# Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermodynamic Conditions for the Liquid–Vapor Phase Transition in a Pure Fluid

## 3. An Isometric Transformation to the Critical Point

## 4. The Scaling Form of the Entropy

## 5. Critical Properties and Exponents

#### 5.1. Order Parameter

#### 5.2. Critical Isotherm

#### 5.3. Specific Heat at Constant Volume

#### 5.4. Isothermal Compressibility

## 6. Final Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${l}_{0}$ | Characteristic length of intermolecular potentials |

m | Characteristic mass of the fluid components. |

h | Planck constant |

${k}_{B}$ | Boltzmann constant |

s | Entropy per unit of volume |

e | Energy per unit of volume |

n | Particle density. |

T | Temperature |

$\mu $ | Chemical potential |

$\alpha $ | Quotient between the chemical potential and temperature $\mu /T$ |

$\beta $ | Inverse of temperature $1/T$ |

${c}_{v}$ | Specific heat at constant volume and number of particles |

${\kappa}_{T}$ | Isothermal compressibility |

p | Pressure |

$\overrightarrow{R}$ | Vector of the entropy surface $(e,n,s(e,n))$ |

$\widehat{n}$ | Normal unit vector |

$\widehat{t}$ | Unit tangent vector |

$\widehat{m}$ | Cross product $\widehat{t}\times \widehat{n}$ |

$\Delta e$ | Energy difference $e-{e}_{c}$ |

$\Delta n$ | Particle density difference $n-{n}_{c}$ |

$\Delta s$ | Entropy density difference $s-{s}_{c}$ |

$\Delta \overrightarrow{R}$ | Vector with components $(\Delta e,\Delta n,\Delta s)$ |

$(x,y,z)$ | local set of coordinates defined in Equation (6) |

${c}_{0}$ | Coefficient characteristic of the coexistence curve |

$\Delta $ | Critical exponent, defined in Equation (11) |

$\eta $ | Dummy exponent |

${f}_{n}({y}^{2})$ | Functions that define an x-power expansion of $z=z(x,y)$. |

${A}_{n},{B}_{n},{C}_{n}$ | Numerical coefficients, elements of different sequences |

${\Gamma}_{n},{\Xi}_{n},{\Omega}_{n}$ | Numerical exponents, elements of different sequences |

${\Gamma}_{0},{\Xi}_{0},{\Omega}_{0}$ | Numerical exponents. |

$\mathcal{F}$ | Scaling function |

X | Argument of the scaling function $\mathcal{F}$, given by $X=x/{y}^{2\Delta}$ |

$\mathcal{G}$ | Scaling function appropriate for $x>0$ |

Y | Argument of the scaling function $\mathcal{G}$, given by $Y={y}^{2}/{x}^{1/\Delta}$ |

${\overrightarrow{\tau}}_{\beta}$ | Auxiliary vector defined as $(1,0,\beta )$. It coresponds to $\partial (e,n,s)/\partial e$ |

${\overrightarrow{\tau}}_{\alpha}$ | Auxiliary vector defined as $(0,1,-\alpha )$. It corresponds to $\partial (e,n,s)/\partial n$ |

${d}_{0}$ | Numerical coefficient characteristic of the isothermal curve |

$\widehat{\beta}$ | Critical exponent, defined in Equation (33) |

$\widehat{\delta}$ | Critical exponent, defined in Equation (37) |

$\widehat{\alpha}$ | Critical exponent, defined in Equation (40) |

$\widehat{\gamma}$ | Critical exponent, defined in Equation (43) |

$\mathcal{B}$ | Auxiliary constant coefficient. |

d | Spatial dimension of the system |

m | Dimensionless magnetization per unit volume |

H | Dimensionless magnetic field |

$\Delta \mu $ | Chemical potential difference $\mu -{\mu}_{c}$ |

$\omega $ | Strictly increasing function |

$\gamma $ | Auxiliary constant |

$\theta $ | Auxiliary constant |

${A}_{ni}$ | Auxiliary coefficient |

${\Gamma}_{ni}$ | Auxiliary exponent |

$\mathcal{D}$ | Auxiliary constant coefficient |

Subscripts | |

c | Critical point value |

l | Coexisting liquid state value |

g | Coexisting gas state value |

$coex$ | Constrained to the coexistence curve |

n | Index used to define elements of a sequence $n=0,1,2,...$ |

$ni$ | The nth element of the ith sequence. |

## Appendix A. A Generalization of the Scaling Form of z, Equation (13)

## Appendix B. Calculation of Critical Exponents

#### Appendix B.1. The Order Parameter

#### Appendix B.2. The Critical Isotherm

#### Appendix B.3. The Specific Heat at Constant Volume

#### Appendix B.4. The Isothermal Compressibility

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**Figure 1.**A level-curve sketch of s as a function of e and n, with ${s}_{1}<{s}_{c}<{s}_{2}<{s}_{3}$. There are no thermodynamic states in the gray zones. Our interest is in the region near the liquid-gas critical point. Figure taken from Ref. [26].

**Figure 2.**A 3D sketch of the function $s=s(e,n)$, in the neigborhood of the critical point, showing the coexistence curve as the edge in red color.

**Figure 3.**(Color online) A sketch of the entropy surface given in Figure 2, in the local, displaced and rotated axes ${\widehat{m}}_{c}$, ${\widehat{t}}_{c}$ and ${\widehat{n}}_{c}$, defining the axes x, y and z, respectively. The coexistence curve is in a continuous (red) line, the curve $y=0$ in a dotted (blue) line and the critical isotherm ${\beta}_{c}$ in a dot-dash (green) line. As discussed in Section 4 and Section 6 , the curves $y=0$ and the coexistence one conform the symmetry-breaking line.

**Figure 4.**(Color online) Sketch of the entropy scaling function $\mathcal{F}$ as a function of its argument X. This function is assumed to be analytic at $X=0$ and defined for $-{c}_{0}\le X<\infty $, with $X=-{c}_{0}$ occurring at the coexistence curve. It has the asymptotic limit $\mathcal{F}\sim {X}^{{\Gamma}_{0}/\Delta}$ for $X\gg 1$. The function is a minimum at $X=-{d}_{0}$ which corresponds to the critical isotherm. In Section 6 we discuss the asymptotic value $\mathcal{F}\sim {(-X)}^{{\Gamma}_{0}/\Delta}$ as $X\to -{c}_{0}$.

**Table 1.**Theoretical critical exponents of the Ising universality class, to which the liquid–vapor critical point belongs, for dimensions $d=2,3,4$. The exponents for $d=2$ and 4 are exact [21], the latter being the mean-field van der Waals exponents. The exponent $\tilde{\alpha}=0$ in $d=2$, indicates a logarithmic divergence. The values for $d=3$ are, so far, the best theoretical predictions, including RG calculations and computer simulations, in close agreement with experimental estimates; see Ref. [19] for a thorough assessment of these exponents.

Critical Exponents | d = 2 | d = 3 | d = 4 |
---|---|---|---|

$\tilde{\alpha}$ | 0 | 0.110(1) | 0 |

$\tilde{\beta}$ | 1/8 | 0.3265(3) | 1/2 |

$\tilde{\gamma}$ | 7/4 | 1.2372(5) | 1 |

$\tilde{\delta}$ | 15 | 4.789(2) | 5 |

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Obeso-Jureidini, J.C.; Olascoaga, D.; Romero-Rochín, V.
Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point. *Entropy* **2021**, *23*, 720.
https://doi.org/10.3390/e23060720

**AMA Style**

Obeso-Jureidini JC, Olascoaga D, Romero-Rochín V.
Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point. *Entropy*. 2021; 23(6):720.
https://doi.org/10.3390/e23060720

**Chicago/Turabian Style**

Obeso-Jureidini, Juan Carlos, Daniela Olascoaga, and Victor Romero-Rochín.
2021. "Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point" *Entropy* 23, no. 6: 720.
https://doi.org/10.3390/e23060720