Unit 3: Solutions and Phase Equilibria

1. Ideal Solutions and Raoult's Law

1.1 Definition of Ideal Solutions

An ideal solution is a solution in which the solute-solute, solvent-solvent, and solute-solvent intermolecular forces of attraction are completely identical. In an ideal binary solution containing components A and B, the A-B intermolecular interactions are equal to the A-A and B-B interactions.

1.2 Raoult's Law

Raoult's Law Statement: The partial vapour pressure of any volatile component in a solution at a given temperature is directly proportional to its mole fraction in the solution.

For a binary solution of two volatile liquids A and B:

P_A = P_A° * x_A
P_B = P_B° * x_B

Where:

• P_A and P_B are partial vapour pressures of components A and B above the solution.
• P_A° and P_B° are the vapour pressures of pure components A and B at the same temperature.
• x_A and x_B are the mole fractions of A and B in the liquid phase.

The total vapour pressure (P_total) above the solution is given by Dalton's Law of Partial Pressures:

P_total = P_A + P_B = (P_A° * x_A) + (P_B° * x_B)

1.3 Thermodynamic Conditions for Ideal Solutions

• Obedience to Raoult's Law: Obeys Raoult's law over the entire range of concentrations and temperatures.
• Enthalpy of Mixing (ΔH_mix = 0): No heat is absorbed or evolved when the components are mixed.
• Volume of Mixing (ΔV_mix = 0): Total volume of the solution equals the sum of the volumes of individual components prior to mixing.

1.4 Examples of Ideal Solutions

• Benzene and Toluene
• n-Hexane and n-Heptane
• Bromoethane and Chloroethane
• Chlorobenzene and Bromobenzene

2. Non-Ideal Solutions and Deviations from Raoult's Law

2.1 Definition of Non-Ideal Solutions

A solution that does not obey Raoult's law over the entire range of concentrations is called a non-ideal solution. For such solutions, the interactions between A-B molecules differ significantly from A-A and B-B interactions.

2.2 Positive Deviation from Raoult's Law

Positive deviation occurs when the intermolecular attractive forces between unlike molecules (A-B) are weaker than those between like molecules (A-A or B-B).

• Vapour Pressure: Higher than predicted by Raoult's law (P_A > P_A° * x_A and P_B > P_B° * x_B).
• Enthalpy of Mixing (ΔH_mix > 0): Endothermic process (heat is absorbed during mixing).
• Volume of Mixing (ΔV_mix > 0): Expansion in volume occurs upon mixing.
• Examples: Ethanol + Acetone, Carbon Disulphide + Acetone, Water + Ethanol.

2.3 Negative Deviation from Raoult's Law

Negative deviation occurs when the intermolecular attractive forces between unlike molecules (A-B) are stronger than those between like molecules (A-A or B-B).

• Vapour Pressure: Lower than predicted by Raoult's law (P_A < P_A° * x_A and P_B < P_B° * x_B).
• Enthalpy of Mixing (ΔH_mix < 0): Exothermic process (heat is released during mixing).
• Volume of Mixing (ΔV_mix < 0): Contraction in volume occurs upon mixing.
• Examples: Acetone + Chloroform (due to hydrogen bonding), Nitric Acid + Water, Hydrochloric Acid + Water.

2.4 Summary Comparison Table

3. Vapour Pressure-Composition and Temperature-Composition Curves

3.1 Vapour Pressure-Composition Curves

Vapour pressure-composition diagrams plot partial and total vapour pressures as a function of liquid composition (mole fraction x) at constant temperature.

• Ideal Solutions: Total vapour pressure is a straight line connecting P_B° and P_A°.
• Positive Deviations: Total vapour pressure curve bulges upward and exhibits a maximum peak at a specific composition.
• Negative Deviations: Total vapour pressure curve sags downward and exhibits a minimum trough at a specific composition.

3.2 Temperature-Composition (Boiling Point) Curves

Since vapour pressure and boiling point are inversely related (higher vapour pressure corresponds to lower boiling point):

• Ideal Solutions: The boiling point changes continuously between the boiling points of pure component A and pure component B. The diagram shows two curves: liquidus (boiling curve) and vapourus (condensation curve).
• Positive Deviation (Minimum Boiling): The curve exhibits a minimum boiling point at the composition where vapour pressure is maximum.
• Negative Deviation (Maximum Boiling): The curve exhibits a maximum boiling point at the composition where vapour pressure is minimum.

4. Distillation of Solutions and Azeotropes

4.1 Fractional Distillation of Ideal Solutions

When an ideal solution is boiled, the vapour formed is richer in the more volatile component (component with higher vapour pressure / lower boiling point) compared to the liquid solution. Repeated vaporization and condensation steps in a fractionating column allow complete separation of the mixture into pure components.

4.2 Azeotropes (Azeotropic Mixtures)

Definition: An azeotrope is a binary liquid mixture having a constant boiling point and distilling over without any change in composition at a given pressure.

Azeotropes cannot be separated into their constituent pure components by fractional distillation because the liquid phase and vapour phase have identical mole fractions at the azeotropic composition.

4.3 Classification of Azeotropes

Minimum Boiling Azeotropes

• Formed by solutions exhibiting large positive deviations from Raoult's law.
• Boil at a temperature lower than the boiling points of either pure component.
• Example: Ethanol-water mixture (95.6% ethanol by mass boils at 78.1 °C, whereas pure ethanol boils at 78.3 °C and water at 100 °C).

Maximum Boiling Azeotropes

• Formed by solutions exhibiting large negative deviations from Raoult's law.
• Boil at a temperature higher than the boiling points of either pure component.
• Example: Nitric acid-water mixture (68% nitric acid by mass boils at 120.5 °C, whereas pure nitric acid boils at 86 °C and water at 100 °C).

5. Basic Concepts: Phase, Component, and Degrees of Freedom

5.1 Phase (P)

Definition: A phase is defined as a homogeneous, physically distinct, and mechanically separable part of a system bounded by a surface and in equilibrium with other parts.

• Gas Phase: All gases are completely miscible; hence, any mixture of gases forms a single phase (P = 1).
• Liquid Phase: Two miscible liquids form 1 phase. Two immiscible liquids form 2 distinct phases.
• Solid Phase: Each distinct solid species constitutes a separate phase. (e.g., Ice = 1 phase; CaCO3(s) and CaO(s) = 2 phases).

5.2 Component (C)

Definition: The component of a system is the minimum number of independent chemical species needed to express the composition of every phase present in the system, directly or via chemical equations.

Mathematical relation for chemically reacting systems: C = N - E, where N is the number of chemical species and E is the number of independent equilibrium conditions/relations.

• Water System (Ice - Water - Vapour): C = 1 (All phases expressed as H2O).
• Dissociation of CaCO3: CaCO3(s) = CaO(s) + CO2(g). Total species N = 3, Equilibrium relations E = 1. Therefore, C = 3 - 1 = 2 components.

5.3 Degrees of Freedom / Variance (F)

Definition: Degrees of freedom is the minimum number of independent intensive variables (such as temperature, pressure, concentration) that must be specified to completely define the thermodynamic state of a system in equilibrium.

• Non-variant / Invariant system (F = 0): The state is completely fixed automatically (e.g., triple point of water).
• Univariant system (F = 1): Only one variable can be varied independently without changing the number of phases.
• Bivariant system (F = 2): Two independent variables must be specified to define the system.

6. Criteria of Phase Equilibrium

For a heterogeneous system consisting of multiple phases in thermodynamic equilibrium at constant temperature and pressure, three conditions must be fulfilled:

6.1 Thermal Equilibrium

The temperature must be uniform throughout all phases in the system:

T_alpha = T_beta = T_gamma ...

6.2 Mechanical Equilibrium

The pressure must be uniform throughout all phases in the system:

P_alpha = P_beta = P_gamma ...

6.3 Chemical Equilibrium

The chemical potential (μ) of each individual component must be equal in all coexisting phases:

μ_i^alpha = μ_i^beta = μ_i^gamma ...

If μ_i^alpha > μ_i^beta, matter of component i will spontaneously transfer from phase alpha to phase beta until chemical potentials equalize.

7. Gibbs Phase Rule

7.1 Statement of Gibbs Phase Rule

Gibbs Phase Rule Formula: F = C - P + 2

Where:

• F = Degrees of Freedom (Variance) of the system
• C = Number of Components
• P = Number of Phases in equilibrium
• 2 = Represents the two variable physical parameters: Temperature and Pressure

7.2 Significance and Application

• Applies strictly to heterogeneous systems in complete thermodynamic equilibrium.
• Determines the maximum number of phases that can coexist at equilibrium for a given number of components.
• It is independent of the nature or amount of substances present in each phase.

8. Phase Diagrams of One-Component Systems (Water and Sulphur)

For any one-component system (C = 1), the phase rule simplifies to:

F = 1 - P + 2 = 3 - P

• If P = 1 (Area): F = 2 (Bivariant). Temperature and pressure can be varied independently.
• If P = 2 (Curve): F = 1 (Univariant). Temperature determines pressure along the line.
• If P = 3 (Triple Point): F = 0 (Invariant). Coexistence occurs at fixed temperature and pressure only.

8.1 Water System (H2O)

The water system is a 1-component, 3-phase system (Solid Ice, Liquid Water, Water Vapour).

Key Features of Water Phase Diagram:

• Curves (P = 2, F = 1):
  • Curve OA (Vapour Pressure Curve): Represents liquid water in equilibrium with water vapour. Terminates at critical point A (374 °C, 218 atm).
  • Curve OB (Sublimation Curve): Represents solid ice in equilibrium with water vapour. Extends down to absolute zero.
  • Curve OC (Fusion/Melting Curve): Represents ice in equilibrium with liquid water. Note: Curve OC has a negative slope (slopes backwards to the left) because ice expands upon freezing (volume of solid > volume of liquid), so increasing pressure lowers the melting point.
  • Curve OA' (Metastable Curve): Represents supercooled liquid water in equilibrium with water vapour.
• Triple Point O (P = 3, F = 0):
  • Point where curves OA, OB, and OC meet.
  • All three phases (Ice, Water, Vapour) coexist in equilibrium.
  • Conditions: Temperature = 0.0098 °C (273.16 K), Pressure = 4.58 mmHg (0.006 atm).
• Areas (P = 1, F = 2):
  • Area AOC: Liquid Water
  • Area AOB: Water Vapour
  • Area BOC: Solid Ice

8.2 Sulphur System

The sulphur system is a 1-component system with 4 possible phases:

1. Solid Rhombic Sulphur (S_R)
2. Solid Monoclinic Sulphur (S_M)
3. Liquid Sulphur (S_L)
4. Sulphur Vapour (S_V)

Polymorphism / Enantiotropy: S_R and S_M are two allotropic solid forms that transform reversibly into each other at the transition point of 95.6 °C at 1 atm.

S_R (Rhombic) ⇌ S_M (Monoclinic) at 95.6 °C

Note on 4 Phases Coexistence: Substituting P = 4 into Gibbs Phase Rule yields F = 1 - 4 + 2 = -1 (impossible). Thus, all four phases can never coexist together at equilibrium.

Key Features of Sulphur Phase Diagram:

• Stable Curves (P = 2, F = 1):
  • Curve AB (Sublimation Curve of S_R): Rhombic sulphur in equilibrium with sulphur vapour. Ends at 95.6 °C (Point B).
  • Curve BC (Sublimation Curve of S_M): Monoclinic sulphur in equilibrium with sulphur vapour. Extends from 95.6 °C to 120 °C.
  • Curve CD (Vapour Pressure Curve of S_L): Liquid sulphur in equilibrium with sulphur vapour.
  • Curve BE (Transition Curve): S_R in equilibrium with S_M. Shows that transition temperature increases with pressure.
  • Curve CE (Fusion Curve of S_M): Monoclinic sulphur in equilibrium with liquid sulphur.
  • Curve EG (Fusion Curve of S_R): Rhombic sulphur in equilibrium with liquid sulphur at high pressure.
• Stable Triple Points (P = 3, F = 0):
  • Point B (95.6 °C, 0.004 mmHg): S_R, S_M, and S_V coexist in equilibrium.
  • Point C (120 °C, 0.02 mmHg): S_M, S_L, and S_V coexist in equilibrium.
  • Point E (151 °C, 1290 atm): S_R, S_M, and S_L coexist in equilibrium.
• Metastable Equilibrium: If S_R is heated rapidly, it bypasses conversion to S_M and melts directly at 114 °C, creating metastable triple point O (S_R, S_L, S_V).