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Fugacity

In thermodynamics, fugacity is a real-valued thermodynamic property that quantifies the "escaping tendency" or effective pressure exerted by a substance in a non-ideal phase, serving as a correction to the mechanical pressure for accurate calculations of chemical equilibria and phase behaviors.^[1] For an ideal gas, fugacity equals the partial pressure, but in real systems, it accounts for intermolecular interactions and deviations from ideality.^[2] Introduced by American chemist Gilbert N. Lewis in his 1901 paper "The Law of Physico-Chemical Change," the concept was developed to simplify the application of thermodynamic principles to real gases, liquids, and solutions by providing an auxiliary function analogous to pressure in ideal cases.^[3] The fugacity f of a pure substance is formally defined through its relation to the chemical potential \mu, such that at constant temperature, d\mu = RT \, d\ln f, where R is the gas constant and T is the temperature; integrating this yields \mu(T, P) = \mu^\circ(T) + RT \ln(f/f^\circ), with f^\circ as a reference fugacity (often 1 bar).^[1] In mixtures, the partial fugacity of component i, f_i, is given by f_i = y_i \hat{\phi}_i P, where y_i is the mole fraction, P is the total pressure, and \hat{\phi}_i is the fugacity coefficient, which encapsulates non-ideality effects and approaches 1 for ideal mixtures.^[4] At phase equilibrium, the fugacities of each component must be equal across phases, enabling precise predictions of vapor-liquid equilibria, solubility, and reaction extents in industrial processes like petroleum refining and chemical manufacturing.^[5] Fugacity coefficients are typically calculated using equations of state, such as the Peng-Robinson or Soave-Redlich-Kwong models, which integrate over pressure to determine deviations from ideality; for example, \ln \phi = \int_0^P (Z-1) \frac{[dP](/page/DP)}{P} at constant temperature, where Z is the compressibility factor.^[6] This framework has become foundational in chemical engineering and geochemistry, influencing models for natural gas processing, supercritical fluid extraction, and geochemical speciation, with ongoing refinements to handle high-pressure and extreme conditions.^[2]

Fundamentals

Definition

Fugacity is a thermodynamic property of a substance that corrects for deviations from ideal behavior in real systems, particularly gases, by serving as an effective partial pressure in equilibrium calculations. Introduced by Gilbert N. Lewis in his 1901 paper on physico-chemical change, fugacity (f) is defined as the pressure that a real substance would exert if it were an ideal gas at the same temperature and chemical potential as the actual substance. This concept extends the utility of ideal gas laws to non-ideal conditions, where intermolecular interactions alter the relationship between pressure and thermodynamic properties.^[3] For an ideal gas, fugacity equals the mechanical pressure, expressed mathematically as f = P. In real gases, however, f ≠ P due to two primary effects: the finite volume of molecules, which reduces the effective space available for movement, and attractive intermolecular forces, which lower the pressure compared to an ideal case. These deviations become significant at high pressures or low temperatures, where the ideal gas assumption fails.^[2]^[1] Fugacity carries units of pressure, such as bar or atm, aligning with its role as a pressure-like quantity. It conceptually represents the "escaping tendency" of a substance—the propensity to transfer between phases—much like vapor pressure indicates the tendency of a liquid to evaporate at equilibrium. For reference in thermodynamic tables and calculations, the standard-state fugacity (f°) is conventionally set to 1 bar, corresponding to the hypothetical ideal-gas state at that pressure.%20Fall%202015%20WORD%202007.pdf)^[7]

Relation to Chemical Potential

The chemical potential \mu of a pure substance at temperature T and fugacity f is given by the relation \mu = \mu^\circ(T) + RT \ln(f / f^\circ), where \mu^\circ(T) is the standard chemical potential at the standard fugacity f^\circ, typically taken as 1 bar, and R is the gas constant. This expression extends the ideal-gas form \mu = \mu^\circ(T) + RT \ln(P / P^\circ) to non-ideal systems by replacing pressure P with fugacity f, ensuring the logarithmic term captures deviations from ideality while preserving the thermodynamic consistency of chemical potential changes.^[3] This relation derives from the Gibbs-Duhem equation for a pure substance, which at constant temperature states d\mu = V_m \, dP, where V_m is the molar volume. For an ideal gas, V_m = [RT](/page/RT) / P, so d\mu = [RT](/page/RT) \, d \ln P. To generalize this for real gases and other phases, fugacity is defined such that d\mu = [RT](/page/RT) \, d \ln f, leading directly to the differential form d \ln f = (V_m / [RT](/page/RT)) \, dP.^[3] This equation highlights fugacity as a pressure-like variable that accounts for non-ideal volume behavior in the pressure dependence of chemical potential. For isothermal processes, integrating the differential equation from state 1 to state 2 yields the relation \ln(f_2 / f_1) = \int_{P_1}^{P_2} (V_m / RT) \, dP. This integrated form allows computation of fugacity changes along an isotherm, provided the equation of state for V_m(P) is known, and it forms the basis for evaluating equilibrium constants in terms of fugacities rather than pressures. Fugacity ensures that the activity a = f / f^\circ serves as a dimensionless measure of the effective concentration or "escaping tendency" in non-ideal systems, generalizing Raoult's law for solvents (where a \approx x for ideal solutions) and Henry's law for solutes (where a \approx k x, with k incorporating solubility deviations). By linking activity directly to chemical potential via \mu = \mu^\circ + RT \ln a, fugacity provides a unified framework for phase equilibria and reaction thermodynamics across ideal and non-ideal conditions. Unlike partial pressure, which assumes ideality and uses P_i = x_i P for mixtures, fugacity incorporates intermolecular interactions through the fugacity coefficient \phi = f / P (or \phi_i = f_i / P_i for components), where \phi = 1 for ideal gases but deviates otherwise to correct for real-gas effects like compressibility and attractions. This distinction is crucial for accurate predictions in high-pressure or dense-phase systems, where partial pressure alone would overestimate or underestimate chemical potentials.

Pure Substances

Gases

For pure gaseous substances, the fugacity f serves as a correction to the pressure P to account for non-ideal behavior, defined through the fugacity coefficient \phi as f = \phi P. This coefficient \phi equals 1 for ideal gases, where the compressibility factor Z = PV/RT = 1, but deviates from unity for real gases due to intermolecular forces and finite molecular volume, with \phi directly related to Z via thermodynamic integration. The fugacity thus provides a measure of the gas's effective pressure that aligns the chemical potential of real gases with that of an ideal gas at the same temperature and fugacity.^[8] The fugacity coefficient for pure gases is typically calculated from equations of state (EOS) that model real gas behavior. A general expression is given by

\ln [\phi](/page/Phi) = \int_0^P (Z - 1) \, d \ln P,

which integrates the departure from ideality along an isotherm. For the van der Waals EOS, \left(P + \frac{a}{V_m^2}\right)(V_m - b) = [RT](/page/RT), the fugacity coefficient takes the form

[RT](/page/RT) \ln [\phi](/page/Phi) = \frac{[RT](/page/RT) b}{V_m - b} - \frac{2a}{V_m} - [RT](/page/RT) \ln \left(1 - \frac{a (V_m - b)}{[RT](/page/RT) V_m^2}\right),

derived from the EOS parameters a and b. Virial expansions, useful at low to moderate pressures, express \ln [\phi](/page/Phi) \approx \frac{B(T) P}{[RT](/page/RT)} + \frac{(C(T) - B(T)^2) P^2}{2 ([RT](/page/RT))^2} + \cdots, where B(T) and C(T) are second and third virial coefficients obtained from experimental data or statistical mechanics. These methods, originating from early EOS developments like the Redlich-Kwong equation, enable accurate predictions across a range of conditions.^[9] At low pressures, real gases approach ideal behavior, so \phi \approx 1 and f \approx P, as intermolecular interactions are negligible. However, at high pressures, \phi often decreases below 1 for gases dominated by attractive forces, reflecting reduced effective pressure due to molecular clustering; repulsive effects can make \phi > 1 near the critical point. For instance, carbon dioxide at 300 K and 100 bar has \phi \approx 0.85, yielding f \approx 85 bar, illustrating significant non-ideality even at moderate supercritical pressures.^[10]^[11]

Condensed Phases

In condensed phases, such as pure liquids and solids, the fugacity is determined primarily by reference to the saturation or equilibrium conditions with the vapor phase, owing to their low compressibility compared to gases. For a pure liquid at temperature T and pressure P, the fugacity f is approximated as f \approx f_{\text{sat}} \exp\left[ \frac{(P - P_{\text{sat}}) V_m}{RT} \right], where f_{\text{sat}} is the fugacity at the saturation pressure P_{\text{sat}}, V_m is the molar volume of the liquid, R is the gas constant, and T is the absolute temperature.^[12] This expression arises from the integration of the Gibbs-Duhem equation, assuming constant molar volume due to near-incompressibility.^[12] The exponential term is known as the Poynting correction factor, which accounts for the effect of pressure deviations from saturation on the liquid's chemical potential; it is typically close to unity because V_m for liquids is small (on the order of 10^{-5} m³/mol), making the correction negligible at moderate pressures below several hundred bars.^[12] At saturation conditions, the fugacity of the liquid equals that of the vapor in equilibrium, f_{\text{liq}} = f_{\text{vap}}, ensuring thermodynamic consistency across phases.^[12] For low saturation pressures, f_{\text{sat}} is often approximated as P_{\text{sat}}, the vapor pressure, since the vapor fugacity coefficient is near 1.^[12] A representative example is liquid water at 25°C and 1 bar. The saturation pressure (vapor pressure) is 0.0317 bar, so f_{\text{sat}} \approx 0.0317 bar; the Poynting correction is minimal (approximately 1.0001), yielding a fugacity of about 0.0317 bar.^[12]^[13] For pure solids, the treatment is analogous to that for liquids, but the effect of pressure is even smaller due to lower compressibility (V_m and isothermal compressibility are typically an order of magnitude less than for liquids).^[12] The fugacity is commonly referenced to the triple point, where the solid, liquid, and vapor phases coexist in equilibrium, providing a standard state for calculations; the same exponential form applies, with f_{\text{sat}} taken as the sublimation fugacity or triple-point vapor pressure.^[12] For instance, in ice (solid water), the Poynting correction remains small even at pressures up to 20 MPa, with first-order approximations accurate to within 0.7 × 10^{-6} relative error near 0°C.^[12]

Mixtures

Gaseous Mixtures

In gaseous mixtures, the partial fugacity of component i, denoted f_i, is given by the expression f_i = y_i \phi_i [P](/page/P′′), where y_i is the mole fraction of i in the gas phase, \phi_i is the fugacity coefficient specific to component i in the mixture, and P is the total pressure of the mixture. This formulation extends the concept of fugacity from pure gases to multi-component systems, where \phi_i corrects for non-ideal behavior arising from molecular interactions. For ideal gas mixtures, \phi_i = 1, reducing f_i to the partial pressure y_i P; however, in real mixtures, \phi_i deviates from unity, particularly at elevated pressures or with dissimilar components. The Lewis-Randall rule offers a practical approximation for partial fugacity in gaseous mixtures exhibiting low non-ideality, positing that \phi_i \approx \phi_i^\pure, where \phi_i^\pure is the fugacity coefficient of pure component i at the same temperature and pressure.^[14] This rule implies f_i \approx y_i f_i^\pure (T, P), simplifying calculations by leveraging pure-component data while assuming minimal influence from composition on the correction factor. It holds reasonably well for dilute mixtures or near-ideal conditions but requires refinement for stronger deviations.^[14] For accurate determination in non-ideal gaseous mixtures, fugacity coefficients are computed using equations of state (EOS) that incorporate mixing rules to capture cross-interactions between components, such as the Peng-Robinson EOS. In this approach, \phi_i = (\phi_i^\pure / y_i) \times \exp[\text{terms for cross-interactions}], where the exponential accounts for deviations due to unlike-pair parameters in the EOS attractive and repulsive terms. These terms arise from the partial derivative of the EOS residual Gibbs energy with respect to composition, ensuring \phi_i reflects the mixture's thermodynamic state beyond pure-component values. The Peng-Robinson EOS, with its van der Waals mixing rules, is widely adopted for hydrocarbon and light gas mixtures due to its balance of accuracy and computational efficiency. A representative example is nitrogen in air, a binary mixture of approximately 78% N_2 and 21% O_2 by mole fraction, at high pressures where ideality fails. Here, the partial fugacity of nitrogen is f_{\ce{N2}} = 0.78 \phi_{\ce{N2}} P, with \phi_{\ce{N2}} \neq 1 due to compressibility effects; for instance, at 50 bar and 298 K, \phi_{\ce{N2}} is approximately 0.964, yielding f_{\ce{N2}} \approx 37.6 bar for P = 50 bar.^[15] Deviations from ideality in gaseous mixtures are often overlooked by approximations like Amagat's law, which assumes additive compressibility factors (Z = \sum y_i Z_i) and neglects interaction effects, leading to inaccurate chemical potentials \mu_i. In contrast, fugacity corrections via EOS-derived \phi_i provide precise \mu_i = \mu_i^\circ (T) + RT \ln (f_i / P^\circ), essential for thermodynamic modeling of real systems.^[16]

Liquid Mixtures

In non-ideal liquid mixtures, the partial fugacity of component i, denoted f_i, is expressed as f_i = x_i \gamma_i f_i^\text{pure}, where x_i is the liquid-phase mole fraction of i, \gamma_i is the activity coefficient (with \gamma_i = 1 for ideal solutions), and f_i^\text{pure} is the fugacity of pure liquid i at the same temperature and pressure. This formulation extends the concept of fugacity from pure substances to mixtures by incorporating deviations from ideality through \gamma_i, which quantifies intermolecular interactions such as hydrogen bonding or dispersion forces that alter the chemical potential. The pure liquid fugacity f_i^\text{pure} serves as the reference state, typically close to the saturation vapor pressure for low pressures but corrected for compressibility effects at higher pressures. Activity coefficients \gamma_i are derived from models of the excess Gibbs energy of mixing G^E, related by the equation

\frac{G^E}{RT} = \sum_i x_i \ln \gamma_i,

where R is the gas constant and T is temperature. Seminal models include the van Laar equation, originally proposed for binary systems to capture asymmetric non-idealities leading to azeotropes; the Wilson equation, which assumes local volume fractions and excels for systems without liquid-liquid immiscibility; and the non-random two-liquid (NRTL) equation, accounting for local composition variations due to differing interaction energies between molecular pairs. These models are parameterized using binary interaction coefficients fitted to vapor-liquid equilibrium (VLE) or calorimetric data, enabling prediction of \gamma_i across compositions. For instance, the Wilson model parameters for ethanol-water indicate \gamma_\text{ethanol} > 1 over much of the composition range, reflecting positive deviations from ideality. At infinite dilution, where x_i \to 0, the activity coefficient approaches \gamma_i^\infty, and the fugacity relates to Henry's law constant H_i via H_i = \lim_{x_i \to 0} (\gamma_i f_i^\text{pure}), describing the proportionality between the solute's partial pressure and its mole fraction in the dilute limit. This limit is crucial for sparingly soluble gases or solutes, as \gamma_i^\infty quantifies the solvation free energy deviation from ideality. In the ethanol-water system at 1 atm and 25°C, hydrogen bonding between ethanol and water molecules disrupts the pure water structure, resulting in \gamma_\text{ethanol} \approx 3.3 near infinite dilution and elevating the ethanol fugacity beyond the ideal Raoult's law prediction.^[17] For consistency in phase equilibria, the fugacity equality f_i^\text{liq} = f_i^\text{vap} holds at VLE for each component i, linking liquid-phase activities to vapor-phase fugacities; under low-pressure assumptions where the vapor behaves ideally, the total pressure satisfies \sum_i y_i P = P_\text{total}, with y_i as vapor mole fractions. This criterion ensures thermodynamic equilibrium, allowing VLE calculations by equating the modified Raoult's law expression from the liquid to the ideal gas form in the vapor.

Thermodynamic Dependencies

Temperature Effects

The temperature dependence of fugacity at constant pressure is described by a thermodynamic relation derived from the Gibbs-Helmholtz equation applied to the chemical potential, yielding

\left( \frac{\partial \ln f}{\partial T} \right)_P = -\frac{\bar{H} - \bar{H}^\circ}{RT^2},

where f is the fugacity, \bar{H} is the partial molar enthalpy of the species, \bar{H}^\circ is its value in the standard state, R is the gas constant, and T is the absolute temperature. This equation indicates that the logarithmic rate of change of fugacity with temperature is inversely proportional to T^2 and proportional to the enthalpy departure from the standard state, providing a direct link between thermal effects on fugacity and enthalpic contributions.^[18] For pure gases, the fugacity coefficient \phi = f/P (where P is pressure) generally decreases toward 1 as temperature rises at fixed pressure, signifying a reduction in deviations from ideal gas behavior due to weakened intermolecular forces at higher thermal energies. This behavior is primarily governed by the temperature dependence of the virial coefficients in the virial equation of state, with the second virial coefficient B(T) typically decreasing in magnitude—often following forms like B(T) \propto T^{-n} for n > 1—which diminishes the non-ideality corrections needed for fugacity. Consequently, at elevated temperatures, real gases more closely approximate the ideal case where f \approx P.^[19] In liquids and solids (condensed phases), the saturation fugacity f^\text{sat} exhibits a strong exponential increase with temperature, driven by the Clausius-Clapeyron relation for the saturation vapor pressure P^\text{sat}:

\frac{d \ln P^\text{sat}}{dT} = \frac{\Delta H^\text{vap}}{RT^2},

where \Delta H^\text{vap} is the enthalpy of vaporization. For liquids, f^\text{sat} \approx P^\text{sat}(T) \times \exp\left( \int_{P^\text{sat}}^P \frac{V_m}{RT} dP' \right), with the exponential term being the Poynting correction that modestly amplifies fugacity above P^\text{sat} at pressures exceeding the saturation value; however, this correction is near unity for many practical conditions below 10 bar. This temperature-driven rise in f^\text{sat} reflects the enhanced volatility of the condensed phase as thermal energy overcomes intermolecular attractions.^[20] For mixtures, temperature influences fugacity through its impact on activity coefficients \gamma_i, which modify the ideal-solution reference fugacity as f_i = \gamma_i x_i f_i^\text{pure}, where x_i is the mole fraction. Excess Gibbs energy models like the Non-Random Two-Liquid (NRTL) equation incorporate temperature-dependent parameters, such as \tau_{ij} = a_{ij} + b_{ij}/T, enabling \gamma_i to capture how molecular interactions evolve with temperature—typically showing decreased non-ideality ( \gamma_i \to 1 ) at higher temperatures for many systems due to increased kinetic energy disrupting associations. This temperature sensitivity is crucial for predicting phase behavior in processes like distillation, where equilibrium constants depend on fugacity ratios. A representative example is benzene vapor at saturation conditions: its fugacity rises from about 0.1 bar at 20°C (where P^\text{sat} \approx 0.1 bar and \phi \approx 1) to 1 bar at 80°C (its normal boiling point, where P^\text{sat} = 1 bar). This increase underscores the exponential temperature effect on volatility for organic liquids.^[21]

Pressure Effects

The isothermal pressure dependence of fugacity for a pure substance arises from the Gibbs-Duhem relation and the definition linking fugacity to chemical potential, yielding

\left( \frac{\partial \ln f}{\partial P} \right)_T = \frac{V_m}{RT},

where V_m is the molar volume, R is the gas constant, and T is temperature. Integrating this expression from a reference pressure P^\circ (typically where f = P^\circ) gives

\ln \left( \frac{f}{P^\circ} \right) = \int_{P^\circ}^{P} \frac{V_m}{RT} \, dP.

This integration accounts for non-ideal volume behavior and requires an equation of state to evaluate V_m(P). For gases, pressure effects become pronounced at elevated levels, where deviations from ideality lead to significant changes in fugacity relative to pressure; the fugacity coefficient \phi = f/P is often determined from compressibility factor charts or cubic equations of state like Peng-Robinson, showing \phi < 1 in regions of attractive intermolecular forces dominance.^[19] For instance, in regions where Z < 1, the effective pressure is reduced. In condensed phases, such as liquids and solids, the small and nearly incompressible molar volume results in V_m / RT \approx 10^{-3} bar^{-1}, causing fugacity to vary little with pressure. The correction is captured by the Poynting factor, approximated as $1 + (P - P^\circ) V_m / RT when applied to the fugacity at saturation pressure P^\circ. For mixtures, pressure influences the component fugacity coefficients \phi_i in the vapor phase or activity coefficients \gamma_i in the liquid phase through changes in non-ideal interactions, though these effects are generally minor compared to composition or temperature impacts on overall phase behavior.

Practical Applications

Calculation and Measurement

Fugacity for pure substances and mixtures can be determined theoretically using equations of state (EOS) or activity coefficient models, which provide the fugacity coefficient \phi_i for gases and the activity coefficient \gamma_i for liquids. For gaseous phases, the Soave-Redlich-Kwong (SRK) EOS is widely applied to compute \phi_i, where the fugacity is given by f_i = \phi_i y_i P, with the EOS parameters adjusted to fit vapor pressure data for improved accuracy in non-ideal conditions.^[22] In liquid phases, the UNIFAC group-contribution method estimates \gamma_i to calculate fugacity as f_i = \gamma_i x_i f_i^\circ, where f_i^\circ is the fugacity of the pure liquid; this approach relies on molecular functional groups to predict interactions without needing mixture-specific data.^[23] Experimentally, fugacity coefficients are often derived directly from pressure-volume-temperature (PVT) data for gases, using the relation \ln \phi = \int_0^P (Z - 1) \, d \ln P, where Z is the compressibility factor obtained from measurements; this integral accounts for deviations from ideality by integrating along an isotherm.^[24] Indirect methods involve vapor-liquid equilibrium (VLE) data, where equality of fugacities between phases (f_i^V = f_i^L) allows back-calculation of \phi_i or \gamma_i from measured compositions and pressures.^[25] Solubility measurements, such as in Henry's law regimes, provide another indirect route, relating gas solubility to fugacity via x_i = f_i / H_i, where H_i is the Henry's law constant defined such that f_i = H_i x_i at infinite dilution and fitted from experiments.^[26] Commercial software facilitates practical fugacity computations in engineering contexts. Aspen Plus employs EOS like SRK or Peng-Robinson for vapor-phase \phi_i and activity models like UNIFAC for liquid \gamma_i, integrating these into process simulations for flash calculations and phase equilibria. Similarly, NIST's REFPROP database uses Helmholtz energy-based EOS to evaluate fugacity for pure fluids and mixtures up to 20 components, supporting high-precision queries via subroutines or wrappers.^[27] Cubic EOS like SRK perform well in fugacity coefficients near ideal gas behavior but exhibit larger deviations at supercritical states due to challenges in capturing dense-phase interactions. REFPROP achieves uncertainties of 0.1–0.5% in derived properties like fugacity for well-characterized fluids under moderate conditions, though higher errors occur in complex mixtures.^[28] Post-2000 advancements incorporate quantum chemistry to refine virial coefficients, enhancing fugacity predictions beyond classical EOS. Path-integral Monte Carlo simulations on ab initio potentials compute quantum-corrected second virial coefficients with uncertainties under 8% across wide temperatures (10–2000 K), improving accuracy for light gases like H₂ in non-ideal regimes.^[29] These methods bridge molecular-scale interactions to macroscopic thermodynamics, outperforming empirical virial expansions in quantum-dominated systems.^[30]

Uses in Phase Equilibria and Processes

In vapor-liquid equilibrium (VLE), the equality of component fugacities between the vapor and liquid phases serves as the fundamental criterion for phase stability, enabling accurate predictions of bubble and dew points in multicomponent systems. For a component i, this condition is expressed as f_i^V = f_i^L, where f_i^V = y_i \phi_i P and f_i^L = x_i \gamma_i f_i^\circ, with \phi_i as the vapor-phase fugacity coefficient, \gamma_i as the liquid-phase activity coefficient, y_i and x_i as mole fractions, P as total pressure, and f_i^\circ as the fugacity of pure liquid i at the system temperature. This equality underpins bubble point calculations, where the temperature or pressure is determined such that the sum of vapor mole fractions equals unity (\sum y_i = 1), and dew point calculations, where the sum of liquid mole fractions equals unity (\sum x_i = 1). The equilibrium ratio, or K-value, K_i = y_i / x_i = \gamma_i f_i^\circ / (\phi_i P), facilitates iterative solutions for these points in non-ideal mixtures, as applied in process simulations for hydrocarbon separations.^[31] Fugacity extends to chemical reaction equilibria by replacing partial pressures in the equilibrium constant to account for non-ideality, yielding K_f = \prod (f_i)^{\nu_i}, where \nu_i are stoichiometric coefficients and f_i are fugacities of species i. The standard Gibbs free energy change relates to this constant via \Delta G^\circ = -RT \ln K_f, providing a thermodynamically consistent framework for reactions involving real gases or mixtures at elevated pressures. This approach is essential for predicting extents of reaction in high-pressure systems, such as ammonia synthesis, where deviations from ideality significantly alter yields. In industrial processes, fugacity drives phase split models for distillation, where iso-fugacity conditions define tray efficiencies and column designs in petroleum refining, ensuring precise separation of close-boiling components like benzene and toluene. For gas processing, such as liquefied natural gas (LNG) production, fugacity coefficients from equations of state inform flash calculations to optimize methane recovery while minimizing energy for refrigeration cycles. In CO2 capture, cryogenic methods leverage solid-vapor fugacity equality to model anti-sublimation processes, enhancing efficiency in post-combustion flue gas treatment by predicting CO2 solidification points under varying pressures.^[32]^[33] Electrochemical systems incorporate fugacity into the Nernst equation for precise cell potential calculations, modifying the standard form to E = E^\circ - \frac{RT}{nF} \ln \left( \prod (f_i)^{\nu_i} \right), where fugacities replace concentrations or partial pressures to correct for non-ideal gas behavior in electrodes involving H2 or O2. This adjustment is critical in fuel cells and electrolyzers operating at high pressures, improving accuracy in predicting reversible potentials and overpotentials.^[34] Recent applications include climate modeling, where ocean surface CO2 fugacity (f_{\ce{CO2}}) quantifies air-sea fluxes of greenhouse gases, with global maps derived from statistical models showing an annual uptake of approximately 2.5 Pg C influenced by temperature and salinity variations. In supercritical extraction processes, solute fugacity equality between solid and fluid phases governs solubility predictions, as in CO2-based decaffeination of coffee, where density-dependent fugacity coefficients from equations of state optimize extraction yields at 10-30 MPa.^[35]^[36]

Historical Development

Origins and Early Concepts

The concept of fugacity was first introduced by Gilbert N. Lewis in 1901 through his seminal paper "The Law of Physico-Chemical Change," published in the Proceedings of the American Academy of Arts and Sciences. In this work, Lewis defined fugacity as an effective pressure that generalizes the notion of partial pressure, extending its applicability beyond ideal gases to non-ideal solutions and mixtures where traditional pressure measures fail to accurately capture thermodynamic behavior.^[3] This introduction marked a key step in refining thermodynamic descriptions for real systems, emphasizing fugacity's role as a measure of a substance's "escaping tendency" from one phase to another.^[37] Lewis's motivation stemmed from the recognized shortcomings of the ideal gas law, which assumes negligible intermolecular forces and zero molecular volume—assumptions that break down in high-pressure environments and complex mixtures, leading to inaccurate predictions of chemical equilibria and phase behavior.^[37] By proposing fugacity, Lewis sought to create an auxiliary thermodynamic function that aligns more closely with experimental observations of real gases and solutions, thereby enabling more precise calculations of equilibrium constants and reaction tendencies under non-ideal conditions.^[37] This approach addressed a critical gap in early 20th-century thermodynamics, where empirical corrections were needed for industrial processes involving compressed gases. The development of fugacity built on foundational prior efforts to model real gas deviations, notably Johannes Diderik van der Waals' 1873 equation of state, which incorporated corrections for attractive forces (a term) and excluded volume (b term) to modify the ideal gas law into (P + \frac{a}{V_m^2})(V_m - b) = RT, where P is pressure, V_m is molar volume, R is the gas constant, T is temperature, and a and b are substance-specific constants.^[38] However, van der Waals' framework, while pioneering in accounting for non-ideality, did not introduce a dedicated fugacity term or integrate it directly into chemical potential expressions, leaving room for Lewis's more thermodynamically rigorous generalization.^[39] This reception was further solidified in Lewis's 1923 textbook Thermodynamics and the Free Energy of Chemical Substances, co-authored with Merle Randall, which formalized the relationship between fugacity f and chemical potential \mu for gases as \mu = \mu^\circ + RT \ln f, where \mu^\circ is the standard chemical potential, R is the gas constant, and T is temperature—providing a cornerstone for subsequent thermodynamic applications.^[40]

Key Advancements and Contributors

In the 1920s, Merle Randall, in collaboration with Gilbert N. Lewis, extended the fugacity concept—originally defined for gases—to condensed phases by incorporating activity coefficients, enabling the treatment of non-ideal solutions where fugacity equals activity times the standard-state fugacity of the pure substance. This advancement allowed for consistent thermodynamic calculations across phases, linking vapor pressures to solution behaviors in liquids and solids. During the 1930s, E. A. Guggenheim established a rigorous statistical mechanics foundation for activity coefficients in mixtures, deriving expressions from quasi-chemical approximations that accounted for molecular interactions in non-ideal solutions. His work emphasized the role of configurational statistics in determining departure functions from ideality, providing a theoretical basis for mixture properties beyond empirical fits. Following World War II, the Benedict-Webb-Rubin equation of state, introduced in the 1940s, was integrated into fugacity computations to yield precise fugacity coefficients for real gases, particularly light hydrocarbons and their mixtures, by capturing volumetric nonlinearities near critical points. This empirical yet thermodynamically consistent model improved accuracy in high-pressure gas-phase predictions, influencing industrial applications like natural gas processing.^[41] From the 1960s to the 1980s, advancements in liquid-phase fugacity modeling emerged through activity coefficient frameworks, notably G. M. Wilson's 1964 equation, which expressed excess Gibbs energy via local composition effects to compute non-ideal liquid fugacities effectively. Building on this, the UNIQUAC model by D. S. Abrams and J. M. Prausnitz in 1975 combined combinatorial and residual contributions from statistical thermodynamics, enhancing fugacity predictions for multicomponent liquid mixtures with diverse molecular sizes and interactions. In the 2000s onward, computational methods like grand canonical Monte Carlo simulations have advanced fugacity calculations in nanomaterials, simulating adsorption isotherms in nanoporous carbons to determine equilibrium fugacities under varying chemical potentials for applications such as hydrogen storage.^[42] These techniques provide nanoscale resolution of fugacity gradients, bridging molecular-level interactions with macroscopic thermodynamic properties in confined systems.^[43]

References

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[PDF] Non-Ideality Through Fugacity and Activity - University of Delaware
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THE LAW OF PHYSICO-CHEMICAL CHANGE. By Gilbert Newton Lewis. Received April 6, 1901. Presented by T. W. Richards, April 10, 1901.
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[PDF] Fugacity coefficients of gases in binary mixtures
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[PDF] EQUATION OF STATE. FUGACITIES OF GASEOUS SOLUTIONS
Sometimes we want to know the fugacity coefficient of a substance in a gaseous mixture below its critical temperature under a total pressure which is higher.<|control11|><|separator|>
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[PDF] Fugacity, Activity, and Standard States
Fugacity (f) is defined by dl = RTdlnf. Activity (ai) is defined as fi/fi. Standard states are ideal gas at 1 bar for gases, and pure solid/liquid at 1 bar for ...Missing: thermodynamics | Show results with:thermodynamics
[8]
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/The_Live_Textbook_of_Physical_Chemistry_(Peverati)/11:_Ideal_and_Non-Ideal_Gases/11.04:_Fugacity
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[PDF] A New Equation of State for Carbon Dioxide Covering the Fluid ...
Oct 15, 2009 · This new equation of state for carbon dioxide covers the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa, ...
[10]
[PDF] CO2-H2O Mixtures in the Geological Sequestration of ... - OSTI.GOV
Fugacity coefficient values for pure CO2 given in the International Union of Pure and. Applied Chemistry (IUPAC) tables (Angus et al., 1976) were fitted to ...
[11]
[PDF] Accuracy of Approximations to the Poynting Correction for Ice and ...
Nov 20, 2017 · Rigorous calculation of the Poynting correction, which describes the effect of pressure on the fugacity of a condensed phase, requires.
[12]
Water Vapor Saturation Pressure: Data, Tables & Calculator
Online calculator, figures and tables with water saturation (vapor) pressure at temperatures ranging 0 to 370 °C (32 to 700°F) - in Imperial and SI Units.
[13]
[PDF] CM 3230 Thermodynamics, Fall 2014 Lecture 28
Fugacity a. Definition: property of a substance, having units of pressure, invented by G.N.. Lewis, to allow calculations of changes in chemical potential that.
[14]
3 Lect 3 Fugacity | PDF | Gases | Physics - Scribd
0.03913 L mol-1. Calculate the fugacity of nitrogen gas at 50bar and 298K. Fugacity (f ) = 48.2 bar. Fugacity ( f ): measure the pressure ...<|control11|><|separator|>
[15]
[PDF] CM 3230 Thermodynamics, Fall 2016 Lecture 24
Lewis proposes a new property called “fugacity”, denoted f̂i, that will yield the same form as that of ideal gas mixtures, but would work for other conditions, ...Missing: formula | Show results with:formula
[16]
Properties of Hydrogen-Bonded Networks in Ethanol–Water Liquid ...
Jun 3, 2021 · A depletion of water-water hydrogen bonding in the first hydration shell of ethanol is compensated by an enhancement in the second hydration ...Introduction · Methods · Results and Discussion · Supporting Information
[17]
[PDF] Ethanol/Water Vapor-Liquid Equilibrium
fugacity coefficient of the pure saturated liquid is near unity, as well as the fugacity coefficient of each species in the mixture. Certainly the ratio of ...
[18]
https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/22:_Helmholtz_and_Gibbs_Energies/22.07:_The_Gibbs-Helmholtz_Equation
[19]
[PDF] Tables for the thermophysical properties of methane
The thermophysical properties of methane are tabulated for a large range of fluid states based on recently formulated correlations.
[20]
Equilibrium constants from a modified Redlich-Kwong equation of ...
View PDF; Download full ... June 1972, Pages 1197-1203. Chemical Engineering Science. Equilibrium constants from a modified Redlich-Kwong equation of state.
[21]
Group‐contribution estimation of activity coefficients in nonideal ...
A group-contribution method is presented for the prediction of activity coefficients in nonelectrolyte liquid mixtures.
[22]
Expressions for Fugacity Calculation | PNG 520
It is clear that, if we want to take advantage of the fugacity criteria to perform equilibrium calculations, we need to have a means of calculating it.
[23]
[PDF] Vapor Liquid Equilibrium (VLE): 10.213 04/29/02 A Guide Spring 2002
use the pure vapor (2) or pure liquid (3) fugacity expressions fugacity coef ... Poynting correction factor (P) usually close to 1 significant only if ...
[24]
REFPROP | NIST - National Institute of Standards and Technology
30-day returnsApr 18, 2013 · NIST Reference Fluid Thermodynamic and Transport Properties Database (REFPROP): Version 10. Download REFPROP 10: $325.00 PLACE ORDER.
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The NIST REFPROP Database for Highly Accurate Properties of ...
Jun 21, 2022 · The NIST REFPROP software program is a powerful tool for calculating thermophysical properties of industrially important fluids.
[26]
Validation with Fully Quantum Second Virial Coefficients - PMC - NIH
We use a new high-accuracy all-dimensional potential to compute the cross second virial coefficient B12(T) between molecular hydrogen and carbon monoxide.
[27]
Virial equation of state as a new frontier for computational chemistry
Nov 15, 2022 · The virial equation of state (VEOS) provides a rigorous bridge between molecular interactions and thermodynamic properties.Missing: post- | Show results with:post-
[28]
[PDF] 7.4 Introduction to vapor/liquid equilibrium
Phase equilibrium, and in particular vapor/liquid-equilibrium (VLE), is important for many process engineering applications. The thermodynamic basis for phase.
[29]
[PDF] Course Title: - Steady-State Modeling of Equilibrium Distillation
Thus, the equations of a distillation column model are obtained from mass and energy balances, mass balances by component and iso-fugacity equations. The ...
[30]
Thermodynamic Framework for Cryogenic Carbon Capture
Aug 7, 2025 · In this system, exiting product gases from the combustion chamber undergoes a CO2 capture process utilizing LNG regasification to avoid CO2 ...
[31]
[PDF] Measurement of oxygen fugacities under reducing conditions
Equation 1 is known as the Nernst equation. By introducing a gas of known oxygen fugacity at A and measuring the cell emf, it is possible through Eqn. 1 to ...<|separator|>
[32]
Global Analysis of Surface Ocean CO2 Fugacity and Air‐Sea Fluxes ...
Apr 23, 2024 · We demonstrate the capacity of statistical models to generate global maps of fCO2 and air-sea flux with a latency reduced to one month A ...
[33]
[PDF] STUDY OF SOLUBILITY IN SUPERCRITICAL FLUIDS - SciELO
The solubility accelerates the initial stages of an extraction process and reduces the time of the process.
[34]
https://geosci.uchicago.edu/~grossman/MBS1998GCA.pdf
[35]
Taking Another Look at the van der Waals Equation of State–Almost ...
Aug 2, 2019 · The van der Waals (vdW) equation of state has long fascinated researchers and engineers, largely because of its simplicity and engineering flexibility.
[36]
https://www.scielo.br/j/bjce/a/Kw4JLYyvb5ZX8Xh8XJjjmTx/?format=pdf&lang=en
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https://www.sciencedirect.com/science/article/pii/S0921319806800220
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Thermodynamics : Lewis, Gilbert Newton, 1875-1946 - Internet Archive
Oct 15, 2010 · First published in 1923 under title: Thermodynamics and the free energy of chemical substances. ... Randall, Merle, 1888-1950, joint author.
[39]
Application of the Benedict‐Webb‐Rubin equation of state to argon
The coefficients of the Benedict-Webb-Rubin equation of state have been developed for argon. By employing these coefficients, the volumetric behavior of ...
[40]
[PDF] Grand Canonical Monte Carlo Simulation of Hydrogen Adsorption in ...
Grand Canonical Monte Carlo (GCMC) simulations are performed to study hydrogen physisorption in different nano carbon porous materials made up of different ...
[41]
Reactive Grand-Canonical Monte Carlo Simulations for Modeling ...
Nov 25, 2021 · A biased grand-canonical Monte Carlo (GCMC) tool is developed, enabling the study of equilibrium conditions at the molecular level.Missing: nanomaterials | Show results with:nanomaterials