Online Lecture 3 (Part I): Gibbs

Phase Rule and Phase diagrams
of one-component systems

Arti Dua
Department of Chemistry
IIT Madras
!
 Phase Equilibrium for one component with two phases
Condition of equilibrium between two phases: 1

One equation, two unknowns: equilibrium curve (Specific case of

the Gibbs phase rule)

p
phase 1
Phase equilibrium curve:
p1
phase 2
Phase 1 (homogenous):

T1 T
Phase 2 (homogenous):
 The Gibbs Phase Rule
At a given temperature and pressure how many phases can exist in equilibrium
with each other?

Number of components = C
Number of Phases = P
 The Gibbs Phase Rule
For solution to exist:
2

The unknown parameters are p, T and C

Number of equations = C(P - 1) mole fractions in each P phase. However, because
of the constraint below, there are C - 1 independent
parameters in P phases
Number of unknowns = P(C - 1) + 2

3
 The Gibbs Phase Rule

f is the number of degrees of freedom required to specify the chemical potential in a

given region in the phase diagram (usually constructed for p versus T for
1-component and T versus Mole-Fraction for 2 components)

One component system with C=1: f = 3 - P

For P = 3 f = 0 represents a coexistence point [both p and T are

simultaneously specified]
For P = 2 f = 1 represents a coexistence line [If p (or T) is specified
then T (or p) can be determined]
For P = 1 f = 2 represents an equilibrium plane [both p and T need to
be specified on the equilibrium plane]

For C = 1, the maximum number of phases that can coexist at

equilibrium is 3
 Phase Equilibrium for one component system with
three phases
For C = 1, the maximum number of phases that can coexist at equilibrium is 3
Let’s call them phase 1, 2 and 3

p
Condition for equilibrium between three phases:
phase 3 phase 2

.
Two equation and two unknowns.
The solution is a specific pair of value
Ptr (Ptr and Ttr)

phase 1

Ttr T
P=3, f=0
Gibbs phase rule: f = 3 - P = 0
(coexistence point)
 Phase Equilibrium for one component system with two phases
Condition for equilibrium between
phases 1 and 2:
µ1 (p, T ) = µ2 (p, T ) 1
P=2, f=1
p
3 Condition for equilibrium between
phase 3 phases 1 and 3:

P1 . phase 2 P=2, f=1

1
µ1 (p, T ) = µ3 (p, T ) 2

Condition for equilibrium between

phases 2 and 3:
2
phase 1
µ2 (p, T ) = µ3 (p, T ) 3

P=2, f=1 T1 T

One equation and two unknowns.

The solution is a coexistence line (pink)
Gibbs phase rule: f = 3 - P = 1 (specify p to determine T).
(coexistence line)
Condition for the existence (stability) of a single (homogeneous)
equilibrium phase in one component system

Condition for the existence of phase 1

µ1 (p, T ) < {µ2 (p, T ), µ3 (p, T )} 1
p
P=1, f=2
Condition for the existence of phase 2
2 phase 2
3 phase 3 µ2 (p, T ) < {µ1 (p, T ), µ3 (p, T )} 2
.
P=1, f=2

P1 Condition for the existence of phase 3

µ3 (p, T ) < {µ1 (p, T ), µ2 (p, T )} 3
P=1, f=2
1 phase 1

T1
Gibbs phase rule: f = 3 - P = 2 (plane)
f = 2 means that we need to specify both p and T to
know the chemical potential
 Phase Equilibrium for One-Component System

f=3-P
We can know the general feature of p-T
P=2, f=1
p phase diagram once we know how p
varies with T for a given substance.
phase 3 phase 2
P=1, f=2
P=1, f=2
P=2, f=1
Need to know the shape (slope)
of the coexistence lines
Ptr
P=1, f=2
phase 1

Need to know dp/dT at coexistence

P=2, f=1 Ttr T
P=3, f=0
 

Online Lecture 3 (Part II):
Determining the sign of the slope
of the coexistence line (dp/dT)
using the Clapeyron Equation

Arti Dua
Department of Chemistry
IIT Madras
!
The Clapeyron Equation

phase ↵
p+dp
p
phase

T T+dT
The Clapeyron Equation

A
 The Clapeyron Equation

2 3

A B

Will be used to get the sign of the Will be used to get the magnitude of the
slope dp/dT slope dp/dT
 Clapeyron Equation

Liquid-gas coexistence curve:

l The slope is always positive

An empirical rule
works for many substances
 Clapeyron Equation
Solid-gas coexistence curve:

The slope is steeper than liquid-gas

 Clapeyron Equation
Solid-liquid coexistence curve:
e.g. carbon dioxide
e.g. water

The slope can be positive

or negative but is very steep

l l

carbon dioxide water

 

Online Lecture 3 (Part III):
Determining (magnitude) of the slope
of the coexistence line (dp/dT)
using the Clausius-Clapeyron Equation

Arti Dua
Department of Chemistry
IIT Madras
!
 Integrated form of Clapeyron Equation:
Clausius-Clapeyron Equation
Calculates the magnitude of slope, dp/dT
Liquid-gas B
coexistence

First assumption

Second assumption
 Integrated form of Clapeyron Equation:
Clausius-Clapeyron Equation

Solid-gas coexistence yield the same expression except for replacing H vap ! H sub
in the above expression.
!
Integrated form of Clapeyron Equation
Solid-liquid coexistence curve:
B
 The Critical Point
Pressure-Temperature curve

Critical point
l At temperatures and pressures above Tc and
Pc Pc there is no difference in phases and the
substance is always homogenous

Tc
At the critical point two phases become
identical.
 Some typical phase diagrams...
Pressure-Temperature curve (three phases)

The slope of Solid-liquid

coexistence curve is negative

Ptrp < 1 atm

water
 Some typical phase diagrams...
Pressure-Temperature curve (three phases)

The slope of Solid-liquid

coexistence curve is positive

Ptrp > 1 atm

Solid carbon dioxide is often known as "dry

ice". You can't get liquid carbon dioxide
carbon dioxide under normal conditions - only the solid or
the vapour.