Phase Equilibria

1.

Lecture 6. Phase Equilibria

2.

What is Phase Equilibria?
Phase equilibrium describes a system in which two or more phases
coexist under conditions where no net change occurs with time.
At equilibrium:
Temperature (T) is constant
Pressure (P) is constant
Chemical potential (μ) of each component is
equal in all phases
Example:
Ice ⇌ Water ⇌ Water vapor at the triple point
A system is in phase equilibrium when:
No macroscopic change occurs with
time
Each phase remains stable

3.

Phase diagrams
Phase diagrams visually represent
phase equilibria. They show
regions of stability for different
phases and boundaries where
phases coexist. Understanding
these diagrams is key to predicting
phase behavior and solving realworld thermodynamic problems.

4.

Understanding Phase Diagram
Each region on the diagram corresponds to
one phase. It’s like chart that tells you
which is gas or liquid or steam.
For example, below zero degree
celsius and normal pressure water is
solid.

5.

Above 100 degrees celsius it’s a gas.
Between it’s liquid.

6.

The lines separating these regions called phase boundaries.
They show conditions where two phases can coexist in equilibrium.

7.

When three phases meet, that’s triple point. A unique
set of temperature and pressure where solid, liquid and
gas exist together.

8.

If you follow the liquid gas boundary far enough, it
ends at the critical point.Where liquid and gas become
indistinguishable.

9.

Key Definitions
Phase
A homogeneous, physically and chemically
uniform part of a system, separated by boundaries.
The minimum number of independent
chemical species required to describe all
phases.
Examples:
Ice (solid)
Liquid water
Water vapor
Examples:
System Types
1.
2.
Component (C)
Pure water → 1 component
NaCl + water → 2 components
Homogeneous system → single phase
Heterogeneous system → two or more phases
Degrees of Freedom (F)
Number of independent intensive variables (T, P, composition) that
can be changed without changing the number of phases.

10.

Phase Rule
The Gibbs Phase Rule is a fundamental principle
in thermodynamics that helps us predict how
many variables can be changed independently
in a system at equilibrium without changing the
number of phases present.
What Does It Mean?
The rule tells us:
How many intensive variables (like temperature,
pressure, or composition) can be changed
independently while maintaining phase equilibrium.
Whether a system is invariant, univariant, or
divariant.
In one of the most elegant calculations of
the whole of chemical thermodynamics,
J.W. Gibbs deduced the phase rule,
which is a general relation between the
variance, F, the number of components,
C, and the number of phases at
equilibrium, P, for a system of any
composition:

11.

One-component system (C = 1)
Example: pure water
1. Single phase (P = 1):
F=1−1+2=2
Both temperature and pressure can vary independently.
2. Two phases (P = 2) (e.g., liquid + vapor):
F=1−2+2=1
Only one variable (T or P) can change independently.
3. Three phases (P = 3) (triple point):
F=1−3+2=0
No freedom - temperature and pressure are fixed.
Triple Point
An invariant system (F = 0)
All three phases coexist at a
unique temperature and
pressure
Example: ice + liquid water +
water vapor

12.

13.

Typical One-Component
Phase Diagram (Water)
Axes:
Pressure (P)
● Temperature (T)
Regions:
Solid
● Liquid
● Vapor

14.

Two-Component Systems (Binary Systems)
A two-component system contains C = 2
chemically independent substances.
Examples:
Ethanol + Water
NaCl + Water
Cu + Ni (alloy)
At Constant Pressure (Most
Common Case)
In real chemistry and materials science, pressure is
often fixed (1 atm).
So we use the reduced phase rule:
Using the Gibbs Phase Rule:
F=C−P+1
F=C−P+2
For two components:
For binary systems:
F=2−P+1=3−P
F=2−P+2=4−P

15.

Case Analysis (At Constant Pressure)
One Phase (P = 1)
Two Phases (P = 2)
F=3-1=2
F= 3-2=1
✔ One degree of freedom
Only one variable can change independently.
✔ Two degrees of freedom
You can independently vary:
F=2−P+1=3−P
Temperature
Composition
This corresponds to area regions in a phase diagram.
Corresponds to lines in a phase diagram
(e.g., liquid + solid region boundary)
Three Phases (P = 3)
F=3-3-0
✔ Invariant system
Temperature and composition are fixed.
Corresponds to a point (eutectic point)

16.

Phase boundaries are the lines (or curves) on a phase
diagram that separate different phase regions.
Along these boundaries, two phases coexist in
equilibrium.
They are fundamental for understanding melting, boiling,
freezing, condensation, and solid–solid transformations.
What Happens on a Phase Boundary?
1.
2.
3.
4.
Two phases are present simultaneously
The system is in dynamic equilibrium
Small changes in conditions cause phase
transformation
Degrees of freedom are reduced
For a two-component system at constant pressure:
On a phase boundary → P = 2, F = 1
Phase Boundaries (Phase
Equilibrium Lines)

17.

Phase Boundaries
Phase Boundary Lines (one component)
1. Solid–Liquid → melting/freezing
2. Liquid–Vapor → boiling/condensation
3. Solid–Vapor → sublimation/deposition
At each boundary:
μphase 1​=μphase 2​

18.

Main Types of Phase Boundaries
Solid-Liquid Boundary
Melting / Freezing line
Separates solid and liquid
Called:
a. Fusion curve (one-component systems)
b. Solidus / Liquidus lines (binary
systems)
Example:
Ice ⇌ Water
Alloy: Solid ⇌ Liquid
➡ Along this boundary:
Solid and liquid coexist
Temperature is fixed for a given composition
The solid-liquid boundary, often called
the fusion or melting curve, is a line on a
phase
diagram
representing
the
temperature and pressure conditions
where solid and liquid phases coexist in
equilibrium. This boundary signifies
where substances melt (solid to liquid) or
freeze (liquid to solid)

19.

Liquid–Gas Boundary
Boiling / Condensation line
Separates liquid and vapor
Ends at the critical point
The liquid–gas boundary (or vaporization curve) is the line on a
phase diagram representing the precise conditions of pressure and
temperature where a substance’s liquid and gas phases exist in
equilibrium, marking boiling or condensation. It separates liquidphase regions from gas-phase regions and terminates at the critical
point, where the phases become indistinguishable.
Example:
Water ⇌ Steam
➡ Beyond the critical point:
No clear distinction between liquid and gas
Supercritical fluid

20.

Solid–Gas Boundary
Sublimation / Deposition line
● Direct transition: Solid ⇌ Gas
● Occurs at low pressure
Example:
● Dry ice (CO₂)
● Naphthalene
The solid–gas boundary, also known as the sublimation curve, is a line on a phase diagram
representing conditions of temperature and pressure where a substance's solid and gaseous
phases coexist in equilibrium. It separates the solid and gas regions, with sublimation
occurring (solid to gas) when heating, and deposition (gas to solid) when cooling.

21.

Solid–Solid Boundary
Polymorphic / allotropic transition
Separates two different solid structures
No liquid involved
Examples:
Graphite ⇌ Diamond
Iron (α-Fe ⇌ γ-Fe)
A solid–solid boundary (interface) is the physical junction separating two distinct
crystalline solid phases, often characterized by complex dislocations, specific atomic
structures, and lower free energies compared to free surfaces. These interfaces are
crucial for analyzing interdiffusion, chemical mapping, and electronic bonding
between materials.

22.

Phase Boundaries in Binary Systems
In a binary system (C = 2), phase boundaries are the
curves or lines on a temperature–composition (T–x)
diagram that separate regions with different phase
stability.
Along these boundaries, two phases coexist in
equilibrium.
At constant pressure, we use the reduced Gibbs phase
rule:
F=C−P+1=2−P+1=3−P
So, on a phase boundary:
P = 2 → F = 1 (univariant system)

23.

24.

25.

Raoult’s Law (Ideal Solutions)
Raoult’s Law describes the vapour pressure of components in an ideal liquid solution.
It states that:
The partial vapour pressure of each component in an ideal solution is proportional to its
mole fraction in the liquid phase.
Physical Meaning (Conceptual Understanding)
Raoult’s Law assumes:
All molecules interact equally
No preference between unlike or like molecules
Escaping tendency of molecules depends only on concentration
Interpretation:
If you reduce the mole fraction of a component → fewer
molecules escape → lower vapour pressure.

26.

Ideal Solutions
Binary Ideal Solutions (Two Components)

27.

LEVER RULE (QUANTITATIVE TOOL)
The Lever Rule is a graphical and mathematical
method used to calculate the relative amounts
(fractions or masses) of coexisting phases in a twophase equilibrium region of a phase diagram.
The Lever Rule applies only when:
It does NOT give composition of phases,
It gives how much of each phase is present.
Common applications:
1.
2.
3.
1.
2.
3.
4.
The system contains two phases
The system is at equilibrium
The system composition lies between phase
boundaries
Solid–liquid equilibria (eutectic systems)
Liquid–liquid equilibria
Vapor–liquid equilibria
Alloy phase diagrams