Phase Rule: Introduction and Basic Terms

Introduction to the Phase Rule

• J. W. Gibbs deduced the Phase Rule 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, influenced by changes in pressure, temperature, and concentration.
• Equilibrium is assumed to be unaffected by gravitational, electrical, magnetic, and surface forces.

Basic Terms of the Phase Rule

Homogeneous Equilibrium

• A system is considered homogeneous when it is uniform throughout its volume, meaning its properties are the same in all parts, excluding single molecular species.
• Equilibrium occurring in a homogeneous system is termed homogeneous equilibrium.
• An example is CH3COOCH3 = CH3COOH + CH3OH.
• This equation shows methylacetate becoming acetic acid plus methyl alcohol.

Heterogeneous Equilibrium

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

Conditions for Heterogeneous Equilibrium

• Thermal equilibrium means there is no flow of heat from one part of a system to another (Tα = Tβ for two phases α and β).
• Mechanical Equilibrium exists when the pressure is constant throughout all parts of the system (Pα = Pβ for two phases α and β).
• Chemical Equilibrium occurs when the rate of each forward reaction equals the rate of corresponding backward reaction (dynamic, reversible interchange of matter).

Types of Equilibrium

• True equilibrium exists when a system is in thermal, mechanical, and chemical equilibrium.
• If external conditions are altered and returned to originals, the same state will occur again
• A salt in contact with its saturated solution is an example.
• Metastable equilibrium exists when a system is in thermal, mechanical, and chemical equilibrium but not in the most stable state.
• Liquid water and water vapor at -1°C is an example where reducing pressure vaporizes liquid and increasing pressure condenses vapor.
• The system is stable if undisturbed (true equilibrium but not most stable).
• If a crystal of ice is added, the water will freeze and temperature will rise to 0°C.
• Apparent equilibrium occurs when the approach to equilibrium is so slow it goes undetected.
• Hydrogen, oxygen, and water in a closed vessel appear to be in equilibrium, unless sparked, then a rapid reaction occurs.
• The Phase Rule cannot distinguish between true and metastable equilibrium.
• The Phase Rule can only be applied to apparent equilibria if the reaction is slow.

System

• It is a substance or mixture of substances isolated (in some way) from all other substances.
• 'The water system' refers to the chemical substance water being 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.
• The term mechanically separable refers to hand-picking crystals by shape, filtration, or separation of two liquid phases without interfering with pressure, temperature, or composition (e.g., fractional distillation or solvent extraction)

Examples of phases

• Ice/liquid water/water vapor is three phases.
• Any number of gases mixing in all proportions is one phase.
• A saturated salt solution, undissolved solid, and vapor is a three-phase solution.
• CaCO3 (s) = CaO (s) + CO2 (g) yields 3 phases as there are two different solids and a gas.
• Mercury/carbon tetrachloride/water is a four-phase system with three immiscible liquids and only one vapor phase.

Components (C)

• It is the smallest number of independently variable constituents by which the composition of each phase present can be expressed.
• The number of components in a system 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 has one component using only one chemical substance, H2O.
• The ionic species 2H2O = H3O+ + OH-. can also be expressed in terms of the chemical H2O.
• CaCO3 (s) = CaO (s) + CO2 (g) uses calcium oxide and carbon dioxide to represent two components, C = 2.
• Na2SO4, Na2SO4.7H2O, Na2SO4.10H2O, Na2SO4 solution, solid ice and water vapor represent a two-component system, C = 2.
• NH4C1 (s) = NH3 (g) + HC1 (g) with vaporizing solid chloride making the gas phase is a one-component system
• 3Fe (s) + 4H2O (g) = Fe3O4 (s) + 4H2 (g), a three-phase system, consisting of three chemicals Fe, O, and H, so C = 3.

Degrees of Freedom or Variance (F)

• The number of degrees of freedom is the minimum number of intensive variables by which the state of the system can be defined.
• 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 one component system:
• If P = 1, F = 2 (bivariant system).
• If P = 2, F = 1 (univariant system).
• If P = 3, F = 0 (invariant system).

Derivation of the Phase Rule

• For a one-component system, pressure and temperature are the only intensive variables.

• In a system where one phase contains two components, there is an additional variable (the ratio of the two components in that phase).

• If one phase contained three components, the additional variable is the proportions of two of them.

• With C components in one phase, there are (C - 1) composition variables.

• In a system of P phases, there must be P (C - 1) variables.

• The pressure and temperature variables are the same throughout the system so that the total number of variables is: P (C – 1) + 2.

• With P phases, it is possible to write (P - 1) equations for each component so that for C components, the number of equations is C (P - 1).

• (No. of variables) - (no. of equations) = no. of independent variables

• P (C - 1) + 2 – C (P - 1) = F

• This simplifies to P + F = C + 2.