Fugacity and Its Applications

When predicting whether a chemical will vaporize, dissolve, or react, the ideal gas law often fails for real-world, high-pressure systems. Fugacity, often described as a "corrected pressure" or "escaping tendency," provides the essential generalization needed to accurately model phase and chemical equilibrium for real fluids. This concept transforms abstract thermodynamic potentials into a practical, pressure-like quantity you can calculate and use, making it indispensable for designing chemical processes like distillation columns, reactors, and pipelines.

Defining Fugacity and the Fugacity Coefficient

The fundamental problem in applied thermodynamics is relating the Gibbs free energy $G$ to measurable properties like temperature and pressure. For an ideal gas, this relationship is simple: the change in molar Gibbs energy with pressure at constant temperature is $dG=RTd(\ln P)$. For a real fluid, this simple proportionality breaks down.

To preserve this mathematically convenient form, G.N. Lewis defined fugacity $f$ for a pure substance. It is defined by the equation: $$dG=RTd(\ln f)$$ at constant temperature. To make this definition complete, fugacity is anchored to reality by the limit: $$\underset{P\to 0}{\lim }\frac{f}{P}=1$$ This states that as pressure approaches zero, all gases behave ideally, and fugacity converges with measurable pressure.

The fugacity coefficient $\varphi$ is the dimensionless ratio that quantifies the deviation from ideal gas behavior: $$\varphi =\frac{f}{P}$$ For an ideal gas, $\varphi =1$ always. For a real fluid, $\varphi$ tells you the story of molecular interactions: $\varphi <1$ typically indicates attractive forces dominating (making the fluid's "escaping tendency" less than its pressure), while $\varphi >1$ indicates repulsive forces dominating. Calculating $\varphi$ is the primary route to obtaining fugacity.

Fugacity for Pure Real Fluids and Ideal Gases

For a pure ideal gas, the derivation is straightforward. Integrating the definition $dG=RTd(\ln f)$ and applying the ideal gas law directly yields $f=P$. Therefore, for an ideal gas, fugacity is the pressure.

For a pure real fluid, we need a way to calculate $f$ or $\varphi$ from an equation of state (EoS) like the Peng-Robinson or Soave-Redlich-Kwong equations, which relate $P$, $V$, and $T$. The derivation starts from the fundamental thermodynamic relation and arrives at a workable formula. The fugacity coefficient for a pure substance is given by: $$\ln \varphi =\ln \left(\frac{f}{P}\right)={\int }_0^P\left(\frac{Z-1}{P}\right)dP$$ where $Z=\frac{PV}{RT}$ is the compressibility factor. This integral can be solved analytically if the equation of state is explicit in $P$ (e.g., cubic EoS). For example, using the Peng-Robinson EoS, you would substitute the expression for $Z$ and evaluate the integral to get a closed-form expression for $\ln \varphi$ as a function of $T$ and $P$.

Fugacity in Mixtures and the Lewis-Randall Rule

For mixtures, the focus shifts to the partial fugacity ${\hat{f}}_i$ of component $i$. This is defined via the partial molar Gibbs energy: $d{\bar{G}}_i=RTd(\ln {\hat{f}}_i)$. The corresponding fugacity coefficient for component $i$ in a mixture is: $${\hat{\varphi }}_i=\frac{{\hat{f}}_i}{y_{i}P}$$ where $y_i$ is the mole fraction of $i$ in the vapor phase (for a liquid phase, $x_i$ and total pressure $P$ would be used appropriately).

A critical simplifying assumption for mixtures is the Lewis-Randall rule. It states that the fugacity of a component in an ideal solution is proportional to its mole fraction and the fugacity of the pure component at the same temperature and pressure as the mixture: $${\hat{f}}_i^{id}=y_i{f}_i$$ where $f_i$ is the fugacity of pure $i$ at $(T,P)$. This rule is valid when the molecular interactions between different species are similar to those between like species. It greatly simplifies calculations because you only need pure-component fugacities. However, for highly non-ideal mixtures (e.g., ethanol-water), this rule fails, and you must use activity coefficient models.

Fugacity from Equations of State

The most rigorous and common method for calculating mixture fugacity in process simulation is using an equation of state. The formula for the fugacity coefficient of a component in a mixture from a pressure-explicit EoS is: $$\ln {\hat{\varphi }}_i=\frac{1}{RT}{\int }_V^{\infty }\left[{\left(\frac{\partial P}{\partial n_i}\right)}_{T,V,n_{j\ne i}}-\frac{RT}{V}\right]dV-\ln Z$$ This looks daunting, but for cubic equations of state, it again simplifies to an analytical expression. The key is that the EoS provides mixing rules for its parameters (e.g., $a$ and $b$ in Peng-Robinson), which are functions of composition. You then differentiate these mixture parameters with respect to $n_i$ as required by the formula. This approach gives ${\hat{f}}_i$ directly and is consistent for both vapor and liquid phases, which is essential for accurate equilibrium calculations.

The Equal Fugacity Criterion for Phase Equilibrium

The ultimate application of fugacity is determining phase equilibrium. The fundamental criterion for phase equilibrium is that the Gibbs free energy is minimized, which translates directly into a powerful, practical rule: for any component $i$, its fugacity must be equal in all coexisting phases.

For vapor-liquid equilibrium (VLE), this means: $${\hat{f}}_i^V={\hat{f}}_i^L$$ where the superscripts denote vapor ($V$) and liquid ($L$) phases. Expanding this using fugacity coefficients: $$y_{i}P{\hat{\varphi }}_i^V=x_{i}P{\hat{\varphi }}_i^L$$ If you assume an ideal gas phase (${\hat{\varphi }}_i^V=1$) and an ideal solution liquid phase (Lewis-Randall rule, leading to ${\hat{\varphi }}_i^L=\frac{f_i^L}{P}$), this simplifies to Raoult's Law. For more realistic cases, you use an EoS to calculate both ${\hat{\varphi }}_i^V$ and ${\hat{\varphi }}_i^L$, solving the equality to find equilibrium compositions ($x_i$, $y_i$) and pressures or temperatures. This is the core calculation inside flash drum and distillation column simulations.

Common Pitfalls

1. Confusing Fugacity with Partial Pressure: Fugacity is not partial pressure, except in the special case of an ideal gas mixture. Partial pressure is $y_{i}P$, while fugacity is $y_{i}P{\hat{\varphi }}_i$. Using partial pressure in equilibrium calculations for non-ideal, high-pressure systems will yield significant errors.
2. Misapplying the Lewis-Randall Rule: This rule is an ideal solution model. Applying it to strongly non-ideal mixtures (e.g., systems with hydrogen bonding or large differences in molecular size) will give incorrect fugacities. For such mixtures, you must use an activity coefficient model (like NRTL or UNIQUAC) or a sophisticated EoS.
3. Ignoring Reference State Consistency: When combining models (e.g., an EoS for the vapor and an activity coefficient model for the liquid), you must ensure the standard state fugacities are defined consistently. A mismatch here is a common source of error in VLE calculations.
4. Overlooking Phase Stability: The equal fugacity condition is necessary but not sufficient. A solution where fugacities are equal might correspond to a metastable or unstable state. Always check phase stability, especially near critical points or in complex multicomponent systems, to ensure you have the correct equilibrium solution.

Summary

• Fugacity is a thermodynamic property representing "escaping tendency" that generalizes the concept of pressure for real fluids, defined to preserve the simple mathematical form of the ideal gas Gibbs energy relationship.
• The fugacity coefficient $\varphi$ measures deviation from ideal gas behavior; it is calculated from an equation of state via a derived integral or analytical expression.
• For mixtures, the Lewis-Randall rule provides a simple estimate for component fugacity in ideal solutions, but rigorous calculation requires mixture fugacity coefficients ${\hat{\varphi }}_i$ from an EoS with proper mixing rules.
• The cornerstone of practical phase equilibrium calculations is the equal fugacity criterion: at equilibrium, the fugacity of each component is identical in all coexisting phases (${\hat{f}}_i^{\alpha }={\hat{f}}_i^{\beta }$).
• Mastery of fugacity involves choosing the correct model (ideal gas, ideal solution, EoS, or activity coefficient) for the system's non-ideality and pressure to apply the equilibrium criterion accurately.