Fugacity & The Fugacity Coefficient

1. Introduction to Fugacity

In classical thermodynamics, the concept of fugacity ($f$) was introduced by G.N. Lewis in 1901 to describe the "escaping tendency" of a substance from a particular phase. While chemical potential ($\mu$) is the fundamental driving force for mass transfer and phase equilibrium, its absolute value cannot be measured, and it approaches negative infinity as pressure approaches zero.

Fugacity solves this mathematical inconvenience. It has units of pressure (e.g., bar, atm, or Pa) and provides a tangible metric for phase equilibrium. In any multi-phase system, thermodynamic equilibrium is reached when the fugacity of each component is equal in all phases: $$f_i^V = f_i^L = f_i^S$$

2. Fugacity vs. Pressure: The Fugacity Coefficient

For an ideal gas, the escaping tendency is exactly equal to its partial pressure. Therefore, the fugacity of an ideal gas component is simply its partial pressure: $$f_i^{ideal} = P_i = y_i P$$

However, real gases exhibit intermolecular forces (attraction and repulsion) and have finite molecular volumes. To account for this non-ideality, we introduce the fugacity coefficient ($\phi_i$), a dimensionless correction factor: $$\phi_i = \frac{f_i}{y_i P}$$

• $\phi_i = 1$: The gas behaves ideally (low pressure, high temperature).
• $\phi_i < 1$: Intermolecular attractive forces dominate, making the gas easier to compress (typical at moderate pressures).
• $\phi_i > 1$: Intermolecular repulsive forces and molecular volume dominate, making the gas harder to compress (typical at very high pressures).

3. Calculating Fugacity from Equations of State (EOS)

To calculate the fugacity coefficient of a component in a gas mixture, engineers rely on rigorous Equations of State (EOS) such as Peng-Robinson (PR) or Soave-Redlich-Kwong (SRK).

The exact thermodynamic relationship derived from the fundamental property relations is: $$\ln \phi_i = \frac{1}{RT} \int_{V}^{\infty} \left[ \left( \frac{\partial P}{\partial n_i} \right)_{T, V, n_{j \neq i}} - \frac{RT}{V} \right] dV - \ln Z$$ where $Z = \frac{PV}{RT}$ is the compressibility factor.

While this integral looks daunting, commercial process simulators (like Aspen HYSYS, VMGSim, or ChemProCal's flash engine) evaluate it analytically using the chosen EOS to instantly predict phase envelopes.

4. Fugacity of a Liquid Phase

For a component in a liquid mixture, fugacity is typically expressed using an activity coefficient ($\gamma_i$) to account for liquid-phase non-idealities (such as hydrogen bonding in water-alcohol mixtures): $$f_i^L = x_i \gamma_i f_i^\circ$$ where $f_i^\circ$ is the standard state fugacity (usually the saturated vapor pressure of the pure component, $P_i^{sat}$).

At low to moderate pressures, where the vapor phase is nearly ideal ($\phi_i \approx 1$) and the liquid fugacity is governed by vapor pressure, the equality $f_i^V = f_i^L$ simplifies to Modified Raoult's Law: $$y_i P = x_i \gamma_i P_i^{sat}$$

5. Worked Example: Pure Gas Fugacity

Estimate the fugacity of pure methane gas at 100 bar and 25 °C. Assume an EOS calculates the compressibility factor as $Z = 0.88$ and the fugacity coefficient as $\phi = 0.85$.

1. System Pressure ($P$): 100 bar
2. Fugacity Coefficient ($\phi$): 0.85
3. Fugacity ($f$): $$f = \phi \times P = 0.85 \times 100 \text{ bar} = \mathbf{85 \text{ bar}}$$

Engineering Insight: Even though the physical pressure on the vessel walls is 100 bar, the "thermodynamic pressure" driving methane to escape or react is only 85 bar due to attractive intermolecular forces keeping the molecules closely packed.