This document discusses binary phase diagrams and the principles that govern them. It introduces the concepts of complete and limited solubility in liquid and solid states, which determine the different types of binary phase diagrams. It also discusses the phase rule and its application to binary systems. Additionally, it explains the theorem of Le Châtelier, Clausius-Clapeyron equation, lever rule, and how to determine the composition and relative amounts of phases that coexist at a given temperature based on a binary phase diagram. Examples are provided to demonstrate determining phase fractions using the lever rule.

1. MSE 2201
Phase Diagrams
and
Transformations
Md. Mafidul Islam,
Lecturer
Dept. of MSE, KUET
Introduction to Phase Diagrams
Lecture 04
References: FC Campbell, PHASE DIAGRAMS: UNDERSTANDING THE BASICS, ASM International, 2012, Ch. 01

2. 2
Binary Phase Diagrams
❑ Binary implies that there are two components.
❑ Pressure changes often have little effect on the equilibrium of
solid phases (unless of course, we apply ‘huge’ pressures).
❑ Hence, binary phase diagrams are usually drawn at 1-
atmosphere pressure.
❑ The Gibbs phase rule is reduced to: Variables are reduced to:
F = C – P + 1. (1 is for T).
❑ T & Composition (these are the usual variables in Materials
Phase Diagrams)

3. 3
Binary Phase Diagrams
We consider the possible binary phase
diagrams. These have been classified
based on:
⮚ Complete Solubility in both liquid &
solid states
⮚ Complete Solubility in liquid state, but
limited solubility in the solid state
⮚ Limited Solubility in both liquid &
solid states.
We have already seen that the reduced phase rule at 1 atm pressure is: F = C – P +
1. The other two variables are: Composition of the liquid (CL) and composition (CS)
of the solid.

4. Binary Phase Diagrams
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5. Significance of DoF
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❑ Suppose that it is desired to ascertain under what conditions a
pure metal can exist with the gas, liquid, and solid phases all present in a
state of equilibrium
❑ DOF (F)=0 ( C=1, P=3)
❑ If it is desired to construct a phase diagram in which the coexistence of the three
phases is represented,
⮚ it becomes apparent that the coordinates of the diagram should be
temperature and pressure.
⮚ and that the coexistence of the three phases must be indicated by a single point
on this diagram.
⮚ If one such set of specific conditions of temperature and pressure is now found
by experiment, it will be unnecessary to look for another set, because the
phase rule shows that only one can exist.

6. Theorem of Le Chậtelier
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Although the phase rule tells what lines and fields should be represented
on a phase diagram, it does not usually define their shapes or the
directions of the lines. Which may be explained by several additional
thermodynamic rules.
The theorem of Le Chậtelier says that,
if a system in equilibrium is subjected to a constraint by which the
equilibrium is altered, a reaction takes place that opposes the constraint,
that is, one by which its effect is partially annulled.
Therefore, if an increase in the temperature of an alloy results in a phase
change, that phase change will be one that proceeds with heat absorption,
or if pressure applied to an alloy system brings about a phase change, this
phase change must be one that is accompanied by a contraction in
volume.

7. Theorem of Le Chậtelier
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Physical significance of
this principle:

8. Clausius-Clapeyron Equation
8
A quantitative statement of the theorem of Le Chậtelier is found in the
Clausius-Clapeyron equation.
This equation leads to the further conclusion that each of the curves
representing two phase equilibrium must lie at such an angle that on
passing through the point of three-phase equilibrium, each would project
into the region of the third phase.
Thus, the sublimination-desublimination line must project into the liquid
field, the vaporization curve into the solid field, and the solid liquid curve
into the vapor field.

9. The Lever Rule
Lever rule says, the relative amount of a given phase is proportional to
the length of the tie- line on the opposite side of the alloy point of the tie-
line.
Chemical Composition of Phases: To
determine the actual chemical
composition of the phases of an alloy, in
equilibrium at any specified temperature
in a two-phase region, draw a horizontal
temperature line, called a tie line, to the
boundaries of the field .
These points of intersection are dropped
to the base line, and the composition is
read directly. 9

10. The Lever Rule
Relative Amounts of Each Phase:
⮚ To determine the relative amounts of the two phases in equilibrium at
any specified temperature in a two-phase region, draw a vertical line
representing the alloy and a horizontal temperature line to the
boundaries of the field.
⮚ The vertical line will divide the horizontal line into two parts whose
lengths are inversely proportional to the amount of the phases present.
This is also known as Lever rule.
⮚ The point where the vertical line intersects the horizontal line may be
considered as the fulcrum of a lever system.
⮚ The relative lengths of the lever arms multiplied by the amounts of the
phases present must balance.
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11. The Lever Rule
Tie line and Lever rule:
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❖ We draw a horizontal line (tie line)at the
temperature of interest (say T0). Let Tie line is
XY.
❖ Solid phase of composition C1 coexists with
liquid of composition C2 .
❖ The portion of the horizontal line in the two-
phase region is akin to ‘lever’ with the fulcrum at
the nominal composition (C0).
❖ The opposite arms of the lever are proportional
to the fraction of the solid and liquid phase
present (this is lever rule).

12. The Lever Rule
Example:
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At C0 = 35 wt% Ni
At TA: Only Liquid(L), Wliquid = 100 wt%,
Wsolid= 0
At TD: Only Solid(S), Wliquid = 0, Wsolid= 100
wt%
At TB : Both Solid and liquid present

13. The Lever Rule
Example: Find the relative amount of Cu, Ni and the percentage of phase present at point C.
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