♨️Thermodynamics of Fluids Unit 10 – Vapor–Liquid Equilibrium

Key Concepts and Definitions

• Vapor–liquid equilibrium (VLE) describes the state where a liquid and its vapor are in equilibrium with each other
• Equilibrium occurs when the rate of evaporation equals the rate of condensation, resulting in no net change in the system
• Saturation pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature
• Dew point represents the temperature at which the saturated vapor pressure equals the ambient pressure, causing the vapor to condense into a liquid
• Bubble point denotes the temperature at which the saturated vapor pressure equals the ambient pressure, causing the liquid to vaporize and form bubbles
• Fugacity measures the tendency of a component to escape from a phase and is used to describe non-ideal behavior in mixtures
  • Fugacity is related to the chemical potential and helps determine the direction of mass transfer between phases

Thermodynamic Principles

• VLE is governed by the equality of chemical potentials of each component in both the liquid and vapor phases
• The Gibbs phase rule, $F = C - P + 2$, relates the number of degrees of freedom ($F$), components ($C$), and phases ($P$) in a system at equilibrium
  • For a single-component system at VLE, there is only one degree of freedom (either temperature or pressure can be specified)
• The Clausius–Clapeyron equation describes the relationship between saturation pressure and temperature for a pure substance
  • $\frac{dP}{dT} = \frac{\Delta H_{vap}}{T \Delta V}$, where $P$ is the saturation pressure, $T$ is the temperature, $\Delta H_{vap}$ is the enthalpy of vaporization, and $\Delta V$ is the volume change during vaporization
• The Antoine equation is an empirical relationship used to describe the vapor pressure of a pure component as a function of temperature
  • $\log P = A - \frac{B}{T + C}$, where $P$ is the vapor pressure, $T$ is the temperature, and $A$, $B$, and $C$ are component-specific constants
• The Poynting correction factor accounts for the effect of pressure on the fugacity of a liquid component in a mixture

Phase Diagrams and Equilibrium Curves

• Phase diagrams graphically represent the equilibrium states of a substance under different conditions of temperature, pressure, and composition
• For a pure substance, the phase diagram consists of regions representing solid, liquid, and vapor phases, separated by equilibrium curves
  • The triple point is the unique condition where all three phases coexist in equilibrium
• The vapor pressure curve (or saturation curve) separates the liquid and vapor regions and ends at the critical point
  • Above the critical temperature and pressure, the substance exists as a supercritical fluid with properties intermediate between those of a liquid and a gas
• For binary mixtures, the phase diagram becomes more complex, with additional regions representing two-phase equilibria (e.g., liquid-liquid, vapor-liquid)
  • The bubble point curve represents the locus of points where the first vapor bubble forms upon heating a liquid mixture
  • The dew point curve represents the locus of points where the first liquid droplet forms upon cooling a vapor mixture
• Azeotropes are special compositions of binary mixtures where the liquid and vapor compositions are equal at a given temperature and pressure
  • Azeotropic mixtures exhibit a maximum or minimum in their vapor-liquid equilibrium curves and cannot be separated by simple distillation

Equations of State and Fugacity

• Equations of state (EOS) are mathematical models that describe the relationship between pressure, volume, and temperature of a substance
• The ideal gas law, $PV = nRT$, is the simplest EOS but is only accurate at low pressures and high temperatures
  • $P$ is the pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the temperature
• More advanced EOS, such as the van der Waals, Redlich-Kwong, and Peng-Robinson equations, account for the non-ideal behavior of real gases and liquids
  • These EOS introduce parameters that capture the effects of molecular size and intermolecular attractions
• Fugacity is a thermodynamic property that describes the tendency of a component to escape from a phase and is used to account for non-ideal behavior
  • For an ideal gas, fugacity equals pressure, but for real systems, fugacity deviates from pressure
• The fugacity coefficient, $\phi$, is defined as the ratio of fugacity to pressure, $\phi = \frac{f}{P}$, and is a measure of the deviation from ideal behavior
  • Fugacity coefficients can be calculated using equations of state or empirical correlations
• The equality of fugacities in each phase is a necessary condition for phase equilibrium, $f_i^{liquid} = f_i^{vapor}$, where $i$ represents a specific component

Gibbs Energy and Chemical Potential

• Gibbs energy (or Gibbs free energy) is a thermodynamic potential that measures the maximum reversible work that can be performed by a system at constant temperature and pressure
• The total differential of Gibbs energy, $dG = -SdT + VdP + \sum_i \mu_i dn_i$, relates changes in temperature ($T$), pressure ($P$), and composition ($n_i$) to changes in Gibbs energy
  • $S$ is the entropy, $V$ is the volume, $\mu_i$ is the chemical potential of component $i$, and $n_i$ is the number of moles of component $i$
• Chemical potential is the partial molar Gibbs energy and represents the change in Gibbs energy when one mole of a component is added to a system at constant temperature, pressure, and composition of other components
  • $\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_j}$, where $j \neq i$
• At equilibrium, the chemical potentials of each component are equal in all phases, $\mu_i^{liquid} = \mu_i^{vapor}$, which is equivalent to the equality of fugacities
• The Gibbs-Duhem equation, $SdT - VdP + \sum_i n_i d\mu_i = 0$, relates changes in chemical potentials to changes in temperature and pressure, ensuring thermodynamic consistency

Raoult's Law and Ideal Solutions

• Raoult's law states that the partial vapor pressure of a component in an ideal solution is equal to the product of its mole fraction in the liquid phase and its pure component vapor pressure
  • $P_i = x_i P_i^{sat}$, where $P_i$ is the partial vapor pressure of component $i$, $x_i$ is the mole fraction of component $i$ in the liquid phase, and $P_i^{sat}$ is the saturated vapor pressure of pure component $i$
• An ideal solution is a mixture in which the interactions between unlike molecules are the same as those between like molecules, resulting in zero enthalpy of mixing and zero volume change upon mixing
• In an ideal solution, the fugacity of each component is proportional to its mole fraction in the liquid phase, $f_i = x_i f_i^{pure}$, where $f_i^{pure}$ is the fugacity of the pure component
• The Gibbs-Duhem equation for an ideal solution simplifies to $\sum_i x_i d\ln f_i = 0$, indicating that the fugacities (and chemical potentials) of the components are not independent
• Ideal solution behavior is often observed in mixtures of chemically similar components, such as hydrocarbons or isomers

Non-Ideal Behavior and Activity Coefficients

• Non-ideal behavior in mixtures arises from differences in molecular size, shape, and intermolecular interactions between components
• Activity coefficients, $\gamma_i$, are introduced to account for non-ideal behavior and relate the fugacity of a component in a mixture to its mole fraction
  • $f_i = \gamma_i x_i f_i^{pure}$, where $\gamma_i$ is the activity coefficient of component $i$
• Activity coefficients are a measure of the deviation from ideal solution behavior, with $\gamma_i = 1$ indicating ideal behavior and $\gamma_i \neq 1$ indicating non-ideal behavior
• The excess Gibbs energy, $G^E$, is the difference between the actual Gibbs energy of a mixture and the Gibbs energy of an ideal mixture at the same temperature, pressure, and composition
  • Activity coefficients are related to the excess Gibbs energy by $RT \ln \gamma_i = \left(\frac{\partial G^E}{\partial n_i}\right)_{T,P,n_j}$, where $j \neq i$
• Models such as Margules, van Laar, and Wilson equations are used to describe the composition dependence of activity coefficients and excess Gibbs energy
  • These models contain adjustable parameters that are fitted to experimental VLE data
• The NRTL (Non-Random Two-Liquid) and UNIQUAC (UNIversal QUAsiChemical) models are more advanced activity coefficient models that account for local composition effects and differences in molecular size and shape

Applications in Chemical Engineering

• VLE is crucial in the design and operation of separation processes such as distillation, absorption, and extraction
  • Understanding VLE helps determine the feasibility and efficiency of separation processes and select appropriate operating conditions
• Distillation columns rely on the differences in volatility between components to achieve separation
  • The composition of the liquid and vapor phases at each stage of the column is determined by VLE
• Flash calculations are used to determine the equilibrium composition and phase amounts when a mixture is partially vaporized or condensed at a given temperature and pressure
  • Flash calculations are essential in the design of flash drums, condensers, and evaporators
• VLE data is used to generate phase diagrams and equilibrium curves, which provide valuable information about the behavior of mixtures under different conditions
  • Phase diagrams help identify azeotropes, miscibility gaps, and critical points, which impact the choice of separation methods
• Equations of state and activity coefficient models are employed in process simulation software to predict the thermodynamic properties and phase behavior of mixtures
  • Accurate VLE predictions are essential for the reliable design and optimization of chemical processes
• VLE is also relevant in environmental engineering, such as in the assessment of air pollution and the modeling of the fate and transport of volatile organic compounds (VOCs) in the atmosphere
  • Understanding the partitioning of pollutants between the gas and liquid phases is crucial for predicting their environmental impact and developing mitigation strategies