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FugacityEdit

Fugacity is a thermodynamic quantity that provides a practical bridge between idealized models and the behavior of real substances. In essence, it is an effective pressure that replaces the actual pressure in many equations when real fluids deviate from ideal behavior. The concept is central to understanding phase equilibria, mixture behavior, and the design of industrial processes where non-ideality plays a significant role. By encapsulating how far a real gas or liquid is from ideality, fugacity allows scientists and engineers to apply familiar equations—such as those that govern chemical potential and phase continuity—without being misled by the limitations of the ideal-gas assumption.

In a phase with temperature T and pressure P, the fugacity f_i of a component i behaves like a corrected pressure that reflects molecular interactions, excluded volume, and other non-ideal effects. For a gas in equilibrium with its liquid, the fugacity of each component is the same in each phase, up to the appropriate phase-specific reference state. In practice, fugacity is tied to a fugacity coefficient φ_i through f_i = φ_i P in the gas phase, and φ_i is a measure of deviation from ideal gas behavior (with φ_i = 1 for an ideal gas). The notion extends from single-component systems to multicomponent mixtures, where each component has its own fugacity f_i and corresponding coefficient φ_i.

Fundamentals

  • Definition and interpretation

    • Fugacity f_i is the effective pressure that makes the chemical potential μ_i behave as if the substance were ideal: μ_i = μ_i^° + RT ln(f_i/P^°), where P^° is a standard pressure. This rewrite preserves the familiar ln-part of the ideal-gas expression while incorporating non-ideality.
    • In the gas phase of a real mixture, f_i ≈ φ_i y_i P, where y_i is the mole fraction of i in the gas and φ_i is the fugacity coefficient. For a pure ideal gas, φ_i = 1 and f_i = P.
    • The relationship f_i = φ_i P generalizes to liquids and solids through their respective reference states, often via activity and fugacity concepts in phase equilibria.

  • Fugacity coefficient

    • The fugacity coefficient φ_i quantifies deviation from ideal behavior: φ_i = f_i/P. Values different from 1 indicate repulsive (φ_i > 1) or attractive (φ_i < 1) non-ideality under the given conditions.
    • φ_i can be determined from experimental data or predicted from models such as equations of state and departure-function analyses.

  • Phase equilibrium

    • At equilibrium, the chemical potential of each component is the same in coexisting phases. This leads to equality of fugacities across phases: f_i^L = f_i^V (or f_i^solid = f_i^liquid, etc.), enabling the calculation of phase diagrams and compositions.

  • Ideal-gas limit and departures

    • In the ideal-gas limit, f_i reduces to the actual pressure contribution, and φ_i → 1. Departure functions quantify the difference between real behavior and the ideal reference and are widely used to present non-ideality in a compact form.

  • Historical and practical context

    • The concept emerged from the need to describe real-gas behavior with equations that resemble those used for ideal gases. It provides a rigorous way to incorporate intermolecular forces and molecular size into thermodynamic calculations, with wide application in chemical engineering, petrochemicals, and materials science.

  • Related concepts

Calculation and models

  • Equations of state and cubic models

    • Real-fluid behavior is commonly modeled with equations of state (EOS). Cubic EOS such as the Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) families provide closed-form expressions for Z, the compressibility factor, from which φ_i can be derived. These models balance accuracy with computational simplicity and are widely used in process design.
    • The fugacity coefficient is obtained from the EOS or from corresponding states methods, often via a relation of the form ln φ_i = (Z - 1) - ln(… ), with integrals or departure functions that measure the difference between the real fluid and the ideal gas.

  • Virial and departure functions

    • For low to moderate pressures, the virial equation of state can be used to compute φ_i from a virial coefficient expansion in powers of pressure. Departure functions provide a way to express non-ideality as a difference between real-property values and ideal-gas values at the same T and P.
    • These approaches yield f_i = φ_i P in the gas phase. In mixtures, each component has its own φ_i, determined by interactions with all other components.

  • Liquids and mixtures

    • In liquids, fugacity is often discussed via activity coefficients (γ_i) and standard-state fugacities f_i^0. The relation f_i^L = x_i γ_i f_i^0 is used to enforce phase-equilibrium criteria when dealing with liquid solutions.
    • For vapor–liquid equilibria, the vapor-phase fugacity of component i is f_i^V = φ_i^V y_i P, while the liquid-phase fugacity is f_i^L = f_i^0 x_i γ_i. Equilibrium requires f_i^V = f_i^L for each i.

  • Practical workflow

    • Choose a model (EOS or activity-coefficient approach) appropriate for the system and conditions.
    • Compute φ_i or γ_i for the phases of interest.
    • Apply the equilibrium conditions to determine compositions, fugacities, and pressures.
    • Validate against experimental data where available.

  • Standard state conventions

    • Fugacity calculations rely on chosen standard states (often 1 bar for gases). The precise reference state affects numerical values but not the fundamental thermodynamic relationships.

Applications

  • Phase behavior and separation processes

    • Fugacity underpins the design and analysis of distillation, absorption, extraction, and other separation techniques. By quantifying non-ideality, engineers can predict phase envelopes, relative volatilities, and component distributions between phases.
    • See Distillation and Phase equilibrium for related topics and methods.

  • Chemical and petrochemical engineering

    • In petrochemical processing, accurate fugacity estimates are essential for modeling hydrocarbon mixtures, refrigerants, and specialty chemicals under high pressure and temperature.
    • EOS-based approaches (e.g., Peng-Robinson or Soave-Redlich-Kwong) are standard tools in process simulators and design calculations.

  • Gas–liquid and gas–solid interfaces

    • Adsorption, heterogeneous catalysis, and gas separation rely on fugacity concepts to describe how species partition between phases and how non-ideality affects equilibrium and capacity.
    • See Adsorption and Catalysis for broader context.

  • Thermodynamic engineering and standards

    • Fugacity is linked to how engineers gauge deviations from ideality in published data, correlations, and standard-state conventions. This helps ensure consistent design across different facilities and regulatory environments.

Historical development and debate

  • Origins and development

    • The fugacity concept arose in the early 20th century as thermodynamics matured and the need to describe real fluids grew more pressing. Foundational work by early thermodynamicists established the mathematical framework that connects chemical potential, pressure, and non-ideality. See Thermodynamics and Chemical potential for historical context.

  • Debates and ongoing refinements

    • A persistent topic in the field is the balance between model complexity and predictive accuracy. Cubic EOS are computationally efficient and work well for many systems, but they struggle with highly non-ideal mixtures, strongly associating fluids, or near critical points. Critics point to the limits of empirical correlations and emphasize data-driven, molecular-based approaches, while proponents highlight the robustness and practicality of EOS in industrial design.
    • Another area of discussion concerns the interpretation of fugacity for liquids and near-solid phases. While the concept remains a powerful mathematical tool, some researchers stress the importance of recognizing its limitations and ensuring that the chosen model respects the underlying physics of intermolecular interactions.

See also