponding expression for the isochoric heat capacity can be obtained by substituting these relationships into Equation (44). In the immediate vicinity of the critical point, the parameter $\theta$ is small; consequently, it is sufficient to retain terms through the corresponding leading order, ${\theta }^2$. Using Equations (41)–(43), we obtain

$$C_{\mathrm{V}}\propto A_0{r}^{-\alpha }\left[1+\left(A_{11}-A_{12}{r}^{\gamma -1}\right)r^{\beta }\theta +\left(1-2\beta \right)b^2{\theta }^2\right],$$

(45)

where $A_{11}\propto c_{\rho {\varphi }_1{\varphi }_1}{c}_{S{\varphi }_1}-c_{S{\varphi }_1{\varphi }_1}{c}_{\rho {\varphi }_1}$. This parameter is therefore directly linked to the complete -scaling framework. In the conventional scaling approach, $A_{11}=0$. Consequently, Equation (45) describes the asymptotic behavior of the isochoric heat capacity within both the complete- and conventional -scaling formalisms.

The loci of the $C_{\mathrm{V}}$extrema can now be determined. Along an isobar, $C_{\mathrm{V}}$ is a function of temperature, $t$, whereas along an isotherm its independent variable is pressure, $\mathsf{\Delta }\overline{P}$. The conditions for the corresponding $C_{\mathrm{V}}$extrema are obtained by setting the derivatives, ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial t}}\right)}_{\mathsf{\Delta }\overline{P}}$ and ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \mathsf{\Delta }\overline{P}}}\right)}_t$, equal to zero. Substitution of the asymptotic expressions given by Equations (43) and (45), retaining only the leading-order terms in the parameter $\theta$, yields the following condition for the $C_{\mathrm{V}}$extrema along both the isobar and the isotherm in the immediate vicinity of the critical point:

$$\begin{gathered} {\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \mathsf{\Delta }\overline{\rho }}}\right)}_{\mathsf{\Delta }\overline{P}}\propto {\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial t}}\right)}_{\mathsf{\Delta }\overline{P}}\propto \left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial r}}\right)\left({\displaystyle \frac{\partial \mathsf{\Delta }\overline{P}}{\partial \theta }}\right)-\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \theta }}\right)\left({\displaystyle \frac{\partial \mathsf{\Delta }\overline{P}}{\partial r}}\right)\propto B_0{r}^{\beta }+B_P{r}^{1-\beta -\alpha }-\theta \\ {\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \mathsf{\Delta }\overline{\rho }}}\right)}_t\propto {\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \mathsf{\Delta }\overline{P}}}\right)}_t\propto \left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial r}}\right)\left({\displaystyle \frac{\partial t}{\partial \theta }}\right)-\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \theta }}\right)\left({\displaystyle \frac{\partial t}{\partial r}}\right)\propto B_0{r}^{\beta }+B_T{r}^{1-\beta -\alpha }-\theta , \end{gathered}$$

(46)

where the coefficient $B_0\propto c_{\rho {\varphi }_1{\varphi }_1}{c}_{S{\varphi }_1}-c_{S{\varphi }_1{\varphi }_1}{c}_{\rho {\varphi }_1}$incorporates the effects associated with complete scaling, whereas the coefficients $B_P$ and $B_T$ are different for the two thermodynamic paths. As evident from Equation (46), the mathematical conditions defining the extrema of $C_{\mathrm{V}}$ along isotherms and isobars are formally identical. The difference is contained only in the corresponding coefficients $B_P$ and $B_T$, which have distinct physical meanings and originate from different thermodynamic constraints and underlying physical mechanisms.

Notably, Equations (45) and (46) are qualitatively consistent with the results reported in Ref. [167] for the extrema of other characteristic thermodynamic quantities, such as ($K_{\mathrm{T}},C_{\mathrm{P}}$, ${\alpha }_{\mathrm{P}}$, etc.), in near-critical fluids. Consequently, the loci of heat capacity$C_{\mathrm{V}}$ maxima coincide along the isochores and isobars in the asymptotic critical region.

If the coefficient $B_0=0$ vanishes, the loci of the heat- capacity maxima along an isobar $\theta =B_P{r}^{1-\beta -\alpha }$and isotherm $\theta =B_T{r}^{1-\beta -\alpha }$ no longer coincide; nevertheless, they retain the same asymptotic functional form. Because the specific functional dependence $\theta \left(r\right)$ determines the thermodynamic path of the corresponding variable in different thermodynamic paths, the characteristic loci of the isochoric heat capacity can be represented and in different thermodynamic planes thermodynamic planes.

6.2. Isochoric Heat Capacity Maxima Loci of Supercritical Fluids Along Isotherms in the $\mathsf{\Delta }\overline{P}-t$Projection

Using the EoS, Equation (38), together with the LPM representation given by Equation (41), an analytical relationship between pressure and temperature can be derived for any characteristic line of a supercritical fluid. In the general case, this relationship can be expressed in the $\mathsf{\Delta }\overline{P}-t$plane as a series expansion in terms of $r\cong t/c_{t{h}_2}$ and $\theta$:

$${\displaystyle \frac{\mathsf{\Delta }\overline{P}}{t}}=D_0+{\displaystyle \frac{a_1}{c_{t{h}_2}^{\gamma +\beta }}}\left(c_{P{h}_1-}{c}_{t{h}_2}{D}_0\right)t^{\gamma -1+\beta }\theta +D_1{t}^{1-\alpha }+D_{2}t+\cdots ,$$

(47)

where the coefficients are given by

$$D_0={\displaystyle \frac{{с}_{P{h}_2}}{c_{\tau h_2}}},D_1={\displaystyle \frac{D_0}{c_{\tau h_2}^{1-\alpha }}}\left({\displaystyle \frac{{с}_{P{h}_3}}{{с}_{P{h}_2}}}-{\displaystyle \frac{{с}_{\tau h_3}}{{с}_{\tau h_2}}}\right),D_2={\displaystyle \frac{D_0}{c_{\tau h_2}}}\left({\displaystyle \frac{{с}_{P{h}_2{h}_2}}{{с}_{P{h}_2}}}-{\displaystyle \frac{{с}_{\tau h_2{h}_2}}{{с}_{\tau h_2}}}\right).$$

(48)

By substituting the expression for $\theta \left(r\right)$from Equation (46) into Equation (47), the analytical pressure–temperature relationship $\mathsf{\Delta }\overline{P}-t$for the loci of the isochoric heat-capacity $C_{\mathrm{V}}$maxima is readily obtained in the form

$$\mathsf{\Delta }{\overline{P}}_{C_{\mathrm{V}}}=\mathsf{\Delta }{\overline{P}}_{\mathrm{W}}+D_{1C_{\mathrm{V}}}{t}^{2-\alpha }+D_{3C_{\mathrm{V}}}{t}^{1+\gamma -\alpha }+\cdots ,$$

(49)

where $\mathsf{\Delta }{\overline{P}}_{\mathrm{W}}$is given by [167]

$$\mathsf{\Delta }{\overline{P}}_{\mathrm{W}}=D_{0}t+D_1{t}^{2-\alpha }+D_2{t}^2+\cdots$$

(50)

The coefficients $D_0$, $D_1$ и $D_2$ are associated with the shape of the coexistence curve in the $\mathsf{\Delta }\overline{P}-t$ plane. The coefficient $D_{1C_{\mathrm{V}}}$is proportional to the determinant $c_{\rho {\phi }_1{\phi }_1}{c}_{s{\phi }_1}-c_{s{\phi }_1{\phi }_1}{c}_{\rho {\phi }_1}$, which vanishes within the framework of conventional scaling, where $c_{\rho {\phi }_1{\phi }_1}=c_{s{\phi }_1{\phi }_1}=0$.

The expansion term in Equation (49) containing the coefficient $D_{3C_{\mathrm{V}}}$ is liquid-dependent and also depends on the thermodynamic path along which the isochoric heat capacity is evaluated. Consequently, the behavior of the $C_{\mathrm{V}}$ - maxima loci along isobars and isotherms in the $\mathsf{\Delta }\overline{P}-t$plane is qualitatively similar to that of the other characteristic lines of supercritical fluids reported in [167]. However, the situation is markedly different when these loci are considered in the $\mathsf{\Delta }\overline{\rho }-t$coordinates.

6.3. Isochoric Heat-Capacity Maxima Loci of Supercritical Fluids Along Isotherms in the $\mathsf{\Delta }\overline{\rho }-t$ Projection

In Ref. [167], only the $\mathsf{\Delta }\overline{P}-t$plane was considered because the Widom line is conventionally defined in these coordinates. The situation, however, is different in other thermodynamic representations, such as the $\mathsf{\Delta }\overline{\rho }-t$plane. Moreover, experimental data for$C_{\mathrm{V}}$maxima are available in this representation in the $\mathsf{\Delta }\overline{\rho }-t$ plane (see Section 3 above, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16).

We first determine the region of the supercritical fluid in which $P\ge P_{\mathrm{c}}$ and $T\ge T_{\mathrm{c}}$ within the framework of complete scaling. In the $\mathsf{\Delta }\overline{\rho }-t$plane, the condition $\mathsf{\Delta }\overline{\rho }-t$ follows directly. For the critical isobar, $\mathsf{\Delta }\overline{P}=0$, the corresponding condition $\theta =-{\displaystyle \frac{1}{b}}+{\displaystyle \frac{a_1}{2b^2}}{\displaystyle \frac{{с}_{P{h}_1}}{{с}_{P{h}_2}}}{r}^{\gamma +\beta -1}$can be obtained from Equation (43). As a result, the asymptotic equation of the critical isobar in the $\mathsf{\Delta }\overline{\rho }-t$ coordinates take the form $-\mathsf{\Delta }\overline{\rho }=B_{\mathrm{cr}}{t}^{{\displaystyle \frac{\beta }{\gamma +\beta }}}$, where $B_{\mathrm{cr}}>0$.

The loci of the $C_{\mathrm{V}}$maxima along isobars and isotherms are determined by the conditions given in Equation (46). Substitution of the asymptotic expressions from Equation (43) then yields

$$\mathsf{\Delta }\overline{\rho }=B_0{\displaystyle \frac{c_{\rho {\varphi }_1}}{c_{t{h}_2}^{2\beta }}}{t}^{2\beta }+\left(a_3{\vartheta }_S^2{\displaystyle \frac{c_{\rho {\varphi }_2}}{c_{t{h}_2}^{1-\alpha }}}+B_{PT}{\displaystyle \frac{c_{\rho {\varphi }_1}}{c_{t{h}_2}^{1-\alpha }}}\right)t^{1-\alpha }+{\displaystyle \frac{c_{\rho h_2}}{c_{t{h}_2}}}t+\cdots =D_{\mathrm{CS}}{t}^{2\beta }+D_{\mathrm{PT}}{t}^{1-\alpha }+Dt+\cdots ,$$

(51)

Here, $D_{\mathrm{CS}}\propto$$B_0\propto c_{\rho {\varphi }_1{\varphi }_1}{c}_{S{\varphi }_1}-c_{S{\varphi }_1{\varphi }_1}{c}_{\rho {\varphi }_1}$, whereas the coefficient $D_{\mathrm{PT}}$ depends on the value of$B_{\mathrm{PT}}$. Therefore, Equation (51) predicts that, within the framework of complete scaling theory, the loci of the isochoric heat- capacity extrema, $C_{\mathrm{V}}$, measured along both isobars and isotherms in the vicinity of the critical point exhibit the same asymptotic behavior, $\propto t^{2\beta }$. This prediction is consistent with the experimental observations, which indicate that $D_{\mathrm{CS}}<0$. In contrast, within the framework of incomplete scaling, for which $D_{\mathrm{CS}}=0$, the asymptotic behavior of $C_{\mathrm{V}}$maxima loci, along isobars and isotherms differs, $\propto t^{1-\alpha }$.Within the framework of complete scaling theory, the scaling equation for the isochoric heat capacity in the asymptotic critical region is expressed as [167]:

$$C_{\mathrm{V}}\propto A_0{r}^{-\alpha }\left[1+\left(A_{11}-A_{12}{r}^{\gamma -1}\right)r^{\beta }\theta +\left(1-2\beta \right)b^2{\theta }^2\right],$$

(52)

In the case of conventional (incomplete) scaling, the parameter $A_{11}=0$ is omitted. By applying the$C_{\mathrm{V}}$-extremum condition, ${\left({\displaystyle \frac{\partial C_{\mathrm{V}}}{\partial \rho }}\right)}_{\mathrm{T}}=0$, an expression for the asymptotic behavior of the Widom line in the $\mathsf{\Delta }\overline{\rho }-t$ projection was derived within the complete- scaling framework [168]:

$$\mathsf{\Delta }\overline{\rho }=D_{\mathrm{CS}}{t}^{2\beta }+D_{\mathrm{PT}}{t}^{1-\alpha }+Dt+\cdots .$$

(53)

The term involving $D_{\mathrm{CS}}{t}^{2\beta }<0$ arises specifically from the complete scaling concept; in the traditional (incomplete) scaling approach, $D_{\mathrm{CS}}=0$. The mathematical form of Equation (53) is formally analogous to the equation describing the liquid–gas coexistence curve predicted by complete scaling theory [175,176,177,178,179]:

$$\mathsf{\Delta }\overline{\rho }=B_4{t}^{2\beta }+B_2{t}^{1-\alpha }-B_{3}t\pm B_0{t}^{\beta }\pm B_1{t}^{\beta +\mathsf{\Delta }},$$

(54)

where $\mathsf{\Delta }=0.5$ is the universal non-asymptotic critical exponent (Wegner correction) [149,150].

Although Equations (53) and (54) have formally analogous mathematical structures, the physical meanings of their coefficients $B_i$ and $C_i$are different. They describe different physical phenomena and therefore have distinct physical interpretations. In Equation (54), the critical amplitudes $B_i$are related to the parameters of the crossover EoS as follows:

$$B_0=k/{(b^2-1)}^{\beta },\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{B}_1=kc/2a{(b^2-1)}^{\beta +\mathsf{\Delta }},$$

(55)

Here, $B_0$ is the asymptotic critical amplitude; $B_1$ is the nonasymptotic critical amplitude associated with Wegner’s correction term; $B_2$ is the singular- diameter amplitude associated with incomplete scaling; $B_3$ is the rectilinear- diameter amplitude; and $B_4$is the complete -scaling correction term. The parameters a, c, and k are the parameters of the crossover EoS (see Section 5.2), whereas e, e₁, e₂, and b² are universal parameters determined by the universal critical exponents (α, β). According to complete scaling theory, the liquid–gas coexistence-curve diameter is defined as [175,176]:

$${\rho }_d-1={\displaystyle \frac{a_3}{1+a_3}}{B}_{02}{t}^{2\beta }-b_2\left[{\displaystyle \frac{{\overline{A}}_0^{-}}{\left(1-\alpha \right)}}{t}^{1-\alpha }-{\displaystyle \frac{{\overline{B}}_{cr}}{k_B}}t\right],$$

(56)

where ${\rho }_{\mathrm{d}}=\left({\rho }_{\mathrm{S}}^{\text{'}}+{\rho }_{\mathrm{S}}^{"}\right)/2{\rho }_{\mathrm{C}}$; ${\rho }_{\mathrm{S}}^{\text{'}}$and ${\rho }_{\mathrm{S}}^{"}$are the liquid and vapor saturated densities along the liquid-gas coexistence curve; $k_{\mathrm{B}}$is Boltzmann’s constant; ${\overline{B}}_{cr}$ is the fluctuation -induced “critical background” parameter; $\alpha$=0.106 and $\beta$=0.326 are the universal asymptotic critical exponents [35] of $C_{\mathrm{V}}$and density $\mathsf{\Delta }\rho$, respectively; $B_0$and ${\overline{A}}_0^{-}$are the asymptotical critical amplitudes of the coexistence curve and isochoric heat capacity, respectively, with the latter defined as [175,176]:

$${\displaystyle \frac{C_{\mathrm{V}}}{k_{\mathrm{B}}}}={\overline{A}}_0^{-}{t}^{-\alpha }\left(1+{\overline{A}}_1^{-}{t}^{\mathsf{\Delta }}\right)-{\displaystyle \frac{{\overline{B}}_{cr}}{k_{\mathrm{B}}}}.$$

(57)

In Equation (56), the parameter $a_3$ is the system- dependent asymmetry coefficient of the liquid+gas coexistence curve and is related to the Yang-Yang anomaly strength parameter [120,122] as $R_{\mathsf{\mu }}={\displaystyle \frac{a_3}{1+a_3}}$. The latter is defined through the two-phase isochoric heat capacity singularity at the critical point as $R_{\mathsf{\mu }}\equiv {\displaystyle \frac{C_{\mathrm{V}\mathsf{\mu }}}{C_{\mathrm{VP}}+C_{\mathrm{V}\mathsf{\mu }}}}$, where $C_{\mathrm{VP}}=VT{\displaystyle \frac{{\mathrm{d}}^2{P}_{\mathrm{S}}}{\mathrm{d}{T}^2}}$ and $C_{\mathrm{V}\mathsf{\mu }}=-T{\displaystyle \frac{{\mathrm{d}}^2\mu }{\mathrm{d}{T}^2}}$are the pressure and chemical- potential contributions, respectively, to the total measured two-phase isochoric heat capacity,$C_{\mathrm{V}2}$, according to the well-known theory of Yang and Yang [180] concerning the singularity of the two-phase isochoric heat capacity near the critical point:

$$C_{\mathrm{V}2}=-T{\displaystyle \frac{{\mathrm{d}}^2\mu }{\mathrm{d}{T}^2}}+VT{\displaystyle \frac{{\mathrm{d}}^2{P}_{\mathrm{S}}}{\mathrm{d}{T}^2}},$$

(58)

It follows directly from Equation (56) that, within the framework of incomplete scaling, the singular part of the coexistence-curve diameter is determined exclusively by the first nonanalytic contribution, $B_2{t}^{1-\alpha }$, to the coexistence-curve asymmetry:

$${\rho }_d-1=-b_2\left[{\displaystyle \frac{{\overline{A}}_0^{-}}{\left(1-\alpha \right)}}{t}^{1-\alpha }-{\displaystyle \frac{{\overline{B}}_{cr}}{k_B}}t\right].$$

(59)

Within the framework of the EoS defined by Equations (38), (41), and (42), an analytical expression for the coexistence-curve diameter can be derived. Under the assumption that ${\theta }^2=1$, it takes the form

$${\displaystyle \frac{\mathsf{\Delta }{\overline{\rho }}_L+\mathsf{\Delta }{\overline{\rho }}_G}{2}}={\displaystyle \frac{c_{\rho {\varphi }_1{\varphi }_1}}{c_{t{h}_2}^{2\beta }}}{t}^{2\beta }-a_3\left(1-{\vartheta }_S^2\right){\displaystyle \frac{c_{\rho {\varphi }_2}}{c_{t{h}_2}^{1-\alpha }}}{t}^{1-\alpha }-\left(b^2-1\right){\displaystyle \frac{c_{\rho h_2}}{c_{t{h}_2}^{1-\alpha }}}t+\cdots$$

(60)

As evident from Equation (60), the derived expression exactly reproduces the liquid–gas coexistence-curve diameter reported in Refs. [175,176,177,178,179]. Furthermore, its mathematical form is identical to that of the $C_{\mathrm{V}}$ maxima locus given by Equation (51), with differences arising solely from the corresponding coefficients and their physical interpretations.

The experimentally determined loci of the isochoric heat- capacity maxima were fitted using Equation (51), derived within the framework of complete scaling theory, for a variety of molecular liquids. The optimized fitting parameters are listed in Table 2, and the quality of the correlations is demonstrated in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. These results provide experimental support for the complete-scaling prediction of the loci of the experimentally observed isochoric heat-capacity maxima (Widom line) in supercritical fluids.

As can be seen from Table 2, the leading complete- scaling coefficient $D_{\mathrm{CS}}$<0 negative for all the fluids studied, as predicted by the theory of complete scaling. The calculated Widom- line ($C_{\mathrm{V}}$-maxima loci) for several of the studied fluids are presented in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 together with the present experimental data. As can be seen, the agreement between the theoretical predictions and experimental results is good, providing further evidence for the applicability of complete scaling theory to the description of ($C_{\mathrm{V}}$-maxima loci, widom-line) in supercritical fluids.

7. Conclusions

It was experimentally found that, along supercritical isotherms, the $C_{\mathrm{V}}$ maximum originates at the critical point, initially shifts toward lower densities, then turns sharply toward higher densities as the temperature increases, and eventually crosses the critical isochore. The locus of the $C_{\mathrm{V}}$maxima (the asymptotic behavior of the Widom line) in supercritical fluids is very sensitive to variations in density (in $T-\rho$ projection) with increasing temperature, whereas some details of the asymptotic behavior of the Widom line are not noticeable in $P-T$ projection.

A detailed analysis of the classical analytical and non-analytical multiparameter reference EoS, widely used for engineering applications, shows that they incorrectly represent the critical-anomaly behavior of the $C_{\mathrm{V}}$maxima loci of supercritical fluids in the immediate vicinity of the critical point, i.e., in the asymptotic region around the critical point, and, consequently, incorrectly describe the asymptotic behavior of the Widom line near the critical point. In particular, the $C_{\mathrm{V}}$ maxima along the critical and supercritical isotherms, calculated from multiparameter reference EoS in the $T-\rho$plane, initially shift toward the low-density region and do not turn toward the high-density region at higher supercritical isotherms or cross the critical isochore. However, nonclassical (scaling-type) EoS predict that the Widom line starts at the critical point, initially shifts toward the low-density side, and then crosses the critical density again at higher supercritical temperatures.

An analytical expression describing the asymptotic behavior of the $C_{\mathrm{V}}$maxima locus along supercritical isotherms has been derived within the framework of complete scaling theory. The analysis demonstrates that, in the $\rho -T$plane, the $C_{\mathrm{V}}$ maxima locus obeys the asymptotic relation $\mathsf{\Delta }\overline{\rho }=D_{\mathrm{CS}}{t}^{2\beta }+D_{\mathrm{PT}}{t}^{1-\alpha }+Dt+\cdots$, with a negative coefficient $D_{\mathrm{CS}}<0$. This result is consistent with the present experimental observations for numerous fluids with diverse molecular structures. Furthermore, the behavior of the $C_{\mathrm{V}}$ maxima locus is determined by the signs of the coefficients $B_{\mathrm{PT}}$ and $c_{\rho h_2}/c_{\tau h_2}$. The condition $c_{\rho h_2}/c_{\tau h_2}<0$ follows directly from the intrinsic asymmetry of the liquid–gas coexistence curve. The coefficient $D_{\mathrm{CS}}\propto$$B_0\propto c_{\rho {\varphi }_1{\varphi }_1}{c}_{S{\varphi }_1}-c_{S{\varphi }_1{\varphi }_1}{c}_{\rho {\varphi }_1}$, while the coefficient $D_{\mathrm{PT}}$ depends on the value of $B_{\mathrm{PT}}$. Therefore, within the framework of complete scaling theory, the loci of the extrema of the $C_{\mathrm{V}}$ measured along both isobars and isotherms in the vicinity of the critical point exhibit the same asymptotic behavior, $\propto t^{2\beta }$, which is in good agreement with the present experimental observations, which clearly indicate that $D_{\mathrm{CS}}<0$. In contrast, within the framework of incomplete scaling, when $D_{\mathrm{CS}}=0$, the asymptotic behavior of the maxima $C_{\mathrm{V}}$loci along isobars and isotherms is described by $\propto t^{1-\alpha }$.

Mathematically, the asymptotic behavior of the $C_{\mathrm{V}}$ maxima locus (Widom line) in the $T-\rho$plane is very close to the relation for the liquid–gas coexistence curve predicted by complete scaling theory, $\mathsf{\Delta }\overline{\rho }\propto B_4{t}^{2\beta }+B_2{t}^{1-\alpha }-B_{3}t$. However, the physical meanings of the coefficients $B_i$ and ($D_{\mathrm{CS}}$, $D_{\mathrm{PT}}$, $D$) are different, and they describe different physical phenomena. In other words, the corresponding coefficients have different physical interpretations and represent different physical effects that depend on different microscopic characteristics of the liquid.