♨️Thermodynamics of Fluids Unit 9 – Fugacity and Activity in Thermodynamics

Key Concepts and Definitions

• Fugacity represents the effective partial pressure of a component in a mixture, accounting for non-ideal behavior
• Activity is a measure of the effective concentration of a component in a mixture, relative to its standard state
• Chemical potential quantifies the change in Gibbs free energy when a component is added to a system at constant temperature and pressure
• Ideal gas assumes no intermolecular interactions and follows the ideal gas law ($PV = nRT$)
• Non-ideal behavior occurs when intermolecular interactions and molecular size affect the properties of a mixture
• Standard state refers to the reference condition at which the activity or fugacity of a component is unity (pure component at the system temperature and pressure)
• Partial molar properties describe the contribution of each component to the total property of a mixture (partial molar volume, partial molar enthalpy)

Historical Context and Development

• The concept of fugacity was introduced by Gilbert N. Lewis in 1901 to account for deviations from ideal gas behavior
• In 1923, Lewis and Merle Randall published the book "Thermodynamics and the Free Energy of Chemical Substances," which laid the foundation for the study of non-ideal mixtures
• The activity concept was developed as an extension of fugacity to describe non-ideal behavior in liquid and solid mixtures
• The development of equations of state (van der Waals, Redlich-Kwong, Peng-Robinson) enabled the calculation of fugacity and activity coefficients
• The introduction of excess functions (excess Gibbs energy, excess enthalpy) allowed for the characterization of non-ideal behavior in mixtures
• Statistical mechanics provided a molecular-level understanding of the factors influencing fugacity and activity (intermolecular forces, molecular size, and shape)

Fugacity: Theory and Applications

• Fugacity is a measure of the tendency of a component to escape from a phase or mixture
• For an ideal gas, fugacity equals pressure; for non-ideal systems, fugacity deviates from pressure
• Fugacity coefficient ($\phi$) is the ratio of fugacity to pressure, quantifying the degree of non-ideality
  • For an ideal gas, $\phi = 1$; for non-ideal systems, $\phi \neq 1$
• Fugacity is related to the chemical potential by $\mu_i = \mu_i^0 + RT \ln(f_i/f_i^0)$, where $\mu_i^0$ and $f_i^0$ are the chemical potential and fugacity at the standard state
• Fugacity is used to calculate vapor-liquid equilibrium (VLE) and chemical reaction equilibrium
• In phase equilibrium calculations, the fugacity of a component is equal in all phases at equilibrium
• Fugacity is essential for the design and optimization of separation processes (distillation, absorption, extraction)

Activity: Principles and Significance

• Activity is a measure of the effective concentration of a component in a mixture, accounting for non-ideal behavior
• For an ideal solution, activity equals mole fraction; for non-ideal solutions, activity deviates from mole fraction
• Activity coefficient ($\gamma$) is the ratio of activity to mole fraction, quantifying the degree of non-ideality
  • For an ideal solution, $\gamma = 1$; for non-ideal solutions, $\gamma \neq 1$
• Activity is related to the chemical potential by $\mu_i = \mu_i^0 + RT \ln(a_i)$, where $\mu_i^0$ is the chemical potential at the standard state
• Activity is used to calculate liquid-liquid equilibrium (LLE), solid-liquid equilibrium (SLE), and chemical reaction equilibrium in condensed phases
• In phase equilibrium calculations, the activity of a component is equal in all phases at equilibrium
• Activity is crucial for understanding and predicting the behavior of electrolyte solutions, polymer solutions, and biological systems

Relationships Between Fugacity and Activity

• Fugacity and activity are related through the fugacity coefficient and the standard state fugacity: $a_i = f_i / f_i^0$
• For an ideal gas mixture, fugacity equals partial pressure, and activity equals mole fraction
• In non-ideal mixtures, fugacity and activity are connected by the excess Gibbs energy: $\ln(\gamma_i) = (G^E/RT)_{T,P,n_j}$
• The Lewis-Randall rule states that the fugacity of a component in an ideal mixture is equal to its fugacity in the pure state at the same temperature and pressure
• The Henry's law constant relates the fugacity of a component in a mixture to its mole fraction: $f_i = H_i x_i$, where $H_i$ is the Henry's law constant
• The activity coefficient can be calculated from the fugacity coefficient using the relationship: $\ln(\gamma_i) = \ln(\phi_i) + \ln(P/P^0)$

Calculation Methods and Equations

• Equations of state (EOS) are used to calculate fugacity coefficients and fugacities in vapor and liquid phases
  • Common EOS include van der Waals, Redlich-Kwong, Soave-Redlich-Kwong (SRK), and Peng-Robinson (PR)
• Activity coefficient models are used to calculate activity coefficients in liquid mixtures
  • Examples include Margules, van Laar, Wilson, NRTL (Non-Random Two-Liquid), and UNIQUAC (Universal Quasi-Chemical) models
• The Gibbs-Duhem equation relates the changes in chemical potential, temperature, and pressure: $\sum x_i d\mu_i = -S_m dT + V_m dP$
• The Gibbs-Helmholtz equation connects the temperature dependence of the Gibbs energy to the enthalpy: $(∂(G/T)/∂T)_P = -H/T^2$
• The Clausius-Clapeyron equation describes the vapor-liquid equilibrium behavior: $dP/dT = \Delta H_{vap}/(T \Delta V)$
• The Poynting correction accounts for the effect of pressure on the fugacity of a liquid component: $f_i^L = x_i \gamma_i f_i^{0,L} \exp(\int_{P^0}^P (V_i^L/RT) dP)$

Real-World Applications and Examples

• Fugacity and activity are essential for the design and optimization of chemical processes, such as distillation, absorption, and extraction
  • Example: In the production of ethanol by fermentation, the activity of ethanol in the liquid phase affects the equilibrium and kinetics of the process
• Fugacity is used to predict the solubility of gases in liquids and the behavior of gas mixtures in pipelines and storage tanks
  • Example: The fugacity of carbon dioxide in water is crucial for understanding its solubility and transport in the context of carbon capture and sequestration
• Activity is important for understanding the behavior of electrolyte solutions, such as in batteries, fuel cells, and electrochemical processes
  • Example: In lithium-ion batteries, the activity of lithium ions in the electrolyte solution affects the performance and safety of the device
• Fugacity and activity are relevant in environmental science for modeling the distribution and fate of pollutants in air, water, and soil
  • Example: The fugacity of persistent organic pollutants (POPs) determines their partitioning between air, water, and organic matter in the environment
• In biochemistry and biotechnology, activity is used to describe the behavior of proteins, enzymes, and other biomolecules in solution
  • Example: The activity of an enzyme in a bioreactor depends on factors such as pH, temperature, and the presence of inhibitors or activators

Common Misconceptions and Pitfalls

• Confusing fugacity with pressure: While related, fugacity is not the same as pressure and accounts for non-ideal behavior
• Assuming ideal behavior: Many real systems exhibit non-ideal behavior, and using ideal gas or ideal solution assumptions can lead to significant errors
• Neglecting the effect of composition on fugacity and activity: The fugacity and activity of a component depend on the composition of the mixture and can vary significantly with changes in concentration
• Ignoring the influence of temperature and pressure: Fugacity and activity are functions of temperature and pressure, and these effects must be considered in calculations
• Using inappropriate standard states: The choice of standard state affects the values of fugacity and activity and must be consistent with the problem at hand
• Overlooking the limitations of models and equations: Equations of state and activity coefficient models have specific ranges of applicability and may not be suitable for all systems or conditions
• Forgetting to check for phase stability: In phase equilibrium calculations, it is essential to verify that the resulting phases are stable and do not undergo further separation or transformation