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Equation of state

An equation of state is a thermodynamic equation that describes the relationship between state variables, such as pressure, volume, and temperature, for a substance in thermodynamic equilibrium under specified physical conditions.^[1] These equations are fundamental in thermodynamics as they fully characterize the macroscopic properties of matter, enabling predictions of system behavior without dependence on the system's history.^[2] The concept of equations of state originated in the 17th and 18th centuries through empirical observations of gas behavior, culminating in the ideal gas law. Robert Boyle first observed the inverse relationship between pressure and volume for gases at constant temperature in 1662, known as Boyle's law.^[3] Jacques Charles later discovered in 1787 that volume is directly proportional to temperature at constant pressure, termed Charles's law.^[4] These findings, combined with Avogadro's law on gas volumes and moles, were unified by Émile Clapeyron in 1834 into the ideal gas law, PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is absolute temperature.^[4] This equation assumes ideal behavior, neglecting intermolecular forces and molecular volume, and serves as the simplest and most widely used equation of state.^[5] Beyond the ideal gas law, more advanced equations account for real gas deviations, such as the van der Waals equation, which incorporates corrections for molecular attractions and finite size: \left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT, where a and b are substance-specific constants.^[6] Other notable examples include the Redlich-Kwong equation for improved accuracy in high-pressure conditions and the Peng-Robinson equation, commonly applied in petroleum engineering for hydrocarbon mixtures.^[1] These semi-empirical models extend applicability to liquids, solids, and supercritical fluids.^[7] Equations of state play a central role in thermodynamics by linking state variables to compute properties like internal energy, enthalpy, and entropy, essential for analyzing processes in engines, refrigeration, and chemical reactions.^[8] They enable engineers to predict phase transitions, compressibility, and work done in thermodynamic cycles, underpinning applications in aerospace, energy production, and materials science.^[9] For instance, in fluid systems, knowledge of the equation of state allows determination of density from pressure and temperature, facilitating simulations of real-world behaviors.^[1]

Introduction

Definition and General Form

An equation of state (EOS) is a thermodynamic equation that relates the state variables of a substance or system, typically pressure P, volume V, temperature T, and composition (such as the number of moles n), under equilibrium conditions.^[7]^[10] This relation describes how these variables are interdependent for a given material, enabling the prediction of one variable from the others without dependence on the system's history.^[11]^[12] The general form of an EOS can be expressed explicitly, such as P = P(V, T, n), where pressure is a function of the other variables, or implicitly as f(P, V, T, n) = 0, encapsulating the interdependence without solving for a single variable.^[10]^[12] Explicit forms are convenient for computations where one variable is isolated, while implicit forms are often more fundamental in derivations, as seen in the simplest example, the ideal gas law PV = nRT.^[5] The choice between explicit and implicit representations depends on the context, with implicit equations preserving symmetry in thermodynamic relations.^[13] In thermodynamic context, an EOS arises from the fundamental relation for the internal energy U, given by the differential form

dU = T \, dS - P \, dV + \mu \, dn,

where S is entropy, \mu is chemical potential, and the equation holds for reversible processes in closed or open systems.^[13]^[14] From this, the EOS can be derived using thermodynamic potentials: for instance, the pressure is obtained as P = -\left( \frac{\partial F}{\partial V} \right)_{T,n} from the Helmholtz free energy F = U - TS, yielding P = P(T, V, n) in the natural variables of F.^[10]^[15] These derivations ensure the EOS is consistent with the first and second laws of thermodynamics.^[16] EOS are typically expressed in standard SI units, with pressure in pascals (Pa) or bars, volume in cubic meters (m³), temperature in kelvin (K), and composition in moles (mol), facilitating universal applicability across substances.^[17]^[18] For generality and comparison, dimensionless reduced variables are often used, defined via critical point properties: reduced pressure P_r = P / P_c, reduced volume V_r = V / V_c, and reduced temperature T_r = T / T_c, where subscript c denotes critical values; this framework underpins the principle of corresponding states, allowing EOS to be scaled across fluids.^[19]^[20]^[21]

Importance and Applications

Equations of state (EOS) play a fundamental role in thermodynamics by providing mathematical relations that connect pressure, volume, temperature, and other thermodynamic variables, enabling the prediction of phase behavior, compressibility factors, and internal energy without relying solely on experimental measurements.^[22] These relations allow for the calculation of key properties such as enthalpy and entropy, which are essential for understanding equilibrium states and phase transitions in substances ranging from gases to solids.^[11] For instance, EOS facilitate the determination of critical points and coexistence curves, crucial for modeling vapor-liquid equilibria.^[23] In engineering, EOS are indispensable for design and optimization across multiple disciplines. In chemical engineering, they underpin process simulations, such as distillation column design and hydrocarbon processing, where cubic EOS like Peng-Robinson predict fluid behavior under varying conditions to ensure efficient separation and reaction systems.^[24]^[25] In mechanical engineering, EOS inform the analysis of thermodynamic cycles in engines, including compression and expansion processes in internal combustion and gas turbine systems, optimizing performance and fuel efficiency.^[22] For materials science, EOS describe high-pressure responses of solids and polymers, aiding in the development of materials for extreme environments like deep-sea or aerospace applications.^[26] Beyond engineering, EOS find critical applications in scientific fields probing natural phenomena. In astrophysics, they model the interiors of stars, where EOS for hydrogen and helium determine pressure-density relations under immense gravitational forces, influencing stellar evolution and supernova dynamics.^[27]^[28] In geophysics, EOS for iron alloys constrain the composition and density of Earth's core, helping explain seismic wave propagation and the planet's magnetic field generation.^[29]^[30] In plasma physics, particularly for fusion reactors, EOS provide the pressure and energy states of compressed plasmas, guiding inertial confinement fusion designs to achieve viable energy production.^[31]^[32] Despite their utility, EOS exhibit limitations in accuracy under extreme conditions, such as ultra-high pressures or temperatures, where semi-empirical models may deviate from quantum or relativistic effects, necessitating advanced theoretical refinements for precise predictions.^[33]^[34]

Historical Background

Early Concepts

The earliest conceptual foundations for understanding the behavior of gases trace back to ancient Greek philosophy, where thinkers like Anaximenes of Miletus in the 6th century BCE proposed air as the fundamental substance composing all matter, capable of rarefaction and condensation to form other elements.^[35] This idea evolved in Aristotle's framework around 350 BCE, which classified air as one of four classical elements—alongside earth, water, and fire—essential to natural phenomena, with air embodying hot and wet qualities that influenced medieval scholastic thought. Throughout the medieval period, European scholars, drawing on Aristotelian and Islamic traditions, viewed air primarily as an elemental medium for respiration and atmospheric phenomena, though without quantitative experimentation, limiting insights to qualitative observations of compression and expansion in everyday contexts like bellows or wind.^[36] The transition to empirical investigation began in the 17th century with advancements in instrumentation, notably Evangelista Torricelli's invention of the mercury barometer in 1643, which demonstrated atmospheric pressure by measuring the height of a mercury column in a sealed tube inverted in a dish of mercury, providing the first reliable means to quantify air's weight and elasticity.^[37] Building on this, Robert Boyle conducted pivotal experiments in 1662 using a J-shaped glass tube apparatus: air was trapped in the closed end, and mercury was poured into the open end to vary pressure, revealing that compressing air reduced its volume while maintaining constant temperature, thus establishing the compressibility of gases through direct observation.^[38] Independently, French physicist Edme Mariotte replicated these findings in 1679 and extended them by noting that temperature variations affected gas volume, laying groundwork for later thermal dependencies. In the 18th century, further empirical generalizations emerged from ballooning and chemical experiments. Jacques Charles observed in 1787 that the volume of gases expanded proportionally with increasing temperature at constant pressure, a relation derived from measurements during hydrogen balloon ascents.^[39] Joseph Louis Gay-Lussac refined pressure-temperature connections in 1808, showing through closed-vessel experiments that gas pressure rose linearly with temperature when volume was fixed.^[40] These observations complemented John Dalton's 1801 law of partial pressures, which posited that in a mixture of non-reacting gases, the total pressure equals the sum of each gas's individual pressure as if alone, based on analyses of vapor and air mixtures. Amedeo Avogadro's 1811 hypothesis further advanced this by suggesting that equal volumes of different gases at the same temperature and pressure contain equal numbers of molecules, reconciling volume ratios in chemical reactions.^[41] Collectively, these pre-modern empirical insights culminated in the ideal gas law, unifying pressure, volume, temperature, and quantity relationships.

Key Developments in the 19th and 20th Centuries

In the late 19th century, significant progress in equations of state emerged with efforts to account for deviations from ideal gas behavior in real gases. Johannes Diderik van der Waals introduced his equation in 1873 as part of his doctoral thesis, marking a breakthrough by incorporating corrections for the finite volume of molecules and attractive intermolecular forces, which allowed for qualitative predictions of liquid-gas phase transitions and critical phenomena.^[42] This model laid the foundation for subsequent real gas theories by bridging empirical observations with molecular interpretations. Toward the end of the century, Conrad Dieterici proposed an alternative equation in 1899, modifying the pressure term with an exponential factor to better capture the asymmetry in real gas isotherms near the critical point, offering improved accuracy for certain substances like carbon dioxide.^[43] Entering the 20th century, Heike Kamerlingh Onnes developed the virial expansion in 1901 as a power series in density, providing a systematic way to quantify non-ideal corrections through virial coefficients derived from experimental data, which proved particularly useful for low-density gases and influenced later theoretical frameworks.^[44] The advent of quantum mechanics in the 1920s profoundly impacted equations of state by introducing quantum statistical mechanics, enabling derivations for degenerate gases where classical statistics failed, such as Fermi-Dirac and Bose-Einstein statistics applied to electron gases and ideal quantum fluids in the 1920s and 1930s.^[45] These developments extended equations of state to low-temperature regimes, accounting for quantum effects like degeneracy pressure. In the mid-20th century, the Benedict-Webb-Rubin equation, formulated in 1940, advanced empirical modeling by incorporating eight adjustable parameters to accurately represent pressure-volume-temperature relations for light hydrocarbons and mixtures, achieving high precision in thermodynamic predictions for industrial applications like natural gas processing. By the 1970s, statistical mechanics provided the basis for more sophisticated theories, with perturbation approaches to chain and associating fluids laying groundwork for advanced models that treated molecular interactions explicitly. This culminated in the late 20th century with refinements to cubic equations, such as the Peng-Robinson equation introduced in 1976, which improved vapor pressure predictions and phase behavior for hydrocarbons through a temperature-dependent attraction term, becoming widely adopted in petroleum engineering. Concurrently, multiparameter equations of state proliferated in the 1980s and 1990s, with NIST developing highly accurate Helmholtz energy formulations—often exceeding 20 terms—for fluids like refrigerants and water, enabling precise thermodynamic property calculations essential for standards and simulations.^[46]

Fundamental Equations for Gases

Classical Ideal Gas Law

The classical ideal gas law describes the behavior of an ideal gas under conditions where intermolecular interactions are negligible. It is expressed as

PV = nRT,

where P is the pressure, V is the volume, n is the number of moles of gas, T is the absolute temperature, and R is the universal gas constant, approximately $8.314 \, \mathrm{J \cdot mol^{-1} \cdot K^{-1}}. On a molar basis, the equation simplifies to Pv = RT, where v = V/n is the molar volume. This form was first synthesized in 1834 by Benoît Paul Émile Clapeyron, who combined empirical observations including Boyle's law (proportionality of pressure and inverse volume at constant temperature, 1662), Charles's law (proportionality of volume and temperature at constant pressure, 1787), and Avogadro's law (proportionality of volume and number of moles at constant pressure and temperature, 1811). The term "ideal gas" to describe systems obeying this equation was introduced by Rudolf Clausius in 1857.^[47] The equation can be derived from the kinetic theory of gases, which posits that a gas consists of a large number of point-like particles in constant, random motion, with pressure resulting from the average momentum transfer imparted to the container walls during elastic collisions. In this model, the pressure P is related to the mean kinetic energy of the particles by P = \frac{1}{3} \frac{N}{V} m \langle v^2 \rangle, where N is the number of particles, m is the particle mass, and \langle v^2 \rangle is the mean square speed; equating this to the kinetic energy expression \frac{3}{2} kT per particle (with k as Boltzmann's constant) yields PV = NkT, or equivalently PV = nRT since R = N_A k (where N_A is Avogadro's number). This derivation was independently developed by August Krönig in 1856 and more rigorously by Clausius in 1857, assuming particles occupy negligible volume, exert no forces on each other except during instantaneous elastic collisions, and move in straight lines between collisions with equal probability in all directions.^[48]^[49] The classical ideal gas law applies accurately to low-density gases at moderate temperatures, where the mean free path between collisions is much larger than molecular size, minimizing effects from finite particle volume and weak intermolecular attractions or repulsions. Deviations become significant at high pressures, where molecular volumes are comparable to the container volume, or at low temperatures near the boiling point, where intermolecular forces influence behavior. For example, it reliably predicts properties of air at standard conditions but fails for compressed gases like CO₂ at room temperature.^[50]

Quantum Ideal Gas Laws

The quantum ideal gas describes a system of non-interacting particles that obey quantum statistics, either Fermi-Dirac for fermions or Bose-Einstein for bosons, where particle indistinguishability and wave nature lead to significant deviations from classical behavior. Unlike the classical ideal gas, which treats particles as distinguishable point masses valid at high temperatures or low densities, the quantum regime becomes relevant when the thermal de Broglie wavelength \lambda = \sqrt{2\pi \hbar^2 / m k_B T} is comparable to or larger than the average interparticle spacing d = n^{-1/3}, where n is the number density, m the particle mass, k_B Boltzmann's constant, and T the temperature; this condition signals the onset of quantum degeneracy effects.^[51] For fermions, such as electrons, the equation of state follows Fermi-Dirac statistics, with the occupation number n(\epsilon) = 1 / (e^{(\epsilon - \mu)/k_B T} + 1), where \mu is the chemical potential. At zero temperature, the gas is fully degenerate, filling all states up to the Fermi energy E_F = \hbar^2 (3\pi^2 n)^{2/3} / (2m), yielding a total energy E = (3/5) N E_F and an energy density u = (3/5) n E_F; the corresponding degeneracy pressure is P = (2/3) u = (2/5) n E_F, arising solely from the Pauli exclusion principle that forces fermions into higher momentum states. At finite but low temperatures, thermal excitations occur near the Fermi surface, but the pressure remains dominated by the zero-temperature term until T approaches the degeneracy temperature T_F = E_F / k_B. This quantum ideal gas serves as the high-temperature limit of more complex fermionic systems but captures essential quantum pressure without interactions.^[51] For bosons, such as photons or helium-4 atoms, the equation of state uses Bose-Einstein statistics, with occupation number n(\epsilon) = 1 / (e^{(\epsilon - \mu)/k_B T} - 1) and \mu < 0 (or \mu = 0 for photons). In the photon gas, a relativistic massless boson system in thermal equilibrium, the energy density is u = (\pi^2 k_B^4 / (15 (\hbar c)^3)) T^4 and the pressure satisfies P = u / 3, reflecting the ultrarelativistic limit where energy and momentum are proportional. For non-relativistic massive bosons like helium-4, the ideal gas exhibits Bose-Einstein condensation below a critical temperature T_c \approx (h^2 / (2\pi m k_B)) (n / \zeta(3/2))^{2/3}, where a macroscopic fraction of particles occupies the ground state; above T_c, the pressure P = (k_B T / \lambda^3) g_{5/2}(z) depends on the fugacity z = e^{\mu / k_B T} and polylogarithm g_{5/2}, while below T_c the pressure becomes independent of density at fixed T, as the condensate does not contribute to pressure.^[51] These quantum equations of state find key applications in astrophysics and low-temperature physics. In white dwarf stars, electron degeneracy pressure P \approx (2/5) n_e E_F (with n_e the electron density) balances gravitational collapse, limiting the maximum mass to the Chandrasekhar limit of about 1.4 solar masses, as derived from relativistic extensions of the non-interacting Fermi gas model. For superfluid helium-4, Bose-Einstein condensation below 2.17 K enables zero-viscosity flow, with the ideal gas providing the foundational understanding of the transition despite weak interactions in reality.^[51]

Equations for Real Gases

Cubic Equations of State

Cubic equations of state constitute a family of thermodynamic models designed to describe the pressure-volume-temperature behavior of real gases by incorporating corrections for molecular interactions and excluded volume effects. These models derive their name from the fact that they yield a cubic polynomial equation when solved for molar volume V_m, facilitating the determination of up to three real roots corresponding to possible phases. The foundational structure, often traced to the van der Waals form, is given by

\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT,

where P is pressure, T is temperature, R is the gas constant, a represents the parameter accounting for attractive intermolecular forces that reduce the observed pressure on the container walls, and b captures the repulsive contributions by representing the effective excluded volume per mole of molecules. The van der Waals equation, introduced in 1873, serves as the archetype for this class and marked a significant advancement in modeling phase transitions in fluids. In this equation, the parameters are related to critical properties as a = 3 P_c V_c^2 and b = V_c / 3, where P_c and V_c are the critical pressure and molar volume, respectively; these relations ensure the model captures the critical point where the distinction between liquid and gas phases vanishes. At the critical point, the van der Waals equation predicts a compressibility factor Z_c = P_c V_c / (R T_c) = 3/8 = 0.375, which, while universal for the model, deviates from experimental values for most substances (typically 0.27–0.29). Subsequent refinements addressed limitations in accuracy, particularly for vapor pressures and densities. The Redlich-Kwong equation, developed in 1949, modified the attractive term to improve predictions at higher temperatures, adopting the form

\left(P + \frac{a}{\sqrt{T} V_m (V_m + b)}\right)(V_m - b) = RT,

with a = 0.42748 R^2 T_c^{2.5} / P_c and b = 0.08664 R T_c / P_c. This adjustment enhanced performance for non-polar gases but still struggled with liquid densities. Further improvements came with the Soave-Redlich-Kwong equation in 1972, which introduced a temperature-dependent attractive parameter to better fit vapor-liquid equilibria data. Its formulation is

\left(P + \frac{a(T)}{V_m (V_m + b)}\right)(V_m - b) = RT,

where a(T) = 0.42747 (R^2 T_c^2 / P_c) [1 + (0.48508 + 1.55171 \omega - 0.15613 \omega^2)(1 - T_r^{0.5})]^2, T_r = T / T_c is the reduced temperature, and \omega is the acentric factor measuring molecular non-sphericity; b = 0.08664 R T_c / P_c. This version significantly improved predictions for hydrocarbons over a wider temperature range. The Peng-Robinson equation, proposed in 1976, offered another enhancement by refining the repulsive term for better liquid volume predictions, expressed as

P = \frac{RT}{V_m - b} - \frac{a(T)}{V_m (V_m + b) + b (V_m - b)},

with a(T) = 0.45724 (R^2 T_c^2 / P_c) [1 + (0.37464 + 1.54226 \omega - 0.26992 \omega^2)(1 - T_r^{0.5})]^2 and b = 0.07780 R T_c / P_c. The a = 0.45724 R^2 T_c^2 / P_c base value, combined with the \alpha(T) function, yields a critical compressibility closer to experimental values for many fluids. A key feature of cubic equations of state is their ability to predict vapor-liquid equilibria through the Maxwell construction, which enforces mechanical and chemical equilibrium by requiring that the areas above and below the horizontal tie line on a subcritical P-V isotherm are equal, corresponding to the saturation pressure. This graphical or numerical method resolves the unphysical van der Waals loop in the two-phase region, providing coexistence densities without additional parameters. These models excel in applications involving hydrocarbons, such as natural gas processing and petroleum reservoir simulations, due to their reliable phase behavior predictions and computational efficiency. However, they exhibit limitations for polar or associating fluids like water or alcohols, where hydrogen bonding leads to inaccuracies in densities and critical properties, often necessitating specialized mixing rules for multicomponent systems; performance also degrades near the critical point without further modifications.^[52]

Virial Equations of State

The virial equation of state provides a theoretical framework for describing deviations from ideal gas behavior in real gases through a power series expansion in terms of density. The compressibility factor Z = \frac{PV}{RT}, where V is the molar volume, P is pressure, R is the gas constant, and T is temperature, is expressed as

Z = 1 + \frac{B}{V} + \frac{C}{V^2} + \frac{D}{V^3} + \cdots,

with B, C, D, and higher terms denoting the second, third, fourth, and subsequent virial coefficients, respectively.^[53] This expansion is particularly useful for dilute and moderately dense gases, as it systematically incorporates molecular interactions via the virial coefficients, which depend only on temperature.^[54] The second virial coefficient B captures the effect of pairwise molecular interactions and arises from the first correction to the ideal gas law. From statistical mechanics, B is given by

B = -2\pi N_A \int_0^\infty \left[ \exp\left(-\frac{u(r)}{kT}\right) - 1 \right] r^2 \, dr,

where N_A is Avogadro's number, k is Boltzmann's constant, u(r) is the intermolecular pair potential, and r is the intermolecular distance.^[54] This integral reflects the temperature-dependent balance between repulsive and attractive forces: at high temperatures, B approaches the hard-sphere limit (positive, reflecting excluded volume), while at lower temperatures, attractive interactions make B negative, leading to gas attraction./16:_The_Properties_of_Gases/16.05:_The_Second_Virial_Coefficient) Higher virial coefficients, such as the third coefficient C, account for three-body interactions and beyond, derived through cluster expansions in the grand canonical ensemble. These Mayer cluster expansions express the coefficients as sums of irreducible cluster integrals over Mayer f-functions, f_{ij} = \exp(-u(r_{ij})/[kT](/page/KT)) - 1, involving combinatorial diagrams for multi-particle configurations.^[55] For practical calculations, the series is typically truncated after the second or third term, as higher coefficients grow rapidly and contribute significantly only at higher densities.^[54] Virial equations are applied to model real gas properties at low to moderate densities, where they provide high accuracy by fitting virial coefficients to experimental data. For instance, virial expansions up to four terms have been used with potential models to predict volumetric and caloric properties of nitrogen gas over wide temperature and pressure ranges.^[56] Cubic equations of state often serve as compact approximations that reproduce the low-density limit of virial expansions.^[53]

Theoretical Equations of State

Perturbation Theory-Based Models

Perturbation theory provides a framework for deriving equations of state (EOS) for real fluids by treating deviations from an ideal reference system as small perturbations. In this approach, the intermolecular potential u(r) is decomposed into a reference potential u_0(r), typically the hard-sphere potential that captures repulsive interactions, and a perturbation term u_1(r) accounting for attractive or softer contributions. The Helmholtz free energy A is then expanded as A = A_0 + \langle u_1 \rangle_0 + higher-order terms, where A_0 is the free energy of the reference system and \langle u_1 \rangle_0 denotes the average of the perturbation in the reference ensemble. This expansion allows thermodynamic properties like pressure and compressibility to be computed perturbatively, bridging microscopic statistical mechanics with macroscopic EOS.^[57] A seminal application is the Weeks-Chandler-Andersen (WCA) theory, which divides the Lennard-Jones (LJ) potential into a repulsive reference u_0(r) for r < 2^{1/6}\sigma and an attractive perturbation otherwise, enabling accurate predictions for simple liquids near the triple point. In WCA, the first-order perturbation yields the EOS pressure as P = P_0 + \rho^2 \int u_1(r) g_0(r) 4\pi r^2 dr, where P_0 is the hard-sphere pressure, \rho is density, and g_0(r) is the reference radial distribution function. This model excels in describing the liquid-vapor coexistence of argon-like fluids with errors under 5% in density.^[58]^[59] For electrolyte solutions, the mean spherical approximation (MSA) serves as a perturbation-based integral equation method, approximating the direct correlation function as the perturbation potential beyond the hard-core diameter while setting the potential inside to zero. Solving the Ornstein-Zernike equation under MSA yields an analytical EOS for charged hard spheres, with the excess free energy derived from charging processes or compressibility routes, capturing Debye-Hückel limiting behavior at low concentrations and finite-size effects at higher densities. The MSA EOS for 1:1 electrolytes like NaCl shows osmotic coefficients within 10% of experimental data up to 1 M.^[60] Perturbed hard-sphere models extend this to soft potentials like LJ fluids by using the Barker-Henderson second-order perturbation, where the effective hard-sphere diameter is temperature-dependent, and pressure includes integrals over the softened repulsion. These models provide EOS for LJ fluids with thermodynamic consistency, predicting critical points and phase diagrams with deviations of 2-3% from simulations. Overall, perturbation theory-based EOS strengths lie in their ability to connect statistical mechanics directly to observable properties like PVT behavior for simple, non-associating fluids, though higher orders are often truncated for practicality.^[61]^[62] Extensions like statistical associating fluid theory (SAFT) build on these foundations for more complex systems.^[63]

Statistical Associating Fluid Theory (SAFT)

The Statistical Associating Fluid Theory (SAFT) is a molecular-based equation of state formulated to describe the thermodynamic properties of complex fluids, with particular emphasis on chain-like molecules and associative interactions such as hydrogen bonding. It provides a rigorous framework for predicting phase equilibria, densities, and other properties in systems where molecular architecture plays a key role, extending beyond simple fluids to handle polydispersity and mixtures. SAFT is grounded in Wertheim's thermodynamic perturbation theory, which treats association and bonding as perturbations on a reference fluid system. In the SAFT framework, the total Helmholtz free energy A is expressed as a sum of distinct contributions capturing different molecular effects:

A = A_{\text{ideal}} + A_{\text{hs}} + A_{\text{chain}} + A_{\text{disp}} + A_{\text{association}} + A_{\text{polar}}

^[64] Here, A_{\text{ideal}} accounts for the ideal-gas behavior of non-interacting molecules, while the remaining terms represent residual contributions from molecular structure and interactions. The hard-sphere reference term A_{\text{hs}} forms the foundation for repulsive core interactions, typically using expressions like the Carnahan-Starling approximation for the free energy of hard spheres. Chain formation is incorporated via A_{\text{chain}}, which models molecules as flexible chains of bonded hard spheres, deriving the entropic penalty of bonding from Wertheim's theory. The dispersion term A_{\text{disp}} accounts for attractive van der Waals interactions. The association term A_{\text{association}} addresses site-specific attractions, such as hydrogen bonding, by integrating over association sites with the Mayer f-function f^{\text{assoc}} = \exp(-\epsilon^{\text{assoc}}/kT) - 1, where \epsilon^{\text{assoc}} is the association energy depth; this allows quantification of dimerization and polymerization effects. Finally, A_{\text{polar}} captures dipole-dipole or multipolar interactions in polar molecules, often using mean-field approximations. This modular structure enables SAFT to derive pressure, chemical potentials, and phase behavior directly from the free energy minimization.^[65] Several variants of SAFT have been developed to enhance accuracy and applicability for specific interaction potentials. The original SAFT, introduced by Chapman et al. in 1989, employs a perturbation expansion around a hard-sphere reference with square-well attractions for dispersion and association, suitable for simple associating liquids.^[65] SAFT-VR, proposed by Gil-Villegas et al. in 1997, replaces square-well potentials with softer, variable-range Mie potentials to better represent realistic dispersion forces in chain molecules, improving predictions for vapor pressures and critical points.^[66] PC-SAFT, developed by Gross and Sadowski in 2001, refines the chain reference by treating dispersion as a perturbation on chains rather than spheres, simplifying parameter fitting with only three to five molecular parameters per component while maintaining high fidelity for non-polar and associating chains.^[67] These variants share the core Helmholtz energy decomposition but differ in reference and perturbation treatments, allowing tailored use across fluid types. SAFT excels in applications to complex systems like polymers, where it models chain length and branching to predict solubility, viscosity, and phase separation in solutions. For surfactants, it captures self-assembly into micelles and microemulsions by accounting for amphiphilic chain associations and hydrophobic tails. Extensions to electrolytes incorporate ion solvation and Debye-Hückel corrections, enabling accurate modeling of aqueous salt solutions. Particularly for associating compounds like water, SAFT variants quantitatively predict vapor-liquid phase diagrams, including the critical point and coexistence curves, with average deviations under 2% in density using transferable parameters derived from molecular simulations or limited experimental data.

Empirical and Multiparameter Equations

Benedict-Webb-Rubin Equation

The Benedict-Webb-Rubin (BWR) equation of state is an empirical multiparameter model designed to accurately represent the pressure-volume-temperature behavior of real gases, particularly at moderate to high densities where deviations from ideality are significant. Developed in 1940 by Manson Benedict, G. B. Webb, and L. C. Rubin, it extends virial-type expansions by incorporating additional terms to account for intermolecular attractions and repulsions beyond the second and third virial coefficients. The equation was originally fitted to experimental thermodynamic data for light hydrocarbons such as methane, with subsequent applications and modifications extending its use to gases like helium, enabling precise predictions of properties like enthalpy and fugacity across gaseous and liquid states.^[68]^[69] The functional form of the BWR equation expresses pressure P as a function of temperature T, molar density \rho = 1/V (where V is molar volume), and the molar gas constant R:

P = \rho RT + \left( B_0 RT - A_0 - \frac{C_0}{T^2} \right) \rho^2 + (b RT - a) \rho^3 + \frac{a \alpha}{T^2} \rho^6 + \frac{c \rho^3}{T^3} \left( 1 + \gamma \rho^2 \right) \exp\left( -\gamma \rho^2 \right)

This equation features eight empirical constants (A_0, B_0, C_0, a, b, c, \alpha, \gamma), which are substance-specific and determined by fitting to experimental P-V-T data; the initial terms provide a virial expansion for low densities, while the higher-order and exponential terms capture attractions, repulsions, and density-dependent interactions. The model excels in reproducing the second virial coefficient (related to pairwise interactions) and third virial coefficient (three-body effects), alongside long-range attractive forces, making it suitable for densities up to the critical point. For methane, the constants yield compressibility factors Z with deviations typically under 1% from experimental values at pressures up to 700 atm and temperatures from 100 K to 500 K.^[68]^[70] To address limitations in temperature extrapolation and accuracy for broader conditions, the modified BWR (MBWR) equation introduces additional polynomial terms in the coefficients (e.g., temperature-dependent expansions of a, b, and c) and up to 32 parameters in some formulations, improving fits over extended ranges—for instance, for helium from near the lambda point (2.17 K) to 1500 K and pressures to 2000 MPa. These extensions maintain the core structure while enhancing flexibility for cryogenic and supercritical applications. In the natural gas industry, the BWR and its variants are routinely employed to compute compressibility factors Z for mixtures at high pressures (up to 5000 psi), aiding in pipeline design, custody transfer, and reservoir simulations with average deviations below 0.5% for typical compositions dominated by methane.^[71]

General Multiparameter Equations

General multiparameter equations of state represent a class of empirical models designed for high-fidelity representation of thermodynamic properties of pure fluids and mixtures, typically incorporating 10 or more adjustable parameters obtained through nonlinear regression to extensive experimental pressure-volume-temperature (PVT) data. These equations prioritize accuracy across broad ranges of temperature, pressure, and density, often serving as reference standards for industrial fluids. For instance, the International Union of Pure and Applied Chemistry (IUPAC) endorses multiparameter formulations for refrigerants, such as those for difluoromethane (R-32) and pentafluoroethane (R-125), which ensure consistent property predictions in engineering applications. Prominent examples include the Lee-Kesler equation, developed in 1975, which applies to hydrocarbons and provides generalized correlations for saturation properties and compressibility factors using corresponding-states principles with acentric factor adjustments. Another key instance is the Span-Wagner equation for carbon dioxide, published in 1996, featuring a Helmholtz energy formulation that covers the fluid region from the triple point to 1100 K and pressures up to 800 MPa with uncertainties below 0.1% in density. Helmholtz-based equations integrated into the NIST REFPROP database exemplify modern implementations, offering multiparameter models for over 140 fluids, including refrigerants and hydrocarbons, with rigorous validation against experimental data.^[72]^[73] These equations commonly adopt a dimensionless Helmholtz free energy as the fundamental relation, expressed as a sum of ideal-gas and residual contributions in terms of reduced temperature \tau = T_c / T and reduced density \delta = \rho / \rho_c, incorporating polynomials, rational functions, and exponential terms to capture complex behaviors like near-critical anomalies. Auxiliary relations, such as polynomials for the ideal-gas isobaric heat capacity c_p^0(T), complement the core equation to derive all thermodynamic derivatives. The Benedict-Webb-Rubin equation serves as an early precursor to this approach.^[74] The primary advantages of general multiparameter equations lie in their exceptional accuracy—often achieving uncertainties under 0.5% for key properties over extended thermodynamic ranges—and their adoption in authoritative standards, such as those maintained by NIST for thermophysical property calculations in chemical engineering and refrigeration. These models enable precise simulations in processes involving phase equilibria and transport, though their complexity demands computational efficiency for practical use.^[75]

Specialized Equations of State

Stiffened Equation of State

The stiffened equation of state (EOS) is a thermodynamic model designed to describe the behavior of compressible liquids, particularly those exhibiting nearly incompressible properties under shock loading, by incorporating a constant reference pressure that accounts for intermolecular repulsive forces. This EOS modifies the ideal gas law to better capture the high-pressure response of fluids like water, where molecular repulsion prevents the pressure from dropping below a certain threshold even at low internal energies. The form of the stiffened EOS is given by

P = (\gamma - 1) \rho e - \gamma P_\infty,

where P is the pressure, \rho is the density, e is the specific internal energy, \gamma is the effective adiabatic index (analogous to the ratio of specific heats), and P_\infty is the stiffening pressure representing the repulsive contribution.^[76] This formulation ensures thermodynamic consistency and convexity, making it suitable for solving Riemann problems in fluid dynamics.^[76] The stiffened EOS was proposed by Menikoff and Plohr in their analysis of the Riemann problem for real materials, specifically to model liquids such as water that exhibit stiff behavior under dynamic compression.^[76] Unlike the ideal gas EOS, which underpredicts shock speeds in liquids due to neglecting repulsive effects, the stiffened form provides a more accurate representation by shifting the Hugoniot curve to higher pressures, improving predictions of shock propagation and wave interactions.^[77] Typical parameters for water are \gamma \approx 4.4 and P_\infty \approx 3 GPa, calibrated to match experimental shock data in the gigapascal range.^[78] This EOS finds primary applications in numerical hydrodynamics simulations involving shock waves, such as underwater explosions, where it models the compressible response of liquids during bubble dynamics and cavitation inception.^[78] It is also employed in simulations of explosive detonation products interacting with surrounding fluids and cavitation phenomena in high-speed flows, offering computational efficiency over more complex multiparameter models while enhancing accuracy in shock speed and pressure predictions compared to ideal gas assumptions.^[79]

Relativistic and Quantum Equations of State

In ultrarelativistic regimes, where particle speeds approach the speed of light and rest mass contributions are negligible, the equation of state for gases composed of massless or effectively massless particles, such as photons or neutrinos, takes the form P = \frac{1}{3} \rho c^2, where P is the pressure, \rho is the energy density, and c is the speed of light.^[80] This relation arises from the isotropic momentum distribution in relativistic kinetic theory, leading to an adiabatic index \gamma = 4/3, and is fundamental in describing radiation-dominated systems where energy density scales as \rho \propto T^4 for blackbody radiation.^[80] For ideal Bose gases, the equation of state extends classical thermodynamics to quantum statistics for bosons, incorporating Bose-Einstein integrals to account for quantum statistics beyond the classical limit, though interactions are typically treated perturbatively in mean-field approximations. The pressure is given by P = \frac{k_B T}{\lambda^3} g_{5/2}(z), where k_B is Boltzmann's constant, \lambda = \sqrt{\frac{2\pi \hbar^2}{m k_B T}} is the thermal de Broglie wavelength, z = e^{\mu / k_B T} is the fugacity with chemical potential \mu \leq 0, and g_{5/2}(z) = \sum_{l=1}^\infty \frac{z^l}{l^{5/2}} is the Bose function; below the condensation temperature, the pressure becomes independent of density due to macroscopic occupation of the ground state.^[81] This formulation captures deviations from the ideal gas law at low temperatures and high densities, with weak interactions modifying the chemical potential and leading to phenomena like superfluidity in extensions such as Bogoliubov theory.^[82] The Morse oscillator model provides an anharmonic description of vibrational contributions to the equation of state in diatomic molecules or solids, using the potential V(r) = D_e \left[1 - \exp\left(-a(r - r_e)\right)\right]^2 - D_e, where D_e is the dissociation energy, a controls the width, and r_e is the equilibrium bond length.^[83] This potential yields quantized vibrational energy levels E_v = \hbar \omega_e (v + 1/2) - \hbar \omega_e x_e (v + 1/2)^2, with v the quantum number, \omega_e the harmonic frequency, and x_e the anharmonicity constant, which influence the partition function and thus pressure-volume-temperature relations through thermal averaging of vibrational modes.^[83] In molecular gases or lattice models of solids, the resulting EOS incorporates these anharmonic effects to predict thermal expansion and compressibility more accurately than harmonic approximations, particularly at high temperatures where higher vibrational states are populated.

Jones-Wilkins-Lee Equation of State

The Jones-Wilkins-Lee (JWL) equation of state is an empirical model tailored for high-explosive detonation products, expressed as P = A \left(1 - \frac{\omega}{V}\right) e^{-R_1 V} + B \left(1 - \frac{\omega}{V}\right) e^{-R_2 V} + \frac{\omega E}{V}, where V is the relative specific volume (V = v / v_0), E is the specific internal energy, and A, B, R_1, R_2, \omega are material-specific parameters fitted to experimental data such as cylinder expansion tests.^[84] This form captures the rapid pressure decay post-detonation, combining unreacted explosive behavior at high densities with expansion-dominated products at lower densities, and is widely implemented in hydrocodes for simulating shock waves and fragmentations.^[84] These specialized equations find applications in extreme environments: the ultrarelativistic EOS is essential for modeling radiation pressure in core-collapse supernovae, where it influences shock propagation and neutrino transport in the proto-neutron star phase.^[85] The Bose EOS underpins interpretations of Bose-Einstein condensate experiments, enabling precise measurements of ultracold atomic interactions via time-of-flight expansions and trap releases.^[82] Meanwhile, the JWL EOS is standard for detonation modeling in munitions and mining, predicting blast overpressures and material damage in numerical simulations of explosive reactions.^[86] The Morse-based EOS aids in computational studies of molecular dynamics under pressure, such as in high-temperature chemistry or material science for vibrational thermodynamics.

Modern Developments

Advances in Computational Methods

Molecular simulations, particularly Monte Carlo (MC) and molecular dynamics (MD) methods, have become essential for deriving equations of state (EOS) by computing pressure-volume-temperature (PVT) data directly from intermolecular potentials. These techniques model fluid behavior at the atomic level, allowing validation and refinement of EOS parameters without relying solely on experimental data. For instance, MC simulations in the isobaric-isothermal ensemble can predict second-order thermodynamic derivatives like compressibility and heat capacities for simple fluids, providing benchmarks for classical EOS models. Similarly, MD simulations using software such as LAMMPS enable the computation of PVT surfaces for complex systems, including validation of SAFT-based EOS by comparing simulated phase diagrams with theoretical predictions for fluids like hydrocarbons under confinement.^[87]^[88]^[89] Recent advancements as of 2025 include the development of differentiable frameworks like DIMOS for end-to-end molecular simulations, enhancing efficiency in MD and MC for EOS derivation.^[90] Ab initio methods offer a quantum-mechanical foundation for EOS development, particularly for systems where classical approximations fail. Density functional theory (DFT) calculations determine the electronic structure and derive quantum EOS for solids and liquids under extreme conditions, such as multiphase transitions in metals like tin, yielding accurate pressure-density relations across phases. For incorporating nuclear quantum effects, such as zero-point motion in light elements, path-integral MD (PIMD) extends classical MD by representing particles as quantum paths, enabling precise computation of thermodynamic properties like the EOS for hydrogen at high densities. These approaches have revealed quantum corrections to the EOS, improving predictions for cryogenic fluids and high-pressure ices.^[91]^[92] Phase equilibrium calculations using EOS are facilitated by specialized simulation techniques to predict vapor-liquid equilibria (VLE). The Gibbs ensemble MC method simulates multi-phase systems by allowing particle swaps and volume exchanges between coexisting phases, solving for VLE compositions and pressures directly from molecular models, as demonstrated for binary refrigerant mixtures and CO2-H2O systems. Complementing this, flash calculations iteratively solve EOS to determine phase fractions and compositions at specified conditions, essential for compositional modeling in petroleum reservoirs where successive substitution or Newton's methods accelerate convergence for cubic EOS. These methods ensure robust VLE predictions, with Gibbs ensemble providing microscopic validation and flash algorithms enabling efficient engineering-scale applications.^[93]^[94]^[95] Process simulation software integrates EOS for practical implementation, supporting model-based design and optimization. Tools like Aspen Plus incorporate advanced EOS such as SAFT and Peng-Robinson for property predictions in flowsheets, handling phase equilibria and transport properties for chemical processes. gPROMS, with its equation-oriented modeling, allows custom EOS implementation and parameter estimation, facilitating dynamic simulations of reactive systems. These platforms often couple EOS with computational fluid dynamics (CFD) solvers to model multiphase flows, such as in reactors or pipelines, where thermodynamic consistency enhances accuracy for high-pressure applications.^[96]^[97]^[98] Recent trends through 2025 emphasize hybrid quantum-classical simulations to probe high-pressure phases inaccessible to classical methods alone, including masterclasses on computational EOS at conferences like the APS SCCM. These approaches combine quantum algorithms for correlated electron treatments with classical MD for nuclear dynamics, deriving EOS for materials like hydrogen under terapascal pressures and revealing phase boundaries in superconducting hydrides. For example, variational quantum eigensolvers integrated with DFT approximate ground-state energies, yielding EOS data for extreme conditions in planetary interiors. Such methods promise scalable accuracy for EOS refinement in geophysics and materials science.^[99]^[100]^[101]

Machine Learning and Data-Driven Equations

Machine learning (ML) and data-driven approaches have revolutionized the development of equations of state (EOS) by leveraging large datasets to model complex thermodynamic behaviors that traditional physics-based methods struggle to capture efficiently. These methods typically involve training models on pressure-volume-temperature (PVT) data, experimental measurements, or simulation outputs to predict EOS properties, often outperforming classical correlations in accuracy for intricate systems like mixtures or extreme conditions. Unlike parametric EOS, data-driven models can adapt to high-dimensional inputs, incorporating molecular structures or environmental variables directly into predictions. Neural networks form a cornerstone of ML-based EOS, particularly feedforward and recurrent architectures that fit PVT surfaces by minimizing prediction errors on empirical or simulated data. Graph neural networks (GNNs) extend this capability for molecular systems, representing molecules as graphs where nodes denote atoms and edges capture interactions, enabling scalable predictions of EOS parameters like compressibility factors for diverse fluids. For instance, GNNs have been applied to generate EOS for alkanes and refrigerants, achieving root-mean-square errors below 1% in density predictions across wide temperature ranges. Surrogate models, such as those built with deep neural networks, serve as efficient replacements for computationally intensive traditional EOS, accelerating evaluations in optimization tasks by orders of magnitude while maintaining fidelity to reference data. Post-2020 studies have demonstrated deep learning's efficacy in predicting parameters for the Statistical Associating Fluid Theory (SAFT), a semi-empirical EOS framework, by training convolutional neural networks on molecular simulation datasets to infer association strengths and chain lengths. These models significantly reduce parameter estimation time and improve phase diagram accuracy for associating fluids like alcohols compared to manual fitting. Gaussian processes (GPs), a probabilistic ML technique, have been employed to quantify uncertainties in EOS for supercritical fluids, providing confidence intervals for density and viscosity predictions under high-pressure conditions relevant to carbon capture applications. GPs excel in sparse data regimes, interpolating EOS behaviors with epistemic uncertainty estimates that highlight regions needing further experimentation. Data-driven EOS offer distinct advantages, including superior handling of non-ideal mixtures and phase transitions through implicit learning of intermolecular potentials from data, without relying on rigid functional forms. They also enable faster computations than ab initio methods for large datasets, facilitating real-time applications in process design. In plasma physics, ML models trained on experimental data from facilities like the National Ignition Facility have derived EOS for inertial confinement fusion, capturing shock compression responses with errors under 5% in Hugoniot curves. In astrophysics, neural networks infer neutron star EOS from gravitational wave observations and multi-messenger data, constraining pressure-radius relations and enabling predictions of merger remnants with improved precision over semi-analytic models, as seen in recent 2024-2025 studies. Computational simulations occasionally provide the training data for these ML surrogates, bridging gaps in experimental accessibility.^[102]^[103] Despite these advances, challenges persist in ML-driven EOS, particularly regarding interpretability, where black-box models obscure the physical mechanisms underlying predictions, complicating validation against thermodynamic principles. Extrapolation beyond training data domains remains problematic, as models may fail in uncharted regimes like extreme densities, leading to unreliable forecasts without robust regularization techniques. In the context of 2025, ethical considerations around data sourcing have gained prominence, emphasizing the need for diverse, unbiased datasets to avoid perpetuating biases in global thermodynamic modeling efforts. Ongoing research focuses on hybrid physics-informed neural networks to mitigate these issues by embedding conservation laws into loss functions.

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[80]
[PDF] Review of Jones-Wilkins-Lee equation of state
The Jones-Wilkins-Lee [1] (JWL) equation of state (EOS) is widely used in reactive hydro simulations to describe thermodynamics of both detonation products (DP) ...
[81]
[nucl-th/9806095] Relativistic Equation of State of Nuclear Matter for ...
Jun 29, 1998 · Relativistic Equation of State of Nuclear Matter for Supernova Explosion. We construct the equation of state (EOS) of nuclear matter at finite ...
[82]
[PDF] Jones-Wilkins-Lee (JWL) Reaction Products Equations of State for ...
Jul 15, 2019 · The JWL equation of state (EOS) calculates the Chapman-Jouguet state of detonation waves and reaction product expansion, and is used to model ...
[83]
Prediction of thermodynamic derivative properties of fluids by Monte ...
Aug 10, 2025 · We compute second order derivatives of the Gibbs energy by Monte Carlo simulation in the isobaric–isothermal ensemble for fluids made of ...
[84]
LAMMPS MD: Equation of State (pressure vs. density) - SPC/E Water
Jul 6, 2017 · We first present output from LAMMPS, and compare it to Monte-Carlo derived results, and then provide a tutorial on how to reproduce that same ...Missing: PVT | Show results with:PVT
[85]
Extension of SAFT equation of state for fluids confined in nano-pores ...
Feb 15, 2022 · The Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [40] was used to run all molecular dynamic simulations. For calcite ...Extension Of Saft Equation... · 2. Theoretical Background · 3. Results And Discussion
[86]
Multiphase tin equation of state using density functional theory
May 7, 2021 · Abstract. We perform density functional theory (DFT) calculations of five solid phases and the liquid phase of tin.
[87]
An analysis of quantum effects on the thermodynamic properties of ...
Apr 4, 2014 · A. Path integral molecular dynamics method. To establish an EOS, the pressure and internal energy of hydrogen are calculated using PIMD method.
[88]
Gibbs Ensemble Monte Carlo simulation of binary vapor-liquid ...
The Gibbs Ensemble Monte Carlo (GEMC) simulation method has been used for vapor-liquid equilibrium calculations of the binary systems R22-R142b and R22-R152a, ...
[89]
Gibbs Ensemble Monte Carlo for Reactive Force Fields to Determine ...
Dec 22, 2020 · Gibbs Ensemble Monte Carlo for Reactive Force Fields to Determine the Vapor–Liquid Equilibrium of CO2 and H2O | Journal of Chemical Theory and ...
[90]
An algorithm for flash calculations using an equation of state
Because of its inherent advantages, the equation-of-state (EOS) approach is preferred to the mixed-model approach for VLE calculations at high pressures.
[91]
Use of the SAFT equation of state in process modelling - Cape-Open
This will make a sophisticated implementation of SAFT technology directly available within standard process modelling tools such as Aspen Plus and gPROMS.
[92]
Thermodynamic Modeling with Equations of State: Present ...
The most accurate EoS available, multiparameter EoS, have a second artificial Maxwell loop in the two-phase region that gives problems in phase-equilibrium ...Introduction · Structure of the... · Present Challenges with... · Conclusion
[93]
Efficient implementation of complex equations of state in a high ...
May 15, 2024 · The coupling of an accurate Computational Fluid Dynamics (CFD) solver with sophisticated thermodynamic models has been investigated, mainly in ...
[94]
Considerations for evaluating thermodynamic properties with hybrid ...
Jan 25, 2022 · This hybrid quantum-classical algorithm aims at finding ground-state solutions of molecular systems based on the variational principle.
[95]
Quantum symmetrization transition in superconducting sulfur ...
Mar 15, 2024 · Quantum symmetrization transition in superconducting sulfur hydride from quantum Monte Carlo and path integral molecular dynamics. Romain ...Results · Quantum 3d Model · Methods<|control11|><|separator|>