### 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 ${C}_{\mathrm{P}}(T,{P}_{\text{c}})$,
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 ${C}_{\mathrm{P}}(T,{P}_{\text{c}})$ branches of these fluids (and of CO_{2} 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 ${C}_{\mathrm{P}}(T,{P}_{\text{c}})$. 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 $t<1$.

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 ${C}_{\mathrm{P}}(\tau )$ 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 $\tau =\mathrm{\infty}$. 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.

### 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],

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 ${C}_{\text{P}}(\tau \to \mathrm{\infty})\sim {A}_{+}{\tau}^{1-1/\delta}$, with exponent and amplitude

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 ${a}_{0}\phantom{\rule{thinmathspace}{0ex}}$,${\alpha}_{0}$ 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.

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],

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

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

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 ${C}_{\text{P}}(\tau \to \mathrm{\infty})\sim {A}_{-}{\tau}^{1-1/\delta}$, with

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