3.3 Cubic equations of state (van der Waals, Redlich-Kwong, Peng-Robinson)

Cubic equations of state improve on the ideal gas law by accounting for real gas behavior. They include the van der Waals, Redlich-Kwong, and Peng-Robinson equations, which consider molecular attraction and volume.

These equations use parameters based on critical properties and the acentric factor. They help predict fluid behavior across a wide range of conditions, especially near the critical point and in vapor-liquid equilibrium.

Cubic Equations of State

Van der Waals Equation

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• Proposed by Johannes Diderik van der Waals in 1873 to improve upon the ideal gas law by accounting for the behavior of real gases
• Modifies the ideal gas law by introducing two parameters: $a$ represents the attraction between molecules and $b$ represents the volume occupied by the molecules
• The van der Waals equation is given by: $(P + \frac{a}{V_m^2})(V_m - b) = RT$
• Captures the behavior of real gases more accurately than the ideal gas law, especially at high pressures and low temperatures (near the critical point)

Redlich-Kwong and Peng-Robinson Equations

• The Redlich-Kwong equation, proposed in 1949, further improves upon the van der Waals equation by introducing temperature-dependent parameters
• The Redlich-Kwong equation is given by: $P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T}V_m(V_m + b)}$
• The Peng-Robinson equation, developed in 1976, is similar to the Redlich-Kwong equation but uses a different attractive term to better represent the behavior of hydrocarbons
• The Peng-Robinson equation is given by: $P = \frac{RT}{V_m - b} - \frac{a(T)}{V_m(V_m + b) + b(V_m - b)}$

Equation Parameters and Compressibility Factor

• The parameters $a$ and $b$ in cubic equations of state are specific to each substance and are determined from critical properties (critical temperature, critical pressure, and acentric factor)
• The acentric factor ($\omega$) is a measure of the non-sphericity of molecules and affects the attractive term in the Peng-Robinson equation
• The compressibility factor ($Z$) is the ratio of the actual molar volume to the molar volume of an ideal gas at the same temperature and pressure: $Z = \frac{PV_m}{RT}$
• Cubic equations of state can be expressed in terms of the compressibility factor, which allows for easier comparison between different equations and substances

Equation Components

Attractive and Repulsive Terms

• Cubic equations of state consist of two main components: an attractive term and a repulsive term
• The attractive term accounts for the intermolecular forces (van der Waals forces) that cause molecules to attract each other, leading to a decrease in pressure
• The repulsive term represents the volume occupied by the molecules and the repulsive forces between them, which cause an increase in pressure
• The balance between the attractive and repulsive terms determines the overall behavior of the fluid

Acentric Factor and Critical Properties

• The acentric factor ($\omega$) is a measure of the non-sphericity of molecules and is used to characterize the shape and polarity of molecules
• Substances with higher acentric factors (e.g., long-chain hydrocarbons) have stronger attractive forces and deviate more from ideal gas behavior
• Critical properties (critical temperature, critical pressure, and critical volume) are used to determine the parameters in cubic equations of state
• The critical point represents the conditions at which the liquid and vapor phases become indistinguishable, and the fluid exhibits unique properties (e.g., infinite compressibility)

Critical Behavior and Equilibrium

Critical Point and Phase Behavior

• The critical point is the highest temperature and pressure at which a substance can exist as a liquid and vapor in equilibrium
• At the critical point, the density and other properties of the liquid and vapor phases become identical, and the fluid exhibits enhanced mass and heat transfer properties
• Cubic equations of state can be used to predict the phase behavior of fluids near the critical point, including the formation of vapor-liquid equilibrium and the calculation of critical properties

Vapor-Liquid Equilibrium and Phase Diagrams

• Vapor-liquid equilibrium (VLE) refers to the condition where a liquid and its vapor are in equilibrium with each other, meaning that the rates of evaporation and condensation are equal
• Cubic equations of state can be used to construct phase diagrams, which show the regions of pressure and temperature where different phases (solid, liquid, and vapor) exist in equilibrium
• Phase diagrams are essential for understanding the behavior of fluids under various conditions and for designing processes that involve phase changes (e.g., distillation, refrigeration)
• The shape of the phase diagram and the location of the critical point depend on the specific substance and can be predicted using cubic equations of state (e.g., the Peng-Robinson equation for hydrocarbons)