Advanced Chemical Engineering Science Unit 6 Review: Non-Ideal Behaviors & Equations of State

Key Concepts

• Non-ideal behavior occurs when gases deviate from the assumptions of the ideal gas law at high pressures, low temperatures, or near the critical point
• Compressibility factor (Z) quantifies the extent of non-ideal behavior and equals 1 for an ideal gas but varies with pressure and temperature for real gases
• Equations of state (EOS) mathematically describe the relationship between pressure, volume, temperature, and composition for real gases
• Virial equation expresses the compressibility factor as a power series expansion in terms of pressure or density
• Cubic equations of state (van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson) model non-ideal behavior using empirical parameters specific to each gas
  • Incorporate critical properties and acentric factor to improve accuracy
• Intermolecular forces (van der Waals forces) and molecular size effects contribute to non-ideal behavior
• Applications in chemical engineering include process design, equipment sizing, phase equilibrium calculations, and thermodynamic analysis

Ideal Gas Law Recap

• Ideal gas law: $PV = nRT$, where P is pressure, V is volume, n is moles of gas, R is the universal gas constant, and T is absolute temperature
• Assumes gas molecules are point particles with no intermolecular forces and elastic collisions
• Valid for gases at low pressures and high temperatures, where intermolecular distances are large compared to molecular size
• Ideal gas law derives from the kinetic theory of gases and combines Boyle's law, Charles's law, and Avogadro's law
• Limitations arise when intermolecular forces become significant, and molecular size is non-negligible compared to intermolecular distances
  • Leads to deviations from ideal behavior
• Molar volume of an ideal gas depends only on temperature and pressure, not on the nature of the gas
• Compressibility factor (Z) equals 1 for an ideal gas at all conditions

Deviations from Ideal Behavior

• Real gases deviate from ideal behavior due to intermolecular forces and molecular size effects
  • Attractive forces (van der Waals forces) reduce the pressure compared to an ideal gas
  • Repulsive forces at short distances increase the pressure
  • Finite molecular size reduces the available volume for molecular motion
• Deviations become significant at high pressures, low temperatures, or near the critical point
• Compressibility factor (Z) quantifies the extent of deviation from ideality
  • Z < 1 indicates dominant attractive forces (e.g., at moderate pressures)
  • Z > 1 indicates dominant repulsive forces (e.g., at high pressures)
• Critical point represents the end of the vapor-liquid equilibrium curve, beyond which distinct liquid and gas phases do not exist
• Reduced properties (reduced temperature, pressure, and volume) normalize the state variables with respect to critical properties
  • Enable comparison of different gases using the principle of corresponding states
• Acentric factor (ω) characterizes the non-sphericity of molecules and affects the shape of the vapor-liquid equilibrium curve

Compressibility Factor

• Compressibility factor (Z) relates the actual molar volume of a gas to the molar volume of an ideal gas at the same pressure and temperature
  • $Z = \frac{PV}{nRT}$ or $Z = \frac{V_{\text{actual}}}{V_{\text{ideal}}}$
• Quantifies the deviation from ideal gas behavior
  • Z = 1 for an ideal gas
  • Z < 1 indicates attractive forces dominate (actual volume is smaller than ideal)
  • Z > 1 indicates repulsive forces dominate (actual volume is larger than ideal)
• Varies with pressure and temperature for real gases
• Generalized compressibility factor charts (Nelson-Obert or Standing-Katz) plot Z as a function of reduced pressure and reduced temperature
  • Allow estimation of Z without knowing the specific equation of state
• Compressibility factor is essential for accurate calculations involving real gases, such as density, molar volume, and fugacity

Virial Equation of State

• Virial equation expresses the compressibility factor as a power series expansion in terms of pressure or density
  • $Z = 1 + B(T)\frac{P}{RT} + C(T)\left(\frac{P}{RT}\right)^2 + ...$
  • $Z = 1 + B'(T)\rho + C'(T)\rho^2 + ...$
• Virial coefficients (B, C, ...) depend on temperature and account for intermolecular interactions
  • B(T) represents two-body interactions, C(T) represents three-body interactions, and so on
• Truncated after the second or third term for most applications
• Suitable for low to moderate pressures and densities
• Virial coefficients can be determined experimentally or estimated from intermolecular potential models (e.g., Lennard-Jones potential)
• Provides a theoretical basis for understanding non-ideal behavior but may not be practical for complex mixtures or high-pressure applications

Cubic Equations of State

• Cubic equations of state (EOS) model the pressure-volume-temperature relationship using a cubic polynomial in terms of molar volume
• General form: $P = \frac{RT}{V-b} - \frac{a(T)}{V^2 + ubV + wb^2}$
  • a(T) and b are empirical parameters related to attractive and repulsive forces, respectively
  • u and w are constants specific to each EOS
• Van der Waals EOS: $P = \frac{RT}{V-b} - \frac{a}{V^2}$
  • First cubic EOS to account for intermolecular forces and molecular size
  • Inaccurate for liquid densities and critical properties
• Redlich-Kwong EOS: $P = \frac{RT}{V-b} - \frac{a}{\sqrt{T}V(V+b)}$
  • Improves upon van der Waals EOS by introducing a temperature-dependent attractive term
• Soave-Redlich-Kwong (SRK) EOS: $P = \frac{RT}{V-b} - \frac{a(T)}{V(V+b)}$
  • Incorporates the acentric factor to better represent the shape of the vapor-liquid equilibrium curve
• Peng-Robinson (PR) EOS: $P = \frac{RT}{V-b} - \frac{a(T)}{V(V+b)+b(V-b)}$
  • Further improves the accuracy of liquid density and vapor pressure predictions
• Cubic EOS require critical properties and acentric factor as input parameters
• Widely used in process simulation and design due to their simplicity and reasonable accuracy

Applications in Chemical Engineering

• Process design and simulation
  • Accurate prediction of thermodynamic properties (density, enthalpy, entropy) for real gases
  • Sizing of equipment (compressors, heat exchangers, separators) based on realistic gas behavior
• Phase equilibrium calculations
  • Vapor-liquid equilibrium (VLE) for distillation, absorption, and stripping operations
  • Liquid-liquid equilibrium (LLE) for extraction processes
  • Vapor-liquid-liquid equilibrium (VLLE) for complex separations
• Equations of state are used in conjunction with mixing rules and activity coefficient models for mixtures
• Thermodynamic analysis
  • Calculation of departure functions (enthalpy, entropy) from ideal gas behavior
  • Estimation of fugacity coefficients and chemical potentials
• Process optimization and control
  • Accurate models for real gas behavior enable better process performance and efficiency
• Pipeline design and compressor selection
  • Accounting for non-ideal gas behavior in pressure drop and compressibility calculations
• Reservoir engineering and enhanced oil recovery
  • Modeling the behavior of gases and gas mixtures in porous media

Problem-Solving Techniques

• Identify the appropriate equation of state (EOS) for the given system and conditions
  • Ideal gas law for low pressures and high temperatures
  • Cubic EOS (SRK, PR) for moderate to high pressures and near-critical conditions
  • Virial equation for low to moderate pressures and densities
• Determine the required input parameters for the chosen EOS
  • Critical properties (temperature, pressure, volume)
  • Acentric factor
  • Mixing rules and binary interaction parameters for mixtures
• Use generalized compressibility factor charts (Nelson-Obert or Standing-Katz) for quick estimations
• Solve the EOS analytically or numerically for the desired property (e.g., molar volume, density)
  • Cubic EOS require solving a cubic polynomial, which may have multiple real roots
  • Select the physically meaningful root based on the phase and stability criteria
• Iterate the solution if necessary for temperature- or pressure-dependent properties
• Validate the results using experimental data or other reliable sources
• Perform sensitivity analysis to assess the impact of uncertainties in input parameters or EOS selection
• Consider the limitations and assumptions of the chosen EOS and interpret the results accordingly