The Phase Rule

• The Phase Rule was first deduced by J. W. Gibbs in 1876
• The Phase Rule is written as P + F = C + 2, where:
  • P = number of phases
  • F = number of degrees of freedom
  • C = number of components of a system in equilibrium
• The Phase Rule applies to macroscopic systems in heterogeneous equilibrium
• Changes in pressure, temperature, and concentration influence the rule
• Equilibrium is assumed to be unaffected by gravitational, electrical, magnetic and surface forces

Homogeneous Equilibrium

• A system is homogeneous when it is uniform throughout its volume
• Properties are the same in all parts, excluding single molecular species
• Equilibrium in a homogeneous system is termed homogeneous equilibrium
• Example: methylacetate = acetic acid + methyl alcohol

Heterogeneous Equilibrium

• A heterogeneous system consists of two or more distinct homogeneous regions
• Example: ice and water, or carbon tetrachloride and water
• Homogeneous regions are separated by surfaces or interfaces with sudden changes in physical and chemical properties
• Equilibrium between physically distinct regions or phases is termed heterogeneous equilibrium

Conditions for Heterogeneous Equilibrium

• Thermal Equilibrium: no flow of heat is present; equilibrium is Tα = Tβ for two phases α and β
• Mechanical Equilibrium: pressure is constant throughout; equilibrium is Pα = Pβ for two phases α and β
• Chemical Equilibrium: the rate of the forward reaction equals the rate of the corresponding backward reaction

Types of Equilibrium

• True Equilibrium: exists when a system is in thermal, mechanical, and chemical equilibrium
  • External conditions can be altered and returned without changing the state
  • Example: A salt in contact with saturated solution
• Metastable Equilibrium: exists when a system is in thermal, mechanical, and chemical equilibrium but is not in the most stable state
  • Example: Liquid water and its vapor at -1°C (super-cooled)
  • System is stable if not disturbed
• Apparent Equilibrium: the approach to an equilibrium position is so slow that it is not detected
  • Example: Hydrogen, oxygen and water in a closed vessel

Other relevant points regarding equilibrium

• The Phase Rule cannot distinguish between metastable and true equilibrium
• The rule can only be applied to apparent equilibria if the reaction is slow

System Definition

• A substance or mixture of substances isolated from all other substances
• The "water system" means water is separated from all other substances
• Changing pressure and temperature on the various phases may be observed

Phases (P) Definition

• Homogeneous, physically distinct and mechanically separable part of a system
• Each phase must be separated from other phases by a physical boundary
• "Mechanically separable" operations:
  • Hand picking crystals
  • Filtration
  • Separation of two liquid phases without interfering with pressure, temperature, or composition (ex: fractional distillation or solvent extraction)

Examples of Phases

• Ice/liquid water/water vapor constitutes three phases
• Any number of gases mixes in all proportions constitutes one phase
• A saturated salt solution constitutes a three-phase solution (undissolved solid and vapor)
• CaCO3(s) ⇌ CaO(s) + CO2(g) represents three phases: two solids and a gas
• Mercury/carbon tetrachloride/water represents a four-phase system (three immiscible liquids and one vapor phase)

Components (C) Definition

• The smallest number of independently variable constituents by which the composition of each phase can be expressed
• The minimum number of molecular species in terms of which the composition of all phases can be quantitatively expressed

Component Examples

• Ice/water/water vapor system is one component (H2O)
• CaCO3(s) ⇌ CaO(s) + CO2(g) consists of two components (C = 2)
• Na2SO4, Na2SO4·7H2O, Na2SO4·10H2O, Na2SO4 solution, solid ice and water vapor is a two component system (C = 2)
• NH4Cl(s) ⇌ NH3(g) + HCl(g) is a one-component system
• 3Fe(s) + 4H2O(g) ⇌ Fe3O4(s) + 4H2(g) is a three-phase system with three components (C = 3)

Degrees of Freedom or Variance (F)

• The minimum number of intensive variables by which the state of the system is completely defined

Intensive variables

• Variables are independent of the mass or size of the system
• Examples are pressure, temperature, concentration, density, refractive index, and molar entropy

Extensive variables

• Variables are dependent of the mass or size of the system.

Significance of Components and Phases

• As the number of components (C) increases, there are more independent variables
• As the number of phases (P) increases, there are fewer independent variables

Degrees of Freedom for One-Component System

• One phase (P = 1): F = 1 - 1 + 2 = 2 (bivariant)
• Two phases (P = 2): F = 1 - 2 + 2 = 1 (univariant)
• Three phases (P = 3): F = 1 - 3 + 2 = 0 (invariant)

Derivation of Phase Rule

• For a one-component system, pressure and temperature are the only intensive variables
• For a system where one phase contains two components, there is an additional variable (component ratio)
• If one phase contains C components, there are (C - 1) composition variables
• For a system of P phases, there are P(C - 1) such variables plus two variables for pressure and temperature
• Total number of variables: P(C - 1) + 2
• In general, for P phases, there can be (P - 1) equations for each component
• For C components, the number of equations is C(P - 1)
• Number of variables - number of equations = number of independent variables
• P(C - 1) + 2 - C(P - 1) = F or P + F = C + 2