2.3 Equations of state and activity coefficient models

Equations of state are mathematical models that describe fluid behavior under various conditions. They're essential for predicting phase changes, calculating properties, and estimating equilibrium compositions in separation processes.

From the ideal gas law to more complex cubic equations, these models account for molecular interactions and non-ideal behavior. Activity coefficient models and parameter estimation techniques further refine our understanding of mixture thermodynamics in real-world applications.

Equations of State

Principles of state equations

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• Equations of state (EOS) describe thermodynamic behavior of fluids through mathematical relationships between temperature, pressure, volume, and composition
• EOS models assume spherical molecules, averaged intermolecular forces, and no chemical reactions occurring
• Used to predict phase behavior, calculate thermodynamic properties (enthalpy, entropy), and estimate equilibrium compositions
• Ideal gas law $(PV = nRT)$ serves as simplest EOS, more complex models account for non-ideal behavior (van der Waals, Redlich-Kwong)

Application of cubic state equations

• Van der Waals equation pioneered cubic EOS incorporating molecular attraction (a) and repulsion (b) parameters: $(P + \frac{a}{v^2})(v - b) = RT$
• Redlich-Kwong equation improved accuracy with temperature-dependent attraction term: $P = \frac{RT}{v - b} - \frac{a}{\sqrt{T}v(v + b)}$
• Peng-Robinson equation further refined liquid density predictions: $P = \frac{RT}{v - b} - \frac{a\alpha(T)}{v(v + b) + b(v - b)}$
• Cubic EOS applications include vapor-liquid equilibrium calculations, critical point estimation, and compressibility factor determination

Activity coefficient models for mixtures

• Activity coefficient models predict non-ideal behavior in liquid mixtures based on excess Gibbs energy
• Models assume local composition concept and utilize binary interaction parameters
• Margules equation offers simplest approach with single adjustable parameter: $ln\gamma_i = Ax_j^2$
• Van Laar equation extends Margules with two parameters: $ln\gamma_1 = \frac{A_{12}}{(1 + \frac{A_{12}}{A_{21}}(\frac{x_1}{x_2}))^2}$
• Wilson equation incorporates local composition concept: $ln\gamma_i = 1 - ln(\sum_j x_j\Lambda_{ij}) - \sum_k \frac{x_k\Lambda_{ik}}{\sum_j x_j\Lambda_{kj}}$
• NRTL model accounts for non-random molecular orientations using three parameters $(\tau_{ij}, G_{ij}, \alpha_{ij})$
• UNIQUAC model combines combinatorial and residual contributions to account for molecular size and shape differences

Parameter estimation for thermodynamic models

• Regression analysis minimizes difference between predicted and experimental data using sum of squared errors as objective function
• Vapor-liquid equilibrium (VLE) data utilized to fit model parameters through P-T-x-y experimental measurements
• Least squares method applied for linear regression (simple models) and non-linear regression (complex models)
• Maximum likelihood estimation accounts for experimental uncertainties in parameter fitting
• Cross-validation technique uses portion of data for fitting and remainder for model validation
• Sensitivity analysis evaluates impact of parameter changes on model predictions and identifies most influential parameters