Isobaric heat capacity of classical and quantum fluids: extending experimental data sets into the critical scaling regime

Abstract

The singular isobaric heat capacity $C_{{\text{P}}} (T,P)$ of nitrogen, methane, water and hydrogen at critical pressure $P_{{\text{c}}}$ is studied over an extended temperature range, from the melting point to the high-temperature cutoff of the experimental data sets. The high- and low-temperature branches (above and below the critical temperature $T_{{\text{c}}}$) of $C_{{\text{P}}} (T,P_{{\text{c}}} )$ can accurately be modeled with broken power-law distributions in which the calculated universal scaling exponent $1 - 1/\delta$ of the isobaric heat capacity at critical pressure is implemented. (The enumerated fluids admit 3D Ising critical exponents). The parameters of these distributions are inferred by nonlinear least-squares regression from high-precision data sets. In each case, a non-perturbative analytic expression for $C_{{\text{P}}} (T,P_{{\text{c}}} )$ is obtained. The broken power laws have closed-form Index functions representing the Log–Log slope of the regressed branches of $C_{{\text{P}}} (T,P_{{\text{c}}} )$. These Index functions quantify the crossover from the experimentally more accessible high- and low-temperature regimes to the critical scaling regime. Ideal power-law scaling (without perturbative corrections and discounting impurities and gravitational rounding effects) of $C_{{\text{P}}} (T,P_{{\text{c}}} )$ occurs in a narrow interval, typically within $\left| {T/T_{{\text{c}}} - 1} \right| < 10^{ - 4}$ or even $10^{ - 5}$ depending on the fluid, and the regressed broken power-law densities provide closed-form analytic extensions of $C_{{\text{P}}} (T,P_{{\text{c}}} )$ to the melting point and up to dissociation temperatures.

Graphical Abstract

1 Introduction

The topic of this paper is the global analytic modeling of thermodynamic functions with critical-point singularities. Specifically, we will calculate the heat capacity at constant pressure along the critical isobar of nitrogen, methane, water and hydrogen. Non-perturbative closed-form expressions will be obtained for the high- and low-temperature heat-capacity branches of the mentioned fluids, by combining least-squares regression (from data sets outside the ideal power-law scaling regime) with the critical scaling predicted by renormalization-group theory [1].

Empirical heat-capacity data for a variety of pure component fluids, stretching from the melting point up to dissociation temperatures, are available in machine-readable synthetic form [2, 3], derived from multiparameter equations of state (EoSs), cf., e.g., Refs. [4,5,6,7]. These EoSs were in turn regressed from a collection of experimental data covering several temperature and pressure intervals, usually well separated from the critical point and outside the two-phase region. Experimental data in the critical scaling regime are only available for a limited number of single-component fluids and mixtures and a limited number of thermodynamic variables such as the isochoric heat capacity or isothermal compressibility [8,9,10,11,12,13,14,15,16,17,18,19]. As for the latter two, simple power-law scaling is typically observed in an interval of width $\left| {T/T_{{\text{c}}} - 1} \right| < 10^{ - 2}$ or $10^{ - 3}$. At temperatures within $\left| {T/T_{{\text{c}}} - 1} \right| < 10^{ - 4}$, a gravitationally generated density gradient causes deviations from power-law scaling, resulting in a rounding of the straight Log–Log slopes, unless the experiments are done at zero gravity, cf., e.g., Refs. [20,21,22,23]. The easy availability of extended data sets makes it attractive to model thermodynamic functions with critical singularities empirically and globally without the use of perturbative expansions, by employing calculated universal scaling properties such as critical exponents and universal amplitude ratios in the vicinity of the critical point where data points are lacking.

In the case of the isobaric heat capacity at critical pressure $C_{{\text{P}}} (T,P_{{\text{c}}} )$, there are virtually no experimental data available in the ideal power-law scaling regime, where critical scaling theory predicts $C_{{\text{P}}} (T,P_{{\text{c}}} )\sim A_{ \pm } \left| {1 - T/T_{{\text{c}}} } \right|^{1/\delta - 1}$, cf. Refs. [24, 25], due to the emergence of long-range correlations as exemplified in Refs. [23, 26]. The $\pm$ subscripts of the amplitude refer to the $T > T_{{\text{c}}}$ and $T < T_{{\text{c}}}$ branches of $C_{{\text{P}}} (T,P_{{\text{c}}} )$, respectively. Nitrogen, methane, water and hydrogen are fluids of the 3D Ising universality class [1], with exponent $1 - 1/\delta = 0.7912$, cf., e.g., Ref. [27].

Synthetic precision data for the isobaric heat capacity at critical pressure are available for the mentioned fluids outside the interval $\left| {T/T_{{\text{c}}} - 1} \right| < 10^{ - 3}$, cf. Refs. [2, 3]. We will demonstrate that the Log–Log slope of the critical heat capacity curve $C_{{\text{P}}} (T,P_{{\text{c}}} )$ in the empirical temperature range $\left| {T/T_{{\text{c}}} - 1} \right| > 10^{ - 3}$ does not exceed 0.7 for any of these fluids. This value is noticeably below the calculated Log–Log slope of $1 - 1/\delta = 0.7912$ in the critical power-law scaling regime of the 3D Ising class. The purpose of this paper is to extend the experimental data range to the ideal power-law scaling regime by means of the calculated scaling exponent $1 - 1/\delta$. To this end, we will use multiply broken power-law densities [28, 29] to model the high- and low-temperature branches of the isobaric heat capacity at critical pressure. These densities are very adaptable and especially suitable for large data sets stretching over several logarithmic decades (in reduced temperature $\left| {T/T_{{\text{c}}} - 1} \right|$ in this case), being composed as multiple products of simple power laws [30,31,32,33,34] and generalized beta distributions [35, 36]. The regressed densities cover the experimental data range from the melting point upward, as well as the critical scaling regime where they admit the above stated power-law asymptotics with calculated exponent $1 - 1/\delta = 0.7912$.

In Sect. 2, we discuss the temperature evolution of the isobaric heat capacity of nitrogen at critical pressure, of methane in Sect. 3, of water in Sect. 4 and of the quantum fluid hydrogen in Sect. 5. In each section, we give an overview of the available experimental data [2, 3], which clearly indicate the singularity of $C_{{\rm P}} (T,P_{{\text{c}}} )$, even though the data sets are still far off the critical power-law scaling regime. The broken power-law densities used for the high- and low-temperature $C_{{\rm P}} (T,P_{{\text{c}}} )$ branches of these fluids (and of CO₂ studied in Ref. [37]) are similarly structured as finite products of power-law factors; the nonlinear least-squares regression of these multiparameter distributions is outlined in Appendix 1.

In Sects. 2, 3, 4, 5, we also study Index functions describing the evolution of the Log–Log slope of the regressed heat-capacity branches over the temperature range covered, cf., e.g., Refs. [30, 35, 38,39,40,41], from the experimental low- and high-temperature regions into the critical scaling regime, where the Index functions reach a constant limit, which is the scaling exponent $1 - 1/\delta$ of $C_{{\rm P}} (T,P_{{\text{c}}} )$. By plotting these Index functions, one can thus obtain a quantitative depiction of the crossover from the high- and low-temperature regimes to the critical scaling regime. In particular, the temperature interval can be estimated in which ideal power-law scaling without perturbative scaling corrections occurs. In Sect. 6, we present our conclusions.

2 Isobaric heat capacity of nitrogen at critical pressure

As a first orientation, synthetic experimental data for the isobaric heat capacity $C_{{\text{P}}}$ of nitrogen, cf. Refs. [2, 3], are plotted in Fig. 1, at critical pressure $P_{{\text{c}}} = {3}{\text{.3958 MPa}}$. Figure 1 shows a Log–Log plot of $C_{{\text{P}}}$ data against reduced temperature $t = T/T_{{\text{c}}}$, from the melting point at $T_{{{\text{melt}}}} = {63}{\text{.15 K}}$ up to 125.6 K and from 126.8 K up to 2000 K. (Log denotes the decadic logarithm.) The critical temperature of nitrogen is $T_{{\text{c}}} = {126}{\text{.19 K}}$. Critical point parameters are denoted by $(T_{{\text{c}}} ,P_{{\text{c}}} ,\rho_{{\text{c}}} ,V_{{\text{c}}} )$, where $\rho$ is the molar density and $V = 1/\rho$ the molar volume, cf. Table 1. Despite the pronounced singularity in Fig. 1, the indicated temperature ranges are still by about two orders separated from the scaling regime, where $C_{{\text{P}}} \sim A_{ \pm } \left| {1 - T/T_{{\text{c}}} } \right|^{1/\delta - 1}$ with critical exponent $\delta = 4.7898$, cf. Ref. [27]. The $\pm$ subscripts refer to temperatures above and below $T_{{\text{c}}}$. That is, $C_{{\text{P}}} \sim A_{ + } (t - 1)^{1/\delta - 1}$ for $t > 1$ and $C_{{\text{P}}} \sim A_{ - } (1 - t)^{1/\delta - 1}$ for $t < 1$.

Fig. 1

Isobaric heat capacity of nitrogen at critical pressure. Data points from Refs. [2, 3] (which are synthetic data based on a multiparameter EoS [4]) covering the low-temperature interval from the melting point $T_{{{\text{melt}}}} = {63}{\text{.15 K}}$ to 125.6 K (155 data points, filled squares) and the high-temperature range from 126.8 K up to 2000 K (460 data points, open squares). The critical temperature of nitrogen is $T_{{\text{c}}} = {126}{\text{.19 K}}$. Depicted is a Log–Log (decadic double-logarithmic) plot of the isobaric N₂ heat capacity $C_{{\text{P}}} (t)$ (at the critical pressure of $P_{{\text{c}}} = {3}{\text{.3958 MPa}}$) versus reduced temperature $t = T/T_{{\text{c}}}$. The lower and upper temperature limits are indicated by the vertical green and red dotted lines. The red and green solid curves show least-squares fits to the depicted heat-capacity data. The least-squares regression of the critical heat capacity is performed with the multiply broken power-law densities $C_{{\text{P}}} (\tau )$ in (2.1) (high-temperature branch, $\tau = 1/(t - 1)$, $t > 1$, red solid curve) and (2.5) (low-temperature branch, $\tau = 1/(1 - t)$, $t < 1$, green solid curve) and fitting parameters in Table 2

Table 1 Critical constants (temperature $T_{{\text{c}}}$, molar density $\rho_{{\text{c}}}$, molar volume $V_{{\text{c}}}$, pressure $P_{{\text{c}}}$) and melting point $T_{{{\text{melt}}}}$ of nitrogen, methane, water and hydrogen, cf. Refs. [2, 3]

To model the crossover from the empirical data in Fig. 1 to the scaling regime, we parametrize the heat capacity with the scaling variable $\tau = 1/\left| {t - 1} \right|$, $t = T/T_{{\text{c}}}$, writing $C_{{\rm P}} (\tau )$ and splitting the temperature range into a low-temperature interval $[T_{{{\text{melt}}}} ,T_{{\text{c}}} ]$ between melting point and $T_{{\text{c}}}$ and a high-temperature interval above $T_{{\text{c}}}$. Thus, in the low-temperature interval, the scaling variable is $\tau = 1/(1 - t)$, $\tau > 1/(1 - T_{{{\text{melt}}}} /T_{{\text{c}}} )$. In the high-temperature interval, $\tau = 1/(t - 1)$, $\tau > 0$. In either case, the critical temperature $T_{{\text{c}}}$ corresponds to $\tau = \infty$. Figure 2 shows Log–Log plots of the isobaric heat-capacity data (the same as in Fig. 1) as a function of $\tau$ instead of reduced temperature $t = T/T_{{\text{c}}}$. The low-temperature ($T < T_{{\text{c}}}$) data points are depicted as filled squares and the high-temperature ($T > T_{{\text{c}}}$) data as open squares, covering the same temperature range as in Fig. 1.

Fig. 2

Isobaric heat capacity $C_{{\text{P}}} (\tau )$ of nitrogen at critical pressure, parametrized with the scaling variable $\tau = 1/\left| {T/T_{{\text{c}}} - 1} \right|$. The open and filled squares are data points from Refs. [2, 3], the same as depicted in Fig. 1. In the high-temperature regime, $T/T_{{\text{c}}} > 1$, the scaling variable is $\tau = 1/(T/T_{{\text{c}}} - 1)$, and the data points are depicted as open squares. In the low-temperature regime, $T/T_{{\text{c}}} < 1$, the scaling variable is $\tau = 1/(1 - T/T_{{\text{c}}} )$, and the data points are plotted as filled squares. In this $\tau$ parametrization, the critical temperature is mapped to infinity, $\tau (T_{{\text{c}}} ) = \infty$. The red and green solid curves are the high- and low-temperature heat capacities $C_{{\text{P}}} (\tau )$ in (2.1) (for $T > T_{{\text{c}}}$) and (2.5) (for $T < T_{{\text{c}}}$) regressed from the depicted data sets, cf. Table 2 and Sect. 2. The $\chi^{2}$ functional used for the regression is stated in Appendix 1; residuals of the least-squares fits of the high- and low-temperature heat-capacity branches are shown in the lower panels. The red and green dashed lines depicting the critical power-law scaling, cf. (2.2) and (2.6), are the asymptotes of the high- and low-temperature heat capacities $C_{{\text{P}}} (\tau )$ in (2.1) and (2.5). These straight lines have a Log–Log slope of $1 - 1/\delta = 0.7912$, which is the critical exponent of the isobaric heat capacity at critical pressure (of a fluid in the 3D Ising universality class)

2.1 High-temperature regime above the critical temperature

The data set $(\tau_{i} ,C_{{{\text{P}},i}} )$, $\tau_{i} = 1/(T_{i} /T_{{\text{c}}} - 1)$, in the high-temperature regime ($T > T_{{\text{c}}}$) comprises 460 data points in the interval between 126.8 K and 2000 K (open squares in Fig. 2, taken from Refs. [2, 3]).

The least-squares fit above $T_{{\text{c}}}$ is performed with the multiply broken power law, cf. Refs. [28, 29, 37],

$$C_{{\text{P}}} (\tau ) = a_{0} \tau^{{\alpha_{0} }} \frac{1}{{(1 + (\tau /b_{1} )^{{\beta_{1} /\eta_{1} }} )^{{\eta_{1} }} }}(1 + (\tau /b_{2} )^{{\beta_{2} /\eta_{2} }} )^{{\eta_{2} }} (1 + (\tau /b_{3} )^{{\beta_{3} /\eta_{3} }} )^{{\eta_{3} }} ,$$

(2.1)

with positive amplitudes $a_{0}$, $b_{k}$, positive exponents $\beta_{k}$, $\eta_{k}$, and real exponent $\alpha_{0}$ as parameters.

The asymptotic limit of (2.1) is $C_{{\text{P}}} (\tau \to \infty )\sim A_{ + } \tau^{1 - 1/\delta }$, with exponent and amplitude

$$1 - 1/\delta = \alpha_{0} - \beta_{1} + \beta_{2} + \beta_{3} ,\;A_{ + } = a_{0} b_{1}^{{\beta_{1} }} /(b_{2}^{{\beta_{2} }} b_{3}^{{\beta_{3} }} ).$$

(2.2)

We can use the scaling exponent $1 - 1/\delta = 0.7912$ and the first equation in (2.2) to eliminate the parameter $\beta_{3}$ in $C_{{\text{P}}} (\tau )$.

The least-squares regression of $C_{{\text{P}}} (\tau )$ is explained in Appendix 1 and is based on supercritical data points $(\tau_{i} ,C_{{{\text{P}},i}} )_{i = 1,...,N}$, $N = 460$, referenced above. The fitting parameters $a_{0} \,$,$\alpha_{0}$ and $(b_{k} ,\beta_{k} ,\eta_{k} )_{k = 1,2,3}$ are recorded in Table 2, including the amplitude $A_{ + }$ in (2.2). (The decadic logarithm ${\text{Log }}b_{k}$ rather than the amplitude $b_{k}$ is listed in this table.) The regressed high-temperature component (2.1) of the isobaric heat capacity $C_{{\text{P}}} (\tau )$ is depicted in Fig. 2 as red solid curve.

Table 2 Fitting parameters of the high-temperature ($T > T_{{\text{c}}}$) and low-temperature ($T_{{{\text{melt}}}} \le T \le T_{{\text{c}}}$) branches of the isobaric heat capacity at critical pressure $P_{{\text{c}}}$ (cf. Table 1) of nitrogen, methane, water and hydrogen. The nonlinear least-squares regression is explained in Appendix 1. The multiply broken power-law density $C_{{\text{P}}} (\tau )$ for the high-temperature heat-capacity branch of nitrogen (column labeled (${\text{N}}_{2}$, $T > T_{{\text{c}}}$)) reads, cf. (2.1), $C_{{\text{P}}} (\tau ) = a_{0} \tau^{{\alpha_{0} }} (1 + (\tau /b_{1} )^{{\beta_{1} /\eta_{1} }} )^{{ - \eta_{1} }} (1 + (\tau /b_{2} )^{{\beta_{2} /\eta_{2} }} )^{{\eta_{2} }} (1 + (\tau /b_{3} )^{{\beta_{3} /\eta_{3} }} )^{{\eta_{3} }}$. The nitrogen heat capacity at low temperature (column labeled (${\text{N}}_{2}$, $T < T_{{\text{c}}}$)) as well as the methane and water heat-capacity branches at high and low temperature (columns labeled (${\text{CH}}_{4}$,$T > T_{{\text{c}}}$), (${\text{CH}}_{4}$,$T < T_{{\text{c}}}$), (${\text{H}}_{2} {\text{O}}$,$T > T_{{\text{c}}}$) and (${\text{H}}_{2} {\text{O}}$,$T < T_{{\text{c}}}$)) are modeled with the broken power law (2.5), $C_{{\text{P}}} (\tau ) = a_{0} \tau^{{\alpha_{0} }} (1 + (\tau /b_{1} )^{{\beta_{1} /\eta_{1} }} )^{{\eta_{1} }} (1 + (\tau /b_{2} )^{{\beta_{2} /\eta_{2} }} )^{{\eta_{2} }}$. The broken power-law density (5.1) is used for the high-temperature heat-capacity branch of hydrogen (column labeled (${\text{H}}_{2}$,$T > T_{{\text{c}}}$)), $C_{{\text{P}}} (\tau ) = a_{0} \tau^{{\alpha_{0} }} (1 + (\tau /b_{1} )^{{\beta_{1} /\eta_{1} }} )^{{\eta_{1} }} (1 + (\tau /b_{2} )^{{\beta_{2} /\eta_{2} }} )^{{ - \eta_{2} }} (1 + (\tau /b_{3} )^{{\beta_{3} /\eta_{3} }} )^{{\eta_{3} }}$. The regression of the low-temperature heat capacity of hydrogen at critical pressure (column labeled (${\text{H}}_{2}$,$T < T_{{\text{c}}}$)) is based on the broken power law (5.4), $C_{{\text{P}}} (\tau ) = a_{0} \tau^{{\alpha_{0} }} (1 + (\tau /b_{1} )^{{\beta_{1} /\eta_{1} }} )^{{ - \eta_{1} }} (1 + (\tau /b_{2} )^{{\beta_{2} /\eta_{2} }} )^{{\eta_{2} }}$. The broken power laws representing the heat-capacity branches are parametrized with the reciprocal reduced temperature $\tau = 1/\left| {T/T_{{\text{c}}} - 1} \right|$. The least-squares functional used for the regression is stated in (7.6). The data points and the regressed high- and low-temperature heat-capacity branches of nitrogen, methane, water and hydrogen are depicted in Figs. 2, 7, 12 and 17. The fitting parameters $a_{0} [{\text{J}}/({\text{mol K}})], \, \alpha_{0} , \, (b_{k}^{{}} ,\beta_{k}^{{}} ,\eta_{k}^{{}} )$ of the enumerated broken power laws for the respective branches are listed in this table. ${\text{Log }}b_{k}$ denotes the decadic logarithm of the amplitude $b_{k}$, and the amplitude $a_{0}$ is in units of ${\text{J}}/({\text{mol K)}}$; all other parameters are dimensionless. Also recorded are the minimum of the least-squares functional $\chi^{2}$, cf. (7.6), and the degrees of freedom of the fit (dof: number $N$ of data points minus number of fitting parameters). The scaling amplitudes $A_{ \pm }^{{}} [{\text{J}}/({\text{mol K)}}]$ of the critical power laws $C_{{\text{P}}} \sim A_{ + } \tau^{1 - 1/\delta }$ (for the $T > T_{{\text{c}}}$ branch) and $C_{{\text{P}}} \sim A_{ - } \tau^{1 - 1/\delta }$ (for the $T < T_{{\text{c}}}$ branch) are recorded as well. The high- and low-temperature Index curves depicting the Log–Log slope of the critical isobaric heat capacities of nitrogen, methane, water and hydrogen in Figs. 3, 8, 13 and 18, respectively, are also defined by the listed parameters $\alpha_{0} , \, (b_{k}^{{}} ,\beta_{k}^{{}} ,\eta_{k}^{{}} )$

In Fig. 1, the isobaric heat capacity is parametrized with reduced temperature $t = T/T_{{\text{c}}}$. The red solid curve in this figure is the high-temperature $C_{{\text{P}}} (\tau )$ in (2.1) with $\tau = 1/(t - 1)$ substituted (shortcut $C_{{\text{P}}} (t)$).

Figure 3 depicts the Index function, cf., e.g., Refs. [30, 35, 38,39,40,41],

$${\text{Index[}}C_{{\text{P}}} (\tau )]: = \frac{{C^{\prime}_{{\text{P}}} (\tau )}}{{C_{{\text{P}}} (\tau )}}\tau = \frac{{{\text{d}}\log C_{{\text{P}}} (\tau )}}{{{\text{d}}\log \tau }},$$

(2.3)

i.e., the Log–Log slope (red solid curve) of the regressed high-temperature heat capacity $C_{{\text{P}}} (\tau )$ in (2.1),

$${\text{Index[}}C_{{\text{P}}} (\tau )] = \alpha_{0} - \beta_{1} \frac{{(\tau /b_{1} )^{{\beta_{1} /\eta_{1} }} }}{{1 + (\tau /b_{1} )^{{\beta_{1} /\eta_{1} }} }} + \beta_{2} \frac{{(\tau /b_{2} )^{{\beta_{2} /\eta_{2} }} }}{{1 + (\tau /b_{2} )^{{\beta_{2} /\eta_{2} }} }} + \beta_{3} \frac{{(\tau /b_{3} )^{{\beta_{3} /\eta_{3} }} }}{{1 + (\tau /b_{3} )^{{\beta_{3} /\eta_{3} }} }}.$$

(2.4)

Fig. 3

Index functions of the critical isobaric heat capacity of nitrogen, parametrized with scaling variable $\tau = 1/\left| {T/T_{{\text{c}}} - 1} \right|$. The red and green solid curves show the Log–Log slope ${\text{Index[}}C_{{\text{P}}} (\tau )]: = {\text{d}}\log C_{{\text{P}}} (\tau )/{\text{d}}\log \tau$ of the high- and low-temperature heat-capacity branches $C_{{\text{P}}} (\tau )$ of nitrogen, cf. Section 2. Log–Log plots of $C_{{\text{P}}} (\tau )$, cf. (2.1) (for $T > T_{{\text{c}}}$) and (2.5) (for $T < T_{{\text{c}}}$), are depicted in Fig. 2 as red and green solid curves. The plotted Index functions ${\text{Index[}}C_{{\text{P}}} (\tau )]$ are stated in (2.4) (red solid curve, $\tau = 1/(T/T_{{\text{c}}} - 1)$, $T > T_{{\text{c}}}$) and (2.7) (green solid curve, $\tau = 1/(1 - T/T_{{\text{c}}} )$, $T < T_{{\text{c}}}$) with parameters in Table 2. The open squares on the (red solid) high-temperature ($T > T_{{\text{c}}}$) Index curve and the filled squares on the (green solid) low-temperature ($T < T_{{\text{c}}}$) Index curve correspond to the heat-capacity data points depicted in Fig. 2. The vertical red and green dotted lines indicate the same temperature intervals as in Fig. 2. The scaling exponent $1 - 1/\delta = 0.7912$ of the heat capacity is indicated by the black dashed horizontal line, which is the asymptote of the depicted Index curves (i.e., the constant $\tau \to \infty$ limit of ${\text{Index[}}C_{{\text{P}}} (\tau )]$ in (2.4) and (2.7)). This horizontal line is also a plot of the constant Index function (Log–Log slope) of the asymptotes of the heat-capacity branches $C_{{\text{P}}} (\tau )$ in Fig. 2 (red and green dashed straight lines in Fig. 2, depicting the asymptotic power laws (2.2) and (2.6) of the isobaric heat capacity at critical pressure)

To better relate Figs. 2 and 3, we have plotted data points $(\tau_{i} ,{\text{Index[}}C_{{\text{P}}} (\tau_{i} )])_{i = 1,...,N}$ (open squares) on the Index curve, using the abscissas $\tau_{i}$ of the data points $(\tau_{i} ,C_{{{\text{P}},i}} )_{i = 1,...,N}$ in Fig. 2 (also indicated by open squares).

2.2 Low-temperature interval between melting point and critical temperature

The data set $(\tau_{i} ,C_{{{\text{P}},i}} )$ used for the regression of the isobaric heat capacity $C_{{\text{P}}} (\tau )$ at critical pressure in the subcritical interval $[T_{{{\text{melt}}}} ,T_{{\text{c}}} ]$ comprises 155 data points between $T_{{{\text{melt}}}} = 63.15{\text{ K}}$ and 125.6 K (filled squares in Fig. 2, taken from Refs. [2, 3]).

The least-squares fit of the low-temperature branch of $C_{{\text{P}}} (\tau )$ is performed with the broken power law

$$C_{{\text{P}}} (\tau ) = a_{0} \tau^{{\alpha_{0} }} (1 + (\tau /b_{1} )^{{\beta_{1} /\eta_{1} }} )^{{\eta_{1} }} (1 + (\tau /b_{2} )^{{\beta_{2} /\eta_{2} }} )^{{\eta_{2} }} .$$

(2.5)

The amplitudes $a_{0}$, $b_{k}$ and exponents $\beta_{k}$, $\eta_{k}$ are positive, and the exponent $\alpha_{0}$ is real. The asymptotic power-law scaling of $C_{{\text{P}}} (\tau )$ in (2.5) reads $C_{{\text{P}}} (\tau \to \infty )\sim A_{ - } \tau^{1 - 1/\delta }$, with

$$1 - 1/\delta = \alpha_{0} + \beta_{1} + \beta_{2} ,\;A_{ - } = a_{0} /(b_{1}^{{\beta_{1} }} b_{2}^{{\beta_{2} }} ).$$

(2.6)

The exponent $\beta_{2}$ in (2.5) can be eliminated via the first identity in (2.6), using the calculated scaling exponent $1 - 1/\delta = 0.7912$.

The least-squares regression of $C_{{\text{P}}} (\tau )$ in (2.5) is analogous to the regression of the high-temperature component of the heat capacity in Sect. 2.1, based on the subcritical data set (τi,C