Fugacity is a key concept in thermodynamics, helping us understand real gas behavior. It's like pressure, but accounts for non-ideal interactions between molecules. Knowing how to calculate fugacity is crucial for predicting phase equilibria and chemical reactions in real systems.

There are several ways to calculate fugacity, each with its pros and cons. From equations of state to graphical methods and generalized correlations, these techniques help us bridge the gap between ideal gas models and real-world behavior.

Equation of State Methods

Virial Equation and Compressibility Factor

• Equation of state method calculates fugacity using an equation that relates pressure, volume, and temperature of a substance
  • Equations of state (ideal gas law, van der Waals equation, Redlich-Kwong equation) describe the behavior of gases and liquids under various conditions
• Virial equation is a power series expansion used to calculate the compressibility factor and fugacity of a gas
  • Virial coefficients ($B$, $C$, $D$, etc.) depend on temperature and account for gas non-ideality due to intermolecular interactions
  • Truncated after the second ($B$) or third ($C$) virial coefficient for most applications
• Compressibility factor ($Z$) measures the deviation of a gas from ideal behavior
  • Defined as the ratio of the actual molar volume to the molar volume of an ideal gas at the same temperature and pressure
  • $Z = 1$ for an ideal gas, $Z < 1$ for a gas with attractive intermolecular forces, and $Z > 1$ for a gas with repulsive intermolecular forces
  • Calculated using the virial equation or equations of state (Redlich-Kwong, Peng-Robinson)

Graphical and Correlation Methods

Graphical Integration and Corresponding States Principle

• Graphical integration involves plotting experimental data (pressure vs. molar volume) and integrating the area under the curve to determine fugacity
  • Requires accurate experimental data and numerical integration techniques (trapezoidal rule, Simpson's rule)
  • Provides a visual representation of the relationship between pressure and molar volume
• Corresponding states principle states that substances at the same reduced temperature, pressure, and volume have similar properties
  • Reduced properties are normalized by their critical values (reduced temperature $T_r = T/T_c$, reduced pressure $P_r = P/P_c$, reduced volume $V_r = V/V_c$)
  • Allows for the prediction of properties for substances with limited experimental data using data from well-characterized substances (methane, carbon dioxide)

Generalized Correlations for Fugacity Calculation

• Generalized correlations are empirical equations that relate fugacity to reduced temperature, pressure, and volume
  • Developed using experimental data for a wide range of substances
  • Examples include the Lee-Kesler correlation and the Pitzer correlation
• Generalized correlations provide a quick and convenient method for estimating fugacity without the need for extensive experimental data or complex equations of state
  • Require only the critical properties and acentric factor of the substance
  • Less accurate than equations of state or graphical methods, particularly for substances with unique intermolecular interactions (hydrogen bonding, polar molecules)

Fugacity Calculation Rules

Lewis and Randall Rule for Ideal Mixtures

• Lewis and Randall rule states that the fugacity of a component in an ideal mixture is equal to its mole fraction multiplied by its fugacity in the pure state at the same temperature and pressure
  • $\hat{f}_i = x_i f_i^0$, where $\hat{f}_i$ is the fugacity of component $i$ in the mixture, $x_i$ is the mole fraction of component $i$, and $f_i^0$ is the fugacity of pure component $i$ at the same temperature and pressure
• Ideal mixtures are characterized by zero enthalpy of mixing, zero volume change on mixing, and no intermolecular interactions between unlike molecules
  • Examples of ideal mixtures include mixtures of isotopes (helium-3 and helium-4) and mixtures of chemically similar substances (benzene and toluene)
• Lewis and Randall rule simplifies fugacity calculations for ideal mixtures by relating the mixture fugacity to pure component fugacities and mole fractions
  • Requires knowledge of pure component fugacities, which can be calculated using equations of state, graphical methods, or generalized correlations
  • Not applicable to non-ideal mixtures, where intermolecular interactions between unlike molecules lead to deviations from ideal behavior