Phase Equilibrium and Fugacity

In one line: When a mixture boils, each component splits between liquid and vapor according to how strongly it “escapes” each phase. Fugacity measures that tendency, and it is the basis for designing every separation.

Core idea

Separating mixtures, such as alcohol from water, solvents from products, or acid gases from natural gas, depends on components distributing differently between phases. Equilibrium is reached when nothing drives any species from one phase to another. Thermodynamically, that driving force is the chemical potential, derived from Gibbs free energy. Chemical potential is awkward to work with, so engineers use fugacity instead, which behaves like a “corrected pressure.” At equilibrium, each species has the same fugacity in every phase.

How it works

Ideal mixtures, Raoult’s law. For a species $i$ in a liquid in equilibrium with vapor:

$y_{i}P=x_i{P}_i^{sat}$

• $x_i$ and $y_i$ are the species’ mole fractions in the liquid and the vapor.
• $P$ is the total pressure, and $P_i^{sat}$ the pure component’s vapor pressure, often estimated with the Antoine equation.

The more volatile component is enriched in the vapor, and that difference is what a separation exploits.

Real mixtures. Molecules attract and repel one another, so correction factors are added:

• a fugacity coefficient for non-ideal gases
• an activity coefficient ${\gamma }_i$ for non-ideal liquids, from models such as Wilson or NRTL

These explain azeotropes, mixtures that boil without changing composition. Ethanol and water form one at about 95.6% ethanol by mass at atmospheric pressure, which is why simple distillation can’t produce pure ethanol.

How many variables can be set? The Gibbs phase rule. For a non-reacting system, the degrees of freedom are

$F=2+C-\Pi$

with $C$ components and $\Pi$ phases. For pure water with two phases present, $F=1$: fix the pressure, and the boiling temperature follows.

Equations of state, such as the ideal gas law and cubic equations like van der Waals and Redlich–Kwong, predict how pressure, volume and temperature relate for real fluids, including where phases form.

Equipment that uses VLE:

• flash drums: a single equilibrium stage
• absorbers: a liquid strips chosen components from a gas
• distillation columns: many stages stacked, with reflux. They separate very effectively but use a lot of energy.

In practice

Separation is often the largest energy user and capital cost in a chemical plant. A leader evaluating a process should ask how hard the separation is: how close the components’ volatilities are, and whether there is an azeotrope. Those two facts largely decide the cost.

Questions to ask the experts

• Which thermodynamic model was used, and has it been checked against data for this mixture?
• Is there an azeotrope or another limit on how pure the product can get?
• How much energy does the separation take, and what are the alternatives?

Connections

• The discipline: Chemical Engineering. Combined with balances: Mass and Energy Balances.
• A familiar separation: Brewing as a Chemical Process. Phase change in machines: Thermodynamic Cycles.
• Capabilities exercised: Inquiry, Connecting Across Disciplines, Decision Making.

Sources

• Course lectures and an instructional comic on vapor-liquid equilibrium and fugacity; no public source.