Fugacity and Activity | Real Gases, Non-Ideal Solutions, Thermodynamics

Fugacity and Activity: Correcting Ideal Behavior in Real Systems

Definitions · Fugacity Coefficient · Activity Coefficient · Chemical Potential · Applications in Thermodynamics & Chemical Engineering

In ideal gases and ideal solutions, the behavior of components is described by simple equations such as \(PV = nRT\) for gases and Raoult’s law for liquid mixtures. However, real systems deviate significantly from ideality due to intermolecular interactions. Fugacity and activity are thermodynamic concepts introduced to correct these deviations, allowing us to apply ideal equations to real systems by using effective pressures or concentrations.

Fugacity: Effective Pressure of a Real Gas

The concept of fugacity (f) was introduced by the American physical chemist Gilbert N. Lewis. Fugacity is defined as the effective partial pressure of a chemical species in a non-ideal gas mixture. For a real gas at temperature \(T\) and pressure \(P\), the fugacity is the pressure that an ideal gas would need to have the same chemical potential as the real gas.

\[ \mu(T, P) = \mu^\circ(T) + RT \ln\left(\frac{f}{P^\circ}\right) \]

where \(\mu\) is chemical potential, \(\mu^\circ\) is the chemical potential at standard state, \(P^\circ\) is standard pressure (usually 1 bar), and \(f\) is fugacity.

Fugacity Coefficient (\(\phi\))

The fugacity coefficient relates fugacity to actual pressure:

\[ \phi = \frac{f}{P} \]

For an ideal gas, \(\phi = 1\) because \(f = P\). For real gases, \(\phi\) deviates from unity. The value of \(\phi\) is less than 1 at moderate pressures (attractive forces dominant) and greater than 1 at high pressures (repulsive forces dominant).

Fig. 1: Variation of fugacity coefficient with pressure. At low pressures, φ < 1 due to attractive forces; at high pressures, φ > 1 due to repulsive forces.

Activity: Effective Concentration in Non-Ideal Solutions

Activity (\(a\)) quantifies the effective concentration of a chemical species in non-ideal conditions. The concept was also developed by Gilbert N. Lewis. Activity is a dimensionless quantity defined as:

\[ a_i = \gamma_i \, x_i \]

where \(x_i\) is the mole fraction (or concentration) and \(\gamma_i\) is the activity coefficient. For an ideal solution, \(\gamma_i = 1\) and \(a_i = x_i\). The activity coefficient accounts for deviations from ideality due to intermolecular interactions.

Factors Affecting Activity

Temperature: Activity changes with temperature as molecular interactions vary.
Pressure: At high pressures, activity can deviate significantly, especially for gases.
Composition: In mixtures, the presence of other species alters the effective concentration of each component.

Standard States for Activity

The value of activity depends on the chosen standard state:

Pure solid or liquid: \(a = 1\) for the pure substance at the same temperature and pressure.
Gas: Activity is defined as \(a = f/P^\circ\), where \(f\) is fugacity and \(P^\circ\) is standard pressure (usually 1 bar).
Solute in solution: Often based on Henry’s law or Raoult’s law reference states.

Fig. 2: Activity coefficient variation with composition for non-ideal solutions. Positive deviation (γ > 1) indicates repulsive interactions; negative deviation (γ < 1) indicates attractive interactions.

Relation Between Fugacity and Activity for Gases

For a gas, the activity is directly related to fugacity:

\[ a = \frac{f}{P^\circ} \]

Thus, the chemical potential becomes:

\[ \mu = \mu^\circ + RT \ln a \]

This unified expression holds for all phases when using appropriate standard states.

Significance in Thermodynamics and Chemical Engineering

Phase equilibrium: For a component to be in equilibrium between two phases, its fugacity (or activity) must be equal in both phases. This is the basis for vapor-liquid equilibrium (VLE) calculations.
Chemical reaction equilibrium: The equilibrium constant \(K\) is expressed in terms of activities: \(K = \prod a_i^{\nu_i}\). Using activities instead of concentrations gives accurate equilibrium constants for non-ideal systems.
Real gas behavior: Fugacity coefficients are used in equations of state (e.g., Peng-Robinson, Soave-Redlich-Kwong) to predict gas properties.
Electrolyte solutions: Activity coefficients are crucial for calculating solubility products, pH, and ionic strength effects.

Interactive: Fugacity Coefficient Calculator (Ideal vs Real)

For a simple real gas model (Van der Waals), the fugacity coefficient can be approximated. Enter pressure (bar) and temperature (K) for a gas like carbon dioxide.

Pressure (bar)

Temperature (K)

Enter pressure and temperature to get an approximate fugacity coefficient using a simplified model.

Note: This is a demonstration; real calculations require accurate EOS parameters.

Activity Coefficient Models

Several empirical and semi-empirical models exist to predict activity coefficients in liquid mixtures:

Margules equations – simple binary mixture models.
Van Laar equation – useful for strongly non-ideal systems.
Wilson equation – temperature-dependent, good for miscible systems.
NRTL (Non-Random Two-Liquid) – handles partially miscible systems.
UNIQUAC and UNIFAC – group contribution methods for predicting activity coefficients without experimental data.

Examples of Fugacity and Activity in Practice

Example 1: Fugacity of steam at high pressure. At 250°C and 50 bar, water vapor is not ideal. Using steam tables, the fugacity can be obtained from the fugacity coefficient \(\phi = f/P\). Typically, \(\phi < 1\) for steam at moderate pressures.

Example 2: Activity of ethanol in water. A mixture of ethanol and water shows positive deviation from Raoult’s law, with activity coefficients greater than 1. This explains the azeotrope formation at about 95% ethanol.

Key Equations Summary

\[ \mu = \mu^\circ + RT \ln \left( \frac{f}{P^\circ} \right) \quad \text{(for gases)} \]

\[ \phi = \frac{f}{P} \quad,\quad \phi \rightarrow 1 \text{ as } P \rightarrow 0 \]

\[ a_i = \gamma_i x_i \quad,\quad \gamma_i = \frac{a_i}{x_i} \]

\[ \mu_i = \mu_i^\circ + RT \ln a_i \quad \text{(general form)} \]

\[ K = \prod_i a_i^{\nu_i} \quad \text{(equilibrium constant in terms of activities)} \]

Complete Lecture: Fugacity and Activity (Urdu/Hindi)

Watch this lecture for a clear conceptual understanding of fugacity, activity, and their applications in real systems.

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Fugacity and Activity: Correcting Ideal Behavior in Real Systems