9.2 Calculation methods for fugacity
3 min read•august 6, 2024
is a key concept in thermodynamics, helping us understand behavior. It's like pressure, but accounts for non-ideal interactions between molecules. Knowing how to calculate fugacity is crucial for predicting and chemical reactions in real systems.
There are several ways to calculate fugacity, each with its pros and cons. From equations of state to graphical methods and , these techniques help us bridge the gap between ideal gas models and real-world behavior.
Equation of State Methods
Virial Equation and Compressibility Factor
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- method calculates fugacity using an equation that relates pressure, volume, and temperature of a substance
- Equations of state (ideal gas law, , ) describe the behavior of gases and liquids under various conditions
- is a power series expansion used to calculate the and fugacity of a gas
- Virial coefficients (, , , etc.) depend on temperature and account for gas non-ideality due to intermolecular interactions
- Truncated after the second () or third () virial coefficient for most applications
- Compressibility factor () measures the deviation of a gas from ideal behavior
- Defined as the ratio of the actual molar volume to the molar volume of an ideal gas at the same temperature and pressure
- for an ideal gas, for a gas with attractive intermolecular forces, and for a gas with repulsive intermolecular forces
- Calculated using the virial equation or equations of state (Redlich-Kwong, Peng-Robinson)
Graphical and Correlation Methods
Graphical Integration and Corresponding States Principle
- involves plotting experimental data (pressure vs. molar volume) and integrating the area under the curve to determine fugacity
- Requires accurate experimental data and numerical integration techniques (trapezoidal rule, Simpson's rule)
- Provides a visual representation of the relationship between pressure and molar volume
- states that substances at the same , pressure, and volume have similar properties
- Reduced properties are normalized by their critical values (reduced temperature , , )
- Allows for the prediction of properties for substances with limited experimental data using data from well-characterized substances (methane, carbon dioxide)
Generalized Correlations for Fugacity Calculation
- Generalized correlations are empirical equations that relate fugacity to reduced temperature, pressure, and volume
- Developed using experimental data for a wide range of substances
- Examples include the and the
- Generalized correlations provide a quick and convenient method for estimating fugacity without the need for extensive experimental data or complex equations of state
- Require only the critical properties and acentric factor of the substance
- Less accurate than equations of state or graphical methods, particularly for substances with unique intermolecular interactions (hydrogen bonding, polar molecules)
Fugacity Calculation Rules
Lewis and Randall Rule for Ideal Mixtures
- states that the fugacity of a component in an ideal mixture is equal to its multiplied by its fugacity in the pure state at the same temperature and pressure
- , where is the fugacity of component in the mixture, is the mole fraction of component , and is the fugacity of pure component at the same temperature and pressure
- are characterized by zero , zero , and no intermolecular interactions between unlike molecules
- Examples of ideal mixtures include mixtures of isotopes (helium-3 and helium-4) and mixtures of chemically similar substances (benzene and toluene)
- Lewis and Randall rule simplifies fugacity calculations for ideal mixtures by relating the mixture fugacity to pure component fugacities and mole fractions
- Requires knowledge of pure component fugacities, which can be calculated using equations of state, graphical methods, or generalized correlations
- Not applicable to non-ideal mixtures, where intermolecular interactions between unlike molecules lead to deviations from ideal behavior
Key Terms to Review (22)
Compressibility Factor: The compressibility factor, denoted as Z, is a dimensionless quantity that describes how much a real gas deviates from ideal gas behavior. It relates the molar volume of a real gas to the molar volume predicted by the ideal gas law under the same temperature and pressure conditions, highlighting the limitations of the ideal gas equation and the nature of real gases.
Corresponding states principle: The corresponding states principle is a concept in thermodynamics that suggests that the properties of gases can be related to one another when they are at the same reduced temperature and reduced pressure, regardless of their specific identity. This principle helps simplify the analysis of gases and fluids by allowing predictions of their behavior based on a universal curve that applies to all fluids, which is particularly useful in the calculation of fugacity and other thermodynamic properties.
Enthalpy of mixing: Enthalpy of mixing refers to the change in enthalpy that occurs when two or more substances are mixed together. This thermodynamic property is crucial in understanding how the interactions between different components affect the overall energy change during mixing, influencing processes such as chemical reactions and phase changes. By analyzing the enthalpy of mixing, one can predict how the properties of mixtures will behave and the energy implications of those behaviors.
Equation of state: An equation of state is a mathematical relationship that describes how the properties of a substance, such as pressure, volume, and temperature, are interrelated. This relationship helps in understanding the behavior of fluids under varying conditions, providing insight into their thermodynamic properties. By establishing connections between these properties, equations of state are essential for calculating fugacity, which is crucial in predicting how real gases behave compared to ideal gases.
Fugacity: Fugacity is a measure of a substance's tendency to escape or vaporize, representing an effective pressure that reflects non-ideal behavior in gases and liquids. It connects the concept of chemical potential with real-world applications, making it crucial for understanding phase equilibria and calculations involving mixtures.
Generalized correlations: Generalized correlations are empirical relationships that relate various properties of fluids, allowing for the prediction of behavior across different substances under varying conditions. These correlations simplify the complex interactions between fluid properties by providing a universal approach that can be applied to gases and liquids, making it easier to estimate critical values, phase behavior, and fugacity without extensive experimental data.
Graphical integration: Graphical integration refers to the process of estimating the area under a curve or between curves using graphical methods, often through techniques like the trapezoidal rule or Simpson's rule. This approach allows for the visual representation of data, making it easier to understand complex relationships and trends in the context of calculating properties like fugacity.
Ideal mixtures: Ideal mixtures refer to a theoretical concept in thermodynamics where components of a mixture behave ideally, meaning they follow Raoult's Law without deviations. In such mixtures, interactions between different species are similar to those within each individual component, allowing for predictable phase behavior and properties. This ideal behavior simplifies calculations for properties like fugacity and vapor-liquid equilibrium, making it easier to analyze systems involving both ideal and non-ideal mixtures.
Lee-kesler correlation: The Lee-Kesler correlation is a method used to estimate the properties of real gases and liquids, particularly focusing on their thermodynamic behavior. This correlation provides a way to relate the compressibility factor of a gas or liquid to its temperature and pressure, allowing for better predictions in processes involving phase changes and non-ideal behavior. It's essential for understanding how gases and liquids deviate from ideal behavior under various conditions.
Lewis and Randall Rule: The Lewis and Randall Rule is a principle used in thermodynamics to relate the fugacity of a component in a mixture to its activity coefficient, particularly in non-ideal solutions. It helps in calculating the fugacity of a species by relating it to the fugacity of an ideal gas at the same temperature and pressure, facilitating more accurate modeling of real fluid behaviors.
Mole fraction: Mole fraction is a way of expressing the concentration of a component in a mixture, defined as the ratio of the number of moles of that component to the total number of moles of all components in the mixture. This concept is essential for understanding how different substances interact and behave in mixtures, particularly in calculations involving fugacity, vapor-liquid equilibrium (VLE), and the properties of solutions. It serves as a crucial parameter in equations related to Raoult's law and Henry's law, which describe the behavior of ideal and non-ideal solutions.
Phase Equilibria: Phase equilibria refers to the condition where different phases of a substance (such as solid, liquid, and gas) coexist in a stable manner without any net change over time. This concept is crucial for understanding how substances behave under varying temperatures and pressures, and it helps in predicting the composition and properties of mixtures. When studying phase equilibria, we often look at how temperature and pressure influence the distribution of components across different phases, which is vital for processes like distillation and extraction.
Pitzer Correlation: The Pitzer correlation is an equation used to calculate the fugacity coefficients of gases and liquids in a mixture, particularly useful for systems with strong interactions between molecules. This correlation incorporates both the temperature and pressure effects, allowing for accurate predictions of phase behavior in non-ideal systems. The Pitzer correlation enhances our understanding of how real gases deviate from ideal gas behavior, especially at high pressures and low temperatures.
Real gas: A real gas is a substance that does not behave ideally due to interactions between its molecules and the volume occupied by the gas itself. Unlike an ideal gas, which follows the ideal gas law strictly under all conditions, real gases exhibit deviations from this behavior at high pressures and low temperatures, where intermolecular forces become significant. Understanding real gas behavior is crucial for accurate calculations in thermodynamics and chemical engineering.
Redlich-Kwong Equation: The Redlich-Kwong equation is a cubic equation of state used to describe the behavior of real gases by relating pressure, volume, and temperature. It improves upon the van der Waals equation by providing better predictions for gases at high pressures and low temperatures. This equation is crucial for calculating properties like fugacity and understanding the phase behavior of substances.
Reduced Pressure: Reduced pressure is a dimensionless quantity defined as the ratio of the pressure of a substance to its critical pressure. It helps in understanding the behavior of substances by allowing comparisons across different states, which is vital for correlating their properties under varying conditions. This concept links closely with how substances behave near their critical points, which is essential for applying generalized correlations and interpreting thermodynamic charts effectively.
Reduced Temperature: Reduced temperature is a dimensionless quantity defined as the ratio of the temperature of a substance to its critical temperature. It provides a way to compare the behavior of different substances under varying conditions, showing how they behave relative to their critical points. This concept is crucial for understanding how fluids behave near their phase transitions and helps in developing correlations for both gases and liquids.
Reduced Volume: Reduced volume is a dimensionless quantity that represents the ratio of the actual volume of a substance to its volume at a reference state, typically at its critical point. It helps in comparing different substances and understanding their behavior under varying conditions by normalizing the volume based on critical parameters. This concept plays a significant role in analyzing fluid properties, particularly in the context of corresponding states, generalized correlations, and fugacity calculations.
Van der Waals equation: The van der Waals equation is a modified ideal gas equation that accounts for the finite size of particles and the interactions between them. It provides a more accurate representation of real gas behavior, particularly under high pressure and low temperature conditions, connecting closely to various thermodynamic properties and state variables.
Virial Equation: The virial equation is a mathematical expression that relates the pressure, volume, and temperature of a gas, allowing for deviations from ideal gas behavior by incorporating interaction between particles. It expands upon the ideal gas law by including terms that account for intermolecular forces and the size of the molecules, making it useful for both ideal and real gases. This equation is especially significant when studying gases under high pressure or low temperature conditions, where real gas behavior becomes pronounced.
Volume change on mixing: Volume change on mixing refers to the alteration in total volume that occurs when two or more substances are combined. This phenomenon can be due to several factors such as intermolecular interactions, differences in molecular size, and the nature of the substances being mixed, leading to either an increase or decrease in volume.
Z: In thermodynamics, 'z' refers to the compressibility factor, which is a dimensionless quantity used to describe the deviation of a real gas from ideal gas behavior. 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. The compressibility factor helps in understanding how interactions between gas molecules affect their behavior under various conditions.