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🧊Thermodynamics II

Vapor-liquid equilibrium is all about balance. It's when the liquid and vapor phases of a mixture are in perfect harmony, with each component's fugacity equal in both phases. This concept is crucial for understanding phase behavior and separation processes.

Fugacity is like a component's "escape tendency" from a mixture. It's affected by temperature, pressure, and composition. In ideal solutions, fugacity follows simple rules, but real-world mixtures often deviate, making things more complex and interesting.

Vapor-Liquid Equilibrium Criterion

Equilibrium Condition

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  • At vapor-liquid equilibrium, the fugacity of each component in the liquid phase is equal to the fugacity of that component in the vapor phase
  • For a multicomponent system, the fugacity of each component in the liquid phase is equal to the fugacity of that component in the vapor phase, which can be expressed as: fiL=fiVf_i^L = f_i^V
  • The fugacity of a pure component is equal to its vapor pressure at the system temperature

Fugacity in Mixtures

Fugacity Calculation

Pure Components

  • For a pure component, the fugacity is equal to the vapor pressure at the system temperature: fipure=Pisatf_i^{pure} = P_i^{sat}
  • The fugacity of a component in an ideal gas mixture is equal to its partial pressure: fi=yiPf_i = y_i P, where yiy_i is the mole fraction of component ii in the vapor phase and PP is the total pressure

Mixtures

  • In a non-ideal mixture, the fugacity of a component is related to its mole fraction, total pressure, and fugacity coefficient: fi=xiϕiPf_i = x_i \phi_i P (liquid phase) or fi=yiϕiPf_i = y_i \phi_i P (vapor phase), where xix_i and yiy_i are the mole fractions of component ii in the liquid and vapor phases, respectively, and ϕi\phi_i is the fugacity coefficient
  • The fugacity coefficient can be calculated using equations of state or empirical correlations, such as the Virial equation, cubic equations of state, or activity coefficient models (Wilson, NRTL, UNIQUAC)
  • The fugacity coefficient for a component in a mixture depends on temperature, pressure, and composition, and it approaches unity as the system approaches ideal behavior

Ideal vs Non-ideal Solutions

Ideal Solutions

  • An ideal solution is a hypothetical mixture in which the interactions between molecules of different components are identical to the interactions between molecules of the same component
  • In an ideal solution, the fugacity of each component is directly proportional to its mole fraction, following Raoult's law: fi=xiPisatf_i = x_i P_i^{sat}, where PisatP_i^{sat} is the vapor pressure of the pure component at the system temperature

Non-ideal Solutions

  • Non-ideal solutions exhibit deviations from Raoult's law due to differences in molecular interactions between components, such as size, shape, or polarity differences
  • Positive deviations from Raoult's law occur when the interactions between unlike molecules are weaker than those between like molecules, leading to higher vapor pressures and increased volatility (ethanol-water)
  • Negative deviations from Raoult's law occur when the interactions between unlike molecules are stronger than those between like molecules, resulting in lower vapor pressures and decreased volatility (acetone-chloroform)
  • The extent of non-ideality in a solution can be quantified using activity coefficients, which relate the actual fugacity of a component to its ideal fugacity: fi=γixifipuref_i = \gamma_i x_i f_i^{pure}, where γi\gamma_i is the activity coefficient of component ii

Analyzing Vapor-Liquid Equilibrium Data

Equations of State and Activity Coefficient Models

  • Equations of state, such as the Virial equation or cubic equations (van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson), can be used to calculate fugacity coefficients and predict vapor-liquid equilibrium behavior
  • Activity coefficient models, such as Wilson, NRTL, or UNIQUAC, can be used to calculate activity coefficients and account for non-ideal behavior in the liquid phase
  • The choice of equation of state or activity coefficient model depends on the system's temperature, pressure, and composition, as well as the availability of experimental data for parameter estimation

Graphical Representation and Interpretation

  • Vapor-liquid equilibrium data can be presented in the form of xyxy diagrams, which plot the mole fractions of components in the vapor phase against those in the liquid phase at constant temperature and pressure
  • Azeotropes, which are mixtures with a constant boiling point and composition, can be identified from xyxy diagrams as points where the equilibrium line intersects the diagonal (ethanol-water, acetone-methanol)
  • The relative volatility, αij=(yi/xi)/(yj/xj)\alpha_{ij} = (y_i/x_i) / (y_j/x_j), can be used to compare the ease of separation of components in a mixture, with higher values indicating a more feasible separation
  • The consistency of experimental vapor-liquid equilibrium data can be assessed using thermodynamic consistency tests, such as the Redlich-Kister or Herington tests, which verify that the data satisfy the Gibbs-Duhem equation

Key Terms to Review (28)

Activity coefficient: The activity coefficient is a factor used in thermodynamics to account for deviations from ideal behavior in a mixture of substances. It relates the chemical potential of a species in a solution to its concentration, allowing for accurate predictions of phase equilibria and chemical reactions under non-ideal conditions. Understanding activity coefficients is crucial for analyzing vapor-liquid equilibrium and fugacity, as they reflect how interactions between molecules affect their effective concentrations in a given phase.
Activity coefficient models: Activity coefficient models are mathematical representations used to estimate the non-ideal behavior of mixtures, particularly in relation to vapor-liquid equilibrium. They provide a way to correct for deviations from ideality in phase behavior by accounting for interactions between different species in a mixture. These models are crucial for accurately predicting properties like fugacity and calculating equilibrium conditions in thermodynamic systems.
Azeotropes: Azeotropes are mixtures of two or more liquids that exhibit a constant boiling point and composition throughout the distillation process, meaning they behave like a single substance. This unique property arises from the specific interactions between the components in the mixture, leading to a situation where vapor and liquid phases have the same composition at that boiling point. Azeotropes are significant because they challenge conventional distillation methods, making it difficult to separate the components completely.
Clausius-Clapeyron Equation: The Clausius-Clapeyron Equation is a fundamental relation in thermodynamics that describes the relationship between the pressure and temperature at which phase changes occur, particularly between the liquid and vapor phases. This equation helps to understand how the vapor pressure of a substance changes with temperature and is essential for analyzing phase diagrams, chemical potentials, and equilibrium states.
Critical Pressure: Critical pressure is the pressure required to liquefy a substance at its critical temperature, beyond which distinct liquid and gas phases do not exist. Understanding critical pressure is crucial for analyzing vapor-liquid equilibrium, as it defines the conditions under which a substance can transition from gaseous to liquid form, influencing phase behavior and thermodynamic calculations in various systems.
Critical temperature: The critical temperature is the highest temperature at which a substance can exist as a liquid, beyond which it becomes impossible to liquefy the substance regardless of the pressure applied. At this temperature, the properties of the liquid and gas phases converge, leading to the formation of a supercritical fluid that exhibits unique characteristics. Understanding this concept is crucial in analyzing phase behavior and vapor-liquid equilibrium in various thermodynamic systems.
Cubic equations of state: Cubic equations of state are mathematical models used to describe the behavior of gases and liquids, specifically their pressure, volume, and temperature relationships. These equations are essential for understanding phase equilibria and the properties of substances, particularly in analyzing vapor-liquid equilibrium and calculating fugacity, which is a measure of a substance's tendency to escape or expand from a phase.
Equilibrium Condition: The equilibrium condition refers to the state in which a system's macroscopic properties remain constant over time, indicating that the rates of opposing processes are equal. In the context of vapor-liquid systems, this condition signifies that the chemical potentials and fugacities of the vapor and liquid phases are equal, leading to no net change in the amount of substance in each phase. Understanding this balance is crucial for analyzing phase transitions and the behavior of mixtures in thermodynamic processes.
Excess gibbs free energy: Excess Gibbs free energy is the difference between the Gibbs free energy of a real solution and that of an ideal solution at the same temperature and pressure. This term helps quantify deviations from ideal behavior in mixtures and is essential for understanding how components interact in solutions, particularly in terms of vapor-liquid equilibrium and the fugacity of species involved. It plays a critical role in determining the stability and spontaneity of phase changes in multi-component systems.
Fugacity: Fugacity is a thermodynamic property that represents the effective pressure of a component in a mixture, allowing us to understand how that component behaves in real systems. It can be thought of as a corrected pressure that accounts for non-ideal behavior, particularly in vapor-liquid equilibrium situations. The concept helps relate the chemical potential of a substance to its behavior under varying conditions, making it essential for analyzing phase equilibria and calculating equilibrium constants.
Fugacity calculation: Fugacity calculation is a method used to determine the effective pressure of a species in a non-ideal mixture, representing its escaping tendency from the mixture. This concept is crucial in understanding vapor-liquid equilibrium, as it allows for the assessment of how real gases deviate from ideal behavior, particularly under varying temperature and pressure conditions. It plays an important role in predicting phase behavior and understanding thermodynamic properties in various chemical processes.
Fugacity in mixtures: Fugacity in mixtures refers to a thermodynamic property that represents the effective pressure exerted by a component in a mixture, particularly in vapor-liquid equilibrium. It is used as a measure of the chemical potential of a species and helps to describe how a substance behaves in the presence of other substances. Fugacity accounts for non-ideal behavior in mixtures, allowing for accurate predictions of phase behavior and equilibrium conditions.
Gibbs Phase Rule: The Gibbs Phase Rule is a fundamental principle in thermodynamics that provides a relationship between the number of phases present in a system, the number of components, and the degrees of freedom available for that system. This rule helps to determine the possible states a system can exist in and plays a crucial role in understanding phase diagrams and vapor-liquid equilibrium, allowing for the prediction of how changes in temperature, pressure, or composition affect the phases present.
Ideal solution: An ideal solution is a mixture of two or more components where the interactions between unlike molecules are equal to the interactions among like molecules. This concept is crucial for understanding vapor-liquid equilibrium, as it simplifies the calculations and predictions of phase behavior in mixtures, assuming that Raoult's law applies perfectly across all concentrations.
Ideal solutions: Ideal solutions are homogeneous mixtures of two or more components where the interactions between different molecules are similar to the interactions among like molecules. In an ideal solution, the enthalpy of mixing is zero, meaning there is no heat absorbed or released when the components are combined. This concept helps in understanding vapor-liquid equilibrium and fugacity, as it simplifies the calculations and predictions of behavior in mixtures.
Molar volume at saturation: Molar volume at saturation refers to the volume occupied by one mole of a substance when it is in equilibrium between its vapor and liquid phases, under specified temperature and pressure conditions. This concept is crucial for understanding the behavior of substances in vapor-liquid equilibrium and directly relates to fugacity, as it provides insights into the distribution of molecules in both phases and their tendency to escape into the vapor phase.
Negative deviations: Negative deviations refer to a situation in vapor-liquid equilibrium where the actual vapor pressure of a solution is lower than what would be predicted by Raoult's Law. This phenomenon indicates that the interactions between different components in a mixture are stronger than those between like components, leading to lower vapor pressures than expected. Understanding negative deviations is crucial for analyzing the behavior of mixtures, particularly in terms of how they influence fugacity and phase equilibrium.
Non-ideal behavior: Non-ideal behavior refers to the deviation of real substances from the predictions of ideal models, particularly in their thermodynamic properties during phase changes. This term is crucial in understanding how real fluids exhibit interactions that differ from those of ideal gases or liquids, leading to differences in vapor-liquid equilibrium and fugacity. Real systems often experience complexities such as molecular interactions, non-uniform distribution of energy, and pressure variations that affect their equilibrium states.
Non-ideal solutions: Non-ideal solutions are mixtures where the interactions between different components lead to deviations from ideal behavior, meaning the properties of the solution do not align with Raoult's Law. These deviations occur due to factors such as differences in molecular size, shape, and the strength of intermolecular forces. Non-ideal solutions often show variations in vapor pressures and concentrations, impacting vapor-liquid equilibrium and fugacity calculations.
Peng-Robinson: The Peng-Robinson equation of state is a thermodynamic model used to describe the behavior of real gases, particularly in relation to vapor-liquid equilibrium. This equation accounts for molecular size and interactions between particles, making it useful in predicting properties like pressure, temperature, and volume for various substances. It's especially relevant in the study of phase equilibria, where it helps in understanding how components separate into different phases under varying conditions.
Positive deviations: Positive deviations refer to the behavior of a real solution where the interactions between different molecules result in a vapor phase that is more abundant than predicted by ideal behavior. This occurs when the vapor pressure of the solution is higher than that calculated from Raoult's Law. Understanding positive deviations helps in analyzing non-ideal solutions, especially in contexts such as vapor-liquid equilibrium and fugacity.
Raoult's Law: Raoult's Law states that the partial vapor pressure of a component in a solution is directly proportional to its mole fraction in the liquid phase. This relationship is crucial for understanding how mixtures behave, particularly when analyzing vapor-liquid equilibria and the stability of different phases in a system. The law highlights the connection between chemical potential and how it influences phase behavior in mixtures, making it fundamental for exploring phase stability criteria.
Redlich-Kwong: The Redlich-Kwong equation is an empirical thermodynamic model that describes the behavior of real gases by accounting for non-ideal interactions between molecules. It improves upon the ideal gas law by including a temperature-dependent volume correction and a term that accounts for the attraction between molecules, making it useful in predicting vapor-liquid equilibrium and fugacity in various systems.
Relative volatility: Relative volatility is a dimensionless number that describes the ratio of the vapor pressures of two components in a liquid-vapor equilibrium mixture. It serves as a measure of how easily one component can be separated from another during processes like distillation, where higher values indicate a greater tendency for separation. This concept connects to the principles of vapor-liquid equilibrium and fugacity, which are critical in understanding how components behave under varying temperature and pressure conditions.
Soave-Redlich-Kwong: The Soave-Redlich-Kwong equation is a modified version of the original Redlich-Kwong equation of state, designed to better predict the behavior of real gases, particularly in vapor-liquid equilibrium situations. This equation incorporates a temperature-dependent parameter that improves its accuracy in estimating phase behavior and fugacity, making it valuable for understanding how gases behave under different conditions, especially in chemical processes.
Van der Waals: Van der Waals refers to the weak intermolecular forces that arise from the interactions between molecules, which include attractions due to dipole-dipole interactions, induced dipole interactions, and London dispersion forces. These forces play a significant role in determining the behavior of real gases and liquids compared to ideal models, particularly in vapor-liquid equilibrium where they influence properties such as fugacity and phase transitions.
Vapor-liquid equilibrium: Vapor-liquid equilibrium refers to the state in which a liquid and its vapor coexist at a certain temperature and pressure, with no net change in the amount of each phase over time. In this state, the rate of evaporation of the liquid equals the rate of condensation of the vapor, leading to a balance between the two phases. This concept is crucial for understanding phase changes and calculating properties like fugacity, which helps predict how substances behave under varying conditions.
Virial Equation: The virial equation is a mathematical relationship that connects the pressure, volume, and temperature of a gas to its molecular interactions, describing how real gases deviate from ideal behavior. It provides a way to express the pressure of a gas in terms of its density and temperature using a series expansion, where coefficients (virial coefficients) account for intermolecular forces. This equation is crucial for understanding vapor-liquid equilibrium and fugacity as it helps to quantify non-ideal behavior of gases in various thermodynamic conditions.
Activity coefficient
See definition

The activity coefficient is a factor used in thermodynamics to account for deviations from ideal behavior in a mixture of substances. It relates the chemical potential of a species in a solution to its concentration, allowing for accurate predictions of phase equilibria and chemical reactions under non-ideal conditions. Understanding activity coefficients is crucial for analyzing vapor-liquid equilibrium and fugacity, as they reflect how interactions between molecules affect their effective concentrations in a given phase.

Term 1 of 28

Key Terms to Review (28)

Activity coefficient
See definition

The activity coefficient is a factor used in thermodynamics to account for deviations from ideal behavior in a mixture of substances. It relates the chemical potential of a species in a solution to its concentration, allowing for accurate predictions of phase equilibria and chemical reactions under non-ideal conditions. Understanding activity coefficients is crucial for analyzing vapor-liquid equilibrium and fugacity, as they reflect how interactions between molecules affect their effective concentrations in a given phase.

Term 1 of 28

Activity coefficient
See definition

The activity coefficient is a factor used in thermodynamics to account for deviations from ideal behavior in a mixture of substances. It relates the chemical potential of a species in a solution to its concentration, allowing for accurate predictions of phase equilibria and chemical reactions under non-ideal conditions. Understanding activity coefficients is crucial for analyzing vapor-liquid equilibrium and fugacity, as they reflect how interactions between molecules affect their effective concentrations in a given phase.

Term 1 of 28



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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.