Phase Rule, Homogeneous and Heterogeneous Equilibrium

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

Homogeneous Equilibrium

• A system is considered homogeneous when it's uniform throughout its volume, meaning its properties are the same in all parts.
• Homogeneous equilibrium is any equilibrium occurring in a homogeneous system.
• An example of homogeneous equilibrium is: CH3COOCH3 = CH3COOH + CH3OH (methylacetate = acetic acid + methyl alcohol).

Heterogeneous Equilibrium

• A heterogeneous system consists of two or more distinct homogeneous regions.
• Examples of heterogeneous systems include ice and water, or carbon tetrachloride and water.
• Homogeneous regions or phases are separated by surfaces or interfaces with sudden changes in physical and chemical properties.
• Heterogeneous equilibrium exists between various physically distinct regions or phases.

Conditions for Heterogeneous Equilibrium

• Thermal equilibrium: no heat flow between parts of the system (Tα = Tβ for two phases alpha and beta).
• Mechanical equilibrium: pressure is constant throughout the system (Pα = Pβ for two phases alpha and beta).
• Chemical equilibrium: the rate of each 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; if external conditions are altered and returned to their originals values, the same state will be given.
• A salt in contact with its saturated solution is an example of a system exhibiting true equilibrium.
• Metastable equilibrium: exists when a system is in thermal, mechanical, and chemical equilibrium but not in the most stable state.
• Liquid water and its vapor at -1°C (super-cooled) is an example of metastable equilibrium.
• When undisturbed, it's stable but not in the most stable equilibrium.
• Apparent equilibrium: arises when approaching an equilibrium position is so slow that it goes undetected.
• The Phase Rule cannot distinguish between metastable and true equilibrium states.
• The rule may only be applied to apparent if the reaction is very slow.

System Definition

• A system is a substance or a mixture of substances isolated (in some way) from all other substances.
• "The water system" means that the chemical substance, water, is separated from all other substances.
• Changing pressure and temperature on the various phases may be observed.

Phases (P)

• A phase is a homogeneous, physically distinct, and mechanically separable part of a system.
• Each phase must be separated from other phases by a physical boundary.
• 'Mechanically separable' covers operations like hand-picking crystals, filtration, and separation of liquid phases without interfering with pressure, temperature, or composition (e.g., fractional distillation, solvent extraction).

Phase Examples

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

Components (C)

• The number of components of a system at equilibrium is the smallest number of independently variable constituents to express the composition of each phase.
• It is the minimum number of molecular species in terms of which the composition of all the phases may be quantitatively expressed.

Component Examples

• Ice/water/water vapor system: one component (H2O), (applies to the ionic species 2H2O =H3O+ + OH-).
• CaCO3(s) = CaO(s) + CO2(g): two components (calcium oxide and carbon dioxide), C=2.
• Na2SO4, Na2SO4.7H2O, Na2SO4.10H2O, Na2SO4 solution, solid ice and water vapor: two components, C=2.
• NH4Cl(s) = NH3(g) + HCl(g): one component because vaporizing the solid chloride makes the gas phase.
• 3Fe(s) + 4H2O(g) = Fe3O4(s) + 4H2(g): three components (Fe, O, and H), C=3.

Degrees of Freedom or Variance (F)

• The number of degrees of freedom is the minimum number of intensive variables needed to completely define the state of the system.
• Intensive variables are independent of the mass or size of the system (e.g., pressure, temperature, concentration, density, refractive index, molar entropy).
• Extensive variables are dependent on the mass or size of the system.
• As the number of components (C) increases, there are more independent variables.
• As the number of phases (P) increases, there are fewer independent variables.
• For a one-component system:
  • If P = 1, then F = 2 (bivariant).
  • If P = 2, then F = 1 (univariant).
  • If P = 3, then F = 0 (invariant).

Derivation of the Phase Rule

• For a one-component system, pressure and temperature are the only intensive variables.
• In a system where phases contain multiple components, additional ratio variables are added.
• C components require (C-1) of composition variables.
• For P phases, there must be P(C-1) such variables.
• Temperature and pressure are the same throughout leading to a total of P(C -1) + 2 variables.
• For P phases, there are (P-1) equations for each component, and for C components, there are C(P-1) total equations.
• Final Phase Rule Equation: P + F = C + 2.