7.2 Residual properties

Residual properties are crucial for understanding real gas behavior. They measure how actual gases deviate from ideal gas models, helping engineers predict and analyze real-world systems more accurately.

Fugacity and compressibility factor are key concepts in this topic. They provide practical tools for calculating properties of real gases, essential for designing and optimizing processes in industries like chemical engineering and HVAC.

Residual Properties

Relationship between Residual Properties and Ideal Gas State

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• Residual properties quantify the deviation of a real gas from the behavior of an ideal gas at the same temperature and pressure
• Calculated by subtracting the property value for an ideal gas from the actual property value of the real gas
• Ideal gas state serves as a reference point for comparing the behavior of real gases
  • In an ideal gas, molecules have no intermolecular forces and occupy negligible volume
  • Real gases deviate from ideal gas behavior due to intermolecular interactions and finite molecular volume

Residual Gibbs Free Energy, Enthalpy, and Entropy

• Residual Gibbs free energy ($G^R$) represents the difference between the actual Gibbs free energy of a real gas and that of an ideal gas at the same temperature and pressure
  • Quantifies the non-ideality of a gas in terms of Gibbs free energy
  • Positive $G^R$ indicates a higher Gibbs free energy than an ideal gas, while negative $G^R$ indicates a lower Gibbs free energy
• Residual enthalpy ($H^R$) is the difference between the actual enthalpy of a real gas and that of an ideal gas at the same temperature and pressure
  • Measures the additional enthalpy due to intermolecular interactions in a real gas
  • Positive $H^R$ indicates higher enthalpy than an ideal gas, while negative $H^R$ indicates lower enthalpy
• Residual entropy ($S^R$) represents the difference between the actual entropy of a real gas and that of an ideal gas at the same temperature and pressure
  • Quantifies the entropy change associated with the non-ideality of a gas
  • Negative $S^R$ indicates a lower entropy than an ideal gas, while positive $S^R$ indicates a higher entropy (less common)

Departure Functions and Their Applications

• Departure functions express the difference between the actual property value of a real gas and that of an ideal gas at the same temperature and pressure
  • Denoted by the superscript "D" (e.g., $G^D$, $H^D$, $S^D$)
  • Related to residual properties: $G^D = G^R$, $H^D = H^R$, $S^D = S^R$
• Departure functions are useful for estimating the properties of real gases when direct experimental data is unavailable
  • Can be calculated using equations of state (EOS) or generalized correlations
  • Example: Peng-Robinson EOS can be used to calculate departure functions for hydrocarbons and their mixtures
• Departure functions find applications in process design, equipment sizing, and thermodynamic analysis of real gas systems
  • Example: Calculating the required compressor power for a natural gas pipeline considering the non-ideal behavior of the gas

Fugacity and Compressibility

Fugacity and Fugacity Coefficient

• Fugacity ($f$) is a thermodynamic property that represents the effective pressure of a real gas, accounting for its non-ideal behavior
  • Has units of pressure (e.g., Pa, bar)
  • For an ideal gas, fugacity equals the actual pressure
  • For a real gas, fugacity can be higher or lower than the actual pressure, depending on the gas and the conditions
• Fugacity coefficient ($\phi$) is the ratio of a gas's fugacity to its actual pressure at a given temperature and pressure
  • Dimensionless quantity
  • For an ideal gas, $\phi = 1$
  • For a real gas, $\phi$ can be greater than or less than 1, indicating positive or negative deviations from ideal behavior
• Fugacity and fugacity coefficient are related by: $f = \phi P$
  • $P$ is the actual pressure of the gas
  • Fugacity coefficient can be calculated using equations of state or experimentally measured

Compressibility Factor and Its Relation to Fugacity

• Compressibility factor ($Z$) is the ratio of the actual volume of a gas to the volume it would occupy if it behaved as an ideal gas at the same temperature and pressure
  • Dimensionless quantity
  • For an ideal gas, $Z = 1$
  • For a real gas, $Z$ can be greater than or less than 1, indicating positive or negative deviations from ideal behavior
• Compressibility factor is related to the fugacity coefficient by: $\ln \phi = \int_0^P \frac{Z-1}{P} dP$
  • This relationship allows the calculation of fugacity coefficient from compressibility factor data
  • Example: Using the virial equation of state to express $Z$ as a function of pressure and calculating $\phi$ by integration
• Compressibility factor is used to characterize the behavior of real gases and to estimate their properties
  • Can be obtained from experimental data (e.g., PVT measurements) or calculated using equations of state
  • Generalized compressibility factor charts (e.g., Nelson-Obert charts) provide $Z$ values for various gases as a function of reduced temperature and pressure