MME213 Phase Diagrams and Transformations
Lecture 5
A. K. M. Bazlur Rashid
Professor, Dept. of Materials and Metallurgical Eng.
Bangladesh Univ. of Eng. and Tech., Dhaka-1000
Thermodynamics and Phase Diagram
2 – Single Component Systems
© Copyright Protected
Lecture Outcome (LO)
At the end of this lecture, students should be able to
1. analyse the dependence of Gibbs free energy on temperature and pressure,
2. differentiate various types order of phase transformations.
3. examine some important unary systems used in materials science and engineering.
2/24
Single-Component Phase Diagram
❑ A single component system is the one containing a pure element or one type
of molecule that does not dissociate over the range of temperature of interest.
pure copper rod
alumina powder
water
3/24
❑ The phase diagrams of single component systems
are generally plotted using (P, T) co-ordinate systems.
Gibbs Phase Rule
F = C-P+2
• For pure copper, the number of phase a system can exist are solid, liquid and gas.
• For silica, however, besides liquid and gas, the system can exist in more than one solid state.
• If a system exhibits more than one phase, the relative amounts of each co-existing phase
cannot be determined using this unary phase diagram.
Phase Diagram for Pure Copper
Phase Diagram for Water
Phase Diagram for Silica
4/24
An elementary principle of chemistry states that
every liquid or solid tends to be in equilibrium with
its vapor.
Piston
Then, what is happened to the
vapour phase of solid and liquid?
Can there be a completely liquid region in the
one component diagram? Will not vapor exist
in equilibrium with the liquid?
P
Cylinder
Metal vapour
Liquid metal
(a)
P
• The explanation lies in the manner in which the one-component system is determined.
• Only the metal being investigated is contained in the cylinder that exerts pressure
on the system; even air is excluded.
• If the external pressure is equal to the vapor pressure of the liquid metal
at the given temperature, both liquid and vapor exist in equilibrium in the cylinder (Fig. a).
Liquid metal
• However, if the external pressure is greater than the vapor pressure of the liquid metal,
the piston is forced down, the vapor condenses, and only the liquid phase remains (Fig. b)
(b)
5/24
❑ For unary systems, the two-phase equilibria curves in (P, T) space
is described mathematically by the function P = P (T).
❑ The Clapeyron equation is a differential form of this equation.
𝑑𝑃
∆𝑆
∆𝐻
=
=
𝑑𝑇
∆𝑉
𝑇∆𝑉
❑ For any pair of coexisting phases in the unary system,
integration of the Clapeyron equation yields a mathematical expression
for the corresponding phase boundary on the phase diagram.
❑ However, to predict the phases that are stable, or mixtures that are in equilibrium
at different temperatures, it is necessary to determine the variation of Gibbs free
energy, G with temperature, T.
6/24
Gibbs Free Energy as a Function of Temperature
How CP is related with H and S?
𝑄𝑃 = ∆𝐻𝑃 = 𝐶𝑃 𝑑𝑇
𝐶𝑃 =
𝛿𝑄
𝜕𝑇
=
𝑃
𝜕𝐻
𝜕𝑇
𝑑𝑆 =
𝑇
∆𝐻𝑃 = 𝐻𝑇 = න 𝐶𝑃 𝑑𝑇
298
𝑃
0
By convention, H of pure material in its
most stable state at 298 K is taken as zero.
By convention, S of homogeneous
material in its most stable state at 0 K
is taken as zero.
Slope = CP
T
0
298
T, K
Entropy, S
H
Slope =
𝜕𝑆
𝜕𝑇
=
𝑃
𝐶𝑃
𝑇
Temperature
7/24
Variation of G with T
𝐺 = 𝐺(𝑇, 𝑃)
Enthalpy / Gibbs free energy
CP
𝐶𝑃
𝑑𝑇 − 𝑉𝛼𝑑𝑃
𝑇 𝑇
𝐶𝑃
∆𝑆𝑃 = 𝑆𝑇 = න
𝑑𝑇
𝑇
𝑑𝐻 = 𝐶𝑃 𝑑𝑇 + 𝑉 1 − 𝑇𝛼 𝑑𝑃
𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃
𝜕𝐺
𝜕𝑇
= −𝑆 ;
𝑃
𝜕𝐺
𝜕𝑃
=𝑉
𝑇
At constant pressures,
𝑇2
∆𝐺 = − න 𝑆𝑑𝑇 ;
𝑇1
Slope =
𝜕𝐺
𝜕𝑇
H
Slope = CP
0
−TS
Slope = -S
G
= −𝑆
𝑃
G decreases with increasing T (at constant P)
at a rate given by –S.
Temperature
Variation Gibbs free energy, G, with T
8/24
G
H (liquid)
d
H (solid)
c
Latent heat, L
b
Tm
a
G (solid)
G (liquid)
f
Liquid stable
T
Variation of enthalpy, H, and free energy, G, with T for solid and liquid
phases of a pure metal. (L = latent heat of melting, Tm = equilibrium
melting temperature)
G
H (liquid)
d
H (solid)
c
Latent heat, L
b
• However, SL > SS and thus GL decreases
more rapidly with increasing T than that for
the solid.
• At Tm, GL = GS and both the solid and liquid
phases can coexist in equilibrium.
• Beyond Tm, the liquid phase has the lowest
free energy and is therefore the equilibrium
state of the system.
9/24
To summarise,
Which is larger, HL or HS ?
• HL > HS at all temperature
Which is larger, SL or SS ?
Tm
a
• Therefore, at low temperatures, GL > GS.
• For temperatures up to Tm, the solid phase
has the lowest free energy and is therefore
the equilibrium state of the system.
e
Liquid stable
• At all temperatures, the liquid phase has a
higher H (or U) than the solid phase.
• SL > SS at all temperature
e
Liquid stable
G (solid)
f
G (liquid)
Liquid stable
T
Variation of enthalpy, H, and free energy, G, with T for solid and liquid
phases of a pure metal. (L = latent heat of melting, Tm = equilibrium
melting temperature)
Which is larger, GL or GS ?
• GL > GS at temperatures below Tm
• GS > GL at temperatures above Tm
This is because Gibbs free energy of liquid
decreases more rapidly with increasing
temperature than that of the solid.
Effect of Pressure on Gibbs Free Energy
❑ The temperature Tm at which both solid and liquid phases coexist
is called the equilibrium melting temperature.
❑ But this equilibrium melting temperature Tm only applies
at a specific pressure (1 atm).
❑ At other pressures, this equilibrium temperature will differ.
11/24
Consider the P-T diagram for pure iron.
❑ Effect of pressure on equilibrium temperatures:
 Increasing pressure raises
L ➔ g-iron equilibrium transformation temperature.
 Increasing pressure lowers
a-iron ➔ g-iron equilibrium transformation temperature.
 Increasing pressure lowers
a-iron ➔ e-iron equilibrium transformation temperature.
Effect of pressure on the equilibrium phase diagram
for pure iron
12/24
❑ The reason for these changes derives from the equation
𝑑𝐺 = −𝑆𝑑𝑇 + 𝑉𝑑𝑃
❑ At constant temperature, the free energy of a phase increases with pressure
such that
𝜕𝐺
𝜕𝑃
=𝑉
𝑇
❑ If the two phases in equilibrium have different molar volumes,
their respective free energies will not change by the same amount at a given temperature
and equilibrium will, therefore, be disturbed by changes in pressure.
❑ The only way to maintain equilibrium at different pressures is by varying the temperature.
13/24
❑ Consider a single-component (unary) system
having two phases, a and b, are in equilibrium.
❑ The free energy of the phases as a function of
temperature and pressure can be written as
From classical thermodynamics,
the conditions for 𝛼 ↔ 𝛽 phase equilibrium :
𝑃 𝛼 = 𝑃𝛽 ➔ 𝑑𝑃 𝛼 = 𝑑𝑃𝛽 = 𝑑𝑃
𝑇 𝛼 = 𝑇𝛽 ➔ 𝑑𝑇 𝛼 = 𝑑𝑇𝛽 = 𝑑𝑇
𝜇𝛼 = 𝜇 𝛽 ➔ 𝑑𝜇𝛼 = 𝑑𝜇 𝛽 = 𝑑𝜇
𝑑𝐺 𝛼 = 𝑉 𝛼 𝑑𝑃 − 𝑆 𝛼 𝑑𝑇
𝑑𝐺𝛽 = 𝑉𝛽 𝑑𝑃 − 𝑆𝛽 𝑑𝑇
❑ For unary systems, it can be shows that G = m
❑ Then, for 𝛼 ↔ 𝛽 equilibrium
𝐺 𝛼 = 𝐺𝛽
𝑑𝑃
𝑑𝑇
=
𝑒𝑞
➔
𝑑𝐺 𝛼 = 𝑑𝐺𝛽
𝑑𝐺 ′ = −𝑆 ′ 𝑑𝑇 + 𝑉 ′ 𝑑𝑃 + 𝜇𝑑𝑛
𝜇=
𝜕𝐺′
𝜕𝑛
=
𝑇,𝑃
𝜕 𝑛𝐺
𝜕𝑛
=𝐺
𝑇,𝑃
𝑆𝛽 − 𝑆 𝛼
∆𝑆 𝛼→𝛽
=
𝑉𝛽 − 𝑉 𝛼
∆𝑉 𝛼→𝛽
14/24
𝑑𝑃
𝑑𝑇
=
𝑒𝑞
𝑆𝛽 − 𝑆 𝛼
∆𝑆 𝛼→𝛽
∆𝐻 𝛼→𝐺
=
=
𝑇𝑒𝑞 ∆𝑉 𝛼→𝐺
𝑉𝛽 − 𝑉 𝛼
∆𝑉 𝛼→𝛽
𝛼 ↔ 𝛾 Equilibrium
∆𝑉 = 𝑉 𝛾(𝐹𝐶𝐶) − 𝑉 𝛼(𝐵𝐶𝐶) = −𝑣𝑒
∆𝐻 = 𝐻 𝛾(𝐻𝑇) − 𝐻 𝛼(𝐿𝑇) = +𝑣𝑒
𝑑𝑃
𝑑𝑇
FCC (0.74)
=
𝑒𝑞
∆𝐻
<0
𝑇𝑒𝑞 ∆𝑉
𝛿 ↔ 𝐿 Equilibrium
BCC (0.68)
∆𝑉 = 𝑉 𝐿 − 𝑉 𝛿 = +𝑣𝑒
∆𝐻 = 𝐻 𝐿 − 𝐻 𝛿 = +𝑣𝑒
𝑑𝑃
∆𝐻
=
>0
𝑑𝑇 𝑒𝑞
𝑇𝑒𝑞 ∆𝑉
Unary phase diagram for pure iron
15/24
❑ Thus, to determine a unary phase diagram,
integration of the following two equations are required to determine P = P (T) curve
of two-phase equilibria of the system.
For equilibrium between
two condensed-phases
• S ↔ L equilibrium
𝑑𝑃
𝑑𝑇
• α ↔ β equilibrium
For equilibrium between
condensed phase and gas phase
• S ↔ G equilibrium
• L ↔ G equilibrium
𝑑𝑃
𝑑𝑇
𝑒𝑞
𝑒𝑞
∆𝑆 𝛼→𝛽
=
∆𝑉 𝛼→𝛽
Clapeyron Equation
∆𝐻 𝛼→𝐺
𝑃∆𝐻 𝛼→𝐺
=
=
𝑇𝑒𝑞 ∆𝑉 𝛼→𝐺
𝑅𝑇 2
Clausius - Clapeyron Equation
16/24
Order of Phase Transformations
❑ Phase transitions are driven by the minimization of the free enthalpy of the
system: if at a certain temperature, the entropy contribution of the Gibbs
enthalpy outweighs the enthalpy contribution in
ΔG = ΔH − TΔS
the high-temperature phase will become thermodynamically stable.
Unfortunately, the term order is
used for two different concepts in
relationship to phase transitions.
❑ The precise nature of this change, i.e. how smoothly or abruptly it occurs,
is different for different types of phase transition.
On the one hand, each phase
transition involves an ordered
(low-temperature) and a disordered
(high-temperature) phase .
❑ To describe this, phase transitions are classified into first-order and
second-order transitions.
On the other hand, the order of the
transition (in the mathematical
sense of the word) determines the
severity of the changes as
described above.
• The order referred to here is the order of the differential of the Gibbs energy
for which a step is observed at the phase transition.
17/24
First Order Transition
Example: Melting, vaporisation, sublimation, etc.
Ice ↔ Water
Water ↔ Steam
Liquid copper ↔ Solid copper
Characteristics of First-Order Transition
• Large heat energy absorbed or liberated during the transition
In all these transitions, T and P remain constant
while S and V change
dG = -SdT + VdP
S = -(𝜕G/𝜕T)P
V = (𝜕G/ 𝜕P)T
CP = T (𝜕S/ 𝜕T)P = T(𝜕 2G/ 𝜕T2)P
a = (1/V) (𝜕V/ 𝜕T)P
b = -(1/V) (𝜕V/ 𝜕P)T
• Gibbs free energy changes continuously and at the point of transition, G remains constant
• First-order derivatives (V, S) and the second-order derivatives (a, b, CP) of Gibbs free energy
changes discontinuously as a function of temperature
18/24
Second Order Transition
Example:
Liquid He-I ↔ Liquid He-II (at Lambda point, TL = 2.19 K)
Ferromagnetic materials ↔ Paramagnetic materials (at Curie point. For Fe, TC = 1043 K)
Characteristics
• No heat energy absorbed or liberated during the transition
• Continuous Gibbs free energy
• Smooth transition in first derivatives of Gibbs free energy but discontinuity in second derivatives
• No change in entropy and volume
• Discontinuity appears in specific heat
• Entropy and volume doesn’t change discontinuously
19/24
FIRST-ORDER PHASE TRANSITION
𝜕𝐺
𝜕𝑇
−𝑆 =
G
𝜕𝐺
𝜕𝑃
𝑉=
𝑃
𝐶𝑃 = 𝑇
𝑇
𝑃,𝑁
To 
DV
DS = L/T
T
T
𝜕𝑆
𝜕𝑇
T
T
SECOND-ORDER PHASE TRANSITION
G
𝑉
S
DS = 0
T
T
CP
DV=0
T
T
20/24
• The Gibbs free energy, G, of both phases involved in the transition is a smooth function
of temperature. At the transition point, the curves of both phases intersect - on crossing
the transition point, the other phase becomes the thermodynamically stable one.
• As a result, there is a kink in the free energy of the system (under equilibrium conditions)
at the transition point of a first-order transition. In a second-order transition, the free
energy of both phases are identical over a limited temperature range before diverging
either side of the transition. Both curves have the same tangent at the transition point.
• For first-order transitions, the kink in G corresponds to a step in its first derivatives at
the transition point. This is a result of the high latent heat associated with the transition.
• In the case of second-order transitions, there is no latent heat and therefore no step in
the entropy at the transition. However, the slope of the curve changes abruptly,
producing a kink similar to that in G itself for first-order transitions.
• For a first-order transition, the heat capacity therefore goes to infinity when the transition
point is approached from either side. For a second-order transition, the kink in S merely
results in a step in its derivative.
21/24
Examples of Unary Phase Diagrams
• Three allotropic forms (d, g and a) with
different crystal structures and different
properties.
• a-iron (BCC), also called Ferrite, exists from
the lowest temperatures up to 910°C, and it's
magnetic up to its Curie temperature of 768°C.
770 K
11 GPa
• g-iron (FCC), also called Austenite, exists from
910-1401°C.
• Above 1401 °C, there is d-iron (BCC). At
normal pressure, the d-iron melts at 1538°C.
Unary phase diagram of pure iron
• At very high P, there exists another solid,
called e-iron (HCP). Although not
technologically useful, is important for
geologists as it is found in Earth’s core.
22/24
• Silica is polymorphic material.
• The polymorphic forms of silica crystals are:
1. At room temperature, the stable form of silica is
a-quartz.
2. At 573°C, a-quartz will change to β-quartz.
3. On cooling, β-quartz will revert to a-quartz.
4. At 870°C, stable β-quartz will change to
β-Tridymite.
5. At 1470°C, β-Tridymite will change to
β- Cristobalite, which melts at 1713°C.
(a)
(b)
Unary phase diagram of silica
6. At high temperature, metastable β-quartz will
change to β-Cristobalite or melt to liquid silica.
(a) Phase diagram for silica system SiO2 at higher pressures
(b) Stability relations in the silica system at atmospheric pressure
23/24
Next Class
Lecture 6
Thermodynamics and Phase Diagrams
3 – Binary Solutions