The document discusses phase diagrams and invariant reactions: 1) It defines phases and phase diagrams, explaining that phases are uniform in composition and state while phase diagrams show the phases present at different temperatures and compositions. 2) It discusses important points on phase diagrams like melting points, eutectic reactions where three phases meet, and lever rules for determining phase percentages. 3) Binary systems can have eutectic or solid solution diagrams depending on solubility. Eutectic diagrams have three distinct phases meeting at the eutectic point below melting temperatures. 4) Invariant reactions involve three phases meeting at a single point or line and include eutectic, eutectoid,

1. Chapter 7
Phase Equilibrium Diagram
Mr. Pem PHAKVISETH
pempvs@gmail.com
Department of Materials Science and Engineering
Souphanouvong University
Faculty of Engineering
1

2. 7.1 Phase and Phase diagram
• A from of matter that is uniform throughout in chemical
composition and physical state
• Homogenous phase is uniform throughout in its chemical
composition and physical state.(no distinction or boundaries)
- Water, ice, water vapor, sugar dissolve in water, etc.
• Heterogenous phase is composed of more than one phase.
These phase are distinguished from each other by boundaries.
- A cube of ice in water.(same chemical) compositions
but different physical states)
- Oil water mixture
- The two phase are said to be coexistent
1. Phase
2

3. 2. Phase diagram
• Phase
- a form of matter that is uniform throughout in
chemical composition and physical state.
• Boundaries
- solid – liquid(Fusion), Liquid – Gas (vaporization),
Solid –Gas (sublimation)
- Also Solid – solid and Liquid – liquid
• Points
- Critical Point – beyond this a gas cannot be liquefied
- Boiling Point – vapor pressure of the gas = atmospheric pressure
- Melting Point – solid and liquid phase coexist (equilibrium)
- Triple Point – solid, liquid and gas phase coexist 3

4. 3. Equilibrium Phase Diagram
• It is actually a collection of solubility limit curves. It is also known as equilibrium or
connotational diagram.
• Equilibrium phase represent there relationships between temperature, compositions
and the quantities of phase at equilibrium.
• important information, useful in materials development and selection, obtainable
from a phase diagram
• It shows phase present at different compositions and temperatures under slow cooling
(equilibrium) conditions
• It indicates equilibrium solid solubility of on element/ compound in another.
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5. 4. Benefit of Phase Diagram
1) Show the phase that appears when the temperature and the ingredients change under
the equilibrium
2) Shows the limits of the solubility of the alloys in basic metals
3) Indicates the temperature of the origin And the end of the metal hardening
(Solidification)
4) Indicates the temperature of the origin And the end of the phase melting (Heating)
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6. 7.2 Phase Rule (Gibb’s Phase Rule)
Usually in a system Alloys usually consist of multiple phases combined. Which are
not all homogeneous characteristics
Factors affecting the occurrence of phase include
1. temperature
2. pressure
3. Chemical mixture
The occurrence of phases and the number of phases is determined by Gibb’s Phase Rule
as follows.
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7. P: Number of phase coexisting in ( the number of phase occurring)
F: Degree of freedom (There are 3 variables, namely temperature,
pressure and composition. The phase will not change.)
C: Number of component (Refers to the number of configuration /
components (What elements are included?))
Phase rule at constant pressure: P+F =C+1
(for example at one atmosphere)
The remaining 2 variables are temperature and composition.
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8. Cooling Curve
Cooling of pure metals at the Equilibrium state
At equilibrium: Changes will
occur slowly. And considered
that the temperature and
pressure are constant. No
energy and matter transfer
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9. Cooling curve of pure metal 100 weight% Element A
Constant pressure, constant composition. As such, the degree of freedom has a single
variable, temperature.
Phase 1 (liquid) and phase 3 (solid) will receive P = 1 and C
= 1, so F = 1 + 1 - 1 = 1 means that there is 1 degree of
freedom. That is, temperature means that temperature
variables can Changeable In which the phase occurred the
liquid is still liquid. (Phase L)
The second phase is the cooling phase. There are both solid
and liquid mixed, will get P = 2 and C = 1. Therefore, F = 1 +
1 - 2 = 0 means no degree of freedom means that the
temperature can not be Changeable If you want two phase
are stay together. If T changes the phase then changes
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10. Cooling curve of alloys 90% A and 10% B
Constant pressure, so the remaining 2 degrees of freedom is temperature and mixture
2-element alloy has a value of C = 2
The phase 1 (liquid) and phase 3 (solid) have two P = 1
values, and so F = 2 + 1 - 1 = 2 means there are 2
degrees of freedom. That is the temperature and
ingredients can be changed. And the phase number will
not change
The second phase, the cooling phase and the mixing
phase of both solid and liquid, will yield P = 2.
Therefore, F = 1 means that the mixed phase can stay
combined even if the temperature changes. (Each
temperature has a different phase mixture)
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11. 7.3 Lever Rule
The lever rule is a rule used to determine the mole fraction or the mass fraction of each phase of
a binary equilibrium phase diagram. It can be used to determine the fraction of liquid and solid phases for a
given binary composition and temperature that is between the liquidus and solidus line.
In an alloy or a mixture with two phases, α and β, which themselves contain two elements, A and B, the
lever rule states that the mass fraction of the α phase is
or
or
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12. phase diagram of germanium -silicon
calculation method:
At the point I consisting of liquid 100%
At the point II consisting of liquid around 100% and solid around 0%
At the point III consisting of liquid =
45−30
45−10
× 100 = 42.86% ;Consisting of solid =
30−10
45−10
× 100 = 57.14%
At the point IV consisting of solid 100%
At the point V consisting of liquid
70−45
70−30
× 100 = 62.5% ; consisting of solid =
45−30
70−30
× 100 = 37.5%
Example: From this picture below
phase diagram of germanium -
silicon. find percentage of each
phase At designated points.
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13. 7.3 Phase diagram of Pure substance
• A substance that has a fixed (homogenous and invariable)
chemical composition throughout is call a pure
substance. It may exist in more than one phase, but the
chemical composition is the same in all phases.
• Pure mean substance of uniform and invariable chemical
composition (but more than one molecular type is
allowed). This allows air to be pure substance.
Example of Pure Substance
o Water (solid, liquid, and vapor phases)
o Mixture of liquid water and water vapor
o Carbon Dioxide
o Homogenous mixture of glass, such as air, as long as there is no change of phase.
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14. Phase diagram of Pure substance
❑ Phase of a Pure Substance
• Solid phase : molecules are arranged in a 3D
pattern(lattice)
• Liquid phase : chunks of molecules float about
each other, but maintain an orderly structure
and relative positions within each chunk.
• Gas phase : random motion, high energy level.
❑ Phase are separated by phase boundaries.
❑ Different phases coexist at triple point.
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15. 7.4 phase diagram of solution substance and Composition
1. Solution substance: is a one homogenous system, when a solvent atom is replaced or combined into a
solvent structure, there is only one phase
2. Composition: is a system to consist of material more then one type and not homogenous .there fore that
occurs in more then one phase. Example Concrete that include from Cement with stone and sand.
In some systems, solvents may not be just
the homogeneous due to the limited solubility,
also known as the solubility limit. For
example, water and sugar systems. It will be
show in figure beside:
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16. In the case that the sugar does not exceed the limit
of dissolution, the phase found will be the only phase
which is Sugar solution in water, also known as Fluid,
when quantity of sugar increase pass over scope of
solution, the phase found will be have two phase such
as fluid and sugar will be make together become to
soluble compounds, sugar will be more soluble, when
the temperature is added to the system is from around
62% by weight at temperature 0℃ become 80% by
temperature at 100 ℃.
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17. 7.4 .1 Binary solid solution
• if a system consist soft components, equilibrium of phase exist is depicted by binary
phase diagram. Pressure is constant, thus in dependently variable parameters are
temperature and composition.
• two components are completely soluble in each other in both solid and liquid phases
• Hume- Rothery’s Rule
o atomic size difference not greater than 15%
o Crystal structure is the same for both components
o Similar electronegativity (no ionic bonding )
o Elements have a similar valance
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18. ➢ Two component systems are classified based on extent of mutual solid solubility
➢ Completely soluble in both liquid and solid phase (isomorphous system)
➢ Completely soluble in liquid phase where as solubility is limited in solid state.
Cooling curve for a binary isomorphous alloys 18

19. At point I : Liquid of Si →
IV: Solid of Si →
𝐶 𝑜(𝐼𝐼𝐼) 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒
At point II : Liquid of Si (tie line) → 𝐶0(𝐼𝐼𝐼) 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒
: Solid of Si (tie line) → 𝐶𝑠(𝐼𝐼) 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒
At point III : Liquid of Si (tie line) → 𝐶𝐼(𝐼𝐼𝐼) 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒
: Solid of Si (tie line) → 𝐶𝑠(𝐼𝐼𝐼) 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒
Example
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20. 7.4 .2 Binary Eutectic diagram
• Often observed for two component systems
• Three single phase region, liquid, solid(α) and solid(𝛽) which are solid solution
• Based on the eutectic reaction L - α + 𝛽, which occurs at a temperature below the
melting temperature of either component
• The eutectic point is an invariant point
• The solubility limit (the maximum amount of the other component) will dissolve is
maximum
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21. Phase diagram of Eutectic system
Example of phase diagram between Lead and Tin
• The alloys can melt together in a
liquid state. But incompatible in
solid state and solid solution.
• Composition of Eutectic 61.95%Sn
• Temperature of Eutectic 183℃
• Eutectic Reaction :
L 𝛼 + 𝛽
Liquid Solid 1 Solid 2+
cooling
Heating
21

22. 7.5 Invariant Reaction
• a horizontal line always indicates an invariant reaction in binary phase diagram
• Involves 3 distinct phases
• Occurs at a single “point”
• Often times is associated with special properties or characteristics for that material
•Eutectic L => S1 + S2
•Eutectoid S1 => S2+ S3
•Peritectic L + S1 => S2
•Peritectoid S1 + S2 => S3
•Monotectic L1 => S1 + L2
•Syntectic L1+L2 => S1 22

23. 23

24. • Example: from figure of phase diagram of Titanium – Nickel. please tell and write
name with equation of reaction when the alloy is slowly cooled?
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25. At Point %Ni and Temperature Name of Reaction Equation of Reaction
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26. 26
Thank you for your attention !!!
감사합니다 !!!
Department of Materials Science and Engineering
Souphanouvong University
Faculty of Engineering