10.4 Compressibility factor and fugacity

Real gases don't always play by the rules. The compressibility factor helps us understand how they deviate from ideal behavior. It's crucial for designing systems that handle high-pressure gases, like natural gas pipelines.

Fugacity is like pressure's cooler cousin for real gases. It accounts for all the messy interactions between molecules that ideal gas laws ignore. This concept is key for predicting how gases will behave in mixtures, reactions, and phase changes.

Compressibility Factor and Real Gas Behavior

Compressibility factor in real gases

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• Compressibility factor ($Z$) measures deviation of real gas from ideal gas behavior
  • Ratio of actual volume of gas to volume it would occupy if it behaved as an ideal gas at same temperature and pressure
    • $Z = \frac{V_{actual}}{V_{ideal}}$
  • $Z = 1$ for ideal gas, $Z < 1$ for gas more compressible than ideal gas, $Z > 1$ for gas less compressible than ideal gas (hydrogen at high pressures)
• Accounts for intermolecular forces and molecular size effects causing real gases to deviate from ideal gas behavior (van der Waals forces, dipole-dipole interactions)
• Allows accurate estimation of real gas properties under various conditions (density, pressure, temperature)
• Essential for designing and operating processes involving high-pressure gases (natural gas pipelines, storage systems, compressed air systems)

Calculation of compressibility factor

• Compressibility chart (Z-chart)
  • Graphical representation of compressibility factor as function of reduced pressure ($P_r$) and reduced temperature ($T_r$)
    • Reduced pressure: $P_r = \frac{P}{P_c}$, where $P_c$ is critical pressure (7.38 MPa for CO2)
    • Reduced temperature: $T_r = \frac{T}{T_c}$, where $T_c$ is critical temperature (304.13 K for CO2)
  • Locate point corresponding to given $P_r$ and $T_r$ values and read corresponding $Z$ value
• Equations of state (EOS)
  • Mathematical models relating pressure, volume, and temperature of substance (van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson)
  • Calculate $Z$ by substituting given pressure, temperature, and substance-specific parameters into equation and solve for $Z$
    • van der Waals equation: $\left(P + \frac{a}{V_m^2}\right)\left(V_m - b\right) = RT$, where $V_m$ is molar volume, $a$ and $b$ are substance-specific constants (CO2: $a = 0.364$ Pa·m6/mol2, $b = 4.267 \times 10^{-5}$ m3/mol)

Fugacity and Real Gas Behavior

Fugacity and real gas behavior

• Fugacity ($f$) represents effective pressure of real gas, accounting for non-ideal behavior
  • Has same units as pressure (Pa, atm)
  • For ideal gas, fugacity equals pressure ($f = P$)
• Measure of chemical potential of real gas, determining its tendency to escape from mixture or phase
• Accounts for intermolecular interactions and non-ideal behavior of real gases (hydrogen bonding, dispersion forces)
• Describes phase equilibria, chemical reactions, and mass transfer processes involving real gases (vapor-liquid equilibrium, gas absorption, adsorption)

Determination of gas fugacity

• Fugacity coefficient ($\phi$) relates fugacity to pressure
  • Ratio of fugacity to pressure: $\phi = \frac{f}{P}$
  • For ideal gas, $\phi = 1$; for real gas, $\phi$ can be greater or less than 1
• Calculating fugacity using fugacity coefficient
  • If fugacity coefficient is known, fugacity can be calculated using: $f = \phi P$
  • Fugacity coefficients determined using equations of state or empirical correlations
    • Soave-Redlich-Kwong equation of state: $\ln \phi = \frac{b_i}{b_m}(Z-1) - \ln(Z-B) - \frac{A}{B}\left(\frac{2\sum_j x_j a_{ij}}{a_m} - \frac{b_i}{b_m}\right)\ln\left(1 + \frac{B}{Z}\right)$, where $a_m$, $b_m$, $A$, and $B$ are mixture parameters, $a_{ij}$ and $b_i$ are component-specific parameters
• Crucial property for understanding and predicting behavior of real gases in various thermodynamic processes and systems (gas sweetening, natural gas processing, refrigeration cycles)