Glossary

2.3 Equations of state and activity coefficient models

2 min read•july 24, 2024

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 to more complex cubic equations, these models account for molecular interactions and . 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 (, ), and estimate equilibrium compositions
  • Ideal gas law (PV=nRT)(PV = nRT) serves as simplest EOS, more complex models account for non-ideal behavior (van der Waals, Redlich-Kwong)

Application of cubic state equations

  • pioneered incorporating molecular attraction (a) and repulsion (b) parameters: (P+av2)(v−b)=RT(P + \frac{a}{v^2})(v - b) = RT
  • improved accuracy with temperature-dependent attraction term: P=RTv−b−aTv(v+b)P = \frac{RT}{v - b} - \frac{a}{\sqrt{T}v(v + b)}
  • further refined liquid density predictions: P=RTv−b−aα(T)v(v+b)+b(v−b)P = \frac{RT}{v - b} - \frac{a\alpha(T)}{v(v + b) + b(v - b)}
  • Cubic EOS applications include calculations, critical point estimation, and determination

Activity coefficient models for mixtures

  • Activity coefficient models predict non-ideal behavior in liquid mixtures based on
  • Models assume local composition concept and utilize binary interaction parameters
  • Margules equation offers simplest approach with single adjustable parameter: lnγi=Axj2ln\gamma_i = Ax_j^2
  • Van Laar equation extends Margules with two parameters: lnγ1=A12(1+A12A21(x1x2))2ln\gamma_1 = \frac{A_{12}}{(1 + \frac{A_{12}}{A_{21}}(\frac{x_1}{x_2}))^2}
  • Wilson equation incorporates local composition concept: lnγi=1−ln(∑jxjΛij)−∑kxkΛik∑jxjΛkjln\gamma_i = 1 - ln(\sum_j x_j\Lambda_{ij}) - \sum_k \frac{x_k\Lambda_{ik}}{\sum_j x_j\Lambda_{kj}}
  • accounts for non-random molecular orientations using three parameters (τij,Gij,αij)(\tau_{ij}, G_{ij}, \alpha_{ij})
  • combines combinatorial and residual contributions to account for molecular size and shape differences

Parameter estimation for thermodynamic models

  • 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
  • applied for linear regression (simple models) and non-linear regression (complex models)
  • accounts for experimental uncertainties in parameter fitting
  • uses portion of data for fitting and remainder for model validation
  • evaluates impact of parameter changes on model predictions and identifies most influential parameters

Key Terms to Review (20)

Activity coefficient models: Activity coefficient models are mathematical representations that describe how the behavior of a mixture deviates from ideality, specifically in terms of the non-ideal interactions between different components in a solution. These models provide a way to quantify how the presence of one substance affects the chemical potential of another, which is critical for understanding phase equilibria and thermodynamic properties of mixtures.
Compressibility Factor: The compressibility factor (Z) is a dimensionless quantity that describes how much a real gas deviates from ideal gas behavior under varying temperature and pressure conditions. It is defined as the ratio of the molar volume of a real gas to the molar volume of an ideal gas at the same temperature and pressure, expressed as $$Z = \frac{PV}{nRT}$$. Understanding the compressibility factor is crucial when applying equations of state and activity coefficient models, as it helps predict gas behavior in non-ideal situations.
Cross-validation technique: The cross-validation technique is a statistical method used to assess the predictive performance of a model by partitioning the data into subsets, training the model on some subsets while validating it on others. This approach helps in evaluating how well the model generalizes to an independent dataset and is crucial for avoiding overfitting, especially when using equations of state and activity coefficient models.
Cubic EOS: Cubic equations of state (EOS) are mathematical models that describe the behavior of real gases by relating pressure, volume, and temperature in a cubic form. These equations help predict the properties of fluids and phase behavior, playing a crucial role in chemical engineering, particularly in designing separation processes and understanding thermodynamic systems.
Enthalpy: Enthalpy is a thermodynamic property that represents the total heat content of a system, defined as the sum of its internal energy and the product of its pressure and volume. This concept is crucial for understanding energy transfers during various processes, especially in chemical reactions and phase changes, where heat is absorbed or released. It plays a vital role in analyzing energy balances in different separation processes.
Entropy: Entropy is a measure of the disorder or randomness in a system and is a key concept in thermodynamics. It reflects the degree of energy dispersion in a physical system and helps determine the feasibility of processes, particularly in separation methods. In separations, understanding entropy is crucial for evaluating energy efficiency, phase behavior, and equilibrium states, which are all significant for effective design and operation.
Excess Gibbs Energy: Excess Gibbs energy is the difference between the actual Gibbs energy of a solution and the Gibbs energy of an ideal solution at the same temperature and pressure. This term helps us understand how non-ideal behavior in mixtures affects thermodynamic properties, which is crucial in separations and understanding interactions between components in a mixture. It plays a significant role in calculating phase equilibria and determining activity coefficients in various separation processes.
Ideal Gas Law: The Ideal Gas Law is a fundamental equation that relates the pressure, volume, temperature, and number of moles of an ideal gas through the formula PV = nRT. This relationship shows how gases behave under various conditions and is essential for understanding thermodynamic principles in separations. The law serves as a foundation for more complex equations of state, connecting to the behavior of real gases and the calculations involved in separation processes.
Least Squares Method: The least squares method is a mathematical optimization technique used to minimize the sum of the squares of the residuals, which are the differences between observed and predicted values. This method is widely employed in regression analysis to fit equations of state and activity coefficient models to experimental data, ensuring that the best-fitting curve accurately represents the underlying relationship between variables.
Maximum likelihood estimation: Maximum likelihood estimation (MLE) is a statistical method used to estimate the parameters of a probabilistic model by maximizing the likelihood function. In the context of equations of state and activity coefficient models, MLE helps in determining the most probable values of parameters that describe the behavior of substances under various conditions, enabling accurate predictions in separation processes.
Non-ideal behavior: Non-ideal behavior refers to the deviations from ideal predictions in physical and chemical processes, particularly in relation to vapor-liquid and liquid-liquid equilibria. This behavior occurs when real mixtures exhibit interactions that differ from those predicted by ideal models, leading to discrepancies in phase compositions, vapor pressures, and other thermodynamic properties. Understanding non-ideal behavior is crucial for accurately modeling and designing separation processes involving different phases.
NRTL Model: The NRTL (Non-Random Two-Liquid) model is an activity coefficient model used to describe the thermodynamic behavior of liquid mixtures, particularly those involving polar and nonpolar components. It focuses on the interactions between unlike molecules and accounts for local composition fluctuations, making it useful for predicting phase equilibria in mixtures. The NRTL model plays a crucial role in separating processes, enabling the calculation of activity coefficients that help in designing efficient separation systems.
Peng-Robinson Equation: The Peng-Robinson equation is an equation of state that is widely used to describe the thermodynamic behavior of gases and liquids, especially for non-ideal systems. It provides a way to calculate properties like pressure, volume, and temperature for substances by accounting for molecular interactions and the size of molecules, which is critical in the context of separations.
Redlich-Kwong Equation: The Redlich-Kwong equation is an equation of state that models the behavior of real gases, taking into account their non-ideal interactions. It improves upon the ideal gas law by introducing parameters that account for the size of molecules and the attraction between them, making it more accurate for predicting phase behavior in various conditions. This equation is particularly relevant in the context of equations of state and activity coefficient models, as it helps in understanding thermodynamic properties and phase equilibria.
Regression analysis: Regression analysis is a statistical method used to determine the relationship between a dependent variable and one or more independent variables. It helps in predicting outcomes and identifying trends by modeling the relationship between variables, allowing for better understanding and analysis in various contexts, including those involving equations of state and activity coefficients.
Sensitivity analysis: Sensitivity analysis is a method used to determine how different values of an independent variable affect a particular dependent variable under a given set of assumptions. This technique is vital for understanding the impact of uncertainty in various parameters on the performance and outcomes of models, especially in areas like process design, modeling, and economic evaluations.
Uniquac model: The UNIQUAC model (Universal Quasi-Chemical) is a thermodynamic model used to predict the activity coefficients of components in non-ideal mixtures, based on their molecular interactions. It employs the concept of both configurational and energetic contributions to the activity coefficient, allowing it to capture complex behavior in liquid-phase systems. This model is crucial for understanding phase equilibria in various chemical engineering applications, particularly when dealing with mixtures that exhibit significant non-ideality.
Van der Waals equation: The van der Waals equation is a modified ideal gas law that accounts for the volume occupied by gas molecules and the intermolecular forces between them. This equation improves the prediction of real gas behavior, especially under high pressure and low temperature conditions, where deviations from ideality are significant. Understanding this equation is crucial for analyzing phase equilibria and the thermodynamic properties of mixtures in separation processes.
Vapor-Liquid Equilibrium: Vapor-liquid equilibrium (VLE) is a state where the rates of evaporation and condensation of a liquid are equal, resulting in a stable balance between its vapor and liquid phases. This concept is essential in various separation processes, as it helps to understand how mixtures behave and how components can be separated based on their volatilities.
Vapor-liquid equilibrium data: Vapor-liquid equilibrium data refers to the set of information that describes the relationship between the vapor and liquid phases of a substance at equilibrium conditions. This data is crucial for understanding how mixtures behave during phase changes, providing insights into boiling points, pressures, and compositions of the vapor and liquid phases. It serves as a foundation for designing separation processes like distillation, where understanding these equilibria is essential for achieving efficient separations.
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