8.1 Criteria for phase equilibrium

Phase equilibrium is crucial for understanding how substances behave under different conditions. It's all about balance - when different phases of a material can coexist without changing. This happens when the chemical potentials of each component are equal across all phases.

Knowing these criteria helps predict phase transitions, like boiling or freezing. It's super useful for designing processes in chemical engineering, understanding geological phenomena, and even cooking! The key is minimizing Gibbs free energy to find the most stable state.

Equilibrium Fundamentals

Types of Equilibrium

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• Thermal equilibrium occurs when two systems have reached the same temperature and there is no net heat transfer between them
  • Achieved through thermal contact and exchange of energy until thermal motion of particles equalizes (conduction, convection, radiation)
  • Example: A hot metal object placed in a room will eventually cool down to room temperature
• Mechanical equilibrium is reached when there is no net force acting on a system and no tendency for movement or deformation
  • Balanced forces such as pressure, tension, and gravity result in a stable, static configuration
  • Example: A book resting on a table experiences equal and opposite forces from gravity and the table's normal force
• Chemical equilibrium is attained when the forward and reverse rates of a chemical reaction are equal, resulting in no net change in concentrations over time
  • Occurs in closed systems where reactants and products coexist at constant amounts determined by the equilibrium constant
  • Example: The reversible reaction of hydrogen and iodine gas to form hydrogen iodide reaches equilibrium when the rates of formation and decomposition are balanced
• Equilibrium conditions are the specific constraints that must be satisfied for a system to be in overall thermodynamic equilibrium
  • Requires simultaneous thermal, mechanical, and chemical equilibrium
  • Characterized by uniform temperature, pressure, and chemical potential throughout the system
  • Represents the most stable state with no tendency for spontaneous change

Achieving Equilibrium

• Equilibrium is reached through the spontaneous processes of energy and mass transfer that minimize differences in intensive properties
  • Systems naturally evolve towards equilibrium to maximize entropy and minimize free energy
  • Non-equilibrium states have gradients or imbalances that drive net flows until uniformity is achieved
• The approach to equilibrium is gradual and asymptotic, with the rate of change decreasing as the system gets closer to the final balanced state
  • Equilibration timescales depend on factors like system size, material properties, and mixing mechanisms
  • Example: Adding a drop of dye to water results in initial concentration gradients that slowly dissipate through diffusion until the color is evenly distributed
• Isolated systems always progress towards equilibrium, while open systems may be maintained in steady-state non-equilibrium conditions by external inputs and outputs
  • Equilibrium represents a dynamic balance with ongoing microscopic fluctuations but no net macroscopic changes
  • Living organisms are examples of open systems that constantly exchange energy and matter with their surroundings to sustain ordered structures and processes

Thermodynamic Criteria

Gibbs Free Energy Minimization

• The equilibrium state of a system is the one that minimizes the total Gibbs free energy ($G$) at constant temperature and pressure
  • Gibbs free energy represents the maximum reversible work that can be extracted from a system
  • Spontaneous processes always proceed in the direction of decreasing $G$ until the minimum is reached at equilibrium
• For a pure substance, equilibrium corresponds to the lowest chemical potential ($\mu$) among all possible phases at the given conditions
  • Chemical potential measures the change in $G$ with respect to the amount of substance added or removed
  • Example: Water at atmospheric pressure exists as solid ice below 0°C, liquid water between 0-100°C, and water vapor above 100°C, each being the stable phase with minimum $\mu$ in its respective temperature range
• In multicomponent systems, equilibrium is determined by minimizing the total $G$ while conserving the amounts of each component
  • The equilibrium composition is calculated by solving a system of equations involving chemical potentials and mass balances
  • Example: The distribution of a solute between two immiscible liquid phases (like iodine between water and carbon tetrachloride) reaches equilibrium when the chemical potentials in both phases are equal

Equality of Chemical Potentials

• At equilibrium, the chemical potential of each component must be uniform throughout the system and across all phases
  • Chemical potential represents the energetic driving force for mass transfer and reaction
  • Differences in chemical potential between phases or locations lead to spontaneous processes that restore equality
• For multiple phases in equilibrium, the chemical potentials of each component are equal in all phases
  • This condition allows the calculation of partition coefficients, solubilities, and vapor pressures
  • Example: The solubility of a gas in a liquid is determined by equating the chemical potential expressions for the gas and dissolved phases
• In reacting systems, equilibrium is reached when the sum of chemical potentials of reactants equals that of products, weighted by stoichiometric coefficients
  • This relationship is equivalent to the minimization of Gibbs free energy and gives rise to the equilibrium constant expression
  • Example: For the synthesis of ammonia from nitrogen and hydrogen ($N_2 + 3H_2 \rightleftharpoons 2NH_3$), equilibrium occurs when $\mu_{N_2} + 3\mu_{H_2} = 2\mu_{NH_3}$

Phase Coexistence

• At specific conditions of temperature and pressure, multiple phases of a substance can coexist in equilibrium
  • Coexistence occurs along lines or curves in phase diagrams where the chemical potentials of the phases are equal
  • Example: On the melting curve of a pure substance, solid and liquid phases have the same chemical potential and can exist together
• Phase transitions happen when the equilibrium shifts from one phase to another due to changes in conditions
  • Transitions are characterized by abrupt changes in properties like density, enthalpy, and entropy
  • Example: Boiling of a liquid occurs at the saturation temperature where the liquid and vapor chemical potentials become equal, resulting in a transition to the vapor phase
• The Gibbs Phase Rule ($F = C - P + 2$) relates the number of components ($C$), phases ($P$), and degrees of freedom ($F$) in a system at equilibrium
  • Degrees of freedom represent the number of intensive variables that can be independently varied without changing the number of phases
  • Example: For a single-component system with two phases in equilibrium (like water and steam), there is only one degree of freedom, so specifying either temperature or pressure automatically fixes the other variable along the coexistence curve